The-Generation-Of-Internal-Stresses-In

THE GENERATION OF INTERNAL STRESSES IN SINGLE AND TWO PHASE MATERIALS

A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering

September 2002

By Edward Charles Oliver Manchester Materials Science Centre Contents

Abstract 6

Declaration 7

Publications 9

Acknowledgements 10

Notation and Nomenclature 11

1 Introduction 14

2 Internal Stress Development in Single and Two Phase Materials 16 2.1 Deﬁnitions and Origins of Internal Stress and Residual Stress ...... 16 2.1.1 Thermal Misﬁt ...... 18 2.1.2 Heterogeneous Plastic Flow ...... 18 2.1.3 Phase transformations ...... 18 2.2 The Concept of Eigenstrain ...... 20 2.3 The Eshelby Theory ...... 20 2.3.1 The Homogeneous Ellipsoidal Inclusion ...... 20 2.3.2 The Equivalent Inclusion Method ...... 21 2.4 Elastic Properties of Heterogeneous Solids ...... 22 2.4.1 Single Crystal Elastic Anisotropy ...... 24 2.4.2 Voigt and Reuss Elastic Averages ...... 25 2.4.3 Eshelby-Based Approximations ...... 25 2.5 Plastic Deformation of Single Crystals ...... 28 2.5.1 Slip in Single Crystals ...... 29 2.5.2 Deformation Twinning ...... 32 2.5.3 Martensitic Transformation ...... 34 2.6 Plastic Properties of Polycrystals ...... 35 2.6.1 Sachs model ...... 35 2.6.2 Taylor model ...... 35 2.6.3 Bishop and Hill Analysis ...... 37 2.6.4 Elastoplastic Self-Consistent Model ...... 37 2.6.5 Crystal Plasticity Finite Element Method ...... 40 2.7 Plastic Properties of Composites ...... 42 2.7.1 Forward and Reverse Yield Stress ...... 43 2.7.2 Plastic Relaxation ...... 43 2.7.3 Finite Element Method ...... 44 2.8 Martensitic Transformation ...... 45

2 2.8.1 Experimental Observations of Martensitic Transformation ...... 45 2.8.2 Crystallographic Theories of Martensitic Transformation ...... 46

3 Measurement of Internal Stress by Neutron Diffraction 50 3.1 Bragg Diffraction as a Strain Gauge ...... 50 3.1.1 The Bragg Condition ...... 50 3.1.2 Fixed Wavelength Diffractometry ...... 51 3.1.3 Time-of-Flight Diffractometry ...... 51 3.2 Determination of Elastic Strains from Diffraction Spectra ...... 52 3.2.1 Elastic Grain Family Strains via Single Peak Fitting ...... 52 3.2.2 Elastic Phase Strains via Rietveld Refinement ...... 53 3.3 The ENGIN Instrument ...... 54 3.3.1 Flight Path ...... 55 3.3.2 Collimation ...... 55 3.3.3 Detector Banks ...... 55 3.3.4 Loading Rig ...... 56 3.3.5 Cooling Grips ...... 56 3.4 Review of Type II Internal Stress Measurements Using Neutron Diffraction . . . 57 3.4.1 Intergranular Stress ...... 57 3.4.2 Interphase Stress ...... 61 3.4.3 Stress-Induced Martensitic Transformation ...... 63

4 Interphase and Intergranular Stress In Carbon Steels 66 4.1 Materials Review ...... 67 4.2 Materials ...... 68 4.3 Neutron Diffraction Method ...... 69 4.4 Macroscopic Response ...... 70 4.5 Interphase Strains ...... 72 4.5.1 Strains Under Applied Loading ...... 73 4.5.2 Residual Strains ...... 76 4.5.3 Unrelaxed Model ...... 78 4.6 Intergranular Strains ...... 80 4.6.1 Strains Under Applied Loading ...... 80 4.6.2 Residual Strains ...... 84 4.7 Reproducibility of Data and Influence of Crystallographic Texture ...... 86 4.7.1 Reproducibility of Data ...... 88 4.7.2 Influence of Crystallographic Texture ...... 90 4.8 Rationalisation of Transverse Intergranular Strains in Ferrite ...... 90 4.9 Investigation into Intergranular Stress in Ferrite using the Elastoplastic Self- Consistent Method ...... 95 4.9.1 Model Specification ...... 96 4.9.2 Comparison of EPSC Predictions With Experimental Data ...... 97 4.9.3 Influence of Elastic Anisotropy ...... 100 4.9.4 Summary of Findings from the EPSC Model ...... 101 4.10 Note on Linear Elastic Response of Transverse Grain Families ...... 102 4.11 Finite Element Model of Interphase Stress in High Carbon Steel ...... 106 4.11.1 FE Model Design ...... 106 4.11.2 Constituent Properties ...... 106 4.11.3 Extent of Constraint ...... 107 4.11.4 Comparison of FE Model Predictions With Experimental Data ...... 108 4.11.5 Summary of Findings from FE Model of Interphase Stress ...... 111 4.12 Combined Model of High Carbon Steel ...... 111 4.12.1 Combined Modelling Strategy ...... 112

3 4.12.2 Combined Model Results ...... 112 4.12.3 Comparison of Combined Model Approach With Two Phase EPSC Model 114 4.13 Summary of Chapter ...... 120

5 Stress-Induced Martensitic Transformation in TRIP Steel 122 5.1 Review of Transformation-Induced Plasticity (TRIP) ...... 122 5.1.1 Overview ...... 122 5.1.2 Observed Features of the TRIP Phenomenon ...... 123 5.1.3 Models of Stress-Induced Transformation in TRIP Steels ...... 126 5.2 Materials ...... 128 5.3 Characterisation of Microstructure and Mechanical Behaviour ...... 129 5.3.1 As-Received Microstructure ...... 129 5.3.2 Ms Temperature ...... 129 5.3.3 Mechanical Behaviour ...... 129 5.3.4 Martensite Volume Fraction ...... 132 5.3.5 Influence of Hot Swaging ...... 133 5.4 Evolution of Crystallographic Texture ...... 138 5.5 Neutron Diffraction Method ...... 138 5.6 Macroscopic Response During Neutron Diffraction Tests ...... 139 5.7 Evolution of Diffraction Spectra and Observation of Preferential Transformation 140 5.7.1 Texture Prior to Testing ...... 140 5.7.2 Development of Martensite ...... 142 5.7.3 Changes in Austenite Texture ...... 144 5.7.4 Summary ...... 146 5.8 Orientation Dependence of Transformation ...... 147 5.9 Rietveld Refinement ...... 150 5.10 Martensite Volume Fraction ...... 151 5.11 Elastic Phase Strains ...... 152 5.11.1 Elastic Strain in Austenite ...... 153 5.11.2 Elastic Strain in Martensite ...... 156 5.11.3 Discussion and Comparison with High Carbon Steel ...... 157 5.12 Intergranular Strain in Austenite ...... 160 5.13 Elastoplastic Self-Consistent Simulation of Deformation in Austenite ...... 163 5.13.1 Simulation of Unswaged Material ...... 163 5.13.2 Simulation of Hot Swaged Material ...... 165 5.14 Summary of Chapter ...... 168

6 Conclusions and Suggestions for Further Work 170 6.1 Summary and Conclusions ...... 170 6.2 Suggestions for Further Work ...... 172

4 Abstract

The subject of this dissertation is the generation of internal stresses arising from mechanical deformation in single and two phase engineering materials. The method of neutron diffraction is employed to study the evolution of both intergranular and interphase stresses in low and high carbon ferritic steels and in an austenitic steel which exhibits stress-induced martensitic transformation. In low carbon steel, intergranular stresses develop because of incompatibilities between grains arising due to single crystal elastic and plastic anisotropy. Insight into the evolution of intergranular stresses is gained by application of the elastoplastic self-consistent method and by arguments based on the pencil glide model of slip in body-centred cubic crystals. The elastic strains developed transversely to an applied uniaxial load are discussed in particular detail. The internal stresses which develop in low and high carbon steels are compared. In the latter material, large interphase stresses develop during yielding, due to greater plastic flow in the ferrite matrix than in the spheroidised cementite inclusions. The load redistribution is particularly dramatic because of yield point softening in the matrix. The influence of this effect is simulated using the finite element method. Intergranular stress evolution in high carbon steel is studied using a combination of the finite element and elastoplastic self-consistent models. Texture changes observed during tensile deformation of the austenitic steel are attributed to a combination of grain rotation and the preferential martensitic transformation of favourably oriented grains. The most favourable orientation is determined and interpreted using a crystallographic theory of martensitic transformation. The formation of martensite plates leads to the development of back stresses in the austenite matrix. As in the case of high carbon steel, these are attributed primarily to the generation of misfit because of greater plastic flow in the matrix phase. The resulting work hardening is identified as an important origin in the phenomenon of transformation-induced plasticity.

5 Declaration

No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institution of learning.

6 Copyright

Copyright in text of this thesis rests with the Author. Copies (by any process) either in full, or of extracts, may be made only in accordance with instructions given by the Author and lodged in the John Rylands University Library of Manchester. Details may be obtained from the Librarian. This page must form part of any such copies made. Further copies (by any process) of copies made in accordance with such instructions may not be made without the permission (in writing) of the Author. The ownership of any intellectual property rights which may be described in this thesis is vested in the University of Manchester, subject to any prior agreement to the contrary, and may not be made available for use by third parties without the written permission of the University, which will prescribe the terms and conditions of any such agreement. Further information on the conditions under which disclosures and exploitation may take place is available from the Head of Department of Manchester Materials Science Centre.

7 Publications

Work included in this dissertation has been presented in the following publications:

E.C. Oliver, T. Mori, P.J. Withers and M.R. Daymond, Measurement of Interphase and In- tergranular Strains in Carbon Steels by Neutron Diﬀraction, In ICRS-6: 6th International Conference on Residual Stresses, Oxford, UK, pages 98-105, IOM Communications, 2000.

E.C. Oliver, P.J. Withers, M.R. Daymond, S. Ueta, T. Mori, Neutron Diﬀraction Study of Stress Induced Martensitic Transformation in TRIP Steel, In ICNS 2001: International Con- ference on Neutron Scattering, Munich, Germany, to be published in Applied Physics A.

E.C. Oliver, M.R. Daymond, P.J. Withers, T. Mori, Stress Induced Martensitic Transformation Studied by Neutron Diﬀraction, In ECRS-6: 6th European Conference on Residual Stresses, Mater. Sci. Forum, Vols. 404-407, pages 489-494, 2002.

8 Acknowledgements

I would like to acknowledge the Engineering and Physical Sciences Research Council for finan- cial support in the provision of a project studentship and beamtime at the ISIS pulsed neutron facility. I am grateful to the Rutherford Appleton Laboratory for providing additional sponsor- ship through the CASE scheme. I would like to thank Dr Carlos Tomé for the use of his EPSC code, and Dr Michio Okabe and Mr Shigeki Ueta for kindly producing the material which is studied in Chapter 5. Thank you to my supervisor, Prof. Philip Withers, for his enthusiastic guidance throughout, and to my co-supervisor, Dr Mark Daymond, for his support on my visits to RAL and for giving me a job! Thanks to all the staff at Manchester Materials Science Centre; in particular to Mark Harris for his patience at last minute requests and for sorting out my ailing car, to Paul Mummery for pints and chats, and to Julie and Lisa, who are as formidable as they are delightful. The Withers group are a fine and diverse bunch - thanks especially to Michael and Axel for their German sophistication, to João for his Portuguese style and to Gaston for his Luxembourgish charm. English eccentricity has been supplied in abundance by Alex M, and in reasonable quantities by Alex O; I thank them both for their considerable help and friendship. Thanks also to all my new friends down South, for making settling in so easy. I have a wonderful and supportive family. My Mum and Dad are absolute stars — thank you for everything, including the worrying! Lots of love and thanks to Gran and Grandad — it’s been lovely to spend a bit more time with you over the past few years — and to James and Claire, Dave and Lou, and Suey and Dave. Love and hugs to Sam for being such a sweetie and for becoming a very special person in my life. Finally, I would like to reserve special thanks for Prof. Ben Mori. Amongst others, he has fulfilled the roles of tutor, mentor, tennis partner and friend. His sustained enthusiasm for materials science is refreshing and inspiring.

9 Notation and Nomenclature

In this dissertation, tensor quantities are represented in two ways. In the full notation, tensors are represented by letters in normal italic type, and suﬃxes are explicitly written. Unless otherwise stated, summation over repeated indices is assumed. In the abbreviated notation, tensors are represented by bold italic letters, without suﬃxes. In this notation, the contracted tensor product written in full notation as Aijklbkl is represented simply by Ab. The uncontracted tensor product aij bkl is also occasionally used. To distinguish from the more commonly used contracted product, this is written as a b. ⊗

According to the common convention, the Miller indices of a crystallographic plane are enclosed by round brackets, those of a set of symmetry-related planes are enclosed by squiggly brackets, the indices of a direction are enclosed by rectangular brackets, and those of a set of directions are enclosed by triangular brackets. When referring specifically to a diffraction reflection or grain family, no brackets are used.

To denote an averaged quantity (over volume or orientation space, for example), a symbol is enclosed by triangular brackets or has a bar written above it. A dot above a symbol denotes the diﬀerential with respect to time.

Symbol Description a, c lattice parameters a transformation matrix between reference frames Almn, Ahkl cubic elastic anisotropy factor A0 degree of anisotropy for a cubic crystal b Burgers vector dhkl lattice spacing D body of integration E Young’s modulus

10 Symbol Description f volume fraction l length d distortion tensor h Planck’s constant hij hardening matrix h unit vector within habit plane L elastic stiffness tensor L elastoplastic instantaneous stiffness tensor M elastic compliance tensor mn neutron mass m Schmid factor M Taylor factor n unit normal vector P mechanical potential energy per unit volume S Eshelby tensor t time t tensile axis T temperature T generic tensor U internal energy per unit volume v speed V volume w work by external force per unit volume α Schmid tensor γ shear strain Γ accumulated shear strain δij Kronecker delta δij identity matrix strain tensor resolved strain ∗ eigenstrain tensor θhkl Bragg diffraction angle λ wavelength ν Poisson’s ratio σ stress tensor Symbol Description σ uniaxial or resolved stress τc, τ critical resolved shear stress τ0, τ1, τˆ, θ0, θ1 hardening law parameters ϕ1, ϕ2, Φ Bunge Euler angles θ, λ, ψ, ξ, φ angles Ω crystal spin tensor

In addition to the quantities given above, some un-named quantities are deﬁned in the text. These may temporarily adopt a symbol used above, but are generally employed in a single section or in subsequent sections which explicitly refer to that in which they are introduced. Subscripts and superscripts

The meanings of some subscripts and superscripts employed are given below. There is no

11 strict convention for the choice of subscripts or superscripts: this depends largely on the main- tenance of clarity in equations and the adoption of symbols used by other authors.

Symbol Description A applied (stress) c constrained (strain) el elastic elong elongational I inclusion M matrix mac macroscopic P plastic y yield (stress) α0 martensite γ austenite parallel to axis k perpendicular to axis ⊥ within an inclusion surrounded by inﬁnite matrix ∞

12 Chapter 1

Introduction

The aim of this dissertation is to further the understanding of internal stress development during the mechanical deformation of single and two phase engineering materials. This subject is important because the mechanical response of a material depends not only upon stress applied externally, but also upon the internal stress state. Even within a single phase engineering material, significant internal stresses may develop due to incompatibility between grains, arising from single crystal anisotropy. This type of internal stress is commonly termed intergranular stress. In a multiphase material, an additional source of internal stress is incompatibility between phases. This type is called interphase stress. In the context of this dissertation, both types are of interest. In order to achieve the above stated aim, the primary experimental method employed is pulsed source neutron diffraction. Diffraction methods have been used for the measurement of internal stress for many decades [1]. The first experiments of this kind were conducted using laboratory X-rays, but such measurements are restricted to the near-surface region, where the stress state is affected by the proximity of the free surface. The neutron method does not suffer from this limitation, owing to the much deeper penetration of neutrons within engineering materials. Neutron diffraction at steady state sources has been applied to the field for about twenty years [2]. However, the pulsed source method has been developed mainly in the past ten years, and is becoming increasingly significant, with the recent advent of dedicated strain measurement instruments [3]. The major advantage of the method over the steady state technique is that a full diffraction profile is automatically acquired. Typically at a steady state source, just one or a few diffraction peaks are monitored. A diffraction peak is produced by reflection solely from a family of similarly oriented grains of a single phase. By monitoring all the diffraction peaks within a profile, the states of different phases and grain families are revealed. This is a great advantage of diffraction over other methods of internal stress measurement. Shifts in peak positions relate to elastic straining of the crystal lattice; thus by comparing shifts of different peaks, both interphase and intergranular elastic strains may be determined, and stress states subsequently calculated. Moreover, by comparison of peak intensities, phase volume fractions and the distribution of grain orientations within a phase may also be determined. These methods are all exploited in this dissertation for the analysis of experimental data. The experimental work in this dissertation is divided into studies of two different types of steel. Although the studies are specific to these materials, the methods used for the analysis and interpretation of results are applicable to a wide range of single and two phase materials. The intention is to extract and explain general trends in the evolution of internal stresses. The neutron diffraction method is applied firstly to the study of the development of internal stresses in carbon steels. Spheroidised high carbon steel provides an example of a material in which a matrix phase deforms elastoplastically in the presence of elastically deforming inclusions. Such a combination of phases is common among many other composite materials, both

13 Chapter 1. Introduction 14

naturally occurring and fabricated, such as metal matrix composites. It is found that the presence of reinforcing cementite inclusions leads to the growth of large interphase stresses during plastic deformation. Comparison of the behaviour of low and high carbon steels enables the influence of reinforcing inclusions upon matrix intergranular stress development to be assessed. At present, the highly non-linear responses of different grain families in the plastic regime remain difficult to predict. In particular, diffraction measurements of transverse lattice strain commonly reveal remarkable shifts in the responses of some grain families when plasticity begins. This has not been adequately explained. Moreover, research has focused on face-centred cubic alloys. In this work, the ferritic carbon steels which are studied have a body-centred cubic crystal structure, giving rise to different trends in intergranular stress development, owing to the operation of different slip modes during plasticity. It is found that these materials also exhibit remarkable transverse strain responses, and the reasons for this are addressed. Recently, interest has grown in the application of the elastoplastic self-consistent method to the prediction of intergranular stress. Computational models based on this method can calculate elastic strain responses which are directly comparable to those measured by neutron diffraction. Such a model is applied for comparison to the experimental data collected on carbon steels. This provides validation of the model when applied to body-centred cubic materials, while also helping to explain intergranular stress development in such materials. The finite element method is also applied for the prediction of interphase stresses and comparison to the experimental results. The neutron diffraction technique is also employed for the study of a steel which exhibits transformation-induced plasticity (TRIP). TRIP is the term given to the property of enhanced ductility arising due to the accompaniment of slip deformation by stress-induced martensitic transformation. It is clear that internal stress plays an important role in the operation of the mechanism, but little experimental work has been reported which demonstrates this. Moreover, since the material presents a system in which a reinforcing second phase is generated dynamically as mechanical deformation progresses, this investigation is a natural extension of that into carbon steels, in which the reinforcing volume fraction remains constant. The experimental data reveal that the evolving martensite phase does indeed perform a similar reinforcing role to the cementite inclusions in high carbon steel. Furthermore, it is seen that certain orientations of austenite grain transform preferentially to martensite. This is related to a crystallographic theory of martensitic transformation based upon consideration of the internal stress developed during transformation. The results and discussion relate to the TRIP mechanism in particular, but also to stress-induced martensitic transformation in general. This phenomenon is fundamental to the mechanical behaviour not only of TRIP steels, but also of superelastic and shape memory alloys [4]. The dissertation is organised as follows. Chapter 2 presents an introduction to the subject of internal stress in single and two phase materials, addressing both intergranular and interphase stress. A review of the subject of martensitic transformation is also presented, in order to provide background for the experimental study of transformation-induced plasticity. Chapter 3 introduces the technique of internal stress measurement by neutron diffraction. The instrument and experimental methods employed for the experimental measurements of internal stress presented in subsequent chapters are described. A review of other relevant work performed using neutron diffraction is also given. Chapter 4 presents the study of internal stress development in carbon steels. This is followed in Chapter 5 by the investigation of transformation-induced plasticity. Finally, Chapter 6 presents conclusions and suggestions for further work. Chapter 2

Introduction to Internal Stress Development in Single and Two Phase Materials

This chapter introduces aspects of the development of internal stress in single and two phase engineering materials which are relevant to the work presented in subsequent chapters. Firstly, deﬁnitions of the terms internal stress and residual stress are presented, and some origins of these types of stress are described. There follows a discussion of the basis of the Eshelby ellipsoidal inclusion theory, which has widespread applicability to problems of internal stress development. Internal stress growth in polycrystals is dependent upon anisotropic properties at the level of the constituent single crystals. For this reason, these properties are discussed before the consideration of internal stress in polycrystals and composites. Methods for the modelling of internal stress growth in such materials are then discussed. Finally, a review is presented of some aspects of martensitic transformation relevant to the work presented in Chapter 5.

2.1 Deﬁnitions and Origins of Internal Stress and Residual Stress

The term “residual stress” describes any stress which remains in a body when all external forces are removed. It is rare, if not impossible, to encounter an engineering material in which no residual stress is present. For example, residual stress is present in any material containing dislocations or vacancies, since such defects give rise to short range stress fields. Often, however, residual stress fields occur in a material over much larger length scales. It is the length scale which is commonly used to classify residual stresses. Residual stress fields which occur over macroscopic distances are classed type I. An example is that present in a bar which has been bent plastically. Residual stress fields which vary over the order of grain size or inclusion size are classed type II. Finally, those with an associated length scale of the order of atomic spacing, such as a dislocation stress field, are classed type III. It is a requirement of equilibrium that the average residual stress, taken over the whole body, is zero. This is commonly referred to as the condition of stress balance, and is written

σij dD = 0 (2.1) ZD where D denotes the body, and σij the residual stress. Clearly, for the stress balance condition to hold, residual stress must vary spatially within the body, unless it is everywhere zero. In broad terms, such a variation may arise either through inherent heterogeneity of the material,

15 Chapter 2. Internal Stress Development in Single and Two Phase Materials 16

or because of a spatial variation in processing history. In the case of the plastically bent bar, for example, residual stress develops because different points experience different applied stresses, so that the extent of plastic deformation varies from point to point. However, this project is primarily concerned with residual stresses which arise through heterogeneity on the type II length scale. Some authors define internal stress in exactly the same way as residual stress (e.g. [5]). However, in this dissertation, a slightly broader working definition is employed. Assuming the applied stress field can be suitably defined (for example, in the case of uniaxial tension, where the applied stress is uniform), then internal stress at a point is defined as the difference between the total stress and the applied stress (note that using this definition, the stress balance condition (2.1) remains valid for internal stresses). When no external forces act, the applied stress is everywhere zero; thus using this definition, internal stress is identical to residual stress. If a residually-stressed, elastically homogeneous body is loaded elastically, then internal and residual stress are again identical. However, there are cases in which the definitions differ. For example, mechanisms of stress relaxation may operate upon unloading, resulting in residual stresses of lower magnitude than the internal stresses existing under load. Evidence of such relaxation is presented in 4.5.2. Another case in which the definitions differ is that of elastic mismatch stresses. Consider§ the composite illustrated in Fig. 2.1. The composite is formed from consecutive slabs of two elastic materials, A and B, with Young’s moduli EA and EB respectively, where EA > EB. Force per unit area σ is applied to the surfaces, as shown, resulting in plane strain deformation. The uniaxial applied stress may be unambiguously identified as σ. However, this is not the average stress experienced by either A or B. Since A is stiffer than B, it bears a greater proportion of the applied load. That is σA > σ > σB , where σA and σB are the (average axial) stresses in materials A and B respectively. Using the above definition, internal stress is present in the composite. However, the residual stress is zero. This is because the deformation is elastic, and thus no stress remains once the external load is removed. Such elastic mismatch stresses are very common, and of relevance to this dissertation. The term “internal stress” is used in the title of the dissertation in order to encompass such types of stress which are excluded by the definition of residual stress, as well as to avoid any implication that measurements are necessarily made after the removal of load. The term “residual stress” is also used throughout the dissertation, to refer to the type of internal stress which does not disappear when external forces are removed. The term “residual strain” is used to refer to the elastic strain which exists due to the presence of residual stress. Some simple conceptual models are now described in order to illustrate some origins of residual stresses.

A B A B

Figure 2.1: Elastic slab composite. The materials A and B have Young’s moduli EA and EB respectively, where EA > EB. External force per unit area σ is applied to the top and bottom surfaces, as illustrated. Chapter 2. Internal Stress Development in Single and Two Phase Materials 17

T0 T1 T1

A B A B l0 lA A lB B l1 A B A B

(a) (b) (c)

Figure 2.2: Schematic illustration of the origin of thermal misﬁt stress.

2.1.1 Thermal Misfit Consider Fig. 2.2, which shows another slab composite, similar to that of Fig. 2.1. The composite is formed at temperature T0, at which temperature the slabs have natural length l0, as illustrated in Fig. 2.2a. Materials A and B are now elastically identical, but differ in that B has a greater coefficient of thermal expansion. On cooling from the stress-free temperature T0 to a lower temperature T1, unconstrained slabs of A and B would contract to natural lengths lA and lB, where lA > lB (Fig. 2.2b). However, since the slabs remain perfectly bonded, the equilibrium length of the composite is l1, where lA > l1 > lB (Fig. 2.2c). Thus slabs of A are in in-plane tension, slabs of B in compression. This is an origin of thermal misfit stress. Since no material is perfectly isotropic and homogeneous, and since many engineering materials are subjected to some form of thermal processing, thermal misfit stresses are commonly present to some extent. They are particularly important in metal matrix composites, where fabrication involves consolidation at high temperature of phases with very different thermal expansion coefficients [6].

2.1.2 Heterogeneous Plastic Flow Consider another slab composite. In this case, materials A and B have identical thermal and elastic constants, but while B remains elastic at all loads, A becomes perfectly plastic at a tensile yield stress of σy (illustrated in Fig. 2.3a). If a tensile stress greater than σy is applied as shown in Fig. 2.3b, then material A will deform plastically. The natural lengths of the slabs of A therefore increase, while those of B remain the same, i.e. lA > lB = l0. Since the composite remains perfectly bonded, after unloading the composite length, l1 is intermediate between lA and lB, i.e. lA > l1 > lB. Thus A is axially compressed, whilst B is forced into axial tension. This type of internal stress is also very common in engineering materials which have been mechanically deformed. The stresses may be large in composite materials, but internal stresses of this kind may also be present in single phase materials, owing to the distribution of orientations of crystallites with anisotropic elastic and plastic properties. Residual stresses in welds originate due to a combination of thermal and plastic mismatch. Temperature gradients during welding give rise to thermal misfit stresses, which are relaxed by plastic flow in hotter regions where the yield stress is lower. Afterwards, misfit remains due to the heterogeneous plastic flow, leaving residual stresses within the weld [7].

2.1.3 Phase transformations Internal stress may also arise due to a phase transformation. Diﬀusional transformations do not tend to introduce deviatoric stress. This is because atoms diﬀuse around or within precipitates in order to minimize the elastic energy, and this leads to hydrostatic stress [8]. There is, Chapter 2. Internal Stress Development in Single and Two Phase Materials 18

B → σy (a)

stress A

strain →

σ > σy After unloading After loading (unconstrained) (constrained)

lA lB A B A B l1 (> l0) A (= l0) B A B A B

(b) (c) (d)

Figure 2.3: Schematic illustration of the introduction of internal stress by heterogeneous plastic ﬂow. Chapter 2. Internal Stress Development in Single and Two Phase Materials 19

however, another class of phase transformation, in which diffusion does not occur. This class is called martensitic transformation. In this case, the transformation occurs by rapid, cooperative movements of atoms, giving rise to a sudden shape change. This may generate misfit between the transformed and the remaining untransformed material. Although the transformation will occur in a manner which minimizes misfit (and thus the elastic energy), internal stresses may still arise.

2.2 The Concept of Eigenstrain

Such phase transformations readily lend themselves to analysis by the theory of eigenstrains, set out in the text by Mura [5]. The eigenstrain is the strain which would occur in a region of material due to transformation, if there were no surroundings to impose constraint. It is sometimes called the “stress-free transformation strain” [6]. For further discussion of the meaning and origin of the term, the reader is referred to the beginning of Mura’s book. The concept of eigenstrain, however, extends far beyond the field of martensitic transformation. In fact, all forms of internal stress may be viewed conceptually as arising from misfit (or equivalently a relative transformation) between different regions of material [9]. For example, the differential thermal contraction explained in 2.1.1 may be viewed as a relative shape change between phases. Similarly, the effect of heterogeneous§ plastic flow is to produce misfit between regions of material, which may be expressed using a transformation strain. The stiffness of composites and the stress fields around dislocations and crack tips may all be predicted, among many other examples, using the theory of eigenstrains [5]. Underpinning the theory is a very important theory by Eshelby, which forms the subject of the next section.

2.3 The Eshelby Theory

The theoretical results contained within Eshelby’s classic paper of 1957 [10] catalysed much work over the following decades. Most important were his analysis of the homogeneous ellipsoidal inclusion and the ellipsoidal inhomogeneity. The major results and concepts of this analysis are summarised in this section. They are of relevance in this dissertation because they form the basis of important models of internal stress development in both polycrystals and composites, which are applied in subsequent chapters for the analysis of experimental measurements.

2.3.1 The Homogeneous Ellipsoidal Inclusion Consider an infinite, homogeneous, elastic solid. Eshelby tackled the problem of the stress and strain fields which result when a region of this solid is subjected to a uniform eigenstrain. The calculations proceed via a number of conceptual steps. Firstly, the region (hereafter called the inclusion) is removed from the surrounding matrix and allowed to undergo the stress-free transformation. Surface tractions are then applied such that the inclusion is deformed elastically back to its original shape and size. These forces are known, since they are equal and opposite to the forces which would give rise to the transformation strain were it to be achieved through elastic deformation rather than a stress-free transformation. The inclusion is then conceptually welded back into place, such that no slipping may occur at the interface. At this stage, there is no stress in the matrix, but there is a layer of body force on the surface of the inclusion. These external forces are then removed by applying equal and opposite body forces. Elasticity theory [11] gives the expression for the displacement at a distance due to a point force. The displacement field is thus given by the integral of this expression around the surface of body force. From this may be derived the strain field and, via Hooke’s Law, the stress field. Eshelby solved this integral explicitly for the inside of an isotropic ellipsoidal inclusion within an isotropic medium. This produced the important result that in such a case, the stress and strain fields within the inclusion are uniform. If the eigenstrain representing the stress-free Chapter 2. Internal Stress Development in Single and Two Phase Materials 20

c transformation is ij∗ , then the total strain in the constrained inclusion ij (relative to its initial, untransformed state) is given by c ij = Sijklk∗l (2.2)

where Sijkl is called the Eshelby tensor. For the case of an isotropic medium, as solved explicitly by Eshelby, the tensor is given in analytical forms, and depends only on the principal axes of the ellipsoid and the Poisson’s ratio of the material [10]. However, Eshelby also showed, without explicit calculation of the Eshelby tensor, that equation (2.2) also holds in an anisotropic medium. In this case, the tensor is usually calculated by numerical integration, but Hori and Nemat-Nasser have recently presented an analytical method [12]. Withers has also presented an analytical solution for the case of a transversely isotropic medium [13]. In order to indicate that the Eshelby tensor is a fourth rank tensor, equation (2.2) employs suﬃx notation with summation over repeated indices. However for clarity, the abbreviated notation, in which tensors are represented by bold letters and suﬃxes are not explicitly written, is now adopted. The stress σ∞ in the inclusion is related, via Hooke’s Law, to the elastic, not total strain. The elastic strain is given by c ∗, since the transformed inclusion is unstressed when it has strain ∗ relative to its initial, un−transformed state. Hence

∞ c ∗ ∗ σ = LM ( ) = LM (S I) (2.3) − − where LM is the elastic stiﬀness tensor of the homogeneous medium, I is the identity tensor and the superscript ∞ indicates that this is an inﬁnite body problem.

2.3.2 The Equivalent Inclusion Method Eshelby’s result for the homogeneous ellipsoidal inclusion is of great importance alone. However, the theory becomes even more powerful when related, via another conceptual step, to the problem of the stress in an ellipsoidal inhomogeneity, i.e. having a different elastic stiffness to the matrix. Consider again the homogeneous problem. The transformed inclusion has uniform stress σ∞ and uniform total strain c. If a uniform stress field σA is now applied to the whole (infinite) body, the stress and strain fields may simply be superposed, since the system is elastically homogeneous. Hence the stress and strain in the inclusion (now denoted by the superscript I ) are given by I ∞ A I c A σ = σ + σ , = + , (2.4) where A 1 A = LM− σ (2.5) gives the elastic strain which develops due to the applied stress field.1 The inclusion stress and elastic strain are related by I I ∗ σ = LM . (2.6) − Now imagine replacing the homogeneous inclusion withan ellipsoid of some ‘tailored’ material (again, welding into place, such that no sliding occurs). The new ellipsoid has the same stress σI , strain I and original shape as the homogeneous inclusion; it thus fits perfectly into the hole left by removing the homogeneous inclusion, without disturbing the matrix field at all. The new inclusion differs only in that the strain is developed fully elastically, without transformation strain. This is possible by tailoring the stiffness of the material LI , according to

I I σ = LI . (2.7)

In the resulting system, the strains both inside and outside the inclusion are fully elastic (and

1Note that if σA = −σ∞, the stress in the inclusion is zero. It is a mere formality to conceptually remove the inclusion; hence solution of the matrix stress field around the misfitting homogeneous ellipsoid automatically gives the solution of the stress disturbance of an applied stress field due to the presence of an ellipsoidal void. Chapter 2. Internal Stress Development in Single and Two Phase Materials 21

thus vanish when the applied stress is removed). The stress field therefore gives the solution to that which arises when a uniform stress is applied to an infinite body containing one ellipsoidal inhomogeneity. The important point to note is that the stress in the inhomogeneity is uniform. In a real composite, of course, the inclusion stiffness is a real material property, and cannot be ‘tailored’. However, we are free to choose instead the eigenstrain, so that the homogeneous inclusion stress matches that in the inhomogeneity. Since the stress in the inhomogeneity is uniform, we are always able to select the appropriate eigenstrain. The problem is illustrated in Fig. 2.4. Application of a uniform applied stress field to the inhomogeneous system results in uniform stress inside the ellipsoid (but not outside), as shown in Fig 2.4a. Fig. 2.4b illustrates the equivalent homogeneous system. The homogeneous inclusion is removed from its surroundings, undergoes a stress-free transformation ∗, and is then welded back into place. The stress inside is uniform, and remains so when the uniform applied stress is exerted. The eigenstrain ∗ is chosen so that the resultant stress field exactly matches that of the inhomogeneous system. The eigenstrain is related to the difference between the inclusion and applied strains by

∗ 1 c 1 I A = S− = S− . (2.8) − Hence from equations (2.3) and (2.4), we have

I A ∗ I A I A ∗ I A σ σ = L or = M σ σ , (2.9) − − − − − − where ∗ 1 ∗ ∗ 1 L = LM (S I) S− and M = L − . (2.10) − − Using (2.5), (2.7) and (2.8), we ﬁnd

∗ I ∗ A (LI + L ) = (LM + L ) , (2.11) ∗ I ∗ A (MI + M ) σ = (MM + M ) σ , (2.12)

1 1 where MM = LM− and MI = LI− are the elastic compliance tensors of the matrix and inclusion respectively. Equations (2.11) and (2.12) give the solutions of the uniform stress and strain states in the inhomogeneous inclusion under a uniform applied stress ﬁeld. It is useful to introduce two further short-hand tensors A and B, where

∗ 1 ∗ A = (LI + L )− (LM + L ) , (2.13) ∗ 1 ∗ B = (MI + M )− (MM + M ) , (2.14) such that the inclusion strain and stress may be neatly written as

I A I A = A and σ = B σ . (2.15)

The equivalent inclusion method is applicable to a wide range of problems, because the equation of an ellipsoid describes shapes as wide ranging as flat plates to spheres to long fibres. The method is not restricted to the calculation of stress disturbance under applied load; for example thermal misfit between phases with different elastic constants may also be solved by envisaging an equivalent homogeneous inclusion. For a discussion of this problem, the reader is referred to the book by Clyne and Withers [6].

2.4 Elastic Properties of Heterogeneous Solids

While the Eshelby theory provides an elegant solution for the response of a single heterogeneity within an inﬁnite matrix, such an idealised system is never found in a real material. The theory may be generalized, however, to provide models of the mechanical behaviour of real heterogeneous solids, such as composites and polycrystalline aggregates. In this section, models Chapter 2. Internal Stress Development in Single and Two Phase Materials 22

(a)

∗

(b)

Figure 2.4: Schematic illustration of equivalent inclusion method of determining stress field inside ellipsoidal inhomogeneity under applied stress, after Clyne and Withers [6]. (a) Stress is applied to a composite containing an inhomogeneous inclusion, resulting in uniform stress and strain within the inclusion. The strain in the inclusion is developed fully elastically. (b) The stress field developed in the inhomogeneous case is reproduced using an equivalent homogeneous inclusion. The natural shape of the homogeneous inclusion transforms. The transformation strain ∗ is chosen wisely, such that when the external stress is applied, the stress field exactly matches that developed in the inhomogeneous case. Chapter 2. Internal Stress Development in Single and Two Phase Materials 23

used to estimate the elastic properties of such solids are described. A polycrystal may be regarded as a type of composite, in which the heterogeneity arises not due to the presence of different phases, but due to the different orientations of grains within the aggregate. The major difference is that composites are generally considered as consisting of inclusions embedded within a single matrix phase, whereas in a polycrystal there are many different types of ‘inclusion’, but no overall matrix. Models which average the properties of grains or inclusions, however, do not commonly consider the microstructural detail, and therefore the models discussed here are applicable both to polycrystals and composites. This should be remembered in the following subsections, although emphasis in each subsection is placed on one or other type of material.

2.4.1 Single Crystal Elastic Anisotropy The reason that the elastic stiffness of a polycrystal differs from that of a single crystal of the same material is that the constituent crystallites are elastically anisotropic. Thus differently oriented crystallites have different elastic properties with respect to an external reference frame. It is therefore appropriate to discuss single crystal elastic anisotropy before considering the elastic properties of polycrystals. Elasticity fundamentally derives from the binding forces between the atoms of a solid. The anisotropic atomic structure of a crystal may thus be reflected in anisotropic elastic stiffness. 2 In terms of the elastic compliance tensor , Mijkl, Hooke’s Law of a general linear elastic solid may be written as ij = Mijklσkl . (2.16)

The symmetry of the stress tensor, σij , and strain tensor, ij , implies the relations Mijkl = Mjikl = Mijlk. This reduces the number of independent coeﬃcients in the compliance tensor from 81 to 36, and makes possible the use of a contracted matrix notation:

i = Mij σj (i, j = 1, 2, ..., 6), (2.17) as explained in Nye [14]. The compliance tensor may be written in terms of the internal energy Uint per unit volume: 2 ∂ Uint Mijkl = . (2.18) ∂σij ∂σkl

This implies Mijkl = Mklij or, in the contracted notation, Mij = Mji, reducing the number of independent coefficients to 21. Further conditions are imposed by the symmetry of particular crystal classes. There remain 21 independent coefficients in triclinic crystals, but only 5 in hexagonal and 3 in cubic crystals. For a cubic crystal, in terms of these three coefficients M11, M12, M44, the stiffness Elmn in a direction described by direction cosines l, m, n is given by

1 1 = M 2 M M M l2m2 + m2n2 + l2n2 . (2.19) E 11 − 11 − 12 − 2 44 lmn 2 2 2 2 2 2 This result is derived in Appendix A. The quantity Almn = l m + m n + l n is called the cubic elastic anisotropy factor. Note that if

2 (M M ) = M (2.20) 11 − 12 44 then the stiffness has no dependence on direction and thus the material is elastically isotropic. The three elastic constants are then not independent; thus only two constants are required to specify the stiffness of an elastically isotropic material. Commonly, the definitions of Young’s

2 The compliance tensor is commonly denoted by Sijkl, and the stiﬀness tensor by Cijkl. However, the Eshelby tensor is also commonly denoted by Sijkl; hence the alternative symbols Mijkl and Lijkl are used in this dissertation to represent the compliance and stiﬀness respectively. Chapter 2. Internal Stress Development in Single and Two Phase Materials 24

modulus and Poisson’s ratio, or the bulk and shear moduli, are used. However, any pair of constants may be expressed in terms of any other pair. The degree of anisotropy for a cubic crystal may be deﬁned as [15]

2 (M11 M12) A0 = − (2.21) M44 such that a value of unity implies isotropy. The greater the deviation from unity, the more anisotropic the material. Cubic crystals vary greatly in their degree of anisotropy: typical values of A0 are 1.0 for tungsten, 1.2 for aluminium, 2.5 for bcc iron, and 3.0 for fcc steel [16].

2.4.2 Voigt and Reuss Elastic Averages The pioneering work in the field of predicting the overall elastic properties of a heterogeneous aggregate from those of its constituents was performed by Voigt [17] and Reuss [18]. The Voigt approximation assumes the strain in all crystallites or phases is the same, equal to the macroscopic average, ij . If the solid occupies a domain D with volume V , the average stress is σ = kl L dD (2.22) ij V ijkl ZD where in the case of polycrystals, the elastic stiffness Lijkl varies spatially because of the different orientations of crystallites. Conversely the Reuss approximation assumes uniform stress, σij throughout. The average strain is then σ = kl M dD . (2.23) ij V ijkl ZD Neither approximation is realistic, since the Voigt model implies discontinuity of stress and the Reuss model discontinuity of strain at phase or crystallite boundaries. Hill [19] proved that if Lijkl and M ijkl are the average stiffness and compliance respectively (i.e. σij = Lijklkl and ij = M ijklσkl), then

1 1 Lijkl Lijkl dD and M ijkl Mijkl dD . (2.24) ≤ V ≤ V ZD ZD Thus the Voigt and Reuss approximations are upper and lower bounds respectively to the aggregate stiﬀness. Hill suggested the arithmetic mean of the Voigt and Reuss elastic constants as a good empirical estimate.

2.4.3 Eshelby-Based Approximations Better approximations are based upon the Eshelby equivalent inclusion method described in 2.3.2. Eshelby himself proposed a method for dilute composites in the same paper in which he § introduced the equivalent inclusion idea. He evaluated the energy change UI per unit volume of inclusion when an ellipsoidal heterogeneous inclusion is introduced into an inﬁnite homogeneous A A solid under applied stress σij (and corresponding strain ij ). This evaluates to

1 A U = σ ∗ (2.25) I 2 ij ij where ij∗ is the equivalent inclusion eigenstrain, as discussed in 2.3.2. For dilute composites, it is assumed there is no interaction between inclusions, and th§us if unit volume of material contains N inclusions each of volume V , the total internal energy per unit volume is

1 A A NV A 1 A A U = σ + σ ∗ = σ + f∗ (2.26) int 2 ij ij 2 ij ij 2 ij ij ij Chapter 2. Internal Stress Development in Single and Two Phase Materials 25

A A where f = NV is the volume fraction of the inclusions. Expressing ij + fij∗ in terms of σij (see 2.3.2), the compliance may be evaluated from (2.18). §

Mean Field Method

The mean field theory introduced by Brown and Stobbs [20] and Mori and Tanaka [21] offers a method for the evaluation of composite stiffness that is not limited to low inclusion volume fractions. Consider first a homogeneous solid of stiffness LM (under zero applied stress). A volume fraction f of homogeneous ellipsoidal inclusions develop uniform eigenstrain ∗. The resulting average stress in the matrix and inclusions are denoted σ M and σ I respectively. Stress balance (equation (2.1)) gives h i h i

f σ + (1 f) σ = 0. (2.27) h iI − h iM Now imagine inserting another inclusion randomly within the matrix. On average, the stress in the new inclusion (or in any inclusion, since they are equivalent) is given by the superposition of the average matrix stress and the stress σ∞ inside a single transformed inclusion within an inﬁnite medium (equation (2.3)):

∞ σ = σ + σ . (2.28) h iI h iM Solving (2.27) and (2.28) simultaneously gives

∞ σ M = fσ , (2.29) h i − ∞ σ = (1 f) σ . (2.30) h iI − Using (2.3) and Hooke’s Law, the average strain in the matrix is

∗ = f (S I) . (2.31) h iM − − If external stress σA is now applied, the stress within the inclusion is given by

A ∞ A σ + σ = σ + σ + σ . (2.32) h iI h iM The total strain in the inclusion is correspondingly

c A + + (2.33) h iM A 1 A c where = LM− σ and is the constrained strain of the single inclusion problem. As in the single inclusion problem, if ∗ is chosen correctly, the stress/strain state of the inclusion can be made to match that of an inhomogeneous inclusion of stiﬀness LI . In the inhomogeneous case, however, the strain is developed fully elastically, without eigenstrain. Equating the inclusion stress in both cases therefore gives

c A c A ∗ LI + + = LM + + . (2.34) h iM h iM − This is known as the equiv alency condition. From(2.31) and (2.2), the eigenstrain ∗ is solved in terms of σA: ∗ A = Zσ (2.35) where 1 1 Z = [(1 f) (S I) (LI LM ) + LI ]− I LI LM− . (2.36) − − − − Similarly, we can imagine replacing all the other homogeneous inclusions with such inhomogeneous inclusions, leaving the average stress and total strain in the matrix and inclusions unchanged. It can be shown [5] that if a volume fraction f of a homogeneous body develops Chapter 2. Internal Stress Development in Single and Two Phase Materials 26

eigenstrain ∗, then the average strain is given by h i ∗ = f . (2.37) h i Hence, together with the strain A which develops in the homogeneous body due to the applied stress, the average strain in the stressed homogeneous body (and hence also in the inhomogeneous body) is A ∗ = + f . (2.38) h i Using (2.36), this is expressed 1 A = LM− + fZ σ . (2.39) h i Hence the composite elastic stiﬀness is given by

1 1 L− = LM− + fZ. (2.40)

Note that Z = 0 when LI = LM , and therefore in this limiting case the composite stiffness is correctly calculated as the matrix stiffness. This is also the case when f = 0. The method is easily generalised to a composite containing many different types of inclusion, by associating a different eigenstrain with each type.

Self-Consistent Method The mean field method is based upon the embedding of inclusions within a matrix phase. A polycrystal, however, does not possess such a matrix. An alternative averaging scheme which is directed more towards the polycrystal case is the self-consistent method, first proposed by Kröner [22]. In this approximation, the embedding medium is assumed to have the moduli not of a matrix phase, but of the solid as a whole. This average stiffness L relates the average stress σ to the average strain :

σ = L. (2.41)

It is to be found from the individual component stiffnesses, where the r-th component has stiffness Lr. In the polycrystal case, the components are single crystals, and Lr varies according to the orientation of the r-th crystallite. Consider a crystallite of the r-th type. The crystallite is assumed to be an ellipsoid. Its stiffness Lr differs from that of the surrounding (infinite) medium which is assumed to have stiffness L. This is simply the problem of the ellipsoidal inhomogeneity, for which the solution is given in 2.3.2. The strain in the inclusion r is related to the strain according to equation (2.41). That§ is r = Ar, (2.42) where Ar is found in terms of the Eshelby tensor of the ellipsoid, using equations (2.10) and (2.13). The stress σr in the inclusion is then

σr = LrAr . (2.43)

Taking the average over all components,

σ = σr = LrAr . (2.44) h i h i Hence, from (2.41), L = LrAr . (2.45) h i This is a non-linear, implicit equation, because Ar is dependent on L. However, it may be solved via an iterative technique. An initial estimate of L is chosen, such as the Hill-proposed average of the Voigt and Reuss moduli. This is used to calculate Ar, which is substituted along Chapter 2. Internal Stress Development in Single and Two Phase Materials 27

with the estimate of L into the right-hand side of the equation, to determine a new estimate of L. Iteration proceeds until the right and left-hand sides coincide to a speciﬁed accuracy; i.e. until self-consistency is achieved.

Comparison of the Mean Field and Self-Consistent Methods

Hashin and Shtrikman [23, 24] presented rigorous bounds to the elastic properties of macroscopically isotropic heterogeneous aggregates, based on a variational principle. The mean field and self-consistent estimates lie within these bounds [25]. Wakashima and Tsukamoto [26] demonstrated this for the mean field method for the case of a composite constituting an isotropic matrix containing one type of randomly oriented isotropic inclusions. They showed that if the inclusions are stiffer than the matrix, then the mean field prediction coincides with the lower bound in the case of spherical inclusions, and the upper bound in the case of randomly oriented flat discs. Conversely, the predictions coincide with the opposite bounds if the inclusions are more compliant than the matrix. Dvorak and Srinivas [25] have demonstrated that there is a formal similarity between the two methods. They differ only in the selection of the medium in which the inclusions are embedded: the matrix phase in the case of the mean field estimate, and the effective medium in the self- consistent estimate. Moreover, the Hashin-Shtrikman bounds may be derived via a similar formulation, by selecting the stiffness of the embedding medium according to certain criteria [27, 28]. Other choices of embedding medium may be chosen, leading to overall stiffnesses which do not violate the bounds. Dvorak and Srinivas suggested some such choices. It is seen then that the embedding medium provides a mechanism for the estimation of elastic properties, while not necessarily having a basis in reality. This may be demonstrated by consideration of equation (2.40) when f = 1. In this case, the embedding matrix is purely fictitious. The composite stiffness is correctly predicted to be the heterogeneous inclusion stiffness, L = LI , with the matrix stiffness, LM , dropping completely from the expression. Noting this role of the embedding medium, it may be argued that, although formulated for composites, the mean field method may also be applied to the case of a polycrystal. It should be noted that in certain cases the self-consistent method does not produce physically reasonable estimates [25, 29]. The estimates of the bulk and shear moduli of a material containing voids become zero when the volume fraction of voids exceeds 0.5. Furthermore, a material is predicted to become completely rigid when a volume fraction of rigid particles greater than 0.4 is introduced. In these cases, the mean field approximation provides more reasonable estimates. The bulk and shear moduli fall to zero only when the volume fraction of voids becomes unity, and the solid becomes rigid only when the volume fraction of rigid spheres becomes unity [25].

2.5 Plastic Deformation of Single Crystals

As illustrated in Fig. 2.3, heterogeneous plastic ﬂow causes internal stress. The heterogeneity may be between grains of a single phase, causing intergranular stress, or between phases, causing interphase stress. In the experimental part of this dissertation, both of these types of internal stress are investigated. While the analysis of interphase stress may be approached by treating the matrix phase as a continuum, the understanding of intergranular stress necessarily requires consideration of the deformation mechanisms of the constituent crystallites. This forms the subject of this section. The term plastic is used in this section to describe all mechanisms which lead to strain which is not recovered upon the removal of load. Although the plastic deformation of single crystals is a widely studied ﬁeld, the purpose of this section is only to highlight essential information relevant to the subjects of intergranular and interphase stress. Chapter 2. Internal Stress Development in Single and Two Phase Materials 28

2.5.1 Slip in Single Crystals Slip is the most common form of non-elastic deformation found in metallic crystals. It is a shear deformation, in which the passage of a dislocation causes one crystallographic plane to move over another. The crystal lattice is undisturbed by the passage of a dislocation and slip is therefore termed lattice invariant. The shear stress required to move a dislocation increases exponentially with falling interplanar spacing [30], and therefore slip tends to occur on the closest packed planes, which have the highest interplanar spacing. Moreover, the stress increases with magnitude of the Burgers vector of the dislocation, and therefore the directions of slip are usually the closest packed directions within the planes.

Slip Systems The crystallographic planes and directions upon which slip occurs depend on the crystal structure. Since face-centred cubic (fcc) metal crystals are close-packed, slip occurs on the close- packed planes in close-packed directions, i.e. on the 12 111 110 slip systems. { } The situation is more complicated in body-centred cubic (bcc) crystals, since there are no close-packed planes. Slip always occurs in the close-packed directions, 111 , but a range of slip planes have been reported, the most common being 110 , 112 and 123h i[31, 32]. Taylor and Elam proposed the mechanism of pencil-glide in whic{ h }an{y plane} con{taining} a 111 direction can act as a slip plane [33, 34]. h i In an ideal hexagonal close-packed (hcp) crystal, only the (0001) basal planes are close- packed and this is indeed the most common slip plane in hcp metal crystals. However, the axial ratio of lattice parameters c/a varies markedly from the ideal value in real hcp metal crystals. As it becomes lower, other planes such as the 1010 prismatic and 1011 pyramidal planes become more close-packed relative to the basal planes,{ }and thus more fa{vourable} for slip. In zinc and cadmium the axial ratio is greater than ideal, and single crystals of these metals deform almost exclusively on the basal plane. In magnesium the ratio is very close to ideal, and slip on non-basal planes is observed, although basal slip predominates. The axial ratio of titanium is lower than ideal, and in the pure metal prismatic slip predominates. Slip is not the only non-elastic deformation mechanism in hcp metals. The limited number of operable slip systems causes mechanical twinning also to be an important mechanism. This is considered in 2.5.2. §

Schmid’s Law The above discussion of slip systems emphasises an important point about plasticity in single crystals: it is highly anisotropic, depending on the direction of the applied stress. Slip will occur on a particular plane in a particular direction if the shear stress resolved on this plane and direction reaches a critical value τc. This is Schmid’s Law. It is most generally stated in tensor notation. Consider the two sets of orthogonal axes shown in Fig. 2.5, X and X 0. The X set speciﬁes the external reference frame in which the components of stress σij are speciﬁed. The X0 axes are chosen such that 10 is parallel to the slip direction b and 30 is parallel to the slip plane normal, n. Then slip occurs when

σ130 = τc (2.46) where σij0 is the stress tensor in the X 0 reference frame. Following the tensor transformation law [14], σ130 = a1ia3j σij , (2.47) where aij is the direction cosine between the i0 and j axes. However, expressing the unit vectors b and n in the X system, bi = a1i, nj = a3j , (2.48) Chapter 2. Internal Stress Development in Single and Two Phase Materials 29

Schmid’s Law is written as binj σij = τc . (2.49) Using the symmetry of the stress tensor, this is sometimes alternatively written as 1 τc = binj σij = (binj + bjni) σij αij σij (2.50) 2 ≡ where the symmetric tensor αij is called the Schmid tensor.

In the case of uniaxial tension along the 3-axis, only the σ33 component of stress is non-zero. In this case, τc = cos λ cos φ σ m σ (2.51) 33 ≡ 33 where cos λ and cos φ are the direction cosines between the tensile axis and the slip direction and slip plane normal respectively. The slip system which requires the lowest tensile yield stress to operate is that with the highest Schmid factor, m.

3 2 X

1 30, n 20

X0 10, b

Figure 2.5: Diagram for the calculation of resolved shear stress on a slip system.

Yield Surface

Note that if the applied stress is hydrostatic, i.e. σij = σδij , then

binj σij = σbini = 0. (2.52)

This shows that only deviatoric stresses cause slip. A general stress can be represented as a vector in a six-dimensional space (one dimension for each independent component). However, deviatoric stresses may be represented in a five-dimensional section of this space, in which σ11 + σ22 + σ33 is constrained to be zero. Just as the equation ax + by + cz = d defines a plane in three dimensions, Schmid’s law (2.49) defines a plane, or more accurately hyperplane, in this deviatoric stress space. There is such a hyperplane for every slip system, and these intersect one another. If there are sufficient slip systems, the hyperplanes enclose a volume within a polyhedron. The surface of this polyhedron is the yield surface. Stresses represented by vectors which lie on this surface lead to slip; stresses represented by vectors which lie below the surface do not. Chapter 2. Internal Stress Development in Single and Two Phase Materials 30

Strain due to slip The shear strain when a system slips may be related to the strain observed in the external reference frame via the tensor transformation law. Using the slip system axes deﬁned in Fig. 2.5, the distortion when the crystal develops a shear strain γ on the speciﬁed system is

0 0 γ dij = 0 0 0 . (2.53)  0 0 0    The symmetric part of this distortion is the strain:

0 0 1 γ = 0 0 0 . (2.54) ij 2  1 0 0    Since in the slip system reference frame bi = (1, 0, 0) and ni = (0, 0, 1), the strain may be written γ = (b n + b n ) = γα . (2.55) ij 2 i j j i ij This is written in general tensor form, and therefore is true in any reference frame, as long as bi and ni (or αij ) are expressed in that frame. Using the symmetry of the stress tensor, (2.49) and (2.55) give

ij σij = γτc . (2.56)

By comparison to the equation for a plane r.n = C, this shows that if ij is expressed as a vector in deviatoric stress space (normalized by the appropriate dimensions), it is normal to the σij hyperplane.

Grain Rotation In uniaxial tension, as a crystal deforms, the slip direction rotates towards the tensile axis. The rotation of the crystal lattice relative to the external reference frame can also be described in tensor notation. The antisymmetric part of the distortion (equation (2.53)) is the rigid body rotation of the crystal relative to the external axes:

0 0 1 γ Ω = 0 0 0 . (2.57) ij 2  1 0 0  −   Similarly to (2.55), this may be written in general tensor form as γ Ωij = (binj bj ni) . (2.58) 2 − Grain rotation plays an important role in the development of crystallographic texture during mechanical deformation of polycrystals. The specification of the orientation of a grain relative to the external axes is important for a description of texture. It is worth introducing here a common system for this specification. In general, a crystal orientation may be obtained from three rotations of the external reference frame, specified by three Euler angles. These angles are commonly defined in the system attributed to Bunge [35]. The reference frame is first rotated by ϕ1 about the z-axis. It is then rotated by Φ about the new x-axis. These two rotations specify the z-axis of the crystal system, but a further rotation ϕ2 about this axis is required to define the orientation of the crystal x- and y-axes. Thus, using the tensor transformation law in matrix notation, the representation of a second rank tensor T0 in the crystal reference frame Chapter 2. Internal Stress Development in Single and Two Phase Materials 31

is given by its representation T in the external frame as

T T0 = aTa (2.59) where the rotation matrix a is given by

cos ϕ2 sin ϕ2 0 1 0 0 cos ϕ1 sin ϕ1 0 a = sin ϕ2 cos ϕ2 0 0 cos Φ sin Φ sin ϕ1 cos ϕ1 0 . (2.60)  − 0 0 1   0 sin Φ cos Φ   − 0 0 1  −       Work Hardening As a crystal is strained, dislocations are generated, and the resulting dislocation interactions raise the critical resolved shear stress for slip. This is a primary cause of work hardening. The rate of increase with shear strain of τ s, the critical resolved shear stress on the i-th slip system, may be parameterized by dτ i = hij dγj . (2.61) j X This expression reflects latent hardening; i.e. the fact that a system may be work hardened by slip on other systems. A range of hardening behaviours can be specified by the hardening matrix, hij . For example, self-hardening only can be specified by hij = θδij , where δij = 1 if i = j and δij = 0 if i = j. Isotropic hardening is specified by hij = θ. In these expressions, the parameter θ is dependen6 t on the accumulated slip, to reflect that the hardening rate varies during straining.

2.5.2 Deformation Twinning A pair of twins consists of two adjacent regions of crystal, the orientations of which are related by a symmetry operation with respect either to a mirror plane or rotation axis [36]. Such adjacent regions commonly develop during crystal growth. For example, twins are seen in the microstructure of austenite shown later in this dissertation (Fig. 5.5), characterised by straight boundaries. The orientation of one twin may be generated from the other by a uniform shear. It is possible for some crystals to deform by such a shear, generating twins mechanically [37, 38]. Like slip, this deformation mode occurs by shearing in particular directions on particular crystallographic planes [39]. However, there are important differences between the two mechanisms. Shear strain may be acquired gradually during slip. In contrast, the generation of a twin causes a rapid reorientation of the lattice and hence change in strain. In a constant strain rate tensile test, this causes sudden stress relief which is manifested in the flow curve as characteristic serrations. The abrupt formation of twins may also give rise to audible sounds. This is the origin of the “tin cry” heard during the deformation of tin. Another important difference between slip and twinning is directionality. Since slip leaves the lattice unchanged, it is equally favoured with respect to the sense of the shear along the slip direction. This means that the yield surface due to slip is centrosymmetric about the origin. Twinning, however is inherently directional. This is illustrated in Fig. 2.6, which shows a (111) close-packed plane in a face-centred cubic metal crystal. The atoms are centred on the positions marked A. The atomic layer contains two sets of holes, marked B and C. Originally the atoms in the next layer above are positioned above the holes marked B. When a twin is generated, this layer is displaced to the C positions, by the passage of a partial dislocation with Burgers vector 1 1 6 211 . However, displacement in the opposite direction, 6 21 1 , is not possible, because the atoms would then sit directly above those in the first layer. A consequence of this directionality is that if twinning is taken into account, the single crystal yield surface is not necessarily centrosymmetric. This can affect the yield surface of Chapter 2. Internal Stress Development in Single and Two Phase Materials 32

y x

1 Burgers vector: 6 211

A A A 101 C C 011 B B B A A A A C C C B B B B A A A A A

110 Figure 2.6: Illustration of movement of atomic layers during twinning in fcc crystal, demonstrating that twinning is directional with respect to the sense of shearing. Chapter 2. Internal Stress Development in Single and Two Phase Materials 33

a polycrystal, if the constituent crystallites have a preferred orientation. For example, the yield stress of wrought magnesium is much greater in tension that in compression, due to the inﬂuence of deformation twinning coupled with strong crystallographic texture [40]. As will be discussed in 2.6, plastic deformation in polycrystals necessarily requires the operation of multiple deformation§ modes. Therefore, twinning is an important deformation mechanism in polycrystals in which the number of operable slip modes is restricted. As stated earlier, this is particularly the case in hcp metals, such as magnesium, titanium and zinc. Deformation twinning also plays an important role in the thermomechanical behaviour of shape memory alloys. For a discussion of this, see [4].

2.5.3 Martensitic Transformation Martensitic transformation (hereafter denoted MT) is related to deformation twinning, in that it occurs by the cooperative movement of adjacent atoms. While twinning simply causes reorientation of the crystal lattice, however, MT involves a change in crystal structure. It is therefore a phase transformation, but one that does not require atomic diffusion. MT occurs in a wide range of materials, and the name martensite is given to any phase which forms by such a transformation. A good account of the characteristics of martensites, and the origins and theories of transformation, is given in the book by Nishiyama [41]. As with other phase transformations, MT occurs because the product phase has lower free energy than the parent phase. Upon cooling below the equilibrium temperature, a driving force for transformation is generated. If cooling is slow, a diffusive transformation may occur. How- ever, if cooling is too rapid relative to the diffusion timescale, then MT occurs. For a particular material, the transformation begins at a well-defined temperature Ms. This is necessarily below the equilibrium temperature, since an energy barrier due to interfacial and strain energy must be overcome in order for the new phase to nucleate. The formation of martensite was originally observed by rapid cooling; specifically, during the quenching of steels [41]. However, in the context of this section, it is important to empha- sise another mechanism by which transformation may be induced in some materials; namely, by the application of stress. An applied stress field alters the free energy change, and hence may promote transformation above Ms. The underlying reason for this is that transformation is accompanied by a shape change. The free energy change therefore contains a term to account for the work done by the externally applied load. Uniaxial tensile stress, for example, will encourage the formation of martensite variants which elongate along the tensile axis. Hydrostatic stress may also induce or inhibit transformation, since in general the phases have different den- sities. For example, in carbon steels, the transformation temperature is reduced with increasing pressure, because the material dilates upon transformation, and thus has to do work against the external force [42]. Since stress-induced MT causes strain, it is viewed in the context of this section as a single crystal deformation mechanism. However, whether or not this deformation may be viewed as plastic depends upon the type of material. TRIP steel [43], which is studied in Chapter 5, exhibits stress-induced MT, but the reverse transformation is not observed upon unloading. Therefore the macroscopic strain due to transformation remains after unloading, and the deformation may be regarded as plastic. However, in another technologically important class of materials, the large strains which develop under applied stress are fully recovered during unloading. Such materials are termed superelastic. The origin of the effect is a reversible stress-induced transformation. An example of the stress-strain curve of a superelastic single crystal (Cu-Al-Ni) is shown in Fig. 2.7. Initially, the crystal deforms elastically. Then, at a certain critical stress, MT begins. In the example shown, the crystal continues to deform without additional stress, to a strain of approximately 6%. When transformation is complete, the stiffness increases sharply, since only further elastic deformation is possible. Upon unloading, the crystal exhibits a small hysteresis, and the strain returns to zero. Alloys which exhibit superelasticity also exhibit the shape memory effect, which is a closely related phenomenon. Chapter 2. Internal Stress Development in Single and Two Phase Materials 34

For a recent account of the research ﬁeld of shape memory materials, the reader is referred to the text edited by Otsuka and Wayman [4].

400

350

300

250

200

150 Tensile Stress [MPa] 100

0 0 1 2 3 4 5 6 7 Strain [%]

Figure 2.7: Tensile stress-strain curve of a Cu-Al-Ni single crystal, exhibiting superelasticity [44].

2.6 Plastic Properties of Polycrystals

Models of polycrystalline plasticity are based upon the properties of the single crystal constituents described above. While single crystals tend to be tested in simple shear or tension, however, the stress states within the grains of a polycrystal will be more complicated, due to the anisotropy and the variation in grain orientations. Although, for completeness, the single crystal deformation modes of twinning and martensitic transformation were presented in 2.5, the polycrystalline models presented in this section are directed primarily towards slip deformation§ which, as stated previously, is the most common form of single crystal plastic deformation.

2.6.1 Sachs model The earliest model of polycrystalline plasticity, presented by Sachs [45], proposed that the tensile yield stress of the polycrystal is the average of the constituent single crystal yield stresses. Averaging equation (2.51) gives for the yield stress

σy = 1/m τc (2.62) h i

where m refers to the highest Schmid factor in each grain, and τc is assumed the same for all systems. In a randomly oriented fcc polycrystal, for example, 1/m = 2.24. By implementing a hardening law, the Sachs model may also be used to predictha plastici ﬂow curve [46]. The Sachs model assumes stress is partitioned among grains in proportion to their yield stress, which is physically unreasonable and not consistent with stress equilibrium. Another assumption implicit in (2.62) is that slip occurs on one slip system only. This causes incompatibility between grains.

2.6.2 Taylor model Taylor introduced a model to satisfy the compatibility requirement [47]. He achieved this by assuming the plastic strain is the same in all grains. This model, then, is analogous to the Voigt Chapter 2. Internal Stress Development in Single and Two Phase Materials 35

model of elasticity. If an applied uniaxial stress σ33 causes a small elongation of strain d along P the 3-axis, with symmetry about this axis, then the plastic strain dij is

1 2 0 0 dP = d −0 1 0 . (2.63) ij  2  0 −0 1   P Note that the trace dii = 0. Since the trace is invariant, this is true in all reference frames, and reﬂects the fact that volume is conserved in slip deformation. The work done by the external force (per unit volume) is dw = σij dij = σ33d . (2.64) This must equate to the work done on all slip systems within all grains in the unit volume. If τ s and dγs are respectively the resolved shear stress and shear strain increment on the s-th system, this work is given by

dw = τ sdγs (2.65) * s + X where denotes the average over all grain orientations. Equating (2.64) and (2.65), and hi s assuming the critical shear stress is the same for all systems, i.e. τ = τc, the yield stress is given as s s dγ σy = τc h i τcM, (2.66) d ≡ P similar to (2.62). M is known as the Taylor factor. Its evaluation requires the determination of the shear strain increments dγs in each grain. This is achieved by equating the strain due to slip on all systems to the macroscopic strain. From equation 2.55, the total strain due to slip is

P s s dij0 = dγ αij . (2.67) s X Using (2.59) and (2.60), when transformed into the crystal system reference frame speciﬁed by Euler angles ϕ1, Φ, ϕ2, the macroscopic strain (2.55) has components

2 2 2 3 sin Φ sin ϕ2 1 3 sin Φ sin ϕ2 cos ϕ2 3 sin Φ cos Φ sin ϕ2 P d 2 − 2 2 d 0 = 3 sin Φ sin ϕ cos ϕ 3 sin Φ cos ϕ 1 3 sin Φ cos Φ cos ϕ . (2.68) ij 2  2 2 2 2  3 sin Φ cos Φ sin ϕ 3 sin Φ cos Φ cos−ϕ 3 cos2 Φ 1 2 2 −   (Note that this expression has no dependence on ϕ1, since this represents a rotation about the external z-axis, about which the strain is symmetric.) Equating (2.67) and (2.68) gives five independent equations for the unknowns dγ s: one for each independent component of strain (11, 22 and 33 are not independent, since ii = 0). A problem arises in that there may be more than five slip systems (e.g. 12 in fcc crystals) and thus the dγs are under-determined. Physically, this means there are many combinations of slip on different systems capable of producing the strain. The work done (and hence yield s stress) will, however, be lowest when the accumulated slip, Γ = s γ , is minimised. Taylor argued that this would correspond to slip on the minimum number of systems required to accommodate a general strain, i.e. five. Therefore, the shear strainsP can be computed by determining the combination of five independent slip systems which give rise to the minimum accumulated slip (for fcc crystals, there are in fact several combinations of slip systems which produce the minimum value — this is known as the Taylor ambiguity problem [48]). In this manner, Taylor computed M = 3.06 for a randomly oriented fcc polycrystal. In comparison to the Sachs prediction:

σy = 2.24τc (Sachs), Chapter 2. Internal Stress Development in Single and Two Phase Materials 36

σy = 3.06τc (Taylor). (2.69)

The Sachs model provides a lower bound to the polycrystal yield stress, while the Taylor model provides an upper bound. Taylor assumed isotropic hardening. Therefore, the critical resolved shear stress will increase by the same amount for a given increment of accumulated slip dΓ whether this is developed on one slip system or five. Taylor could thus predict the polycrystal flow curve from a simple shear test on a single crystal. From the single crystal shear stress - shear strain curve (τ, γ) the polycrystal uniaxial flow curve is predicted as (σ, ) where σ = Mτ and = γ/M. The prediction was in reasonable agreement with the experimental flow curve of an aluminium polycrystal.

2.6.3 Bishop and Hill Analysis Bishop and Hill performed an equivalent analysis to Taylor by considering the stress states required to activate five slip systems [49, 50]. Assuming all systems have the same critical resolved shear stress, the yield polyhedron (see 2.5.1) of a fcc single crystal has 56 vertices. It is only at these vertices that the hyperplanes of §at least five systems meet. Thus there are only 56 discrete deviatoric stress states which meet the Taylor compatibility criterion for polycrystal slip. In fact, at each vertex, the yield hyperplanes of either six or eight systems meet – this explains the Taylor ambiguity problem noted above. A vertex stress state can lead to a range of possible strains, bounded by the normals to the hyperplanes which meet at the vertex, as P explained in 2.5.1. For a prescribed dij (equation (2.68)), however, the vertex stress state σij∗ § P which produces it is that which maximises the plastic work done, σij∗ dij . Bishop and Hill used this fact to determine which vertex stress state operates, and thus the work done in straining, as a function of crystal orientation. The yield stress follows from finding the average work done over all crystal orientations and applying equation (2.65). Although this approach seems very different from Taylor’s analysis, it is in fact mathematically equivalent. Models based on Taylor’s theory have proved reasonably successful in the prediction of texture development [51]. However, the faceted single crystal yield surface and the resulting indeterminacy of slip system selection causes problems in numerical implementations. For this reason, strain rate sensitivity of slip is usually invoked: that is, the concept of the critical resolved shear stress – at which shearing begins and may occur at any rate – is replaced by the assumption that shear rate varies smoothly with resolved shear stress [52]. Strictly, the concept of the yield surface is then no longer well defined, and must be replaced by the more sophisticated concept of the plastic potential (for an explanation, see the book by Kocks, Tomé and Wenk [51]). Effectively, however, the consequence of the introduction of rate sensitivity is to smooth the sharp vertices of the yield surface into rounded corners, removing the ambiguity problem and allowing a stable numerical formulation. This is the primary motivation for its incorporation into models of texture development, but the phenomenon does indeed have a physical basis.

2.6.4 Elastoplastic Self-Consistent Model Although the Taylor model ensures compatibility by assuming homogeneous straining, this causes discontinuities of stress at grain boundaries which are not consistent with the requirement for equilibrium. In reality, the operation of five slip systems in all grains is rarely observed [53]. This is not inconsistent with the requirement for compatibility, because plastically soft grains may help to accommodate neighbouring hard grains through extra deformation. Another flaw in both the Sachs and Taylor models is that they are rigid-plastic, taking no account of the elastic properties of the crystallites. It is clear that elastic anisotropy must play a role in the early stages of plasticity, because grains oriented with a stiff direction parallel to the tensile axis will bear a greater proportion of the applied load. The importance of this effect shall be Chapter 2. Internal Stress Development in Single and Two Phase Materials 37

discussed in the experimental part of this dissertation ( 4.9.3). § An alternative model based on Eshelby’s equivalent inclusion theory is the elastoplastic self- consistent (EPSC) model, which is a generalisation of the elastic self-consistent model described in 2.4.3. The model is described in some detail here, because it is employed in Chapters 4 and§ 5. The first self-consistent model of polycrystalline plasticity was introduced by Kröner [54] and Budiansky and Wu [55]. Hutchinson used this formulation to predict the tensile flow curves of fcc and bcc polycrystals [56, 57]. This model does not require five slip systems to be active in each grain, but assumes isotropic elasticity. Hill subsequently presented a treatment in which elastic anisotropy was accounted for [58]. With this improvement, and with increases in computational power, the Hill model has become useful in the prediction of internal stress in polycrystals. Its validation through neutron diffraction will be reviewed in Chapter 3. Turner et al. [59] and Clausen [46] have written computer implementations of the Hill model, following the formulation by Hutchinson [60]. A summary of this formulation is given here. The basis of the Hill model is the elastic self-consistent model described in 2.4.3. However, in order to account for plasticity, the additional concept of instantaneous moduli§ is introduced. That is, at any stage in the deformation, a small increment in the average stress is related to the resulting increment in the average total (as opposed to elastic) strain by the instantaneous stiffness tensor L. That is σ˙ = L˙ (2.70) where σ and are the average (hence macroscopic) stress and strain in the polycrystal. A dot above a variable indicates the rate of change of that variable. However, note that the elastoplastic model is not time-dependent; hence the notation is simply employed as a shorthand to describe how an increment in one variable relates to an increment in another. The instantaneous stiffness L is represented in calligraphic type to distinguish it from the elastic stiffness, L, which relates stress to elastic strain.3 In the elastic regime, the instantaneous and elastic stiffnesses are identical. However, when plasticity occurs, the material effectively becomes much more compliant. Thus the instantaneous stiffness falls dramatically. The elastic stiffness, however, does not change (in practice, factors such as changing grain shape and grain rotation may alter the elastic stiffness slightly, but these factors are neglected in the model). Similarly, the single crystal instantaneous stiffness Lr relates the increments of stress and total strain in the r-th type of crystallite:

σ˙r = Lr˙r . (2.71)

The individual crystallites are again assumed to be ellipsoidal, and to interact with an (inﬁnitely- extending) homogeneous eﬀective medium (HEM) which has the average properties of the bulk solid. Consider the early stages of deformation, when the system is fully elastic. Equation (2.42) relates the elastic strain in a crystallite to the average elastic strain of the body. Moreover, since the equation is linear, the strain rates are also related by the tensor Ar. That is,

˙r = Ar˙ . (2.72)

Now consider the material during plastic deformation. At any stage, it is possible to conceive of a fully elastic (albeit very compliant) material with the same distribution of crystallites as the elastoplastic material, and with single crystal elastic stiffnesses Lr which match the instantaneous stiffnesses Lr of the elastoplastic material. For the fully elastic material, equation (2.72) remains valid. However, for a given increment of stress, the increments of elastic strain which develop in the fully elastic material will be equal to the increments of total strain in the elastoplastic material. Therefore, equation (2.72) correctly relates the increment of total strain in a crystallite to average total strain in the elastoplastic material, if the tensor Ar is calculated using the instantaneous, rather than elastic stiffnesses. Specifically, from equations (2.13) and

3Note: this is the opposite of the notation used in Hutchinson’s paper [60]. Chapter 2. Internal Stress Development in Single and Two Phase Materials 38

(2.10), r r ∗ 1 ∗ A = (L + L )− (L + L ) (2.73) where ∗ r r 1 L = L (S I) S − (2.74) − − and Sr is the Eshelby tensor of the crystallite, depending upon its shape and the instantaneous stiﬀness of the bulk medium, L. Commonly, crystallites are assumed all to have the same shape, in which case the Eshelby tensor is the same for all crystallites and is denoted simply by S. Combining (2.71) and (2.72) and averaging gives

σ˙ = σ˙r = LrAr ˙ . (2.75) h i Hence from (2.70), L = LrAr . (2.76) h i Since the right-hand side of this equation depends on L through Ar, this equation is solved by iteration until self-consistency is achieved. In order to proceed with solution of (2.76), an expression for the single crystal instantaneous moduli Lr must be specified. This is achieved using the expressions presented in 2.5 for the description of single crystal deformation. A slip system is only potentially activ§e if the single crystal stress σr satisfies Schmid’s Law (2.50) and the rate of increase of resolved shear stress matches the rate of increase of the critical shear stress due to work hardening. That is, differentiating (2.50), σ˙ rαi = τ˙ i. 4 (2.77) This assumes α˙ i = 0, which implies no grain rotation. Since grain rotation is neglected, the EPSC formulation is restricted to small plastic strains. τ˙ i is given by the hardening law, (2.61). Re-stating this in the present rate notation:

τ˙ i = hij γ˙ j (2.78) j X where the sum is over the active slip systems. The stress rate σ˙ r is related to the elastic strain rate by the elastic stiﬀness Lr. That is,

σ˙ r = Lr (˙r ˙r,p) , (2.79) − where the plastic strain rate, ˙r,p, is given by equation (2.67). In the present notation:

˙r,p = γ˙ jαj , (2.80) j X summing over the active slip systems. Combining (2.77), (2.78), (2.79) and (2.80) gives

αiLr˙r = γ˙ j αiLrαj + hij γ˙ j Xij . (2.81) ≡ j j X X This is a straightforward matrix equation. If the square matrix X ij is non-singular, then it may be inverted to solve for the shear rates on the active systems in terms of the crystal strain rate: i i r i 1 ij j r γ˙ = f ˙ where f = (X− ) α L . (2.82) j X 4αi, τ i and γ˙ i are also speciﬁc to the r-th type of crystallite, but the superscripts are dropped for the purpose of clarity. Chapter 2. Internal Stress Development in Single and Two Phase Materials 39

If Xij is singular, then a diﬀerent combination of potentially active slip systems may be chosen such that it is non-singular. Substituting into (2.80) and then (2.79) gives for Lr:

Lr = Lr I αj f j 5 (2.83)  − ⊗  j X   where I is the identity tensor. It is seen from (2.83) that when no slip systems are active, the instantaneous single crystal stiffness is identical to the elastic stiffness. Having established this relation for Lr, the following iterative procedure may be applied to determine the polycrystal flow curve. At any stage in the calculation, the stress and strain of ∗ each crystallite is known. An initial estimate is made for L, from which S and L are found. Within each crystallite, slip is assumed on a subset of the potentially active slip systems. From this subset, the f i are determined, and hence Lr and Ar. From Ar, the single crystal strain and stress rates are determined, enabling the slip conditions (2.77) to be checked. If they are not satisfied, the procedure must be repeated for a different subset of slip systems. Otherwise, a new estimate for L is determined from (2.76). The whole procedure is repeated until the difference between successive values of L is small enough to be neglected. Having determined the self- consistent instantaneous stiffness, a small strain increment is prescribed, and the stress-strain states of the individual crystallites are updated. A limitation of the EPSC model is that it does not adequately account for the grain size dependence of yield stress. The validity of the Hall-Petch relationship has been established for a wide variety of materials: that is, yield stress is linearly related to the inverse square root of the grain size. This has been rationalised in terms of dislocation pile ups. The larger the grain size, the longer the pile ups and the greater the stress concentration at grain boundaries, promoting yield at lower applied stress. The EPSC model is independent of scale, and does not account for this dependence. Hutchinson stated that grain size dependence is accounted for only by interpreting the critical resolved shear stress as that in situ rather than that expected for an unconstrained single crystal. Since the model is based upon the properties of individual crystallites, this interpretation is not entirely satisfactory. It should be recognised that the EPSC model is statistical, in the sense that it estimates the average response not only of the bulk polycrystal, but of each grain orientation family. In reality, each individual grain does not interact with a homogeneous medium and does not have a simple shape like an ellipsoid. Rather, its stress-strain response is influenced by its true shape and the properties of its neighbours, and is almost certainly not uniform. However, in a real polycrystal there will be many grains of a particular orientation, each with a different shape and different neighbours. The average shape will likely be well approximated by an ellipsoid, and the average properties of the surroundings will equal those of the bulk medium. These considerations justify the use of the model for the prediction of average responses. Diffraction measurements of elastic strain in polycrystals also provide the average over many grains of a common orientation. Therefore the EPSC model is well suited for comparison to experimental data gained in this way. For this reason, the model is applied in Chapters 4 and 5 for comparison with and interpretation of neutron diffraction measurements of intergranular stress.

2.6.5 Crystal Plasticity Finite Element Method The statistical nature of the EPSC model may be regarded as a beneﬁt in that it estimates average intergranular stresses without requiring detailed microstructural modelling. However, this is also a limitation, because the detailed eﬀects of particular neighbouring grains are not accounted for. Therefore, the distribution of stress within a grain orientation family is not determined. An alternative model which may be used to account for near neighbour interactions

5The symbol ⊗ represents the uncontracted tensor product: i.e. a ⊗ b is the abbreviated notation form of aij bkl. Chapter 2. Internal Stress Development in Single and Two Phase Materials 40

is the crystal plasticity finite element method (CPFEM). The finite element (FE) method is well established for the modelling of thermomechanical deformation. The basis of the method is to solve for the response of a complicated body to imposed constraints by dividing it into smaller elements whose responses are more readily evaluated. Each element has a simple shape, such as a triangle or quadrilateral for a 2D problem, or a tetrahedron or “brick” for a 3D analysis. The vertices of the elements (and possibly additional points on the edges) are called nodes; the complete body is formed by a mesh of elements, joined at nodes. Associated with each element is a stiffness which is dependent upon the constitutive law of the material within the element, and in non-linear problems the present state of the material. The element is subjected to a force, dependent upon the element stiffness and the displacements of the nodes upon its boundaries. The overall response of the body is determined by simultaneously solving for the node displacements, so that the conditions of equilibrium and boundary constraints are met. Zienkiewicz and Taylor describe the method of solution in detail [61]. For many engineering stress problems, it is sufficient to assume that the deforming material is homogeneous, and thus to attribute the same constitutive law to all elements. In the CPFEM approach, however, the discretization which is inherent in the FE method is exploited to predict the influence of heterogeneity in the substructure of the material. There are different approaches to this. In some works, each element is regarded as an agglomeration of typically a few hundred differently oriented crystallites. The element response is determined by averaging the single crystal response over all these orientations. This is usually performed using the Tay- lor assumption that all crystallites develop the same plastic strain. This approach is suited to the prediction of texture development during processing procedures such as plane strain forging [62] and sheet forming [63]. For the prediction of internal stress, however, the Taylor model is inadequate because the stress state does not satisfy equilibrium at grain boundaries. The alternative approach is to discretize the material on a scale equal to or smaller than the grain size, and assign a single crystal constitutive law to each individual element. Bate has applied the method to the prediction of intragranular deformation microstructures [64]. Due to the computational expense and the need to discretize a single grain into many elements, the analysis was restricted to a small number of grains in 2D. Sarma and Dawson used parallel computing to improve computational efficiency and power [65]. They performed 3D simulations using brick-shaped elements in a one-brick-per-grain scheme. This scheme allows a larger grain population to be simulated, but clearly intragranular microstructure evolution can not be predicted, and the grain shape is simplistic. Mika and Dawson addressed these issues in an extension to the earlier work [66]. They represented grains by rhombic dodecahedra, discretised into tetrahedral elements, performing simulations of 1099 grains each composed of 48 elements, as well as simulations using fewer, more finely discretized crystallites. This demonstrated that plastic flow is more heterogeneous around the periphery of a grain than in the core, due to the constraint imposed by the surrounding grains. Both the brick and dodecahedral schemes predicted texture development under simple loading conditions more accurately than Taylor-based models. Despite the greater computational cost incurred, the dodecahedral scheme predictions were not discernably superior to the brick scheme predictions. Moreover, the application of such schemes to texture prediction in real forming processes remains computationally impractical at the present time. Dawson, Boyce, MacEwen and Rogge [67, 68] have compared CPFEM predictions of intergranular strain to measurements made by neutron diffraction (to be discussed in Chapter 3). They used the one-brick-per-grain scheme to simulate tensile deformation of a cuboidal sample. The agreement with experimental results is comparable to that achieved by predictions using the EPSC method, and a comparable number of grain orientations are simulated. The CPFEM method has the advantage that it can predict the distribution of strains in a grain family due to the influence of grain neighbours. That is, grains with exactly the same orientation may develop strain differently due to different constraints imposed by neighbouring grains. The EPSC model can not capture this effect since all grains are assumed to have exactly the same environment. Chapter 2. Internal Stress Development in Single and Two Phase Materials 41

The CPFEM work indicates that the effect is significant, the spread in elastic strains being of the order of half the axial strain at the yield point [67]. However, in the brick model each grain shares faces with only six other grains, whereas grains in a real material will typically have a greater number of neighbours. Moreover, the simplistic shape and representation of a grain by a single element restricts its freedom to deform differently to its neighbours. Therefore, the CPFEM model may overestimate the influence of neighbour interactions. An advantage of the EPSC model is that it predicts average intergranular strain development with a much lower degree of computation, which may be performed quickly on any standard desktop computer.

2.7 Plastic Properties of Composites

As noted in the discussion of elastic properties, polycrystals may be regarded as a type of composite. Therefore, much of what has been written in the previous section about polycrystals applies also to plasticity in composites. For example, the EPSC model may be applied to predict the composite flow curve. In the paper in which he presented the formulation described in 2.6.4, Hutchinson applied the method to the case of rigid spheres within a matrix of elastically §isotropic, plastically non-hardening fcc crystallites [60]. However, the analysis of plasticity in composites may be simplified by treating each phase as homogeneous. In many composites this is a reasonable assumption, since properties such as yield stress and elastic stiffness differ much more greatly between phases than between grains of a particular phase. The advantage of assuming each phase to be homogeneous is that the results of the Eshelby and mean field theories can be simply applied. This approach was introduced by Brown and Stobbs [20] and Mori and Tanaka [21]. To illustrate the approach, consider the example of a composite consisting of a random dispersion of fully elastic, spherical inclusions within a ductile matrix. The volume fraction of inclusions is f. Both phases are assumed to be elastically identical and isotropic, with Young’s modulus E and Poisson’s ratio ν. The results of this example are applied in Chapter 4. Firstly consider a single inclusion within an infinite matrix. The presence of the inclusion disturbs the matrix plastic flow. Since no plastic strain develops in the inclusion, a misfit strain ij∗ is generated. Thus, the internal stress is given by the solution of the homogeneous inclusion problem ( 2.3.1). By explicit evaluation of the isotropic Eshelby tensor for the case § of a spherical inclusion [5], the stress σij∞ in the inclusion is given as

E (1 + 5ν) E (7 5ν) σ∞ = ∗ δij − ∗ . (2.84) ij −15 (1 + ν) (1 ν) kk − 15 (1 + ν) (1 ν) ij − − If the matrix plastic strain is assumed uniform, the misﬁt is equal and opposite to the plastic P strain: ij∗ = ij . Then, since the plastic strain conforms to the condition of volume constancy P − ii = 0, the ﬁrst term in equation (2.84) vanishes, so that the inclusion stress components are simply proportional to the corresponding plastic strain components:

E (7 5ν) P σ∞ = − . (2.85) ij 15 (1 + ν) (1 ν) ij − Consider now the complete composite. The mean ﬁeld method addressed in 2.4.3 provides § solutions for the average residual stress in the matrix, σij M , and in the inclusions, σij I , in terms of the solution for the single misﬁtting inclusion. hThati is, repeating equations (2.29)h i and (2.30):

σij = f σ∞ , (2.86) h iM − ij σij = (1 f) σ∞ . (2.87) h iI − ij Although this analysis has been presented for an elastically homogeneous composite, a similar Chapter 2. Internal Stress Development in Single and Two Phase Materials 42

analysis may be followed for a composite in which the inclusions have diﬀerent stiﬀness to the matrix, using the equivalent inclusion method.

2.7.1 Forward and Reverse Yield Stress Consider the composite described above when the plastic strain is introduced by uniaxial tensile stress. The symmetry of the problem implies that the uniform plastic strain tensor must be invariant with respect to rotations about the tensile axis. Taking the reference frame 3-axis to lie parallel to the tensile axis, this condition together with that of volume constancy implies that the plastic strain has the form

1/2 0 0 P P − ij = 0 1/2 0 (2.88)  0 −0 1    where P is a positive scalar. Thus the average axial matrix and inclusion stresses are respectively

σ = f A P and (2.89) h 33iM − σ = (1 f) A P , (2.90) h 33iI − where E (7 5ν) A = − (2.91) 15 (1 + ν) (1 ν) − is a positive constant.6 Note that the matrix axial internal stress is compressive, and proportional to the axial plastic strain P . Hence, during tensile loading, the applied tensile stress must increase with plastic strain, in order to maintain a constant matrix tensile stress and continue yielding. That is, the partitioning of load provides an eﬀective source of hardening, known as back stress hardening. However, this hardening is directional, since the internal stress aids matrix yielding when the sense of the stress is reversed. For simplicity, assume that yielding occurs when the magnitude of the average axial matrix stress reaches a critical value σy. During forward (tensile) straining, this occurs when the applied stress σf satisﬁes

σy = σf + σ . (2.92) h 33iM When the stress is reversed, the yielding criterion is met when the applied compressive stress magnitude σb satisfies σy = σb σ . (2.93) − h 33iM Hence σf σb = 2 σ . (2.94) − − h 33iM Since σ33 M is negative, the reverse yield stress is lower than the forward yield stress. This is one caseh ofi the phenomenon known as the Bauschinger effect. Mori and Narita demonstrated the importance of back stress to the origin of the effect in dispersion hardened alloys [69].

2.7.2 Plastic Relaxation It is clear that the back stress hardening mechanism is not sustained indeﬁnitely. If it were, the diﬀerence in forward and reverse yield stresses would continue to increase linearly with prior plastic strain. This is not observed; see, e.g. [70]. As discussed by Brown and Stobbs [71], the

6Subject to the conditions E > 0 and −1 < ν < 1. A material violating these conditions would provide a very interesting research topic! Chapter 2. Internal Stress Development in Single and Two Phase Materials 43

back stress is limited by relaxation involving mechanisms such as cross slip and secondary dislocation processes. Local plastic relaxation invalidates the assumption of homogeneous plastic strain in the matrix, and therefore the misﬁt strain is no longer simply related to the macroscopic strain. Wilson and Bate have observed back stress saturation in a high carbon steel using X-ray diﬀraction [70].

2.7.3 Finite Element Method

While analytical models are very useful for understanding average properties and trends in composite behaviour, numerical methods enable more quantitative predictions to be made at a greater level of microstructural detail. The FE method in particular has become widely used for the modelling of composite materials. Whereas CPFEM requires single crystal plasticity to be incorporated within the constitutive relations, composite modelling may be conducted using simpler representations of material behaviour. For example, in many cases it is adequate to assume isotropic elastic and plastic properties of the phases. Simple material models are incorporated within commercial FE packages such as ABAQUS. More detailed models can be incorporated in user-supplied routines (e.g. see the thesis by Daymond [72]). Even treating each phase as a homogeneous continuum, it still remains generally beyond computational capabilities to solve a full three dimensional FE model which accurately represents the microstructure of a real composite. For this reason, simplifying assumptions are commonly made, applying symmetry considerations and a choice element arrangement in order to reduce the computational task. 2D modelling, under the condition of plane strain or plane stress, is appropriate for some cases. For example, when a continuous fibre composite is loaded in the transverse plane, it is appropriate to assume zero displacement in the fibre axis direction and apply plane strain boundary conditions within a 2D model. Brockenbrough et al. modelled a real fibre distribution in this manner [73]. Many problems can not be adequately represented in two dimensions and must therefore be solved using a full 3D model. In order to remain within computational limits, it is common to model only a building block of the full composite, known as a representative volume element, or unit cell. The composite is deemed to be constructed by stacking together unit cells, all of which deform in exactly the same way. Boundary conditions are applied to the modelled cell so that it remains possible to tessellate adjacent cells as deformation progresses. Commonly, the unit cell contains only one or two inclusions or parts of inclusions, if symmetry conditions may be applied. A limitation of unit cell models is that they represent a composite containing a regular array of inclusions, rather than a random distribution. Various authors have demonstrated that the composite macroscopic response depends on the structure of the array – as determined by the choice of unit cell. Thus it is clear that a regular array model is not a fully adequate representation of a composite containing a random distribution of inclusions. Levy and Papazian have compared a primitive cuboidal arrangement of aligned short fibres with a body-centred arrangement [74]. They argued that these arrangements represent the extremes of interaction between the fibres. Using these arrangements, Daymond and Withers have performed simulations of Al/SiC aligned short fibre composites, and demonstrated that the primitive array overestimates, while the body-centred array underestimates the work hardening of the real composite [75]. They attributed the softer response of the body-centred array to the promotion of matrix plastic flow between corners of the inclusions and the relative ease of plastic flow in shear bands between the fibres. The FE method is used in Chapter 4 for the prediction of interphase stresses in high carbon steel. Chapter 2. Internal Stress Development in Single and Two Phase Materials 44

2.8 Martensitic Transformation

This chapter concludes with a review of martensitic transformation (MT). This is of particular relevance to Chapter 5, which presents a study of stress-induced transformation in TRIP steel.

2.8.1 Experimental Observations of Martensitic Transformation Since MT is a phase transformation, thermodynamic and kinetic factors must be taken into account for a complete description of transformation behaviour. However, given that transformation is energetically favoured, attention may be focused on purely crystallographic considerations, in order to minimise the strain energy developed during transformation. Hence crystallographic theories of transformation have been widely used. Such theories are based upon experimentally-determined facts about MT. The important observations are now discussed briefly, with reference to the MT in steels, to which the theories were originally directed. The MT in a typical steel changes the crystal structure from fcc in the parent austenite phase to body-centred tetragonal (bct) in the martensitic phase. Bain proposed a lattice deformation to give rise to this change [76]. This is illustrated in Fig. 2.8. It consists of contraction along one principal axis of the fcc unit cell, and uniform expansion perpendicular to this axis. The a and b axes of the bct unit cell lie at 45◦ to those of the original fcc unit cell. Such a distortion could generate a bcc unit cell; however, some tetragonality is generally maintained due to the ordered arrangement of interstitial atoms, such as C or N, known as Zener ordering [77]. Evidence of the validity of the Bain distortion has been established using electron microscopy [78]. For MTs other than the fcc-bct type, different lattice distortions are associated with the transformation. However, the correspondence between the initial and final lattices is generally referred to as the Bain correspondence.

Figure 2.8: Illustration of fcc-bct Bain distortion [76]. Atoms are omitted from the front and back faces for the sake of clarity.

Microstructural observation of partially transformed steels reveals that martensite tends to form as plates, where the planar interface lies parallel to well-defined crystallographic planes in the parent crystal. The interface plane is called the habit plane. By observing the trace made by the habit plane on two sides of a single crystal of parent phase of known orientation, it is possible to index the habit plane. The product phase has a well-defined orientation relative to the parent. Orientation relationships have been determined using X-ray diffraction. They may be expressed by specifying a plane in the parent phase which lies parallel to a plane in the product phase, together with directions in the two planes which also lie parallel. For carbon steels, the Kurdjumov-Sachs relations [79] are often quoted:

(111)γ (011)α0 , [101]γ [1 1 1]α0 , (2.95) || || where γ denotes austenite, α0 denotes martensite and means “is parallel to”. Experimentally, the relative orientation deviates from this relationship||by up to a few degrees [79, 80]. Chapter 2. Internal Stress Development in Single and Two Phase Materials 45

It will be explained shortly that the accommodation of martensite plates in an austenite matrix requires shear deformation in addition to the Bain distortion. Striations on etched surfaces of ferrous martensites indicate that shear occurs on 112 α0 planes [41]. Many electron microscopy studies have been carried out, and these reveal{that}both twinning and slip deformation can be responsible for the shear (e.g. [81]), depending on alloy composition. Both can give rise to shear on 112 α0 planes in 111 0 directions. { } h iα 2.8.2 Crystallographic Theories of Martensitic Transformation (i) WLR-BM Theory Phenomenological theories of MT were developed independently by Weschler, Lieberman and Read [82] and Bowles and Mackenzie [83, 84]. These theories are mathematically equivalent, and will now be referred to as the WLR-BM theory. The object of the WLR-BM theory is to ﬁnd the shape deformation which occurs when a region of parent material transforms to martensite, and hence predict habit planes and orientation relationships. The approach is based upon the fact that since martensitic transformation occurs by a cooperative movement of atoms, the interface between the product and parent phases must be highly coherent. This is supported by the observation that the habit plane is well-deﬁned crystallographically. The theory therefore proceeds by constructing a deformation such that the interface is an undistorted, unrotated plane (that is, an invariant plane). The deformation must include the experimentally-determined Bain distortion. However, as will be shown shortly, this does not result in an invariant plane. If it is combined with a simple shear, however, an undistorted plane may result (as noted above, there is experimental evidence for such a shear). Then, by a body rotation, this plane can be rotated back to its original orientation, thus making it an invariant plane. The WLR-BM theory seeks the shear and rotation which are required. The formulation is based upon matrix algebra, and will not be described here. For details, see the book by Wayman [85]. However, the essential ideas of the theory can be demonstrated by a graphical method which is now considered, with reference to the fcc-bct transformation.

g K0 − →

K 2 K α α 20

shear direction, s

(a) (b)

Figure 2.9: (a) Deformation of a unit sphere by the Bain distortion, resulting in an ellipsoid. Undistorted lines (illustrated by arrows) lie on cones. There is no undistorted plane. (b) Deformation of a hemisphere by simple shear. There are two undistorted planes: the shear plane, K1, and the plane marked K20 (originally K2).

Initially, the Bain distortion is considered. The fact that the fcc-bct Bain distortion does not give rise to an undistorted plane is illustrated by Fig. 2.9a, which shows the eﬀect of the distortion on a sphere of material. The sphere is transformed into an ellipsoid. If one considers Chapter 2. Internal Stress Development in Single and Two Phase Materials 46

vectors from the centre to the surface of the ellipsoid, then only those that point to the circles of intersection with the sphere are unchanged in length by the distortion. Therefore, undistorted lines lie on cones, not on a plane, i.e. the Bain distortion does not leave any plane undistorted. A simple shear does, however, leave certain planes undistorted. This is illustrated by Fig. 2.9b, in which a hemisphere of material is sheared in the direction d within the plane marked K1. The shear plane, K1, is trivially undistorted and unrotated. However, it is clear that there is also an undistorted plane where the sheared hemisphere intersects the original hemisphere. This is marked K20 . It is rotated relative to its original position, marked K2, by an angle 2α, where α is the angle between either of these planes and K0, the plane whose normal is the shear direction, s. The theory therefore considers the possibility of an invariant plane arising from a combination of simple shear and Bain distortion. From the experimental observations, shear is assumed to occur on 112 α0 planes in the 111 0 directions. This corresponds to the 110 γ planes and { } h iα { } 110 γ directions in the austenite lattice. The combination of deformations is shown graphically onh thei stereographic projection in Fig. 2.10. The green traces represent the undistorted plane

before (K2) and after (K20 ) shear along s = 101 γ. The inner red circle represents the Bain cone of undistorted lines, prior to the Bain distortion. After distortion, these lines are displaced to the outer red circle.

_ s=[101]

′′ b

K b 1

[001]

′ K ′ 2↔ α a α ↔ ′′ K a 0 K 2 a

Figure 2.10: Stereographic projection analysis of deformation during martensitic transformation.

Taking the shear first, consider the vector a on the K2 plane, which is chosen such that 0 after shear, it lies at a , on the intersection of the K20 plane and the inner red circle. The Bain distortion then displaces a0 to a00, on the outer red circle. During this deformation a is unchanged in length. Similarly, consider the vector b. This lies on the shear plane, and is therefore unchanged by shear. It also lies on the inner red circle, and so its length is also unchanged by the Bain distortion, which changes its direction to b00. Thus a and b are undistorted by the deformation. Now, if the shear angle α is chosen correctly, then the angle between a and b can be made equal to that between a00 and b00. Then, the plane defined by a and b is undistorted by the total deformation, and represents a possible habit plane. In order to fulfil the criterion of being unrotated, the body rotation is found which brings a00 and b00 into coincidence with a and b respectively. This then also gives the orientation relationship. Other combinations of undistorted lines can be found in order to predict other habit plane variants. The problem may be solved graphically by varying α Chapter 2. Internal Stress Development in Single and Two Phase Materials 47

in the stereographic projection, or exactly by matrix algebra. The predicted habit planes and orientation relationships commonly lie within a few degrees of those found experimentally.

(ii) Inﬁnitesimal Deformation Theory Approach

The WLR-BM approach is based on finite deformation theory. An appealing alternative approach is based on infinitesimal deformation theory [86, 87]. To demonstrate the difference, consider the uniform deformation of a material. This may be represented by a matrix operator A, such that the new position of a point x1 is related to the original position by

x1 = (I + A) x0 (2.96) where I is the identity matrix. Two consecutive uniform deformations A1 followed by A2 thus cause displacement given by

x2 = (I + A2) (I + A1) x0 = (I + A2 + A1 + A2A1) x0 . (2.97)

Since matrices are generally non-commutative, a different displacement would arise if the deformations were carried out in reverse order. However, if the deformations are small, the term A2A1x0 may be neglected. In this case, the displacement is the same regardless of the order of application of the deformations. It is this approximation which is the basis of infinitesimal deformation theory, as opposed to finite deformation theory. The theory of elasticity is based on infinitesimal deformation theory, and thus allows the superposition of deformation states, independent of the order of application. More specifically, the theory of eigenstrains (and in particular the Eshelby theory) makes use of the superposition of strain fields. Application of this theory to the problem of martensitic transformation simplifies the analysis and provides an intuitive understanding. The disadvantage is that transformation strains may be large enough that the infinitesimal approximation is questionable. Agreement between the two theories is good when the transformation strain is small, but gets poorer as the transformation strain increases. For example, for the case of NiTi shape memory alloy, which has a maximum principal transformation strain of 10.1%, the predicted habit plane normals differ by just 3◦ [87]. However for the case of Fe-31%Ni, which has a maximum absolute principal transformation strain of 17.7% [41], the habit plane normal predictions differ by approximately 10◦. Mura, Mori and Kato have shown that in the limit of infinitesimal transformation strains, the predictions of the two theories are identical [86]. The remainder of this section adapts the infinitesimal deformation analysis presented by Liang et al. [87] for the transformation in NiTi to that in steels. The basis of the infinitesimal approach is Eshelby’s solution of the homogeneous inclusion problem for the case of a flat plate [5]. That is, the ellipsoid has principal axes a1 = a2, and a3 0. Consider a coordinate frame in which the 3-axis is parallel to the plate normal (i.e. habit →plane normal). If the components of strain within the plane are zero, i.e.

11∗ = 22∗ = 12∗ = 0 (2.98) then the Eshelby solution yields zero internal stress (and elastic energy) both inside and outside the plate. This is equivalent to the undistorted plane condition of the ﬁnite deformation analysis. Determining the eigenvalues of the strain tensor, it is easily seen that conditions (2.98) imply that the principal strains 1, 2, 3, satisfy

= 0, 0 . (2.99) 3 1 2 ≤ This implies the relations

det ∗ = 0, ∗ ∗ ∗ ∗ 0. (2.100) ij ii jj − ij ij ≤ Chapter 2. Internal Stress Development in Single and Two Phase Materials 48

These expressions are written in terms of invariant quantities, and therefore hold in any reference frame, greatly simplifying the analysis. For the fcc-bct transformation in steels, the Bain strain for contraction along the austenite 1-axis is written in the austenite reference frame as

c 0 0 ∗ (1) = 0 a 0 diag (c, a, a) (2.101) ij   ≡ 0 0 a   where a is positive, c is negative, and the shorthand diag notation is introduced for diagonal tensors and matrices. The two other Bain correspondence variants (BCVs), consisting of contraction along the 2- and 3-axes respectively are written similarly, as ij∗ (2) = diag (a, c, a) and ij∗ (3) = diag (a, a, c). Clearly, a single BCV does not satisfy conditions (2.100). This corresponds to the argument made earlier that the Bain distortion does not leave any plane undistorted. If, however, a plate consists of a combination of two twinned BCVs, then conditions (2.100) may be satisﬁed by the average eigenstrain. For example, a combination of BCVs 1 and 2 gives an average eigenstrain of

= f∗ (1) + (1 f) ∗ (2) (2.102) ij∗ ij − ij where f is the volume fraction of BCV(1). Substitution of equation (2.102) into the condition upon the determinant (2.100) yields a quadratic equation for f. Solution of this yields the average eigenstrain. For the fcc-bct transformation, a solution is

ij∗ = diag (0, a + c, a) , (2.103)

with other solutions given by permutation of the diagonal elements. Thus there are six solutions. This is because there are three pairs of BCVs, and for each pair there are two possible plate structures, the majority variant being one or other of the pair. Once the average eigenstrain is determined, the habit plane normal may be found by transforming to a reference frame in which conditions (2.98) are satisﬁed. For example, for the fcc-bct solution given in equation (2.103), this is achieved by a rotation about the 1-axis. When the appropriate reference frame is found, the habit plane normal is given by the 3-axis. Transformed back into the austenite reference frame, the solution corresponding to the eigenstrain given in (2.103) is

ni = 0, 1 + a/c, a/c (2.104) − It is assumed above that all BCVs cpan form twinnedp pairs with one another. This is only necessarily the case if the BCVs can join along a stress free interface. Fortunately, this is exactly the same problem as that of finding a stress free habit plane. In this case, however, the eigenstrain which must obey conditions (2.100) is the relative eigenstrain between the BCVs; that is, the difference between the eigenstrains. It is easily seen that this satisfies the conditions for all BCV pairs of the fcc-bct transformation (however, in the more complicated cubic to monoclinic transformation in NiTi, only 42 of the 66 combinations satisfy the conditions [87]). Again, the interface plane may be found by transforming to the appropriate reference frame. This provides an elegant way of determining coherent twin boundaries, regardless of whether they are associated with martensitic transformation. Chapter 3

Measurement of Internal Stress by Neutron Diﬀraction

Diffraction offers a unique tool for measurement of the types of internal stress described in Chapter 2. Unlike other techniques used to measure residual stress (such as hole drilling, ul- trasonic and magnetic methods), diffraction measurements are specific to individual phases and grain families. This is essential for research into intergranular and interphase stress development. Among the diffraction methods, neutron diffraction has the deepest penetration in engineering materials ( 10mm in iron, compared to 10µm for X-rays of similar wavelength [88]), enabling sampling∼ within the bulk material rather∼ than the surface. This is a major advantage because the stress state at the surface often differs from that in the bulk. The scattering intensities achieved with neutron diffraction are low in comparison to synchrotron X-ray diffraction, and thus the diffracting volume must be relatively large ( 10mm3, in comparison to 0.1mm3 for synchrotron X-rays). However, this carries the advan∼tage that a large number of grains∼ are sampled, providing a statistically representative average. For the reasons presented above, the internal stress measurements presented in this dissertation were obtained using the neutron diffraction method. This chapter presents the basis of the method, and describes the instrument used for the experimental work covered in the following chapters. A review of internal stress research using neutron diffraction is then presented.

3.1 Bragg Diﬀraction as a Strain Gauge

Strictly, diffraction methods do not measure internal stress, but rather interplanar spacings within the crystal lattice. Elastic strain is calculated from changes in these spacings. In this dissertation, the term lattice strain is also used to refer to elastic strains measured by diffraction. Stress is calculated from lattice strain by application of Hooke’s Law. This section summarises the basis of lattice strain measurement by neutron diffraction.

3.1.1 The Bragg Condition Diffraction occurs when radiation incident upon a regular array of scattering centres is scattered such that in certain directions there is constructive interference. In the case of crystalline solids, a diffraction maximum occurs if the Bragg condition is satisfied:

λ = 2dhkl sin θhkl , (3.1) where λ is the wavelength of the radiation, dhkl is the interplanar spacing of lattice planes with Miller indices hkl, and θhkl is the angle at which the radiation is incident upon the set of lattice planes (see Fig. 3.1). A single crystal may be oriented such that this condition is not

49 Chapter 3. Measurement of Internal Stress by Neutron Diﬀraction 50

satisfied by any set of lattice planes. A polycrystal, however, contains grains of a whole range of orientations, and thus — provided the radiation has a suitable wavelength and a sufficient quantity of grains lie within the diffracting volume — Debye-Scherrer cones of diffraction are produced.

θ hkl

dhkl θ 2 hkl

Figure 3.1: Bragg diﬀraction from a set of hkl lattice planes.

3.1.2 Fixed Wavelength Diffractometry The measurement of lattice strain by neutron diffraction relies upon the precise measurement of dhkl or, more accurately, the precise measurement of changes in dhkl. The most common method is to use monochromatic radiation, and to detect such changes via the shift in diffraction angle, θhkl. Differentiating equation (3.1) at constant λ gives for the elastic strain:

δdhkl = = δθhkl cot θhkl . (3.2) dhkl − This method is suited to a reactor source, where there is a constant ﬂux of neutrons.

3.1.3 Time-of-Flight Diffractometry The lattice strain measurements presented in this dissertation were collected at the ISIS pulsed neutron source, Rutherford Appleton Laboratory, UK. At this and similar pulsed neutron sources, protons are accelerated to high energies (e.g. 800MeV at ISIS [89]), and then directed towards a heavy metal target, e.g. tantalum. Neutrons are produced by a nuclear reaction between the high energy protons and target nuclei, known as spallation. On average, an accelerated proton produces approximately 15 neutrons. The neutrons pass through moderating material, such as liquid hydrogen or methane, in order to reduce the energies into the thermal range, thereby increasing the neutron wavelengths to suitable length scales for diffraction from atomic planes. The emerging neutrons have a continuous distribution of wavelengths. The entire process is repeated many times per second, e.g. 50Hz at ISIS. The time-integrated flux of such a source is generally much lower than at a reactor source. However, pulsed sources are well suited to the time-of-flight (TOF) method, which employs a Chapter 3. Measurement of Internal Stress by Neutron Diffraction 51

whole range of wavelengths and thus utilises the available neutrons much more efficiently. In the TOF method, the neutron pulses are directed a distance l along a beamline. A neutron’s time of flight along this distance, t, depends upon its momentum and, through the de Broglie relation (equation (3.3)), its wavelength λ. Detectors are placed at a fixed diffraction angle θ, and intensity is recorded as a function of time. Diffraction maxima are observed when a wavelength arrives which satisfies the Bragg condition for a certain lattice spacing. Since the moderated neutrons have speeds of the order of the speed of sound, they behave non-relativistically. The de Broglie wavelength is thus given by h ht λ = = (3.3) mnv mnl where h is Planck’s constant, mn the neutron rest mass, and v the neutron speed. Substituting for λ in the Bragg equation gives 2m l sin θ t = n hkl d (3.4) h hkl In this case, strain measurement depends upon the precise measurement of shifts in the time of flight. The counts from many pulses must be summed to produce a sufficiently intense diffraction profile. Since dhkl is proportional to t, elastic strain is given simply by δd δt = hkl = (3.5) dhkl t 3.2 Determination of Elastic Strains from Diffraction Spectra

At a pulsed source, a full diffraction profile is collected, containing many diffraction peaks. The profile contains a wealth of information about the diffracting material, including information on the internal stress state.

3.2.1 Elastic Grain Family Strains via Single Peak Fitting Note that for a particular hkl reflection, the Bragg condition is only satisfied if the hkl plane normal lies parallel to the scattering vector (the bisector of the incident and diffracted beams). Therefore, within a polycrystal, only crystallites with a particular set of orientations will contribute to a particular diffraction peak. Such a set of crystallites is referred to as a grain family. Due to single crystal anisotropy, as discussed in Chapter 2, the average elastic strain measured by diffraction will differ between grain families, and therefore between diffraction peaks. There- fore, by measuring shifts in individual diffraction peaks, the intergranular stress state within a single phase may be determined. This method is used for the investigation of intergranular stresses in Chapters 4 and 5. In order to determine accurately the position of a peak, a peak shape function is fitted to the experimental data using least squares minimisation. The peak shape at a pulsed source is asymmetric, due to the asymmetric pulse shape emerging from the moderator. Various functions may be used to describe the peak shape. In this dissertation, the function used is a convolution of two back-to-back exponential decays, with a pseudo-Voigt function (a linear combination of Gaussian and Lorentzian functions). This produces a good fit to the ISIS peak shape. More detail about the function is given in reference [90]. The peak width is influenced by a range of broadening factors, including both instrumental contributions (e.g. imperfect collimation) and sample contributions (e.g. the fluctuation of lattice spacing within a grain due to type III microstresses). These contributions are manifested in the peak shape function through the parameters which describe the decays and widths of the basic exponential, Gaussian and Lorentzian components. These parameters are themselves dependent on lattice spacing. In practice, the parameters arising from instrumental contributions are determined by Chapter 3. Measurement of Internal Stress by Neutron Diffraction 52

calibration of the instrument, and are not varied in the least squares minimisation procedure. The parameter of primary interest is the peak position, and for accurate determination of this, the number of other fitting parameters is kept to a minimum. However, the sample contribution to the peak width is also of interest. Sample factors usually add symmetric contributions to the peak shape (see Bacon [91]). Therefore, the Gaussian or Lorentzian width may also be fitted in the least squares minimisation. The algorithm used for single peak fitting in this dissertation is incorporated within the GSAS (General Structure Analysis System) software [90]. However, this package is primarily intended for Rietveld refinement of diffraction spectra. This is addressed in the following section.

3.2.2 Elastic Phase Strains via Rietveld Refinement Rietveld refinement was developed in the late 1960s and the 1970s for the determination of crystal structures using diffraction data [92, 93]. From a proposed crystal structure, a theoretical diffraction profile is predicted. The variation of intensity with time of flight (or angle) is dependent on a large number of parameters, including multiplicity and structure factors, peak shape parameters (as discussed above), and lattice parameters. By using a set of these as fitting parameters, the predicted profile is fitted to the experimental profile by least squares minimisation. The algorithm may also be used for accurate lattice parameter determination, when the crystal structure is already known. It is in this manner that Rietveld refinement is used to measure elastic strain. Since the entire profile is fitted, the positions of all diffraction peaks are taken into account. This greatly improves the counting statistics relative to a single peak fit, leading to more precise strain determination using a shorter count time. As explained above, strain varies between grain families, due to single crystal anisotropy. Therefore, if peak positions are determined by varying only lattice parameters, the predicted profile cannot perfectly match the experimental profile of a strained polycrystal, if there is significant single crystal anisotropy. Nevertheless, since some peak positions are overestimated while others are underestimated, it has been shown by Daymond et al. [94] that the refinement gives a good estimate of the average elastic phase strain. This offers a considerable advantage over conventional residual stress measurements made at steady state neutron sources, which usually rely on the collection of a single diffraction peak. Commonly in such experiments, the chosen reflection is a ‘good actor’ in that it is known to exhibit a stress–elastic strain response which is close to that of the bulk material, and remains nearly linear even in the plastic regime. Nevertheless, a single reflection is inevitably affected by intergranular stress, and taking many reflections into account undoubtedly improves the reliability of the measurement. By providing a measure of elastic phase strain, Rietveld refinement is very useful for the determination of interphase strain in a multiphase material. The lattice parameters of each phase are determined independently, since different peaks within a profile correspond to different phases. The method can also be used to provide a measure of phase fractions. This is exploited in Chapter 5, to measure the variation of phase fractions in a TRIP steel during stress-induced martensitic transformation. Rietveld refinement can also be used to determine the preferred orientation of grains, called texture. Texture causes the relative intensities of diffraction peaks to differ from those expected by scattering from a randomly oriented polycrystal. By incorporating a parameterised description of the orientation distribution function (ODF) into the Rietveld refinement, the parameters may be found by fitting the observed intensities, thereby estimating the ODF. For a full texture determination using this method, spectra must be collected with the sample positioned in many different orientations relative to the incident beam and detectors. However, fewer spectra are required if the texture is assumed to have a particular symmetry. This is exploited in Chapter 4 for a simple estimation of texture in a cylindrically symmetric sample. Even if texture is not of primary interest, it is still necessary to include texture parameters in the Rietveld refinement of a sample which exhibits preferred orientation, in order to obtain a good fit between the predicted and experimental profiles. Chapter 3. Measurement of Internal Stress by Neutron Diffraction 53

To the author’s knowledge, there has been no work to demonstrate that the strain determined by Rietveld refinement remains a good estimate of the average elastic phase strain in the case of a strongly textured material. Therefore, caution should be applied when analysing the diffraction spectra of such a material in terms of bulk elastic strains. One of the original motivations for the introduction of Rietveld analysis was to be able to extract information from overlapping diffraction peaks [92]. This is indeed a great advantage of the method, but it should be noted that if there are few diffraction peaks and these are overlapping, there may be a systematic error in the refined lattice parameter. This is particularly likely in a strongly textured, multiphase material. This is considered further in Chapter 5. The elastic phase strains presented in Chapters 4 and 5 were determined by Rietveld refinement using the GSAS program [90].

3.3 The ENGIN Instrument

The lattice strain measurements presented in this dissertation were made using the ENGIN instrument at ISIS [89]. ENGIN is a custom-built strain measurement diﬀractometer [3]. A schematic view of the instrument is shown in Fig. 3.2. This section summarises the relevant features of the instrument. Greater detail is given in reference [95].

Incident beam Scattering

(right) vecto r

r vecto (left) Slits - Scattering

Radial collimator Radial collimator - -

Detector bank Detector bank (left) L R (right)

Positioning table Sample (x,y,z translation & rotation)

rig axis

ensileloading T

optional r fo

Figure 3.2: Schematic diagram of ENGIN instrument, overhead view. Chapter 3. Measurement of Internal Stress by Neutron Diﬀraction 54

3.3.1 Flight Path The wavelength resolution of a time-of-flight instrument increases with flight path length. This is because the time interval between the arrival at the detector of two neutrons of different wavelength is proportional to the length of the flight path. It is not beneficial to increase the moderator-sample distance indefinitely, however, because of beam loss and because the faster neutrons in a pulse would begin to overlap the slower neutrons from the previous pulse. This is overcome on the highest resolution instruments (such as HRPD at ISIS, which has a flight path length of approximately 100m) by using chopper devices to reduce the pulse frequency, at the cost of reduced time-integrated intensity. In order to maintain intensity, a chopper is not used on ENGIN. The length of the ENGIN flight path, from the neutron moderator to the detector banks, is approximately 15m. This provides a compromise between wavelength range (0.8-6.6A),˚ intensity and resolution which is suitable for engineering strain measurements.

3.3.2 Collimation Neutrons travel approximately horizontally from a moderator aperture to the sample. The incident beam is defined using horizontal and vertical cadmium slits which are positioned as close to the sample as practical (typically 10-50cm). A range of slits are available, varying in width from 0.5mm to 10mm. Due to the relatively long distance ( 14m) between the moderator aperture and the incident slits, the beam divergence is low in ≈comparison to instruments at steady state sources. In order that diffraction angles are well determined, only neutrons scattered within a small section of the incident beam must be detected. This is achieved on ENGIN using radial collimators, positioned in front of the detectors, at horizontal angles of +/-90◦ to the incident beam (as shown schematically in Fig. 3.2). The collimators define the direction of travel in the horizontal plane. Computer simulations have shown that the collimators define an approximately 1.5mm section of the incident beam from which scattered neutrons may travel to the detectors [96]. Thus, for example, using incidents slits of 5 x 10 mm defines an approximately cuboidal gauge volume with dimensions approximately 5 x 10 x 1.5 mm from which scattered neutrons are detected. If the gauge volume is not immersed fully within the sample, systematic shifts of the diffraction peaks may develop in the recorded spectra. The origin of these shifts has been discussed by Webster et al. [97]. If unaccounted for, such shifts may be attributed to elastic strain within the material. Strain scanning experiments — in which the elastic strain in an engineering component is mapped by translating the component with respect to the gauge volume — are prone to systematic errors of this type if the strain is measured in the vicinity of the component surface. In such experiments, accurate positioning of the sample is essential in order that the position of the surface relative to the gauge volume is well known and systematic peak shifts may be avoided or accounted for. In the type of experiment described in this dissertation, however, the sample position is fixed with respect to the gauge volume and, moreover, strains are calculated relative to reference lattice spacings determined on the same sample before the application of stress. It is reasonable to assume that any systematic shifts in the diffraction peaks remain constant throughout the experiment, and to choose wide incident slits such that the gauge volume is not fully immersed. The advantage of bathing the sample in this manner is that the scattered intensity rises in proportion to the volume of scattering material, reducing the count time required to record a spectrum of sufficient accuracy.

3.3.3 Detector Banks

ENGIN has two fixed-angle detector banks centred behind the radial collimators at +/- 90◦ to the incident beam. This set-up is advantageous because it enables two orthogonal strain components to be measured simultaneously in approximately the same gauge volume. Moreover, the approximately cuboidal shape of the gauge volume provides good spatial resolution. The fixed Chapter 3. Measurement of Internal Stress by Neutron Diffraction 55

angle geometry is particularly appropriate for intergranular and interphase strain measurement because the scattering vector is the same for every reflection, even though each is produced by a different grain family or phase. In elastic scattering, the scattering vector is the bisector of the incident and diffracted beams, and lies parallel to the normal of the reflecting planes. Each bank consists of 3 horizontal rows each of 43 individual scintillator elements, spanning an angular range of 14◦. This is a small enough range that the scattering vector remains well-defined, but greatly increases the collected intensity. There are time shifts between the diffraction spectra recorded in each detector element, since the path length and diffraction angle vary from element to element. To produce the final diffraction spectrum, the relative time shifts are subtracted and the intensity from all detectors is summed. This is called time focusing [95].

3.3.4 Loading Rig In Chapters 4 and 5, results are presented of elastic strain measurements acquired during in situ tensile loading. The tensile loading was achieved using the custom-built 50kN Instron loading rig which may be mounted on ENGIN. The tensile axis is aligned in the horizontal plane at +45◦ to the incident beam (see Fig. 3.2). With this set-up, the right and left detectors measure strains parallel and transverse to the applied load, respectively. For tensile tests, universal joints are used to ensure uniaxiality of loading. An extensometer with a gauge length of 12.5mm monitors the macroscopic strain.

3.3.5 Cooling Grips

The study of a TRIP steel presented in Chapter 5 required samples to be cooled to -20◦C during tensile loading. For this purpose, cooling grips were designed, shown schematically in Fig. 3.3. Cooling is achieved by pumping refrigerated oil through copper tubing surrounding the grips. Thermal contact is made with the sample through the screw threads which join it to the grips. Conducting grease is used to improve the thermal contact. The temperature is monitored by K-type thermocouples which are ﬁxed in a hole in the sample shoulder and to the surface in the centre of the sample. Convection of air around the sample is minimised by surrounding with a sheath of aluminium foil.

K-type thermocouples

Aluminium foil 6

6 Copper tubing Cooled oil IN

Cooled oil OUT

Figure 3.3: Schematic diagram of grip design for in situ cooling of samples to -20◦C.

Using this design, cooling to -20◦C has been achieved. Using a TRIP steel sample, a Chapter 3. Measurement of Internal Stress by Neutron Diﬀraction 56

temperature gradient of several degrees Celsius from the shoulder to sample centre was observed, but the temperature was stable to within 1◦C. The sample temperature increased systematically with elongation. This was attributed to the dependence of the rates of heat conduction and convection on the changing sample dimensions (length, cross sectional area, and surface area). 6 However, the maximum change was 6◦C, corresponding to thermal strain of about 100 10− , which is small relative to the measured lattice strains. ×

3.4 Review of Type II Internal Stress Measurements Using Neutron Diﬀraction

The first diffraction experiments to measure stress in materials were conducted using X-ray radiation as long ago as 1925 [1]. However, due to the low penetration of laboratory X-rays in engineering materials, X-ray stress measurements are restricted to the near-surface region, where the stress state is influenced by the proximity of the free surface. This review considers only work conducted using neutron diffraction, in which much greater penetration is achieved, as noted in the introduction to this chapter. Recently, synchrotron X-ray radiation has proved to be another very useful method of measuring residual stress. This is not considered in this review, but the reader is referred to the thesis of Owen for further information on this topic [98]. Neutron diffraction has been used as an experimental method for the measurement of residual stress since the 1980s. The state of the art was documented in 1992 in the publication of the proceedings of a workshop on the “Measurement of Residual and Applied Stress Using Neutron Diffraction” [99]. At that time, dedicated strain measurement instruments had not been built at pulsed sources, and the vast majority of work was conducted at reactor sources. In recent years, more and more results have been published from work conducted at pulsed sources, exploiting the benefits of the TOF technique. The neutron diffraction method has commonly been used to measure type I residual stress fields, arising from such processes as cold expansion of fastener holes [100], plastic bending of a bar [2], and modern welding techniques, such as tungsten inert gas welding [7], electron beam welding [101], inertia friction welding [102], and friction stir welding [103]. However, in the context of this dissertation, measurements of average type II internal stresses, such as intergranular and interphase stresses, are more relevant. This review concentrates on these types of measurement.

3.4.1 Intergranular Stress When stress is applied to a polycrystal, the stress and strain varies from grain to grain, due to single crystal anisotropy. This has been discussed in Chapter 2, and in particular in 2.6.4 in the context of the elastoplastic self-consistent (EPSC) model. Neutron diffraction, and§ in particular the TOF technique, has proved a valuable experimental method for the validation of this model. Using a numerical formulation of the model, it is straightforward to study the average elastic strain development along a particular sample direction in the subset of grains which are oriented for diffraction from hkl planes, with the scattering vector parallel to that direction. This corresponds directly to the strain measured by the shift in the hkl diffraction peak. Since a full diffraction profile is collected in a TOF measurement, the responses of a number of grain families can be measured, and compared to the EPSC predictions.

Intergranular Stress in fcc Polycrystals An excellent dataset of grain family strain responses has been presented by Clausen et al. for uniaxial tensile loading of austenitic stainless steel (fcc crystal structure) [104]. Some of this data is shown in Fig. 3.4. The figure shows the axial strain response of different grain families Chapter 3. Measurement of Internal Stress by Neutron Diffraction 57

having a particular hkl plane normal pointing along (or near to) the tensile axis, determined by single peak ﬁtting of the axial diﬀraction spectra. The data was collected at the LANSCE pulsed neutron source, using the TOF technique.

Figure 3.4: Elastic strain response parallel to uniaxial applied stress of hkl grain families in stainless steel. Experimental data is represented by datapoints; lines are EPSC model predictions. From Clausen et al. [104].

At low applied stresses, the material deforms elastically, and the effects of elastic anisotropy, as discussed in 2.4.1, are evident. Since for fcc steel 111 is the stiffest crystallographic direction, the 111 grain family§ shows the stiffest elastic response. The 200 family shows the most compliant response, 200 being the least stiff crystallographic direction. Engineers commonly refer to the grain family slopes in the elastic regime as the “diffraction elastic constants”. It is worth reminding that these values differ from the single crystal elastic constants, as determined for a cubic crystal by equation (2.19), since the strain developed in each grain is influenced by the constraint imposed by its neighbours. This tends to reduce the apparent difference in stiffness between the stiffest and most compliant crystallographic directions. At higher applied stresses, plasticity begins. The shift to greater elastic strain in the 200 family indicates that it bears greater stress; this suggests that it continues to deform elastically, taking up extra load while grains in other families begin to relax due to plasticity. For example, the deviation towards lower lattice strain in the 331 family indicates early yielding. EPSC model predictions are also plotted. The EPSC simulation employed a grain population of 5700 grains, with orientations chosen to match the weak texture of the sample. The yield and hardening parameters were used as fitting parameters to gain agreement between the experimental and predicted macroscopic stress-strain responses. Good agreement is achieved between the experimental and predicted grain family responses, both in the elastic and plastic regimes. In particular the “double bend” of the 200 response is almost perfectly captured. Poorer agreement between model and experiment is achieved in the transverse direction (not shown). This has also been reported by other authors (e.g. [105]). An explanation, forwarded by Clausen et al. in a separate paper [106], is that a grain with a certain hkl plane normal lying perpendicular to the loading axis can have a range of crystallographic directions lying parallel to the tensile axis. Therefore, grains in a transverse family can actually behave very differently to one another, making the average response very sensitive to the exact distribution of grain orientations. Clausen et al. presented model calculations which show that the standard deviation of strain in a transverse grain family is an order of magnitude greater than that in an axial family. Daymond et al. [107] have also demonstrated that in austenitic stainless steel, transverse elastic strains are susceptible to texture. They attributed this to the reason given above. Their Chapter 3. Measurement of Internal Stress by Neutron Diffraction 58

work concerned the comparison of untextured and rolled stainless steel, studied by neutron diffraction and EPSC modelling. They concluded that texture evolution during rolling has a relatively minor effect on the mechanical properties of the material, which are dominated by hardening and residual stress development. Lorentzen et al. have performed a neutron diffraction study of the cyclic loading of austenitic stainless steel [108]. Their data reveal a gradient change in the lattice strain response of the axial 200 grain family as the loading direction is reversed. This suggests a relaxation of residual stress due to early reverse plasticity in these grains. Although the EPSC model predicts a Bauschinger effect due to the promotion of reverse yielding by back stress, the experimental 200 response deviates from linearity far earlier than the modelled 200 response. This suggests that the EPSC model suffers from too simplistic a criterion for the onset of plasticity, and that an additional mechanism of plastic strain recovery is required. This may also help to explain the early onset of reverse plasticity in the rolled steel studied by Daymond et al. [107]. Pang et al. have also studied rolled plate austenitic stainless steel with neutron diffraction [109]. They too noted that the unloading response of the axial 200 family is not parallel to the initial elastic response. They performed a series of tensile loads and unloads up to 8% macroscopic plastic strain. Residual strain development was greatest in the first load-unload cycle (up to 2% strain), after which intergranular strains saturated, further deformation being accommodated plastically in all grains. They observed changes in diffraction peak intensities, with the axial 111 reflection growing stronger with plastic strain. This is consistent with previous observations of texture development in fcc alloys due to grain rotation [110]. A common theme noted in papers which compare tensile loading results of fcc materials to EPSC predictions is the poor agreement between the model and experimental 200 transverse responses. Daymond et al. [107] published calculations which suggest that for fcc materials this response is particularly sensitive to texture. The model predicts a dramatic tensile shift at the onset of plasticity, the extent of the shift depending on texture. Shifts are indeed seen in their experimental data, but not of the predicted magnitudes. However, Holden et al. have observed tensile 200 transverse residual strains in Inconel-600 [105, 111, 112]. This material has an fcc structure with similar elastic anisotropy to stainless steel. EPSC model calculations predict the correct sense of these strains, while overestimating the magnitudes. Again, the EPSC model predicts the axial strains more accurately. Comparison of measurements made on stainless steel to those made on a less elastically anisotropic fcc material demonstrates that the evolution of intergranular stress depends upon an important interplay between elastic and plastic anisotropy. Allen et al. [113] have measured grain family responses during tensile loading of aluminium, which has a much lower degree of elastic anisotropy than stainless steel (see 2.4.1). Their experiment was performed at a reactor source rather than a pulsed neutron source.§ The deviations from linearity at the onset of plasticity are far less marked than in stainless steel, and the elastic responses of the 200, 311 and 222 grain families are all similar in both the elastic and plastic regimes. Pang et al. [114] reported similar results for a different aluminium alloy. Calculations by Clausen et al. [106] actually indicate that the sense of the deviations is opposite in aluminium to those in stainless steel, with the 200 axial response deviating towards lower elastic strain, while the 111 axial family develops greater elastic strain. This suggests that 200 type grains are better oriented for slip. However, when elastic anisotropy is significant, as in stainless steel, the extra stress borne by the elastically stiffer 111 grains causes them to yield first.

Intergranular Stress in non-fcc Polycrystals Although the majority of neutron diffraction intergranular strain measurements have been conducted upon fcc metals, materials with other crystal structures have also been investigated. Pang et al. [115] have studied a ferritic steel, with bcc crystal structure. Although the single crystal elastic anisotropy follows a similar trend to that in an fcc steel — with 200 the most compliant and 111 the stiffest direction — different slip systems operate, as discussed in 2.5.1. § Chapter 3. Measurement of Internal Stress by Neutron Diffraction 59

The data of Pang et al. is rather scattered. The influence of elastic anisotropy is evident in the axial grain family strains, but there are not clear deviations from the elastic responses during yielding. However, in the transverse direction, the 200 family develops a large tensile shift during yielding, resulting in a tensile residual strain. As noted above, similar shifts of the transverse 200 response are sometimes observed in fcc materials, but caution should be maintained when drawing a link between these findings, since different slip systems operate in the two structures. Pang et al. do not offer a satisfactory explanation for the transverse 200 response, because they do not clearly distinguish between the 200 transverse and axial families. As explained above, these do not consist of the same set of grains, since grains with a 200 direction pointing transversely may have a range of crystallographic directions pointing along the tensile axis. As stated in 2.5.1, the slip planes in bcc materials are not clearly defined like those in fcc materials, due§ to the absence of close-packed planes. In their EPSC model implementation, Pang et al. assumed slip on 123 , 110 and 112 planes. They specified the slip direction 111 , as has been established{ unam} {biguously} {for b}cc alloys. They used the strategy employedh in otheri papers, of fitting the predicted macroscopic response to the experimental flow curve, then comparing internal strains. However, plastic deformation in the ferritic steel is initially by Luders¨ band propagation (the meaning of the term Luders¨ band is described in 4.1). Their EPSC model implementation was unable to simulate this type of heterogeneous deformation§ accurately. Nevertheless, they fitted the calculated macroscopic curve to the experimental curves as well as possible. The model successfully identified major features of intergranular strain development; in particular the tensile shift of the transverse 200 elastic strain.

Various authors have also studied hcp materials. As noted in 2.5.1, these materials have fewer operable slip systems than cubic materials. Therefore, they exhibit§ more marked plastic anisotropy, which may lead to more significant intergranular stresses. Furthermore, deformation by mechanical twinning is commonly observed, in order to accommodate deformation which cannot be accommodated by slip. MacEwen et al. have measured residual strains in zircaloy-2, a zirconium-based alloy [116, 117]. More recently, Pang et al. have re-visited this material, performing uniaxial tensile load-unload cycles [118]. They used swaged material with strong rod texture, such that the majority of grains have the a-axis aligned parallel to the tensile axis, with the c-axis distributed randomly around the transverse plane. Due to the axial symmetry and rod texture, the grains all deform plastically to a similar extent along the tensile axis, so that axial intergranular strains are minimal. However, there exist strong transverse interactions. This is because it is more difficult to accommodate plastic strain parallel to the c-axis than perpendicular to it. Therefore, each grain possesses a plastically hard and a plastically soft transverse direction, but the alignment of these directions differs from grain to grain, causing incompatibility. This is manifested very clearly in the transverse elastic strain data, with the 0004 (c-axis) response being forced further into elastic compression due to the plastic contraction along this direction of surrounding grains. Correspondingly, the 2020 transverse response is forced into tension. Pang et al. applied the EPSC model. They gained excellent agreement with their data when they assumed that slip occurs on prismatic, pyramidal and basal systems, noting that the latter type improves agreement with experiment but is neglected in the earlier work of Turner and Tomé [119]. Pang et al. ignored mechanical twinning, but this is quite reasonable, since the most easily activated twinning mode in zirconium at room temperature requires tension along the c-axis [120]. Since the c-axis of most grains lies in the transverse plane, it tends to be compressed, so that the directional twinning mechanism is not activated.

Tom´e et al. [121] have studied compressive loading of hcp beryllium. Although they used untextured material, they too observed strong transverse strain deviations. They varied the active deformation mechanisms and resolved shear stresses within the EPSC model, but reported best agreement with their experimental data when pyramidal slip or c-axis compressive twinning was assumed, neither of which are reported to operate in Be. Chapter 3. Measurement of Internal Stress by Neutron Diﬀraction 60

3.4.2 Interphase Stress

Neutron diffraction measurements of interphase stress have been performed both on naturally occurring and man-made composites, such as metal matrix composites (MMCs). Studies have varied from fundamental research of internal stress growth and relaxation mechanisms, to measurements on real engineering components. For example, Withers et al. have followed the development of interphase stress during processing of a continuous SiC fibre-reinforced titanium aeroengine ring [122]. The residual stress in this component is of thermal origin, arising due to the difference in thermal expansion coefficients of the two phases. Maeda et al. have measured such thermal stresses generated by quenching an Al/SiC particulate composite [123]. They found tensile residual stress in the Al matrix, as expected due to its greater contraction upon cooling. The residual stresses agreed with mean field predictions (see 2.4.3 and 2.7) assuming no stress relaxation. Fitzpatrick et al. have also measured internal strains§ in an§Al/SiC particulate composite, in which the type II stresses were superposed with a type I, macroscopic residual stress field [124]. They successfully extracted the internal stress contributions arising from thermal misfit and elastic mismatch. Withers, Lilholt, Juul Jensen and Stobbs [125, 126] have investigated the relaxation of thermal strains in MMCs, by studying the evolution of the matrix lattice parameter with time at various annealing temperatures. Comparison of whisker and particle-reinforced composites revealed that larger thermal stresses develop in the former, but that these relax more rapidly and at lower temperatures than in the latter. The dependence of the magnitude on inclusion shape is easily explained on the basis of the Eshelby and mean field theories (see Chapter 2). The authors argued that the more rapid relaxation in the whisker-reinforced composite occurs because the hydrostatic component of matrix stress varies around the inclusion interface. The stress gradient leads to interfacial diffusion of atoms, which occurs more rapidly than diffusion through the bulk, largely due to a shorter diffusion length scale. Interfacial diffusion is not active in the particulate composite, owing to the lack of strong stress gradients around the inclusion interfaces. The authors also noted that bulk diffusion is more rapid in composites with more finely dispersed inclusions, since for a given reinforcement volume fraction, this reduces the distance between inclusions and therefore the characteristic diffusion distance. Lorentzen et al. have also published results concerning the relaxation of thermal strains, but their data is of rather poor quality and they did not propose a relaxation mechanism [127]. Daymond and Withers developed a stroboscopic technique to overcome the problem of long neutron count times when measuring thermal cycling of MMCs under a small load [72, 128]. They collected data over many cycles, dividing each cycle into a number of time windows, and summing the data from corresponding windows in each cycle. They applied the method to an Al/SiC whisker composite. They reported hysteresis loops in the matrix and reinforcement elastic strains, which they rationalised according to elastic and thermal mismatch combined with reduction in the matrix yield stress and promotion of matrix creep at elevated temperatures. They developed a finite element model which produced good agreement with the diffraction data. Madgwick et al. have used neutron diffraction to improve the understanding of creep in MMCs [129, 130]. By measuring the matrix internal stress, they showed that the power law creep stress exponent of the matrix is the same as for the unreinforced material, even though the bulk composite has a much higher apparent exponent. They also demonstrated the importance of stress relaxation for continuation of creep, and identified damage as an important source of stress relaxation around particles. In the context of the experimental work to be presented in this dissertation, in situ loading experiments of composites are of particular relevance. Allen et al. performed such a study on particulate and whisker-reinforced Al/SiC composites [113]. The elastic response of the SiC reinforcement is much stiffer than that of the Al matrix. Due to scatter in the data, deviations from linearity in the plastic regime are difficult to identify reliably. However, clear measurements of residual strain after unloading were reported. Interestingly, the sense of the residual strain in Chapter 3. Measurement of Internal Stress by Neutron Diffraction 61

each phase differs between the particulate and whisker-reinforced composites. In the latter case, the axial matrix strain is compressive, while the axial reinforcement strain is correspondingly tensile. This is consistent with matrix plastic flow around elastically-deforming inclusions during tensile straining, as explained in Chapter 2. In contrast, the particulate composite develops a tensile axial residual strain in the matrix, and correspondingly compressive residual strain in the reinforcement. In order to explain this, Allen et al. proposed diffusional stress relaxation, in which matrix material is transported around the particles in order to lower the regions of tensile stress concentration. However, they were vague as to why this mechanism should be more active in the particulate rather than whisker-reinforced composite. Nevertheless, using the equivalent inclusion and mean field theories, they produced predictions of the elastic phase strain responses which agreed well with the experimental data. Withers and Clarke investigated tensile loading of a continuously reinforced Ti/SiC composite [131]. When tension is applied parallel to the axis of the continuous fibres, the axial diffraction elastic constants of all the measured Ti grain families are similar, and close to that of the measured SiC response (220). This demonstrates that the continuous fibres constrain both matrix and reinforcement to develop approximately the same strain (as is assumed exactly in the Voigt elastic average — see 2.4.2). When plastic deformation begins, the matrix is clearly seen to shed stress, while the §fibres gradually bear more of the load. The residual strains are of the sense expected if relaxation does not occur; the matrix develops a compressive axial residual strain, which increases as plastic deformation progresses. When tension is applied perpendicular to the axis of the fibres, both the macroscopic and internal strain behaviour are very different. The matrix diffraction elastic constants are much lower than the fibre constants, since much less constraint is imposed by the fibres in this direction. Moreover, the elastic strain responses indicate that the matrix bears an increasing proportion of the load as the stress is increased: the opposite trend to that usually observed. Macroscopically, the composite appar- ently yields at a stress which is far below that expected of the unreinforced material. These surprising observations are succintly explained by matrix-fibre debonding. With stress applied perpendicular to the axis of the fibres, the weak interface is damaged at low stresses, so that load transfer to the fibres becomes less effective as the stress is increased. The early deviation of the macroscopic response from linearity is not due to slip plasticity, but rather due to the matrix bearing an increasing proportion of the applied stress, and thus deforming to a greater extent elastically than would otherwise be the case. Evidence for this is that the strain is recovered upon unloading, and the internal strain responses do not exhibit residual strain after unloading. Bonner et al. have measured load partitioning in a pearlitic high carbon steel [132]. This is of particular relevance to Chapter 4 of this dissertation, which also addresses internal stress in a high carbon steel, but with a spheroidised rather than pearlitic microstructure. Bonner et al. found similar diffraction elastic constants for the ferrite and cementite phases, indicating that cementite does not provide reinforcement in the elastic regime. However, during plastic tensile straining, extremely high tensile axial elastic strains of approximately 1% were developed in the cementite phase, while the ferrite strain dropped correspondingly. Using the nominal cementite volume fraction of 13.5%, the calculated stresses in both phases satisfied the stress balance condition. Bonner et al. attributed the broadening of cementite peaks to plasticity in this phase; however, it continued to bear an increasing proportion of the applied stress as loading progressed. Large residual strains developed in both phases, with the ferrite forced into axial compression and the cementite into axial tension, as expected due to misfit caused by ferrite plastic flow. The residual strains saturated after approximately 6% macroscopic strain. Although this could be due to the onset of cementite plasticity, it could equally be attributed to local plastic relaxation of the ferrite matrix, as discussed in 2.7.2. § Daymond and Priesmeyer have also recently published a tensile loading study of pearlitic steel [133]. They used Rietveld refinement to determine the elastic phase strains, overcoming the problem of poor neutron scattering by cementite through taking multiple peaks into consideration. Their results are similar to those of Bonner et al, indicating almost identical elastic Chapter 3. Measurement of Internal Stress by Neutron Diffraction 62

constants of the two phases, and load transfer from ferrite to cementite in the plastic regime. The elastic strains generated in cementite are, however, much lower than those reported by Bonner et al. At higher loads, the axial cementite elastic strain falls, indicating the breakdown of the load transfer mechanism, and possibly plasticity or cracking in the reinforcing phase. Daymond and Priesmeyer used a two phase EPSC model to predict both interphase and intergranular strain development, obtaining reasonable agreement with the experimental data. Their data and modelling strategy are considered further in 4.12.3. § 3.4.3 Stress-Induced Martensitic Transformation The subject of Chapter 5 is stress-induced martensitic transformation studied by neutron diffraction. The material of study is a steel which exhibits transformation-induced plasticity (TRIP). The mechanical properties of such steels are discussed in Chapter 5. To the author’s knowledge, no other neutron diffraction studies of internal stress development in TRIP steel have been published, although there have been some X-ray studies [134, 135]. However, there have been neutron diffraction studies of shape memory alloys (SMAs), the mechanical behaviour of which also depends on stress-induced martensitic transformation. Vaidyanathan, Dunand and Bourke have performed uniaxial compression tests on the SMA NiTi at a pulsed neutron source [136, 137]. As the applied stress is increased, the development of martensite is evident from the emergence of new peaks in the diffraction spectra, accompanied by intensity reductions in the existing austenite peaks. Austenite texture develops as grains of certain orientations transform preferentially. The authors used Rietveld refinement to quantify this, and observed that austenite grains having [100] pointing along the tensile axis transform preferentially. They pointed out that since there is a crystallographic relationship between the parent and daughter phases, preferential transformation also causes the developing martensite phase to have strong texture. However, they did not enter into an explicit discussion about the transformation crystallography, nor offer an explanation as to why it is that austenite grains with [100] pointing axially are the first to transform. Rietveld refinement was also used to determine the austenite elastic strain. This deviates slightly from the initially linear response as transformation progresses, suggesting load transfer to the evolving martensite phase. In contrast to the case of slip plasticity, the responses of individual austenite grain families (determined by single peak fitting) do not vary markedly from one another when non-elastic deformation initiates. The same authors have extended their work to NiTi reinforced with TiC particles [138, 139]. The presence of the stiff TiC particles inhibits transformation. The elastic phase strains indeed reveal that TiC bears the greater proportion of the applied stress, thus reducing the mean matrix stress, and hence the transformation driving force. Clearly this contributes to the inhibition of transformation, although some other reasons were also forwarded by Vaidyanathan et al. As transformation progresses, the TiC response remains linear. This indicates that apart from that due to elastic mismatch, misfit between the matrix and particles does not develop, even though the matrix deforms superelastically. This may possibly be explained by the matrix material surrounding each particle transforming in a manner so as to accommodate the misfit. Evidence for this has not, however, been presented. The austenite response does, however, appear to deviate from linearity, possibly due to load transfer to martensite. The group of Sittnerˇ and Lukáˇs have studied another SMA of technological interest, CuAlZnMn [140, 141]. Unlike in NiTi, the austenite grain family elastic strain responses differ greatly, and exhibit large hysteretic loops. After an initial tensile load-unload cycle, a small amount of martensite is retained, and the austenite remains in a state of tensile residual stress. Sittnerˇ et al. have postulated that this aids transformation in subsequent cycles, thus explaining an observed lowering of the transformation onset stress. Concerning the phenomenon of preferential transformation, Sittnerˇ et al. stated that no significant change of austenite texture occurs because large scale stress-induced transformation occurs approximately simultaneously for most grain families, with the exception of grains with 111 parallel to the tensile axis, which are the Chapter 3. Measurement of Internal Stress by Neutron Diffraction 63

last to transform. The group also presented high resolution measurements of peak profile. The austenite peaks are initially symmetric, but develop a marked asymmetry prior to the onset of large scale transformation. This asymmetry diminishes as transformation occurs and the peak intensities fall. The authors claimed that this behaviour can be interpreted on the basis of the complex internal stress state, and is predicted by a self-consistent model developed by themselves. However, the model results and interpretation remain unpublished. The group of Sittnerˇ and Lukáˇs have also collaborated with the group of Tomota, in a tensile study of FeMnSiCr [142]. This SMA is different to NiTi and CuAlZnMn, in that the martensitic transformation has a large temperature hysteresis ( 100◦C compared to 10◦C). Also, the transformation is from fcc to hcp, by the motion of∼Shockley partial dislo∼cations. In order for a sample of this material to exhibit an effective shape memory effect, it must be subjected to a training treatment, consisting of several thermomechanical cycles. Tomota et al. used neutron diffraction to follow two tensile load-unload-anneal cycles in order to better understand the mechanism of training. Upon initial loading, the austenite axial elastic strain increases sharply in the elastic regime, and then plateaus when transformation begins. Upon unloading, the remaining austenite is forced into residual compression, owing to the retainment of the martensite phase. The absence of reverse transformation during unloading is related thermodynamically to the large temperature hysteresis. The residual strain is almost completely removed during annealing, as the reverse transformation is activated. On the second loading step, the austenite axial elastic strain increases less rapidly than previously, indicating an earlier onset of transformation. Tomota et al. attributed this to the introduction of austenite stacking faults during the first cycle. These act as nucleation points for transformation. Chapter 3. Measurement of Internal Stress by Neutron Diffraction 64 Chapter 4

Interphase and Intergranular Stress In Carbon Steels

The review presented in Chapter 2 has demonstrated that theories of internal stress development during mechanical deformation are well established. The experimental validation of such theories has been ongoing for many years. However, as discussed in Chapter 3, the recent advent of engineering instruments at pulsed neutron sources offers new opportunities to obtain more accurate and representative measurements of internal stress in real engineering materials. Therefore, the technique is applied in this chapter to the study of standard engineering materials, exploiting the benefits of this modern experimental method for the validation of concepts and theories of both interphase and intergranular stress evolution. The materials chosen for this study are carbon steels, which possess a range of advantageous characteristics. Amongst these are the excellent neutron scattering properties of iron. In addition, selection of both a low carbon and a high carbon steel makes it possible to compare the response of ferrite in a single phase material to that when it is accompanied by a reinforcing cementite phase. Moreover, the morphology of the reinforcement can be controlled by heat treatment, in order to produce a dispersion which can be realistically modelled. Another appealing characteristic of cementite is that it has very similar elastic constants to ferrite, as shown in this chapter. It is therefore possible to isolate the influence of load transfer due to differences in plastic properties from that due to differences in elastic stiffness. This chapter is structured as follows. A brief materials review of some relevant aspects of the properties of carbon steels is given first. This is followed by a characterisation of the materials used, and the processing steps employed in order to prepare the materials for the neutron study. A short description of the neutron diffraction method is given in order to supplement the general description provided in Chapter 3. Results and discussion of the neutron diffraction measurements are divided between two sections, focusing on the development of interphase and intergranular strains respectively. Comparison is made between the development of internal stress in low carbon and high carbon steel. A further section presents a wider dataset, considering the issues of reproducibility of results and the influence of crystallographic texture on internal stress evolution. The neutron diffraction measurements reveal a marked variation in the responses of different ferrite transverse grain families. Following the discussion of the measurements, a simple rationalisation of the observed variation is forwarded, based on the pencil glide model of slip in bcc crystals. More quantitative analysis of intergranular strain development is then presented by application of the elastoplastic self-consistent method. A note is then presented relating to a particular issue arising from this analysis; namely the relative sensitivity to texture of the linear elastic responses of transverse grain families. The subject of interphase stress is then re-addressed using the finite element method. Finally, both the self-consistent and finite element computational methods are employed in a combined analysis of internal stress development in high carbon steel.

65 Chapter 4. Interphase and Intergranular Stress In Carbon Steels 66

4.1 Materials Review

Due to their widespread usage, carbon steels have naturally been widely studied. The purpose of this section is not to provide a comprehensive review of their characteristics, which may be found in standard text books, e.g. [143]. Rather, the intention is only to highlight some characteristics and previous studies which are of particular relevance to the work presented in this chapter. The solubility of carbon in ferrite at room temperature is extremely low: less than 0.00005wt.% [143]. Therefore, during slow cooling of a plain carbon steel, carbon is precipitated into the iron carbide phase Fe3C, commonly known as cementite. A typical microstructure of such a steel consists of grains of ferrite interspersed with pearlite, which itself consists of fine lamellae of ferrite and cementite. However, by heat treating for a number of hours at a temperature just below the eutectoid temperature, the cementite coalesces into larger, spheroidal particles dispersed within a ferrite matrix. Wilson and Konnan studied spheroidised carbon steels in one of the earliest studies of residual stress by a diffraction method [144]. They measured shifts in X-ray diffraction peaks during and after uniaxial tensile straining. In high carbon steel, the diffraction elastic constants of both the ferrite and cementite phases were found to be similar. During plastic deformation, an axially compressive residual stress developed in the ferrite matrix, balanced by tensile stress in the cementite particles. Wilson and Bate later returned to study a similar spheroidised high carbon steel using X-ray diffraction, their purpose being to relate changes in internal stress during load reversal to the Bauschinger effect [70]. They found that the build up of back stress in ferrite is initially rapid, but begins to saturate after about 3% macroscopic strain. They argued that the back stress is insufficient to fully explain the observed Bauschinger effect. However, they also observed that the diffraction peak width falls upon load reversal, indicating a reduction in local stress fluctuations, which they attributed to a reduction in the non-directional component of work hardening in order to fully account for the observed softening. Wilson and Bate acknowledged that the limitation of near-surface measurement limits the accuracy of their results and the extent to which they may be interpreted as representative of the bulk material. Nevertheless, the studies by Wilson and co-workers are worthy investiga- tions which demonstrate the value of ferrite-cementite as a system for the study of interphase stress growth during plastic deformation. This provides a motivation for the present work: to overcome the limitations of earlier research by use of the modern neutron diffraction method. As discussed in the previous chapter, Bonner et al. [132] and Daymond and Priesmeyer [133] have recently published neutron diffraction measurements of internal stress in carbon steels with pearlitic microstructures, but to the author’s knowledge, no such studies have been reported for steels containing spheroidised cementite. However, such a microstructure is similar to that in other particulate composites and readily lends itself to analysis by the Eshelby inclusion formalism and other methods. Therefore the spheroidised microstructure was chosen for this study, in order to maintain originality and applicability, whilst building upon the work of Wilson and co-workers. A point of note about the mechanical behaviour of carbon steels is that it is commonly observed during tensile testing that plastic strain initially develops heterogeneously along the tensile specimen.In the engineering stress-strain curves published in Wilson’s papers, this is evident from flat regions directly after yielding. Such regions are characteristically observed when a tensile sample develops a neck which stabilises and then propagates along the gauge length at a constant applied stress. The strained region is known as a Luders¨ band. In general terms, the origin of this phenomenon is an initially low work hardening rate which increases during straining. The initial neck forms because of tensile plastic instability. That is, if a small part of the sample elongates slightly, the increase of stress due to the drop in cross-sectional area (CSA) exceeds the increase in yield stress due to strain hardening. Thus the region continues to elongate, and the CSA continues to fall. In terms of the curve of tensile yield stress σy versus Chapter 4. Interphase and Intergranular Stress In Carbon Steels 67

Steel C Mn Si Cr Low carbon 0.07 1.0 0.05 trace High carbon 1.00 0.35 0.30 0.40

Table 4.1: Compositions of the low and high carbon steels (wt. %).

strain , this condition may be expressed as [16]

dσ y < σ . (4.1) d y In many cases, this condition remains fulfilled until the material eventually fails. However, if the strain hardening rate increases sufficiently, equality between the above quantities is achieved, and plastic stability is restored in the elongated region. At the boundaries of the strained region, the material which has yet to work harden sufficiently continues to deform; thus the deformed region grows, forming a Luders¨ band. For this to occur, there must be some mechanism by which the strain hardening increases. Clearly, if the rate of change of yield stress with strain is negative, condition (4.1) is fulfilled. This is in fact commonly the case directly after yielding in carbon steels. Initial yield occurs at what is known as the upper yield stress, but the stress required to continue plastic straining is frequently observed to then drop dramatically to a smaller level known as the lower yield stress. The explanation for this phenomenon is provided by the work of Cottrell and Bilby, who showed that interstitial atoms such as carbon diffuse towards dislocation strain fields, in order to minimise elastic energy [145]. The binding energy is great enough that these atoms are effective in pinning dislocations, consequently increasing the stress required for dislocation motion. The upper yield stress is that required to unpin dislocations or generate new, unpinned dislocations. Unpinning or the generation of new dislocations results in a substantially greater mobile dislocation density, enabling plastic flow to continue at the lower yield stress. Due to this softening mechanism, tensile specimens are initially plastically unstable. After the initial softening, the material begins to work harden in a similar manner to other engineering materials. If the rate is sufficient to restore plastic stability, a Luders¨ band is formed.

4.2 Materials

This section describes the materials investigated in the neutron diffraction study, and the processing steps followed to prepare the required microstructures. Two materials were selected for comparison: a low carbon steel (designation EN1A), and a high carbon steel (known as “silver steel”). The low and high carbon steels are hereafter referred to by the abbreviations LC and HC respectively. The alloys were purchased from a commercial steels supplier, West Yorkshire Steel Co Ltd in the form of 12mm diameter cylindrical rods. The compositions of the materials are given in Table 4.1. In order to achieve the desired spheroidal microstructure in the HC steel, the material was first oil-quenched from 900◦C to avoid carbide network formation. It was then heat treated at 700◦C for 10 hours and allowed to furnace cool in order to minimise the generation of thermal residual stresses. The LC steel was heat treated according to the same procedure. Micrographs of the final microstructures are shown in Fig. 4.1. The LC steel has a ferrite grain size of 10µm. There is a small volume fraction of MnS, but this is a soft, ductile phase which does not∼ act as a reinforcement. In contrast, the HC material contains about 20 vol. % cementite particles. It is demonstrated in this chapter that these particles play an important role in the mechanical behaviour. Individual ferrite grains are not visible in the HC steel micrograph, but have been observed by eye. The grain size is similar to the size of the cementite inclusions – Chapter 4. Interphase and Intergranular Stress In Carbon Steels 68

i.e. a few microns; signiﬁcantly smaller than in the LC steel.

(a) (b)

Figure 4.1: Optical micrographs of (a) low carbon; (b) high carbon steel. Etchant: 2% nital solution.

As noted in 3.4.1, the presence of crystallographic texture may influence the development of internal stress.§ In order to investigate this, further processing was performed for the purpose of introducing a varying extent of texture within different samples. As discussed in Chapter 2, plastic deformation alters texture due to the effect of grain rotation. Therefore, rods were plastically deformed by swaging. Rods of both the LC and HC steels were swaged to reductions in cross sectional area of 50% and 70%, hereafter denoted medium and heavily swaged, respectively. The swaged material was spheroidised at 700◦C for a further 10 hours, and furnace cooled. No cracking of the cementite particles was observed under optical microscopy.

4.3 Neutron Diﬀraction Method

The time-of-flight neutron diffraction method, as described in Chapter 3, was employed for the collection of diffraction spectra. The experiments were performed using the ENGIN instrument at the ISIS pulsed neutron facility. This instrument has been described in 3.3. As explained there, the instrument enables simultaneous collection of diffraction spectra§ with scattering vectors pointing along two perpendicular, horizontal directions. The axis of an online Instron stress rig is aligned parallel to one of these directions, enabling the measurement of lattice strain parallel and transverse to an applied uniaxial stress. This set-up was adopted in order to test samples in uniaxial tension. Tensile test-pieces were cut from rods of both LC and HC steels in the unswaged, medium and heavily swaged conditions, the tensile axis lying parallel to the rod axis. The test-pieces had threaded ends and cylindrical gauges of length 50mm. Samples machined from unswaged rods had a gauge diameter of 8mm. This dimension was chosen to optimise the diffracted beam intensity, which increases with the volume of scattering material, but falls with increasing path length through the material due to attenuation. The gauge diameters of samples machined from the medium and heavily swaged material were necessarily smaller — at 6mm and 4mm respectively — due to the reduction in rod diameter caused by the swaging process. The test-pieces were screwed into grips attached to universal joints in order to maintain uniaxial loading. Macroscopic strain was monitored over a gauge length of 12.5mm using an extensometer. The incident beam was defined using cadmium slits. The horizontal slit size was 4mm; the vertical slit size was large enough to bathe the sample in the vertical direction: 10mm for the Chapter 4. Interphase and Intergranular Stress In Carbon Steels 69

unswaged and medium swaged samples, and 5mm for the heavily swaged samples. As discussed in 3.3.2, bathing is justified using the set-up described here, since any systematic shifts in diffraction§ peaks remain constant throughout a test and therefore do not give rise to systematic errors in the determination of elastic strains. Diffraction spectra were recorded at a series of applied uniaxial tensile loads. Count times were approximately 45 minutes. The load was held constant throughout each measurement; during measurements taken in the plastic regime, a small amount of room temperature creep 6 1 occurred (at a rate of the order of 10− s− ). Each test included several unloads in order to measure the evolution of residual strain as a function of accumulated plastic strain. The spectra were analysed according to the methods described in 3.2. That is, single peak fitting was performed for the determination of single phase grain family§ elastic strains, and Rietveld refinement was used to determine volume-averaged elastic phase strains. For both procedures, the stress-free lattice spacings were taken as those measured prior to loading; i.e. it was assumed that no residual stress was present before testing. An example of a typical spectrum recorded for HC steel is shown in Fig. 4.2. The green line passing through the datapoints is the Rietveld fit to the spectrum. Note that the cementite reflections are much weaker than those of ferrite. Two factors contribute to this. Firstly, the volume fraction of cementite is much smaller than that of ferrite. Secondly, due to its orthorhombic crystal structure, the diffraction spectrum of cementite consists of many reflections, but of low multiplicity. Due to the low intensity of the cementite reflections, it was found to be impractical to increase the measurement time sufficiently in order to perform single peak fits of adequate accuracy. Therefore, single peak fitting was performed only on the ferrite reflections. However, it was found that the volume-averaged elastic phase strain of cementite could be reliably determined, since Rietveld refinement takes into account all reflections and therefore benefits from improved counting statistics.

4.4 Macroscopic Response

The following sections on interphase and intergranular strains focus on the data collected on the medium swaged LC and unswaged HC samples. For this reason, the macroscopic stress-strain curves of these samples are presented in this section. However, the swaging and subsequent heat treatment was not observed to significantly affect the macroscopic response, so the LC and HC flow curves presented here may be regarded as typical of all samples of the respective materials. The macroscopic stress-strain curves of the medium swaged LC and unswaged HC materials — recorded during the neutron measurements — are shown in Fig. 4.3. No significant difference in stiffness is evident between the materials, both of which have a Young’s modulus of 210GPa. The high carbon steel has a far greater yield stress than the low carbon steel. A common reason for the yield stress of a composite being greater than that of the unreinforced matrix material is that for a given applied load, the presence of stiff inclusions reduces the stress borne by the matrix. However, there is no evidence to suggest that the yield stress of the HC steel is increased due to elastic reinforcement provided by the cementite inclusions. If this was the case, the macroscopic stiffness of the HC steel would be significantly higher than that of LC steel. Moreover, as stated earlier, work by other authors has indicated that cementite has similar stiffness to ferrite. The difference in yield stress is more likely associated with the difference in ferrite grain size. In 4.2 it was stated that the ferrite grain size in HC steel is substantially smaller than that in LC§ steel. As discussed in 2.6.4, yield stress is linearly related to the inverse square root of the grain size, as expressed in the§ Hall-Petch relationship. Thus it appears that the cementite inclusions only indirectly influence the yield stress, by inhibiting ferrite grain growth. The flat regions directly after yielding in both the LC and HC curves are manifestations of Luders¨ band propagation. As discussed in 4.1, Luders¨ bands develop because of a change § Chapter 4. Interphase and Intergranular Stress In Carbon Steels 70

110

211 321 310 200 220 222

Figure 4.2: Typical neutron diffraction spectrum recorded for unswaged HC steel. Ferrite reflections are indexed. The red markers directly beneath the plot indicate the positions of cementite reflections; the black markers below these indicate the positions of ferrite reflections. The green line passing through the datapoints is a Rietveld fit; the purple line at the bottom is the difference curve between the data and the fit. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 71

700 700 600 600 500 500 400 400 300 300

Stress [MPa] 200 Stress [MPa] 200 100 100 0 0 0 3 6 9 12 0 3 6 9 12 Strain [%] Strain [%] (a) Low carbon (b) High carbon

Figure 4.3: Applied stress versus macroscopic strain response during collection of neutron data.

from tensile plastic instability to stability which, in carbon steels, is associated with yield point softening. Distinct upper and lower yield points are not evident in the curves because the tests were performed in load control. Note that the steps in the plastic regions of the flow curves reflect this nature of the loading. Specifically, rather than imposing a constant strain rate, the load was increased over a relatively short time scale (approximately 30 seconds to 1 minute) and then held constant for periods of about 45 minutes during neutron data collection. The rising edge of each step corresponds to one of the short time intervals when the load was increased between neutron measurements; the flat part of the step corresponds to one of the longer periods during which the load was held constant for neutron data collection. Note that the steep gradient of each edge indicates that after holding at load, the stress has to be increased significantly before further plastic flow occurs. This suggests that some strain ageing occurs during holding. That is, according to the theory of Cottrell and Bilby discussed earlier [145], carbon atoms tend to diffuse to dislocation cores, pinning the dislocations. Thus a new upper yield point develops; the stress must be raised to this level before further plastic straining occurs. It was observed during loading that when the stress was raised beyond a critical level, an increment of plastic strain developed almost instantaneously. This may be explained on the same basis which explains the formation of Luders¨ bands. At constant load, the softening associated with an increase in mobile dislocation density must be compensated by an equal increment of hardening, developed by a rapid increment of plastic strain. Such a strain increment, plus a small increment due to room temperature creep developed during the hold period, are responsible for the flat part of each step.

4.5 Interphase Strains

As noted above, this and the following section focus on measurements made on the medium swaged LC and unswaged HC samples. These were the samples for which the best datasets, in terms of counting statistics and number of datapoints, were obtained. The full dataset of measurements is presented later in the context of assessing the reproducibility of measurements and the inﬂuence of crystallographic texture. As seen from Fig. 3.2, strains parallel to the tensile axis are determined from the spectra recorded in the right-hand detector bank, while transverse strains are determined from those recorded in the left-hand bank. The lattice parameters determined from Rietveld reﬁnement of the spectra recorded in both banks prior to loading are tabulated in Table 4.2, for both the medium swaged LC and unswaged HC samples. For all lattice parameters shown, the values determined from the left-hand bank data are systematically lower than those determined Chapter 4. Interphase and Intergranular Stress In Carbon Steels 72

Right Left LC ferrite 2.86654 0.000036 2.865848 0.000039 HC ferrite 2.86621 0.000029 2.865557 0.000031 HC cementite (a) 4.52119 0.000555 4.51966 0.000566 HC cementite (b) 5.078431 0.000661 5.077344 0.000687 HC cementite (c) 6.743696 0.000871 6.741212 0.000912 Table 4.2: Lattice parameters in medium swaged LC and unswaged HC samples prior to tensile loading, as determined by Rietveld reﬁnement of the spectra recorded in the right (axial) and left (transverse) detector banks.

from the right-hand bank data. This suggests a systematic error, possibly arising because the scattering volume is not fully immersed within the sample, as discussed in section 3.3.2. However, as also noted in that section, this should not present a problem, since the systematic shift should remain the same in each bank for all measurements, and therefore should not aﬀect the strains calculated with respect to the initial lattice spacings. Error bars plotted on the graphs in this section are statistical ﬁtting uncertainties; where unseen, they are too small to represent. Measurements made under applied loading and after the removal of load are presented and discussed in separate subsections below. A short analysis based on the unrelaxed model discussed in 2.7 is then presented. §

4.5.1 Strains Under Applied Loading

The development of elastic phase strains with applied stress is shown in Fig. 4.4. The LC and HC responses are plotted together, for the purpose of comparison. Strain is presented in the 6 units of microstrain, where 1 microstrain 1 10− . This unit is used commonly throughout the remainder of this dissertation, and is sometimes≡ × written µe.

700 700

600 600

500 500 HC yield stress HC yield stress

400 400

LC yield stress LC yield stress 300 300 Applied stress [MPa] Applied stress [MPa] 200 200 Ferrite (LC) Ferrite (HC) 100 Cementite (HC) 100

0 0 0 2000 4000 6000 8000 −3000 −2000 −1000 0 Lattice strain [microstrain] Lattice strain [microstrain] (a) (b)

Figure 4.4: Lattice strains determined by Rietveld reﬁnement as a function of applied stress: (a) parallel to applied stress (axial); (b) transverse to applied stress. The dashed lines passing through the origin are best ﬁts to the elastic response of the ferrite phase in the high carbon steel. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 73

LC Steel Consider firstly the LC axial and transverse ferrite strains. Prior to yielding, the axial elastic strain develops linearly with applied stress, the slope corresponding to a Young’s modulus of 213GPa, in good agreement with the macroscopically-determined stiffness. Correspondingly, a negative transverse strain also develops linearly in the elastic regime, due to Poisson contraction. The ratio of axial to transverse slopes gives a Poisson’s ratio of 0.28. Note that if a single phase material obeys Hooke’s Law in the elastic regime, then the linear relationship between applied stress and average elastic phase strain should hold even after the onset of plastic flow. This is because no re-partitioning of load between phases is possible. The axial and transverse ferrite responses in the LC steel do indeed remain approximately linear through the yield point, even though 2.5% plastic strain is introduced between the measurements taken just before and just after yielding (due to the Luders¨ band propagation). There are however slight deviations from linearity – a drop of about 200µe in the axial strain and a positive shift of about 120µe in the transverse strain. These shifts may possibly be due to the presence of a small amount of cementite: the carbon content of the LC steel corresponds to a cementite volume fraction of 1%, although this is not visible in the micrograph of 4.1. However, in their X-ray study, Wilson and Konnan also observed a slight shift from linearity during yielding [144]. They noted that axially compressive residual stresses have also been detected by diffraction experiments on other single phase metals [146, 147], and suggested that these may be balanced by tensile stresses in highly distorted regions which fail to diffract, such as dislocation cell walls.

HC Steel Whatever the explanation for the deviations from linearity observed in the LC steel, they are very small when compared to the shifts observed during yielding of the HC steel. Consider firstly, however, the elastic phase strains which develop in the HC steel within the elastic regime. Both in the axial and transverse directions, the cementite response follows closely the (linear) ferrite response, indicating that both phases have similar Young’s moduli and Poisson’s ratios. Moreover, the gradients of the axial and transverse ferrite responses are almost identical to those in LC steel. This also supports the assertion that the elastic moduli of the phases are similar; if they were significantly different then the ferrite would be expected to bear a greater or smaller proportion of the applied load, and thus exhibit a different diffraction elastic constant in the composite to that when present as a single phase. The observed similarity of the elastic moduli agrees with the results reported by Bonner et al. [132] and Daymond and Priesmeyer [133], as discussed in 3.4.2. It also explains the observation that the stiffness of ferritic steel is hardly affected by cemen§ tite content, as reported by Bohnenkamp and Sandström [148]. In fact, from fitting straight lines to the present data, the diffraction elastic constant of cementite is seen to be slightly lower than that of ferrite: 211GPa in comparison to 220GPa. This is in agreement with a recent study of bulk cementite, which also reported the phase to be slightly less stiff than ferrite [149]. However, the difference is small enough that in order to simplify analysis the phases may justifiably be assumed to have identical Young’s moduli. The ratio of gradients of the axial to transverse slopes gives -0.28 for ferrite and -0.29 for cementite; again these are sufficiently close to assume for the purpose of analyis that the Poisson’s ratios of both phases are identical. In contrast to the almost linear response of the LC steel through yielding, dramatic shifts in lattice strain are seen in the HC steel between the measurements taken before and after Luders¨ straining. The axial ferrite response deviates away from the elastic line towards lower lattice strain. As discussed for similar composites in 3.4.2, such a shift indicates that load is transferred away from this phase, as it begins to deform§ plastically. However, whereas in other systems the axial response of the plastically deforming phase is commonly reported to shift gradually from the elastic line and maintain a positive gradient, in this case a large negative shift is seen. This observation relates to the yield point softening which is responsible for the development of Luders¨ bands, as discussed in 4.1. Directly after yielding, the material cannot § Chapter 4. Interphase and Intergranular Stress In Carbon Steels 74

bear the stress that it was able to bear directly before yielding. In order to continue to bear this applied stress, the material must work harden. If it was present as a single phase, the ferrite alone would have to work harden sufficiently to exactly compensate the yield point softening. It would then continue to bear all of the applied stress; hence the lack of a significant lattice strain shift during yielding of the LC steel. However, in the two phase HC steel, some of the work hardening may be attributed to back stress hardening, due to the presence of reinforcing particles — as discussed in 2.7. If this is the case, then the ferrite yield stress may remain below the applied stress and§thus the elastic strain below that prior to yield. Confirmation of this hypothesis requires evidence that cementite does indeed act as a reinforcing phase. This evidence is provided by the cementite elastic strain. The axial strain increases dramatically during macroscopic yielding. This indicates that the cementite bears a greater proportion of the applied load when the ferrite yields, by continuing to deform elastically. As described in 2.7, the resulting shape misfit between the phases causes on average axially tensile residual §stress in the cementite and compressive residual stress in the ferrite, providing the required back stress hardening. The transverse strain information supplements that observed in the axial direction. During yielding, the cementite transverse elastic strain grows in magnitude while the ferrite strain falls. However, even under the assumption that both materials have identical elastic constants, after yield the ratio of transverse to axial strain magnitudes in each phase is not expected to equal the Poisson’s ratio, because although the macroscopic stress state remains one of uniaxial tension, this is not so for the average stress in either phase, due to the introduction of residual stress. Estimates of the cementite volume fraction may be determined from the relative lattice strain shifts, under the approximation that the phases are elastically identical and isotropic. Further it is assumed that the inclusions are randomly distributed with shapes which are randomly oriented. With these assumptions, it is straightforward to show from the stress balance equation (2.27) and the standard stress-strain relationships for elastically isotropic materials [16] that for uniaxial applied stress σA, the elastic strain tensor component parallel to the tensile axis, el, satisfies k σA f el + (1 f) el = (4.2) k I − k M E D E D E and the transverse component el (assumed isotropic in the transverse plane) satisfies ⊥ A el el σ f I + (1 f) M = ν . (4.3) ⊥ − ⊥ − E

In these equations, f is the cementite volume fraction, E and ν are the Young’s modulus and

Poisson’s ratio, and I denotes the average in the cementite inclusions, while M denotes that in the ferrite matrix.hi hi Taking E = 215GPa and ν=0.28, and applying equations (4.2) and (4.3) to pairs of ferrite and cementite datapoints in both the axial and transverse datasets gives estimates of f between 15% and 20%. This range is very reasonable, encompassing the value determined by Rietveld refinement of 19%, and a value of 16% determined from the specified composition. The gradient of the axial cementite response after yield is shallower than the gradient beforehand (a straight line fit to the datapoints after the yield point shift gives a gradient of 70GPa, in comparison to 211GPa before yield). This indicates that as plastic flow continues, the phase continues to bear a greater proportion of the applied load. That is, back stress hardening continues to contribute to the overall hardening of the composite. This indicates that the plastic misfit continues to grow, suggesting that the cementite continues to deform elastically or at least develops less plastic strain than the ferrite. The slope of the transverse cementite curve is also shallower after yield (-150GPa, in comparison to -730GPa in the elastic regime), providing supplementary evidence of the continuation of back stress hardening. Correspondingly, the stress balance condition requires that the ferrite gradients should also change. Although this Chapter 4. Interphase and Intergranular Stress In Carbon Steels 75

is not clearly evident from the experimental data, the cementite gradient changes are expected to be more easily evident because of the smaller volume fraction of the reinforcing phase.

4.5.2 Residual Strains The generation of residual stress is evident from the elastic strains which remain when the applied stress is removed. Fig. 4.5 shows the development of the residual lattice strain with plastic strain in the LC steel, determined through Rietveld reﬁnement of the spectra recorded after the removal of load. Small compressive and tensile residual strains develop in the axial and transverse directions, respectively: approximately 200µe in the axial direction and 150µe in the transverse direction. These are consistent with the− lattice strain shifts seen under applied loading, and possible origins have been brieﬂy discussed above.

400 Axial 300 Transverse

200

100

−100

−200 Residual lattice strain [microstrain] −300

−400 0 2 4 6 8 10 12 Macroscopic plastic strain [%]

Figure 4.5: Rietveld-determined residual lattice strain of ferrite in LC steel, as a function of macroscopic plastic strain. Fitted curves through the datapoints are included as a guide to the eye.

Fig. 4.6 shows the Rietveld-determined residual lattice strain versus macroscopic plastic strain in both phases of the HC steel. Two sets of measurements are presented. The datapoints joined by blue dashed lines represent residual strains determined from the measurements made under applied loading, after subtraction of the elastic strain due to the applied stress (found by extrapolation of the linear elastic response). The datapoints joined by continuous red lines represent residual strain measurements made after the removal of applied stress. In terms of chronology, each blue datapoint was measured directly after the corresponding red datapoint. Corresponding measurements made under load and after load removal would be expected to be equal if unloading is fully elastic. However, the cementite strains appear to relax by approximately 15% during unloading. This relaxation may arise due to a significant Bauschinger effect, as discussed in 2.7.1. That is, the local compressive residual stress in a region of matrix near to an inclusion ma§ y be of such magnitude that the reverse yield stress is reached even before the forward load is fully removed. This would cause reverse plasticity which would act to reduce the plastic misfit between phases and relax the residual strains. Evidence that some reverse plasticity occurs is seen by examining the macroscopic stress-strain curve in greater detail. Fig. 4.7 shows the macroscopic stress-strain response during the unloading and subsequent re-loading steps performed at about 3% strain. During unloading the gradient becomes shallower as load is removed, indicating that some reverse plasticity does indeed occur. The re-loading curve begins more steeply, due to the absence of plasticity. The curve also shows that during the 45 minute dwell in the unloaded state, the sample contracts slightly over time. Thus it appears that some time dependent relaxation occurs. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 76

5000

4000

3000 Key: Squares − cementite; Triangles − ferrite, 2000 Unfilled − axial; Filled − transverse, Blue dashed line − at load; Red solid line − after unloading 1000

−1000 Residual lattice strain [microstrain] −2000

−3000 0 1 2 3 4 5 6 7 8 9 Macroscopic plastic strain [%]

Figure 4.6: Rietveld-determined residual lattice strain of ferrite and cementite phases of HC steel, as a function of macroscopic plastic strain. The datapoints joined by blue dashed lines represent measurements made under applied loading; datapoints joined by continuous red lines represent measurements made after the removal of load. Axial and transverse strains are presented for both the ferrite and cementite phases, as indicated in the key.

600

500

400

300

Applied stress [MPa] 200

100

0 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5 3.55 Macroscopic strain [%]

Figure 4.7: Macroscopic stress-strain response during unloading, a dwell of approximately 45 minutes, and subsequent re-loading. The arrows indicate the unloading and re-loading stages. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 77

The ferrite strains recorded under load and after load removal are almost identical. This appears to conflict with the evidence of relaxation indicated by the cementite strains. If the average residual stress (and strain) in one phase changes, then the average residual stress (and strain) in the other phase must also change, in order that the stress balance condition remains fulfilled. A possible explanation is that the ferrite strains measured by diffraction are not fully representative of the phase average. This might be so if highly deformed regions fail to diffract because the crystal structure is too distorted by dislocations and type III microstresses. The most highly distorted regions are expected to be close to the matrix-inclusion interfaces, where the matrix plastic flow is most disturbed. These are also the regions in which local plastic deformation may operate in order to reduce the plastic misfit. Such plastic deformation would further increase the dislocation density and distortion of the crystal lattice in these regions. Hence the matrix regions where the residual stress is expected to relax most during unloading are also those regions which may not diffract effectively. Since the evidence suggests that cementite deforms only elastically, in this phase the diffraction-determined strains should be representative of the phase average. This may help to resolve the apparent conflict between the observations of relaxation in the cementite strains but not in the ferrite strains. Whether or not some relaxation occurs during unloading, the trends in residual strain development remain clear. The cementite inclusions develop axially tensile and transversely compressive residual strains, while the ferrite matrix correspondingly develops axially compressive and transversely tensile strains. Although Luders¨ straining precludes measurements at low plastic strains, it is clear from the measurements at 2.2% plastic strain and subsequently that there is an initially high rate of change of residual strain with plastic strain, but that saturation begins to occur after a few percent strain. This agrees well with the assertion by Wilson and Bate that back stress hardening saturates after about 3% pre-straining [70], noted in the materials review at the beginning of this chapter. A possible explanation for saturation is that the cementite inclusions begin to deform plastically. Cementite plasticity has been previously been reported, particularly in pearlitic steels [150, 151]. However, since the evidence presented above suggests that plastic relaxation mechanisms operate, it is likely that saturation occurs because the plastic misfit is limited by the operation of such mechanisms. Further estimates of cementite volume fraction f may be determined from the strains determined after load removal, using equations (4.2) and (4.3). Unlike those determined from the at-load data, these estimates do not require values of Young’s modulus and Poisson’s ratio to be specified. Pairs of ferrite-cementite datapoints give estimates of f between 18% and 22%. This range is slightly higher than that determined from the at-load data, and does not encompass the value of 16% determined from the specified composition. However, the accuracy of the manufacturer’s specification is not known.

4.5.3 Unrelaxed Model In 2.7, the Eshelby and mean ﬁeld theories were applied to the prediction of phase-averaged residual§ stresses in a composite consisting of a random dispersion of fully elastic spherical inclusions within a ductile matrix. The phases were assumed to be elastically identical and isotropic. This model provides a good representation of the microstructure of the HC steel, since the cementite inclusions are spheroidal and it has been demonstrated in this chapter that the cementite elastic stiﬀness is very similar to that of ferrite. The model is applied in this section to facilitate further discussion about the residual strains presented in 4.5.2. P § If the matrix plastic strain ij is assumed to be uniform, then the model composite analysis presented in 2.7 gives for the average stress in each phase §

σij = f σ∞ , (4.4) h iM − ij σij = (1 f) σ∞ , (4.5) h iI − ij where f is the volume fraction of inclusions, subscript M denotes the matrix, I denotes the Chapter 4. Interphase and Intergranular Stress In Carbon Steels 78

inclusions, and σij∞ is given by

E (7 5ν) P σ∞ = − . (4.6) ij 15 (1 + ν) (1 ν) ij − Note that the phase-averaged stress components are directly proportional to the corresponding components of the plastic strain tensor. For comparison to the experimental data, the average stresses given by equations (4.4) and (4.5) must be related to the average elastic residual strains. For an isotropic material, Hooke’s el Law relating the elastic strain ij to the stress σij may be written as [5]

el (1 + ν) ν = σij σkk δij . (4.7) ij E − E As discussed in 2.7, the fact that volume is conserved during plastic deformation implies that § P the trace of the plastic strain tensor vanishes, i.e. ii = 0. Since in this example the components of the phase-averaged stress tensors are proportional to the corresponding plastic strain tensor

components, the traces of these tensors vanish also, i.e. σii M,I = 0. Hence, the second term h i el in equation (4.7) vanishes and the average matrix and inclusion residual elastic strains ij M and el respectively are given by ij I (7 5ν) el = f − P and (4.8) ij M − 15 (1 ν) ij − (7 5ν) el = (1 f) − P . (4.9) ij I − 15 (1 ν) ij −

As discussed in 2.7.1, if the model composite is subjected to uniaxial tensile stress along the 3-axis of the coordinate§ reference frame, the conditions of rotational symmetry about the tensile axis and volume constancy imply that the uniform plastic strain tensor must have the form

1/2 0 0 P P − ij = 0 1/2 0 , (4.10)  0 −0 1    where the scalar P equals the matrix axial plastic strain. It may be shown that when an elastically homogeneous body develops a distribution of eigenstrain, the macroscopic strain equals P the average of the eigenstrain over the whole body [5]. In this example, uniform eigenstrain ij is developed in the matrix phase, which has volume fraction 1 f. Hence the macroscopically- measured tensile plastic strain mac is related to the axial comp− onent of matrix plastic strain P by mac = (1 f) P . (4.11) − From (4.8) to (4.11), the phase-averaged residual elastic strains parallel ( ) to the tensile axis are given in terms of the macroscopic axial plastic strain mac as k

f (7 5ν) el = − mac , (4.12) k M −15 (1 f) (1 ν) − − D E (7 5ν) el = − mac , (4.13) k I 15 (1 ν) D E − and the (transversely isotropic) components perpendicular to the tensile axis ( ) are related to the axial components by a factor of 1/2: ⊥ −

el 1 el M, I = . (4.14) ⊥ −2 k M, I D E Chapter 4. Interphase and Intergranular Stress In Carbon Steels 79

These model predictions are directly comparable to the measured residual phase strains. The axial strain predictions are plotted as dashed lines on Fig. 4.8, assuming a Poisson’s ratio of 0.28, as determined from the data measured under applied loading, and a cementite volume fraction f of 20%, which is within the range of values determined by consideration of stress balance. The measured axial and transverse phase-averaged residual strains are also reproduced and labelled on Fig. 4.8. Since the model assumes that the matrix develops uniform plastic strain, related to the macroscopic strain by equation (4.11), it takes no account of relaxation. Hence the residual strains develop linearly with applied stress, and may be regarded as upper bounds to the real lattice strains. The experimental data are consistent with this statement. On Fig. 4.8, curves are fitted through the axial strain experimental datapoints, to serve as guides to the eye. The initial slopes of the curves are of comparable magnitudes but slightly shallower than the gradients predicted by the unrelaxed model. However, as plastic strain increases, the measured strains begin to saturate and are therefore increasingly overestimated by the model predictions. Whatever its origin (e.g. cementite plasticity or local matrix plastic flow), saturation indicates that the misfit between phases does not continue to grow linearly with plastic strain. Nevertheless, the origin of the misfit remains plastic deformation which conforms to volume constancy, and the rotational symmetry about the tensile axis is maintained. Therefore, the average misfit between an inclusion and the surrounding matrix must still be of the form given by equation (4.10), although the scalar P no longer relates to the macroscopic plastic strain according to the simple relationship of equation (4.11). Apart from this point, the analysis remains valid. Thus, the ratio of transverse to axial residual strains (equation 4.14) should remain 1/2, regardless of relaxation of the misfit. The experimental data agree well with this assertion.− This is shown on Fig. 4.8 in the following manner. The strains determined by fitting curves through the measured axial strains are multiplied by 1/2 in order to give predictions of the transverse strains. These curves match the measured transv− erse strains very well, particularly in the ferrite phase.

4.6 Intergranular Strains

In this section, the evolution of ferrite intergranular strains is presented for the medium swaged LC and unswaged HC materials. The results and discussion are divided into sections concerning the strains developed under applied loading ( 4.6.1) and the residual strains remaining after the removal of load ( 4.6.2). § § 4.6.1 Strains Under Applied Loading Fig. 4.9 shows the strain responses of individual grain families in ferrite, determined by fitting of single diffraction peaks. The families are labelled according to the hkl plane normal which lies parallel to the scattering vector (either the tensile axis or transverse direction). For clarity, only the responses of the 110, 200 and 310 axial and transverse families are presented. These have been chosen because, by inspection of the entire dataset (see 4.7), it is seen that they cover the range of behaviour demonstrated by all the families. It should§ be noted that in the LC steel the axial 200 reflection was rather weak, but the response of the corresponding grain family is of particular interest because it is significantly different from that of other families, both in the elastic and plastic regime. The volume-averaged elastic phase strains, as determined by Rietveld refinement, are also shown on Fig. 4.9. The elastic anisotropy of ferrite is evident in the axial loading data. The anisotropy factor (see equation (2.19)) may be written in terms of the Miller indices hkl as Ahkl = 2 2 2 2 2 2 2 2 2 2 1 h k + k l + l h / h + k + l , with higher values indicating stiffer directions. Ahkl

1 Higher values of Ahklindicate stiﬀerdirections only if the single crystal compliance tensor components satisfy 2 (M11 − M12) > M44. This is the case for ferrite [16]. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 80

5000 Cementite axial − unrelaxed model 4000 Cementite axial

3000

2000

1000 Ferrite transverse

0 Ferrite axial −1000

Residual lattice strain [microstrain] Cementite transverse −2000 Ferrite axial − unrelaxed model −3000 0 1 2 3 4 5 6 7 8 9 Macroscopic plastic strain [%]

Figure 4.8: Elastic residual strains of ferrite and cementite phases of HC steel, as functions of macroscopic plastic strain. The straight, dashed lines are predictions made using the mean field, assuming the cementite deforms elastically and there is no relaxation of the plastic misfit strain. Datapoints show Rietveld-determined residual lattice strains. The curves fitted through the axial strain datapoints serve as guides to the eye. The corresponding transverse strain curves are predicted by multiplying the fitted axial strains by 1/2. − Chapter 4. Interphase and Intergranular Stress In Carbon Steels 81

(b) 300 (a) 300 [MPa] 200 200 stress

110 200 100 100 310 Applied Applied stress [MPa] Rietveld

0 0 0 500 1000 1500 2000 500 0 Lattice strain [microstrain] Lattice strain [microstrain] 700 700 (c) 600 600 (d) 500 500 400 400 300 300 200 200

Applied stress [MPa] 100 Applied stress [MPa] 100 0 0 0 1000 2000 3000 −1000 −500 0 Lattice strain [microstrain] Lattice strain [microstrain] Figure 4.9: Strain responses of individual hkl ferrite grain families and Rietveld- determined strain: (a) LC, axial; (b) LC, transverse; (c) HC, axial; (d) HC, transverse. The horizontal dashed lines indicate the yield stresses. Error bars represent statistical ﬁt uncertainties range; for clarity, some are omitted from (b). The legend in (a) applies to all plots. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 82

evaluates to 0.00, 0.09 and 0.25 for the 200, 310 and 110 reflections, agreeing with the observed trend in diffraction elastic constants in both the LC and HC axial strain data (Figs. 4.9a and 4.9c respectively). The yield stresses of the HC and LC steels are indicated on the plots of Fig. 4.9 by horizontal dashed lines. After yielding, the axial grain family responses in the LC steel (Fig. 4.9a) are seen to diverge to a greater extent than in the elastic regime. However, the deviations from the linear, elastic responses are smaller than those typically observed in fcc steels (see Fig. 3.4) [104, 107]. Hutchinson’s self-consistent analysis, described in 2.6.4, shows that the incremental difference of plastic strain between grains, and consequently§ that of internal stress, becomes smaller as the macroscopic plastic strain increases. Therefore, the changes in intergranular strain are most clearly seen in the low plastic regime. In this study, the Luders¨ elongation precludes measurements at low plastic strains, but the final results are seen in the data measured after Luders¨ straining. The axial 200 curve shifts to greater lattice strain, while the 110 and 310 curves shift in the opposite direction. As discussed in 3.4.1, this implies that the latter grain families deform plastically first, leaving the non-plastic§ grains, such as the 200 family, to bear extra load. Although the datapoints are joined by lines as an aid to the eye, these may falsely give the impression that the shifts occur gradually as the load is increased. In fact, measurements made close to the yield point on other samples (as shown in 4.7) confirm that the shifts are very sharp, occurring simultaneously with Luders¨ elongation. § Fig. 4.9b shows the development of transverse grain family strains in the LC steel. Before consideration of these transverse responses, a point made in 3.4.1 should be emphasised once more. That is, grains belonging to a single transverse diffraction§ family may possess a range of different orientations with respect to the tensile axis. This complicates the interpretation of the transverse strains. For instance, the variation of slopes in the elastic regime cannot be simply related to the elastic anisotropy factor, since grains in the same transverse family may have very different elastic stiffnesses parallel to the tensile axis. The transverse 200 family, for example, consists of grains which may have any crystallographic direction in the set hk0 aligned with the tensile axis; the elastic anisotropy factors of which vary from 0.0 to 0.25. hThusi it is not trivial to predict which grain families should have the steepest gradients. Predictions using the self-consistent model are given in 4.9, however. § Since grains in a transverse family possess a range of orientations with respect to the tensile axis while all grains in an axial family share a common orientation, one might intuitively expect that the variation between transverse strain responses would be less distinct than the variation between axial responses. Comparison of Figs. 4.9a and 4.9b shows that this is far from true! Upon yielding, the grain families develop markedly different transverse lattice strains. The 200 transverse response is of particular note. During Luders¨ elongation, it develops a large positive residual strain. This is consistent with the result reported by Pang et al. [115], which has been discussed in 3.4.1. After yielding, the slope of the 200 transverse slope is almost vertical. These results suggest§ that although there are a range of orientations within the family, on average grains which belong to the 200 transverse family tend to yield early relative to other grains. As noted in the discussion of their paper, Pang et al. did not provide an adequate explanation for this. The purpose of 4.8 is to provide some insight into the problem. During Luders¨ elongation, the 310 transverse family§ also develops a large positive shift in strain, although smaller than the 200 shift. In contrast, the 110 response hardly deviates from linearity during yielding. These results are also discussed further in 4.8. § In the axial response of the HC steel (Fig. 4.9c), plastic intergranular effects are obscured by the dominant influence of stress redistribution between the phases. While all the ferrite grain families exhibit a yield point shift to lower lattice strain, the relative magnitudes of the shifts cannot be easily related to the behaviour in the LC steel. The data is not sufficiently accurate to distinguish between the slopes of different grain families after yielding. However, the HC transverse data (Fig. 4.9d) reveals similar intergranular strain trends to those in the LC material. During Luders¨ elongation, all grain families exhibit a positive lattice strain shift, arising due to the load transfer to cementite discussed in 4.5. Variations § Chapter 4. Interphase and Intergranular Stress In Carbon Steels 83

in the magnitudes of the shifts between grain families, however, imply intergranular stress development. The observed magnitudes are in the order 110 < 310 < 200, with the 200 shift being particularly large. This trend concurs with that observed in the LC steel. Finally in this section, a note is raised concerning the strains determined by Rietveld refinement. Figs. 4.9a and 4.9c show that for both the LC and HC materials, the axial strains determined by Rietveld refinement are close to those determined for the 110 grain family. A possible reason for this is that the material is textured. Indeed it is shown in 4.7.2 that the medium swaged LC steel has a 110 fibre texture. Thus the phase average is exp§ ected to be biased towards the strain in the 110 family, and the Rietveld refinements may accurately reflect this. However, there is also evidence shown in 4.7.2 that the unswaged HC steel is untextured, and therefore there is a possibility that the Rietv§ eld-determined strains are not truly representative of the phase average. If so, this may explain the small apparent residual phase strain observed in ferrite in LC steel. However, the discrepancy between the Rietveld-determined strain and the true phase average should be no greater than the magnitude of the intergranular residual strains, and therefore should be negligible in comparison to the large interphase strains developed in HC steel.

4.6.2 Residual Strains In this section, a comparison of the intergranular strain development in the LC and HC steels is made using the residual strain measurements recorded after the removal of load. In order to carry out this comparison, the Rietveld-determined residual interphase strains (Figs. 4.5 and 4.6) are subtracted from the reflection-specific responses. The resulting curves of intergranular residual strains versus macroscopic plastic strain are shown in Fig. 4.10. The axial intergranular residual strains in the LC steel (Fig. 4.10a) are small, with the 200 family developing a small tensile strain, while the 110 and 310 strains do not deviate significantly from the Rietveld-determined values. This is consistent with the strains developed under loading. The most noticeable feature is not the variation among grain families, but that no family develops appreciable residual strain. This point is addressed in 4.9.3, where simulations using the elastoplastic self-consistent method are used to demonstrate§ that elastic anisotropy tends to counteract the effects of plastic anisotropy in ferrite. However, greater variations between families are seen in the HC axial strains (Fig. 4.10b). The 200 and 310 families develop negative and the 110 positive axial intergranular residual strains. Allen et al. [113], who studied Al/SiC metal matrix composites, also observed greater axial grain family variations in the composite matrix than in the single phase parent material. The transverse intergranular residual strain curves in the two materials are, however, strik- ingly similar (Figs. 4.10c and 4.10d). In concurrence with the loading strains of Fig. 4.9, the variation in transverse intergranular residual strains in the LC steel is much greater than that in the axial residual strains. The 200 transverse family develops a large strain of +400µe, and the 310 family develops a smaller positive strain. The 110 strain is slightly negative. The trends and absolute magnitudes of the transverse intergranular residual strains developed in the HC steel are very similar. The influence of elastic anisotropy helps to explain the difference between the axial intergranular residual strains in the two materials. Whereas there is almost no average residual stress in the ferrite phase in the LC steel, the ferrite in HC steel remains in a state of axial compression after unloading. Due to elastic anisotropy, the grain families should respond to this average state by developing compressive strains of different magnitudes. Of the three families considered, the 110 family is stiffest and should thus develop the smallest compressive strain, while the 200 family is most compliant and should develop the largest compressive strain. The 310 stiffness lies closer to that of 200 than 110; thus its compressive strain should be nearer to that of 200. These are indeed the trends seen in Fig. 4.10c. This argument neglects the fact that the ferrite average residual stress is in fact triaxial. However, the Poisson effects of residual tensile stresses perpendicular to the (former) applied stress axis should act to enhance Chapter 4. Interphase and Intergranular Stress In Carbon Steels 84

600 600 110 400 200 400 310 200 200

0 0

−200 −200

−400 (a) −400 (b) Strain difference [microstrain] Strain difference [microstrain] 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Macroscopic plastic strain [%] Macroscopic plastic strain [%]

600 600

400 400

200 200

0 0

−200 −200

−400 (c) −400 (d) Strain difference [microstrain] Strain difference [microstrain] 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Macroscopic plastic strain [%] Macroscopic plastic strain [%] Figure 4.10: Ferrite intergranular residual strain as a function of macroscopic plastic strain. The contribution from interphase strain is removed by subtracting the Rietveld- determined phase strain from the reﬂection-speciﬁc strain. (a) LC, axial; (b) LC, transverse; (c) HC, axial; (d) HC, transverse. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 85

the variation, although proper justification of this statement requires detailed consideration of Poisson contraction in single crystals, which is not given here. A simple estimate can be made of the magnitude of the strain variation expected due to elastic anisotropy. From Fig. 4.8, the average axial and transverse ferrite residual strains saturate at values of approximately 900µe and +450µe respectively. Taking the Young’s modulus as E = 215GPa and Poisson’s− ratio as ν = 0.28, these strains give an axial residual stress of 150MPa. Using fitted diffraction elastic constants for the axial 110 and 200 reflections of 227GP−a and 178GPa respectively, the difference in grain family elastic strains due to the application of a uniaxial compressive stress of 150MPa is 180µe. The measured difference is approximately 450µe. However, as stated, the −stress is not uniaxial and the tensile transverse stresses should tend to increase the estimated difference towards the measured value. From the above estimate, it seems reasonable to suggest that this effect at least partly explains the greater axial intergranular strains observed in the HC steel. An additional effect may be that plastic flow is more heterogeneous in the composite material, due to the requirement for the matrix to flow around inclusions. Close to the inclusions, the local direction of the maximum principal plastic strain may deviate significantly from the tensile axis (although the possibility of such local disturbance was ignored in the model presented in 4.5.3). Hence the intergranular strains in these regions may differ from those developed due to§simple elongation along the tensile axis. Although the local plastic strain tensor averaged over the whole body must equal the macroscopic plastic strain tensor, this does not imply that residual grain family strains due to local disturbances must average to zero. Despite the possibility of heterogeneous plastic flow, for the most part plastic deformation in the matrix is likely to be reasonably similar to the simple axial elongation developed in the LC steel. Since this deformation gives rise to significant transverse intergranular residual strains, on average these should dominate over those developed due to the disturbance of flow near to inclusions. Moreover, the effect of elastic anisotropy due to interphase stress should be less evident in the transverse strain data. There are two reasons for this. The tensile stress resolved along a particular transverse direction has only half the magnitude of the axial compressive stress. Additionally, since one of the other principal stresses is tensile while the second is compressive, the Poisson effects counteract each other. These points help to explain why plots 4.10b and 4.10d are so similar.

4.7 Reproducibility of Data and Inﬂuence of Crystallographic Texture

The previous sections have focused on the results obtained from the medium swaged LC and unswaged HC materials, for which the best datasets, in terms of counting statistics and number of datapoints, were obtained. Measurements were made, however, on unswaged, medium and heavily swaged specimens, for both the LC and HC steels. In this section, a summary of the results obtained is presented. Fig. 4.11 shows the axial neutron diffraction spectra obtained from the unswaged and heavily swaged LC and HC materials. The spectra are fitted by Rietveld refinement without the inclusion of texture parameters. Therefore, the refinement (green line) shows the predicted spectrum for a randomly oriented polycrystal, and the discrepancies in peak intensities between this and the experimental spectrum (red datapoints) provide an indication of the texture in the sample. The difference in intensity as a function of d-spacing between the experimental and Rietveld refined spectra is shown by the purple line beneath the spectra. From this, it is seen that the untextured refinement predicts too low an intensity for the 110 reflection in the unswaged LC steel; thus the LC steel has a significant 110 fibre texture even prior to swaging (Fig. 4.11a). This is a commonly observed texture in extruded or drawn bcc metals [152, 153], and clearly arises from the mechanical processing of the steel prior to being received. Its presence limits the extent to which the swaging process may further introduce texture. Upon Chapter 4. Interphase and Intergranular Stress In Carbon Steels 86

110 − →

(a) (c)

(b) (d)

(e)

Figure 4.11: Axial neutron diffraction spectra. (a) to (d) are prior to deformation: (a) unswaged LC; (b) heavily swaged LC; (c) unswaged HC; (d) heavily swaged HC. (e): unswaged HC after 8% plastic strain. Red datapoints show the neutron data. Green lines are Rietveld refinements of the spectra with no texture parameters included. The purple lines show the difference between data and fit. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 87

heavy swaging and subsequent heat treatment (Fig. 4.11b), the 110 fibre texture is slightly accentuated, but there is little difference between the unswaged and heavily swaged spectra. Although this means that the datasets from the differently processed LC steels are not effective in showing the influence of texture on intergranular stress development, they may be used in an alternative manner to assess the reproducibility of the data. The swaging process is more effective, however, when applied to the HC steel. The untextured Rietveld refinement provides a very good fit to the unswaged spectrum (Fig. 4.11c), indicating that this material is untextured. Upon swaging, a strong 110 fibre texture develops, as seen by the large peak in the difference curve of Fig. 4.11d). Therefore, the HC datasets can be used to assess the influence of texture on lattice strain evolution. Fig. 4.11e shows the unswaged HC spectrum after 8% tensile plastic strain is introduced. This demonstrates that some texture development also occurs during tensile testing.

4.7.1 Reproducibility of Data It is important to ascertain that the trends in lattice strain evolution presented in previous sections are representative of the materials studied. If measurements on different samples of the same material yield significantly different results, then doubt may be cast upon the generality of the observed trends (due to the possibility of sample-to-sample variations in microstructure, etc.) or the reliability of the measurement technique. Since the swaging and subsequent heat treatment of the LC steel did not significantly alter its texture or microstructure, the measurements made on the various LC samples can be used in order to assess the reproducibility of the results. For this purpose, the measurements are presented in this section. The opportunity is also taken to present and briefly discuss the responses of grain families which were omitted from the previous sections for the purpose of clarity. Fig. 4.12 shows axial and transverse ferrite grain family lattice strains versus applied stress in the various LC samples which were tested. For the purpose of clarity, error bars representing the statistical uncertainty in peak fitting are omitted, but in all cases these are consistent with the scatter between datapoints. This scatter varies considerably from family to family, due to the significant differences in intensity of the corresponding diffraction peaks. Qualitatively, the development of each individual grain family response is similar in most samples, and this confirms that the measurements do not suffer unduly from sample-to-sample variations, and that data gained through the neutron diffraction method is reproducible. In the axial lattice strain plots (b) to (d), the 200 and 310 grain families are consistently the most elastically compliant, while the 110, 211 and 321 families are the stiffest. This is reasonable, because these three families all have the same elastic anisotropy factor of 0.25. At yielding, the 200 response consistently deviates towards greater lattice strain while the 110 and 321 responses deviate towards lower lattice strain. The 211 and 310 responses consistently lie between these extremes. The Rietveld-determined strain always lies close to the 110 response. This point has been discussed in section 4.6.1. Although good reproducibility is observed in plots (b) to (d), there are discrepancies between these and plot (a), which shows the axial lattice strains recorded in unswaged sample 1. In this sample, the elastic 200 response appears stiffer than those of the other grain families. This may arise solely from unreliability in the first datapoint. However, in the plastic region, the 200 response is very different from that observed in the other samples, significantly deviating towards lower lattice strain. This is not reproduced in unswaged sample 2, which was tested for the purpose of verifying this observation. All other responses shown in plot (a) are in good agreement with those in (b) to (d). Further reassurance of reproducibility is given by the transverse lattice strain plots, (e) to (h). Although the elastic responses are too noisy to draw any firm conclusions, the characteristic tensile jump of the 200 and 310 grain families is evident in all datasets. One further test of reproducibility is given by comparison of the 110 and 220 curves. Since these reflections arise from the same grain family, the strains should be consistent, within the Chapter 4. Interphase and Intergranular Stress In Carbon Steels 88

Unswaged Unswaged Sample 1 Sample 1 400 400 (a)

300 yield point 300

yield point

200 200 110 110 200 200 211 211 Applied load [MPa] 220 Applied stress [MPa] 220 100 310 100 310 321 321 Rietveld (e) Rietveld

0 0 0 1000 2000 −700 −500 −300 −100 100 Lattice strain [microstrain] Lattice strain [microstrain] Unswaged Unswaged Sample 2 Sample 2 400 400 (b) (f)

300 300 yield point yield point

200 200 Applied load [MPa] Applied stress [MPa] 100 100

0 0 0 1000 2000 −700 −500 −300 −100 100 Lattice strain [microstrain] Lattice strain [microstrain] Medium swaged Medium swaged

400 400 (c) (g)

300 300

yield point yield point

200 200 Applied load [MPa] Applied stress [MPa] 100 100

0 0 0 500 1000 1500 2000 2500 −700 −500 −300 −100 100 Lattice strain [microstrain] Lattice strain [microstrain] High swaged High swaged

400 400 (d) (h)

300 300 yield point yield point

200 200 Applied load [MPa] Applied stress [MPa] 100 100

0 0 0 1000 2000 −700 −500 −300 −100 100 Lattice strain [microstrain] Lattice strain [microstrain]

Figure 4.12: Applied stress vs lattice strain for all LC samples tested. (a) to (d): axial; (e) to (h) transverse. The legend in (a) applies to all plots. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 89

limits of accuracy of the data. It is interesting to note that this is indeed the case for all samples apart from unswaged sample 1. No reason has been identiﬁed for the discrepancies between this sample and the others. However, the consistency of the other datasets suggests that, in general, lattice strains obtained by this method are reliable and reproducible.

4.7.2 Influence of Crystallographic Texture As demonstrated by the spectra shown in Fig. 4.11, the swaging process introduced a significant 110 fibre texture into the ferrite phase of the HC steel. In this section, ferrite grain family lattice strain development is compared in the differently textured HC samples. As with the previous section, a secondary purpose is to present the responses of grain families which were not discussed in earlier sections. Fig. 4.13 shows axial and transverse ferrite lattice strains versus applied stress for all the HC samples which were tested. These plots are remarkable for their similarity. The differences in elastic stiffness between grain families are more clearly seen than in the LC data. The improvement may be explained by the greater yield stress of the HC steel. Since greater lattice strains are developed prior to yielding, the slopes of different grain families are more easily distinguished. It is clearly seen in the axial lattice strain plots (Figs. 4.13 (a)-(d)) that 200 is elastically most compliant, followed by 310. The stiffnesses of the 110, 211, 220 and 321 families are very similar. This is to be expected, since all these grain families have the same elastic anisotropy factor of 0.25. There is no discernible evidence that the variation in texture alters the diffraction elastic constants, with values of 230GPa and 180GPa for the 110 and 200 reflections respectively, in both the unswaged and heavily swaged materials. There is little evidence of divergence of the axial responses after yielding. In all the materials, the tensile strain drops as load is transferred to the cementite phase, but the drops in all grain families are of similar magnitude and again, the change in texture has little effect. The transverse lattice strains are also very similar in all cases. The characteristic yield point shift of the 200 family is approximately 1000µe in all cases. The next largest shift is consistently seen in the 310 family. In summary, it is seen that the introduction of a 110 ferrite fibre texture has little influence on the generation of grain family lattice strains. One would expect texture to have some influence, since it alters the volume fractions of the different families, and hence the average properties of the polycrystal with which individual grains interact. However, within the levels of texture introduced into the HC steel, this is clearly a minor effect. One further point should be mentioned with regard to the datasets plotted in Figs. 4.13(c) and (g). The purpose of measuring on a second medium swaged specimen was to characterise the lattice strain close to the yield point. It can be seen that, as asserted in 4.6, the lattice strain shifts do indeed occur simultaneously with yielding. §

4.8 Rationalisation of Transverse Intergranular Strains in Fer- rite

The discussion of intergranular strains in section 4.6 has highlighted a marked variation in transverse grain family responses in ferrite. The 200 response is of particular note: it exhibits a dramatic tensile shift during yielding. This feature is also evident in the data published by Pang et al. [115] and Daymond and Priesmeyer [133]. In section 4.9, the lattice strain responses will be compared to predictions using the elastoplastic self-consistent method. In this section, however, a simple rationalisation of the variation in transverse responses is presented. Grains contributing to a particular axial reflection all have the same stiffness along the tensile axis, which differs to that of other axial grain families. Therefore, elastic anisotropy is likely to be an important factor in explaining the axial responses. However, the effects of elastic anisotropy are partially averaged out in the transverse response because, as noted several times Chapter 4. Interphase and Intergranular Stress In Carbon Steels 90

Unswaged Unswaged

700 700 (a) (e) 600 yield point 600 yield point

500 500

400 400

300 110 300 200 Applied load [MPa] 211 Applied load [MPa] 200 220 200 310 321 100 100 Rietveld

0 0 0 1000 2000 3000 −1000 −800 −600 −400 −200 0 Lattice strain [microstrain] Lattice strain [microstrain] Medium swaged Medium swaged Sample 1 Sample 1 700 700 (b) (f) 600 600 yield point yield point

500 500

400 400

300 300 Applied load [MPa] Applied load [MPa] 200 200

100 100

0 0 0 1000 2000 3000 −1100 −900 −700 −500 −300 −100 Lattice strain [microstrain] Lattice strain [microstrain] Medium swaged Medium swaged Sample 2 Sample 2 700 700 (c) 600 yield point 600 yield point

500 500

400 400

300 300 Applied load [MPa] Applied load [MPa] 200 200

100 100 (g) 0 0 0 1000 2000 3000 −1000 −800 −600 −400 −200 0 Lattice strain [microstrain] Lattice strain [microstrain] High swaged High swaged

700 700 (d) (h) 600 600 yield point yield point

500 500

400 400

300 300 Applied load [MPa] Applied load [MPa] 200 200

100 100

0 0 0 1000 2000 3000 −700 −500 −300 −100 100 300 Lattice strain [microstrain] Lattice strain [microstrain]

Figure 4.13: Applied stress vs lattice strain for all HC samples tested. (a) to (d): axial; (e) to (h) transverse. The legend in (a) applies to all plots. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 91

b,3 h,3

t b ξ θ

2 2

ψ φ

n(ψ) t(φ) 1 1 (a) (b)

Figure 4.14: Coordinate systems deﬁned for the calculation of pencil glide Schmid factors. The unit vectors t, b, n and h represent respectively the tensile axis, slip direction, slip plane normal and an hkl plane normal lying perpendicular to the tensile axis. Pencil glide assumes slip may occur on any plane with normal n perpendicular to b. In (a), the 3-axis lies parallel to b and the 1-axis lies in the plane of b and t, which subtend angle ξ. The vector n subtends an angle ψ with the 1-axis. In (b), the 3-axis lies parallel to h and the 1-axis lies in the plane of h and b, which subtend angle θ. The vector t subtends an angle φ with the 1-axis.

previously, the grains corresponding to a transverse hkl reflection have a range of directions of different stiffnesses lying parallel to the tensile axis. A discussion of the widely different transverse responses may thus focus on the plastic properties, which are not necessarily averaged out in the same way. In uniaxial tension, the Schmid factor of a slip system (see equation (2.51)) is maximised when the tensile axis subtends angles of 45◦ with both the slip direction and the slip plane normal. However, in bcc crystals, only the former angle is important: if there is a well oriented 111 slip direction, then there is bound also to be a well oriented slip plane, since slip may occurh onia whole range of planes in the 111 zones. With this in mind, compare the cases when 110 and 100 lie in the transverse plane.h i In the former case, it is possible for a 111 directionh toi lie parallelh i to the tensile axis, thus inhibiting slip in this direction. If the crystalh isi so oriented, slip is possible along the other 111 directions, but these lie at 70.5◦ to the tensile h i axis: far from the optimum orientation of 45◦. The crystal may alternatively be oriented such that one slip direction lies in the transverse plane. Slip is then inhibited in this direction, and the other directions subtend angles with the axis of 61.9◦ and 19.5◦ – again, not optimal. In contrast, with 100 transverse, only directions in the range 0V W lie along the tensile axis; i.e. excluding theh i111 directions. The extremes are 001 andh 011i ; in the former case, all h i h i h i slip directions are oriented at 54.7◦ to the tensile axis; in the latter, two of the slip directions lie in the transverse plane, inhibiting slip, but the other directions subtend the axis at 35.2◦. For any orientation, at least two slip directions subtend the axis at an angle within the range 35.2◦ to 54.7◦ . In summary, with 100 in the transverse plane, there are always slip systems which are well oriented for slip, whileh withi 110 in the transverse plane, there exist plastically hard orientations. h i Chapter 4. Interphase and Intergranular Stress In Carbon Steels 92

0.5 111

0.49

0.48 112

0.47

213 0.46

0.45

001 101 103

Figure 4.15: Stereographic projection of basic crystallographic triangle, showing variation with transverse direction [hkl] of the maximum pencil glide Schmid factor, averaged over all tensile axes perpendicular to [hkl]. The calculations were performed by sampling 50 evenly distributed tensile axes in the (hkl) plane. The positions corresponding to transverse grain families studied by neutron diﬀraction are marked on the projection.

Maximum Pencil Glide Schmid Factor

This discussion may be quantified by computation of the variation in Schmid factor with orientation. In order to achieve this, the bcc slip modes must be specified. As discussed in 2.5.1, one suggested slip mechanism is pencil glide, in which slip may occur on any plane in a§ 111 zone. Although this mechanism may not be physically realistic, it facilitates computationh andi reflects the large number of possible slip planes in a bcc crystal. Using the coordinate system defined in Fig. 4.14a, the Schmid factor m = t.b t.n is evaluated as | | m = cos ξ sin ξ cos ψ . (4.15) | | For given t and b, this is maximised when ψ = 0, i.e. when the slip plane normal lies in the plane of the slip direction and tensile axis. Comparing the values of m calculated for each of the 111 slip directions, the maximum pencil glide Schmid factor may be calculated for a given tensileh axisi t. In order to relate this to the transverse response of a particular (hkl) grain family, the average of this value must be evaluated over all possible tensile axes t perpendicular to the (hkl) plane normal. That is, the average must be taken over all directions t which lie within the (hkl) plane. A Matlab routine was written to evaluate this average (assuming random texture) by calculating the maximum pencil glide Schmid factor for N equally distributed tensile axes in the (hkl) plane. Adjacent axes were separated by (180/N)◦. For certain high symmetry (hkl) (such as (100) and (110)), it is not necessary for the tensile axes to span 180◦. However, since the routine was written for general (hkl), this fact was not utilised. It was found that the calculations converged with increasing N and that N = 50 was sufficient to calculate the averaged quantity to 3 significant figures. The calculated dependence of the average maximum pencil glide Schmid factor upon the specified transverse direction is shown in Fig. 4.15. The difference between the cases when 100 and 110 lie in the transverse plane is clearly evident from the plot. While the 110 h i h i h i Chapter 4. Interphase and Intergranular Stress In Carbon Steels 93

orientation has the lowest average maximum Schmid factor, 100 lies close to the transverse direction with the highest value. The plot shows that the 310h transvi erse grain family is also relatively soft. h i This computation considers only the system most favourably oriented for slip, and thus derives from the Sachs approach. As discussed in 2.6.1, the assumption that slip occurs on one system is an over-simplification which does not ob§ ey the requirements of compatibility and stress continuity. The Taylor model, which assumes slip on five independent systems, is an improvement (although it too violates stress continuity). Rather than the maximum Schmid factor, the ease of slip in the Taylor model is determined by the Taylor factor M, defined in 2.6.2. The construction of a Taylor model is, however, beyond the scope of this section, which §seeks only to illustrate the origin of the characteristic transverse lattice strain response.

Average Pencil Glide Schmid Factor An indication of the eﬀect of non-optimally oriented systems may be gained, however, by averaging the Schmid factor over all pencil glide slip planes, as well as all possible tensile axes. For a given slip direction and tensile axis, the average over all slip planes n is found by integrating (4.15) over all ψ, giving (2/π) cos ξ sin ξ . Using the coordinate system deﬁned in Fig. 4.14(b), the average over all tensile axes| in the (|hkl) plane is then

1 2π 2 m = cos ξ sin ξ dφ. (4.16) h i 2π π | | Z0 By evaluating t.b in both coordinate systems of Fig. 4.14,

cos ξ = cos φ sin θ. (4.17)

Substituting for ξ in (4.16) and integrating, the expression for the average pencil glide Schmid factor is 2 2 1 m = sin θ + √2 cos θ tanh− (sin θ) (4.18) h i π2 h i where θ is the angle between the slip direction b and transverse direction [hkl]. This considers one slip direction only, however. The average over all slip directions is 1 m = m (111) + m (111) + m (111) + m (111) . (4.19) h i 4 h i h i h i h i where m (111) denotes the average of m evaluated for the [111] slip direction. The full average is evaluatedh i over the basic crystallographic triangle and shown in Fig. 4.16. The lowest average Schmid factor is with 111 transverse, while the highest is with 100 transverse. Figs. 4.15 and 4.16h bothi support the assertion that orientationsh iwith 100 in the transverse plane are, on average, particularly well oriented for slip. Fig. 4.15 hdemonstratesi that there are transverse directions close to 100 with a higher average maximum Schmid factor, but the average over non-optimal orientationsh i suggests that the range of orientations with 100 transverse are, on average, plastically softest. The computations also demonstrate that orien-h i tations with 310 in the transverse plane tend to be relatively soft. On average, the ranges of orientations withh i 110 , 211 or 321 lying in the transverse plane all exhibit lower optimal and non-optimal Sch hmidi hfactors.i h i In summary, the work presented in this section demonstrates that grains within the 200 and 310 transverse families tend to be well oriented for slip and therefore tend to yield earlier than other grains. The neutron diffraction data indicates that this early yielding causes significant tensile residual stresses to develop perpendicular to the tensile axis in these families. In previous sections, the discussion of residual stress generation due to plastic misfit has tended to focus on the axial direction, but in this paragraph the generation of transverse residual stresses is discussed from a slightly different viewpoint. For a given elongation along the tensile axis, on Chapter 4. Interphase and Intergranular Stress In Carbon Steels 94

0.27 111

0.26

0.25

112 0.24

0.23

0.22 213

0.21

0.2

001 101 103 0.19

Figure 4.16: Stereographic projection of basic crystallographic triangle, showing variation with direction [hkl] of the Schmid factor, averaged over all pencil glide slip systems and tensile axes perpendicular to [hkl].

average the natural transverse contraction of a grain is greater if it deforms plastically than if it deforms elastically. This statement is justified as long as the Poisson’s ratio of the bulk material ν satisfies ν < 0.5 (which is of course true for ferrite), since the ratio of transverse to axial strain is equal to -0.5 for uniaxial plastic elongation (as discussed in 4.5.3). Therefore, if a grain yields while its surroundings remain elastic, its tendency for greater§ contraction in the transverse direction causes significant transverse tensile stress to develop. This provides an explanation of why the 200 and 310 transverse families exhibit large tensile shifts in lattice strain during yielding.

4.9 Investigation into Intergranular Stress in Ferrite using the Elastoplastic Self-Consistent Method

In this section, a study of internal stress development in single phase ferrite is conducted using the elastoplastic self-consistent (EPSC) method of Hill, described in 2.6.4. The aims of this study are twofold. Firstly, as reviewed in 3.4.1, while a significant body§ of work has been published on the comparison of EPSC predictions§ to measured lattice strains in fcc polycrystals, only a few studies have been reported concerning bcc polycrystals [115, 133]. The different crystal structures are expected to give rise to different trends in residual stress development, due to the operation of different slip modes. This study therefore provides validation of the ability of the EPSC method to predict average grain family stresses in bcc polycrystals. Secondly, by varying the input parameters such as elastic constants and crystallographic texture, the model is employed to study the influence of these parameters on internal stress generation, and thus provide a means by which the experimental results reported earlier in this chapter may be further interpreted and investigated. The model has been implemented in Fortran by Turner, Tomé et al [59, 119], following the formulation of Hutchinson [60], as summarised in 2.6.4. Their code was supplied for the use of the author by Dr Mark Daymond of the Rutherford§ Appleton Laboratory, who formerly worked Chapter 4. Interphase and Intergranular Stress In Carbon Steels 95

in collaboration with Dr Tom´e at the Los Alamos National Laboratory, USA.

4.9.1 Model Speciﬁcation In this section, the single crystal characteristics and model parameters employed are speciﬁed.

Choice of Slip Systems As explained earlier, the slip systems in bcc crystals are not as uniquely defined as those in fcc crystals, depending upon temperature and the specific material. In general, the slip direction is 111 , and a number of slip planes in the 111 zone may operate. The greater this number, the closerh i the behaviour to pencil glide slip, which hiwas used for the analysis of 4.8. Using an earlier self-consistent model which does not account for elastic anisotropy, Hutchinson§ [57] employed 10 and 20 slip planes per slip direction, the results of which differed by no more than 0.7%. Chin and Mammel [154] assumed slip was equally possible on 110 , 112 and 123 planes, giving 12 distinct slip planes per slip direction. This selection w{as adopted} { }in the{recen}t works by Pang et al. [115] and Daymond and Priesmeyer [133]. It is also adopted here. However, as argued by Hutchinson, if the number of slip planes is large enough and they are well distributed about the slip direction, then the model should be insensitive to the planes explicitly specified.

Hardening Law A useful strategy to assess whether the EPSC model is successful in predicting internal stress development is to select a hardening law and adjust the hardening parameters so that the model’s macroscopic response fits the observed macroscopic stress-strain curve. The predicted intergranular strains may then be compared to those measured by neutron diffraction. However, in this instance, the softening observed during yielding of the ferrite cannot be emulated in the current model implementation: the specification of negative hardening causes instability in the computer code. Therefore, the macroscopic behaviour cannot be fully captured, particularly in the low strain region. Nevertheless, the macroscopic fitting strategy remains the best way to relate the applied stress to the macroscopic strain, upon which the intergranular stresses are strongly dependent. The strategy is therefore adopted in this work. Since a close fit may not be achieved to the macroscopic curve whatever the hardening law, the simplest law is applied, i.e. linear hardening. Moreover, isotropic hardening is assumed. This hardening law is parameterized by equation (2.61), with hij = θ where θ is a constant:

dτ i = θ dγj . (4.20) j X In accordance with the assertion that slip is equally possible on all slip planes, the initial critical resolved shear stress is set equal for all slip systems, and adjusted to ﬁt the macroscopic yield point. There are thus only two ﬁtting parameters used to optimise agreement between the model and experimental data.

Crystallographic Texture The GSAS structural analysis software [90] allows the simultaneous Rietveld refinement of diffraction spectra obtained at a number of scattering angles. This enables crystallographic texture to be determined in the refinement. Although this generally requires spectra to be recorded at a large number of scattering angles, if the texture is assumed to be cylindrically symmetric, then it may be reasonably estimated from only axial and transverse spectra. Since the carbon steel samples are expected to be cylindrically symmetric, this refinement procedure is used in this section as a simple but effective method to simulate the observed texture. The Chapter 4. Interphase and Intergranular Stress In Carbon Steels 96

11 12 1 L11 L12 L44 (10 Pa) M11 M12 M44 (10− Pa− ) 2.37 1.41 1.16 7.6 -2.8 8.6

Table 4.3: Components of the single crystal elastic stiﬀness tensor Lij and corresponding compliance tensor Mij of ferrite [16].

texture information is exported from GSAS as a series of pole ﬁgures of diﬀerent crystal directions. The orientation distribution function (ODF) is then calculated using popLA, a texture analysis package [155]. From the ODF, popLA is used to generate a discrete grain population for use with the EPSC code.

Elastic Constants The single crystal elastic stiﬀness tensor components of ferrite were taken from the literature [16]. They are given in Table 4.3.

4.9.2 Comparison of EPSC Predictions With Experimental Data The medium swaged LC dataset is again selected as the focus of this work. A grain population of 3066 orientations was generated from the spectra of the untested material, using the procedure described in 4.9.1. A 110 pole figure of this discrete population is shown in Fig. 4.17a. The central axis of§ the projection corresponds to the tensile axis used in the diffraction experiments. The 110 fibre texture is apparent from the concentration of poles around this axis. A concentric ring of poles is also seen at 60◦ to the tensile axis, corresponding to the other 110 directions in each grain. For comparison, a similar plot is shown in Fig. 4.17b for a computer-generatedh i population of 1000 grain orientations, representing a randomly oriented polycrystal. The distribution of points is more even in this pole figure; the variation from the centre to the edge simply arises from the fact that the stereographic projection is not an equal area projection. The fitted macroscopic response of the model under uniaxial tensile stress — with the tensile axis corresponding to the central axis of the pole figure in Fig. 4.17a (and thus the tensile axis in the diffraction experiments) — is shown in Fig. 4.18. As noted in the previous section, the model can not successfully capture the Luders¨ band region observed in the experimental flow curve, because it is incapable of simulating softening. However, the calculated macroscopic elastic stiffness is in excellent agreement with the experimental stiffness, and the flow curve is well captured at large plastic strains. The test of the success of the EPSC model in simulating internal stress generation lies in the comparison of the predicted grain family lattice strains with the measured strains. In order to carry out such a comparison, a selection must be made of the discrete grains in the model which correspond to each reflecting grain family in the diffraction experiments. In these simulations, it is assumed that all grains with an hkl direction lying within 7.5◦ of an axis contribute to the hkl reflection along that axis. Theh elastici strain resolved along the axis is averaged over this subset of grains. The resulting elastic strains are shown as a function of applied stress in Fig. 4.19, alongside the experimental strains. In the axial direction (Fig. 4.19a), the model performs well in capturing the elastic anisotropy of the grain families and the deviations from linearity during yielding. The deviations are small (in comparison to those of some of the transverse lattice strain curves — in absolute as well as relative terms). Hence the resulting residual strains after unloading are also small. This is in good agreement with the data shown in Fig. 4.10. The model does not perfectly capture the gradients of the lattice strain curves in the greater plastic region, but overall the correspondence is very good, considering the inability of the model to closely fit the macroscopic flow curve. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 97

(a) (b)

Figure 4.17: 110 pole figures from discrete grain populations employed in EPSC model. The area of each point is proportional to the weight of the grain population it represents. (a) grain population produced using diffraction spectra of medium swaged low carbon steel (central axis corresponds to the tensile axis in the diffraction experiments); (b) grain population produced by computer simulation of randomly oriented polycrystal.

400 Model 350 Experimental

300

250

200

150 Applied stress [MPa]

100

0 0 1 2 3 4 5 6 7 8 9 10 Macroscopic strain [%]

Figure 4.18: Fit of EPSC macroscopic response to experimental ﬂow curve. The critical resolved shear stress and hardening parameters are used as ﬁtting parameters. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 98

The EPSC-calculated strains along two orthogonal transverse directions are shown in Fig. 4.19b. Although the creation of a grain population assumed cylindrical symmetry, these responses are not identical because a discrete population was sampled from the symmetrical texture. The difference between the two responses has no significance for the real material, but gives an indication of the sensitivity of lattice strains in the transverse direction to the exact grain population. Given that the responses in the two transverse directions differ considerably, this sensitivity is clearly high and this helps to explain the variation in the magnitudes of the tensile jumps observed in different samples experimentally (Fig. 4.12). Moreover, given this sensitivity, it is unsurprising that the responses in neither direction closely match the experimental strains in the medium swaged LC sample. Nevertheless, the model successfully captures the “fingerprint” transverse 200 curve, exhibiting a dramatic tensile jump during yielding, with the 310 curve also showing a tensile shift. It is encouraging that the EPSC model picks up these shifts, which were rationalised in simple terms in 4.8. As noted in 3.4.1, in their application of the EPSC model, Pang et al. also successfully§captured the tensile§ shift of the transverse 200 elastic strain [115].

Note that the two 200 transverse responses only diverge at the onset of plasticity: the linear responses in the elastic regime are essentially identical. The two 310 transverse responses are also very close in the elastic regime. However, the linear responses of the 110 families appear to be very sensitive to the exact population of reﬂecting grains. The reason for this sensitivity is explored in 4.10. § In summary, the EPSC model demonstrates modest success in the prediction of average lattice strains in ferrite, and can certainly be used to qualitatively predict the trends of lattice strain growth.

1 2 1 2 2 1

300 300

200 200

110 expt 110 model Applied stress [MPa]

Applied stress [MPa] 200 expt 100 100 200 model 310 expt 310 model

0 0 0 1000 2000 −600 −400 −200 0 Lattice strain [microstrain] Lattice strain [microstrain] (a) (b)

Figure 4.19: EPSC model prediction of lattice strain evolution with applied stress in medium swaged LC steel, compared with experimental data: (a) axial direction; (b) transverse directions. The average responses of grains reflecting close to two orthogonal transverse directions (1 and 2) are shown. The responses differ due to the discrete specification of the grain population, even though this is sampled from a texture which is assumed axially symmetric. The legend in (a) applies to both figures; for clarity, error bars are not shown in (b). Chapter 4. Interphase and Intergranular Stress In Carbon Steels 99

4.9.3 Inﬂuence of Elastic Anisotropy

The successful validation of the EPSC model presented in the last section is exploited in this and the subsequent section in order to assess the influence of elastic anisotropy on internal stress generation. An important observation noted in the previous section is that the axial residual strains both observed experimentally and predicted by the EPSC model are small in comparison to those of some grain families in the transverse direction. In 4.8 it was argued that the large average residual strains observed in the transverse direction ma§ y be explained primarily in terms of the single crystal plastic properties, because the influence of elastic anisotropy is partially averaged out. This is because grains in a transversely reflecting family have a range of crystallographic directions of different stiffnesses aligned along the tensile axis. In contrast, all grains in an axially-reflecting family have the same direction pointing along the tensile axis, and it is clear that elastic anisotropy may not be neglected in explaining the (lack of) intergranular strain development in the axial direction. The role played by elastic anisotropy may be investigated by modelling a theoretical polycrystal composed of elastically isotropic crystallites. This is easily achieved by choosing the M44 compliance component according to M44 = 2 (M11 M12), ensuring that the degree of anisotropy (equation (2.21)) is unity. Fig. 4.20 shows results− from an EPSC simulation of uniaxial tension of such a polycrystal, alongside an equivalent simulation of ferrite. The two simulations differ only in the specification of the M44 stiffness component. In the ferrite simulation, only small deviations from the elastic linear response are seen in the axial lattice strain versus applied stress curves of individual grain families (Fig. 4.20a). The residual strains after unloading (Fig. 4.20c) are therefore also small, and are indeed almost completely absent in the 111 and 200 families. Of the grain families considered, the 111 and 110, being elastically stiffest, develop the smallest tensile strains during loading. 200, being elastically most compliant, develops the greatest tensile strain. The removal of elastic anisotropy completely alters the trends in intergranular strain generation. Now all families have exactly the same elastic stiffness and thus the lattice strain responses to loading (Fig. 4.20b) diverge only when plasticity begins. It is seen that 111 and 110 develop the greatest tensile strains. This implies that these families bear extra stress when other families yield, and are thus plastically hardest. 200 now develops the smallest tensile strain, implying it yields earlier than the other families and is plastically softest. Whereas in the ferrite simulation residual strains are virtually absent in the 111 and 200 families, in the isotropic crystallite simulation, all families develop large residual strains. This analysis demonstrates that to some extent the influence of plastic anisotropy is counteracted in ferrite by elastic anisotropy. Grains oriented with the tensile axis along 111 or 110 are not well oriented for slip. However, these are also among the elastically stiffesth grains,i hand iso bear greater stress in the elastic regime. Grains with the tensile axis along 200 are more favourably oriented for slip, but also tend to bear less stress due to this being an elasticallyh i compliant direction. The result is that these grain families tend to yield at approximately the same applied external stress, and thus the generation of large axial residual strains is inhibited. This explanation is illustrated in Fig. 4.21. The stereographic projection in Fig. 4.21(a) shows the variation with tensile axis orientation of the directional Young’s modulus (equation (2.19)) of ferrite, calculated using the elastic constants in Table 4.3. The projection in (b) shows the variation with tensile axis orientation of the Schmid factor, averaged over the four slip systems comprising one of the four 111 slip directions and the optimally oriented pencil glide slip plane containing that direction.h Althoughi not rigorous, this certainly gives an indication of how plastically hard the orientation is. Comparing the two figures, it is clear that there is a strong correlation between the directional elastic stiffness and ease of plastic flow, with the elastically stiffest directions also being least favourably oriented for slip. This supports the above observations made with the EPSC model, and the conclusion that elastic and plastic anisotropy counteract one another, leading to reduced average axial grain family residual strains Chapter 4. Interphase and Intergranular Stress In Carbon Steels 100

in ferrite.

400 400

111 110 310 200 200 310 110 111

300 300

200 200 Applied stress [MPa] Applied stress [MPa] 100 100

0 0 0 1000 2000 0 1000 2000 3000 Lattice strain [microstrain] Lattice strain [microstrain] (a) (b)

111 110 200 310 111 110 200 310 400 400

200 200

0 0

−200 −200

−400 −400 Residual strain [microstrain] Residual strain [microstrain]

−600 −600 (c) (d)

Figure 4.20: Inﬂuence of elastic anisotropy on EPSC intergranular strain predictions. In (a), the model is run with the elastic constants of ferrite given in Table 4.3. In (b), the same values of M11 and M12 are used, but M44 is set according to M44 = 2 (M11 M12), to ensure single crystal elastic isotropy. (c) and (d) show the residual strains which− result from the runs in (a) and (b) respectively.

4.9.4 Summary of Findings from the EPSC Model In summary, the EPSC model has been successfully validated for the qualitative prediction of the trends in intergranular strain growth in ferrite, based upon fitting of the macroscopic flow curve. Quantitative prediction is unsurprisingly more difficult. The EPSC model is limited in its assumption that all grains interact with an effective medium with the average properties of all grains, whereas in the real material, each grain’s response is dependent on the properties of its immediate neighbours. However, this assumption allows fast calculation of average internal stresses and strains. The method has been used to identify the importance of elastic anisotropy in reducing the build up of axial internal stress when ferrite is deformed in uniaxial tension. The simulations have also indicated that transverse grain family strains may be sensitive to crystallographic texture. The particular issue of the sensitivity to texture of the linear elastic responses of transverse grain families is addressed in the next section. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 101

111 111 0.45 260

240 0.4

220

0.35 200

180 0.3

160

001 101 140 001 101 0.25

(a) (b)

Figure 4.21: Stereographic projections of the standard triangle showing variation with tensile direction of: (a) Young’s modulus of ferrite; (b) Schmid factor averaged over the four slip systems comprising a 111 direction and the optimally-oriented pencil glide slip plane containing that direction.h Thei stiﬀness constants used to calculated the direction- dependent Young’s modulus are given in Table 4.3.

hkl TD (GPa) ND (GPa) 200 -530 +/- 23 -480 +/- 34 112 -702 +/- 80 -707 +/- 38 220 -651 +/- 34 -771 +/- 46 222 -873 +/- 43 -853 +/- 58

Table 4.4: Grain family linear elastic responses in transverse direction (TD) and normal direction (ND) of rolled ferritic steel, subjected to uniaxial tension along the rolling direction. Reproduced from [115].

4.10 Note on Linear Elastic Response of Transverse Grain Families

This section discusses a particular point revealed in the EPSC analysis presented in 4.9.2. That is, according to EPSC calculations, the slope of the transverse 110 family linear elastic§ response is very sensitive to the exact population of reflecting grains, while that of other families is not. In fact, experimental data presented by Pang et al. [115] also suggests that the 110 transverse linear elastic response is more sensitive to texture than other families. Table 4.4 reproduces from their paper the initial gradients of applied stress versus lattice strain along the transverse and normal directions of a rolled ferritic steel plate, when uniaxial stress is applied along the rolling direction. Note that for the 220 reflection, the difference between the gradients of the two responses is significantly greater than that for any other reflection. The result may be rationalised by considering the elastic response of a single cubic crystal to uniaxial stress. In Appendix A, it is shown that if uniaxial stress σ is applied to a cubic crystal along a unit vector [l, m, n], then the elastic strain developed along a unit vector [u, v, w] perpendicular to [l, m, n] is given by ⊥

1 ⊥ = M + M M M F (4.21) σ 12 11 − 12 − 2 44 where Mij is the elastic compliance tensor in contracted matrix notation and F is a factor given Chapter 4. Interphase and Intergranular Stress In Carbon Steels 102

by F = l2u2 + m2v2 + n2w2 . (4.22) Consider the case that [u, v, w] = [1, 0, 0]. Then, since the directions are perpendicular, [l, m, n] = [0, m, n]. Thus F = 0 for all possible [l, m, n]. Hence,

= σM12 . (4.23) ⊥ That is, if uniaxial stress is applied along a direction perpendicular to [1, 0, 0], the strain developed along [1, 0, 0] is independent of the particular direction chosen. This is not the case if [u, v, w] = 1/√2 [1, 1, 0]. Then [l, m, n] = [l, l, n], giving F = l2 where l 1/√2. Hence, 0 F 1 , and the extremes of (4.21) are − | | ≤ ≤ ≤ 2

= σM12 when [l, m, n] = [0, 0, 1] , and (4.24) ⊥ σ 1 1 = M11 + M12 M44 when [l, m, n] = [1, 1, 0] . (4.25) ⊥ 2 − 2 √2 − Inserting the compliance tensor components of ferrite into (4.25) gives the surprising result that if tension is applied along 1 1 0 , then the elastic strain along the perpendicular direction [1 1 0] is positive! This result helps to explain the observed and modelled sensitivity of the 110 transverse linear elastic response to texture. Grains contributing to the 110 transverse reflection have directions between 100 and 110 lying parallel to the tensile axis. While those with the tensile axis lying near to 100h itend toh coni tract along the (transverse) scattering vector direction, those in which it lies nearh toi 110 tend to expand (the strains are, of course, modified by the constraint of the surrounding medium).h i Thus the average response depends on the distribution of orientations: if 110 poles tend to be aligned along the tensile axis, then the 110 transverse response will be steeper than if 100 poles are aligned along the axis. A further self-consistent calculation demonstrates this point. Fig. 4.22 shows the transverse linear elastic response of two individual grains, embedded in a medium composed of grains of 1000 other randomly selected orientations. In both grains, the elastic strain is resolved along a 110 direction which lies perpendicular to the tensile axis. In grain A, the tensile axis lies parallelh ito a 100 direction, while in grain B it lies parallel to another 110 direction. The resolved strainh ofigrain B is not positive, due to the constraint imposed hby thei effective medium. However, the gradient of its response is much steeper than that of grain A, supporting the argument given above.

Grain A: Grain B 25 100 axial, 110 axial, 110 transverse 110 transverse

10 Applied stress [MPa]

0 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 Lattice strain [microstrain]

Figure 4.22: Linear elastic response perpendicular to applied stress axis of two ferrite grains, embedded in an eﬀective medium of grains of 1000 random orientations. Grains A and B are aligned with 100 and 110 parallel to the tensile axis, respectively. In both grains, the elastic strain his resolvedi alongh i a 110 direction lying perpendicular to the stress axis. h i Chapter 4. Interphase and Intergranular Stress In Carbon Steels 103

Note that if [u, v, w] = 1 [3, 1, 0], then [l, m, n] = [l, 3l, n], giving F = 9 l2 with l 1 . √10 5 √10 9 − | | ≤ Hence 0 F 50 . Thus varies with the choice of tensile axis, but to a much smaller extent than in the≤ case≤ of [u, v, w⊥] = 1 [1, 1, 0]. This explains why the transverse 310 linear elastic √2 response varies slightly with the grain distribution, while the 110 response varies markedly (see Fig. 4.19). Also note that if [u, v, w] = 1 [1, 1, 1] then F = 1 l2 + m2 + n2 = 1 . That is, does √3 3 3 ⊥ not vary with the tensile axis direction, as for the case of [u, v, w] = [1, 0, 0]. These directions have in common that they are axes of rotational symmetry. The independence on tensile axis direction of the transverse strain developed along these directions is in fact related to this symmetry. Although there is also rotational symmetry about [110], this is only a 2-fold (diad) axis. The following statement is now proven:

Uniaxial stress σ is applied to a crystal along a direction which lies in the plane to which an n-fold rotation axis is normal, where n > 2. The elastic strain developed along this axis is independent of the particular direction in the plane along which the stress is applied.

The proof is as follows. Consider that uniaxial stresses σ are applied along n evenly distributed directions in the plane (i.e. separated by 360/n◦). Denote the elastic strain resolved along the rotation axis as . The linear theory of elasticity allows the superposition of solutions. Therefore is the sum of the contributions due to the stress along each axis when it is applied alone. Since the applied stress axes are symmetry-related, all of the contributions are equal. Therefore, if stress σ is applied along one of the axes alone, the strain resolved along the rotation axis is given by /n. Consider once more the situation when stress is applied along all n axes. It is straightforward to show that, for n > 2, this is a simple biaxial stress state. That is, taking the rotation axis parallel to the reference frame 3-axis, the stress tensor σij is of the form

Σ 0 0 σij = 0 Σ 0 (4.26)  0 0 0    (in fact, it may be shown that Σ = nσ/2). It follows that the stress state is equivalently represented by uniaxial stresses σ applied along a new set of evenly distributed axes, generated from the first set by an arbitrary rotation about the symmetry axis. Clearly, this has no effect on the strain resolved along the symmetry axis, which remains . Following the argument given above, if stress σ is applied along one of the new axes alone, the elastic strain resolved along the symmetry axis is given by /n. That is, it is equal to that developed by the application of stress σ along one of the original axes, proving the above statement. Note that the argument breaks down for n = 2, because the two uniaxial stresses are applied along the same axis, and do not give rise to a biaxial stress state of the form of (4.26). This argument considers only a single crystal. Within a polycrystal, each crystallite is constrained by the surrounding medium. However, the above argument is easily modified to demonstrate the following, related statement:

An (anisotropic) elastic crystallite is embedded within an inﬁnite, isotropic, homogeneous elastic medium. The crystallite is an ellipsoid of revolution, the revolution axis lying parallel to an n-fold axis of rotational crystal symmetry, where n > 2. If uniaxial stress σ is applied to the medium along a direction in the plane to which this axis is normal, the elastic strain resolved parallel to the axis inside the crystallite is independent of the particular direction in the plane along which the stress is applied.

This may be shown as follows. Taking the 3-axis parallel to the axis of rotational symmetry, consider the application of a uniform biaxial stress in the 1-2 plane (i.e. of the form of Chapter 4. Interphase and Intergranular Stress In Carbon Steels 104

(4.26)). As discussed above, for n > 2, this may be regarded as arising from uniaxial stresses applied along n evenly distributed axes in the 1-2 plane. Equivalently, it may be regarded as arising from a different set of n axes, generated from the first set by a rotation about the symmetry axis by an arbitrary angle. The uniform applied stress is disturbed in the vicinity of the inclusion. However, the entire body — including the crystal structure — has n-fold rotational symmetry about a line lying parallel to the crystal rotational symmetry axis, and passing through the centre of the inclusion. Therefore, along this line, the above argument based upon symmetry and the principle of the superposition of solutions remains valid. That is, if the strain resolved along the symmetry axis due to the biaxial stress state is , that due to one of the uniaxial stresses applied alone is /n. This is true regardless of whether the axis belongs to the first or second set, proving the above statement. Note that the above arguments are not valid if the deformation is plastic. The reason for this is that the principle of the superposition of solutions does not hold. For example, while plastic slip strain may develop due to the application of uniaxial stress, the stress state resulting from the superposition of three equal uniaxial stresses along mutually orthogonal directions is hydrostatic, and thus may not give rise to slip. In summary, this section has demonstrated why the average linear elastic responses of some transverse grain families are much more sensitive to crystallographic texture than those of others. Families for which the plane normal is an n-fold axis of rotational symmetry, with n > 2, are insensitive to texture, because in a single crystal, the elastic strain developed along such an axis is independent of the particular direction perpendicular to it along which uniaxial stress is applied. However, other families may be extremely sensitive to texture — in particular those for which the plane normal makes a large angle with any such rotation axes. The example has been given of the 110 transverse family in ferrite. In the single crystal, the strain which develops along [110] depends dramatically upon the direction perpendicular to it along which stress is applied: the strain is negative for some directions, but positive for others! Similar dependence is expected in other anisotropic cubic materials: the F factor introduced in this section relates to all cubic crystals, not specifically bcc crystals. The analysis presented in this section has some implications for residual stress measurements on engineering components. If a single diffraction peak is used to measure lattice strain, then caution should be exercised in converting the measured strains to stresses. It is a common procedure to use published values of diffraction elastic constants to convert measured elastic strains to stresses. The analysis presented here indicates that the published values may not be appropriate if the material in the component exhibits high single crystal elastic anisotropy and strong crystallographic texture. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 105

4.11 Finite Element Model of Interphase Stress in High Car- bon Steel

This section returns to the subject of interphase stress development. A feature of interest in the evolution of interphase strains in HC steel presented in 4.5 is the dramatic phase strain shifts observed during yielding. It was asserted that these shifts§ could be explained in terms of matrix softening. A question which arises from this assertion is: can these strains be emulated in a computer model in which matrix softening is present? It has already been noted that the current EPSC model implementation is incapable of effectively simulating softening. However, the finite element (FE) method, which is well suited to the study of multiphase systems, may be employed for this purpose. The application of the FE method to modelling of composites has been briefly introduced in 2.7.3. In this section, a simple FE model is applied in an attempt to emulate the effect of matrix§ softening on interphase stress growth in the HC steel.

4.11.1 FE Model Design The HC steel consists of a random dispersion of inclusions within a heterogeneous matrix. As discussed in 2.7.3, to build a complete, representative model of such a material would require a great amoun§ t of time to design, and would be far too costly in processing time to be practical. Therefore, certain simplifications need to be made. Since in this section only the interphase stresses are of interest, the first simplification is to treat both phases as homogeneous and isotropic. Thus the FE model takes no account of the intergranular stresses considered in the previous sections. To further simplify the task, the geometry is simplified through modelling a regular array of inclusions rather than a random dispersion. As noted in 2.7.3, this approach has become popular for the modelling of composites since, by the imposition§ of suitable boundary conditions, only a single unit cell of the array need be modelled. This vastly reduces the computational task. In this work, the performances of models based on two different unit cells are compared in order to assess how well such an array represents the real material. In addition to these simplifications, two assumptions are made. Firstly, the cementite inclusions are assumed to be perfectly elastic spheres. The approximation of spherical shape is well justified by the spheroidal microstructure shown in Fig. 4.1b, and there is no evidence for cementite yielding in the lattice strain data of Fig. 4.4. Secondly, the interface between these phases is assumed to remain perfectly bonded. The two arrays of inclusions investigated are a simple cubic array, and a body-centred cubic (bcc) array. The unit cells of these arrays are shown in Fig. 4.23. The inclusion volume fraction is 20% in both cases. This figure was assumed and justified in the application of the mean field method in 4.5.3. The unit cells are meshed with a density of elements which optimises accuracy and processing§ time — i.e. at the level at which increased density makes negligible difference to the model output. Boundary conditions are imposed such that the unit cells are constrained to remain cuboidal as they deform. In this way it is ensured that adjacent cells tessellate and the model remains representative of an infinite array of cells.

4.11.2 Constituent Properties The ABAQUS finite element code [156] is used for processing of the models. Elastic and plastic properties of the constituent phases are specified in the ABAQUS input file. As supported by the diffraction data, it is assumed that both phases have the same elastic constants. Using the fitted slopes of the linear elastic responses of Fig. 4.4, the Young’s modulus and Poisson’s ratio are assigned values of 210GPa and 0.28 respectively. While the cementite is constrained to deform elastically, ferrite is attributed an isotropic yield stress of 580MPa. Yielding is according to the von Mises criterion [16] and hardening is also isotropic. The flow behaviour under uniaxial stress is specified in the ABAQUS input file as a series of datapoints of the type Chapter 4. Interphase and Intergranular Stress In Carbon Steels 106

Y Y

Z X (b) (a)

Figure 4.23: Unit cells of Fe/Fe3C arrays investigated using the FE method.

(stress, plastic strain, temperature), with interpolation being performed between the datapoints. Only a monotonic curve can be specified at a given temperature, since there is no chronology in the datapoints, so that specifying two values of plastic strain at a given stress would be ambiguous. This presents a problem for the incorporation of matrix softening. However, this may be circumvented by the specification of different flow curves at two nominal temperatures. By switching the nominal temperature after yielding, matrix softening may be introduced. An example of a matrix flow curve achieved in this manner is shown in Fig. 4.24. As illustrated in the figure, the drop in stress directly after yielding is referred to as the softening.

4.11.3 Extent of Constraint In 4.5, the observed yield point drop in ferrite axial elastic strain in HC steel was explained in terms§ of softening and back stress hardening. To reiterate: in order to maintain the same applied stress after yielding, unreinforced material has to work harden to an extent equal in magnitude to the yield point softening. However, in the reinforced HC steel, the matrix does not need to work harden to this extent, because load transfer to the cementite inclusions provides an additional source of hardening. Therefore the matrix does not need to plastically deform to the same extent as if it were alone. In other words, the elastically-deforming inclusions act to constrain matrix plastic flow. For a given volume fraction of inclusions, the extent of this constraint depends on the inclusion shape and distribution. The flow is most greatly constrained if the inclusions are continuous fibres aligned along the tensile axis (see, for example, the paper by Withers and Clarke [131], discussed in 3.4.2). In this section, the constraints imposed by the simple and body-centred cubic arrays of§spherical inclusions are investigated by comparison to a continuous fibre model. The continuous fibre model is illustrated in Fig. 4.25. It is an axisymmetric 2-D model with a cementite volume fraction of 20%, equal to that of both spherical inclusion models. The models are subjected to a monotonically increasing tensile stress (across opposite faces of the cubic unit cells, and along the fibre axis in the continuous fibre model). The constraint is quantified in Fig. 4.26 by plotting the difference in the volume-averaged matrix axial stress component just before and just after yielding, as a function of the matrix softening. In all cases, the matrix isotropic hardening rate is 70MPa/%. In the continuous fibre model, the reinforcement is able to fully bear the increased stress arising from matrix softening. The matrix softening – stress drop curve is thus almost perfectly Chapter 4. Interphase and Intergranular Stress In Carbon Steels 107

600 softening 400

Stress [MPa] 200

0 0 1 2 3 4 5 6 7 Strain [%]

Figure 4.24: Example of softening matrix ﬂow curve. The yield point stress drop is referred to as the softening.

linear, with a gradient of 0.93. The gradient is not exactly unity because the model must extend slightly in order for the reinforcement to bear the extra stress, and this causes some work hardening in the matrix. By contrast, in the particulate models there is an asymptotic matrix stress drop that can be achieved, whatever the extent of softening. This is because the reinforcement is not continuous, so that the matrix must bear some load. Since the matrix is not capable of supporting the imposed load directly after yielding, the model must extend considerably so that the matrix may work harden. The greater the softening, the more the model must extend until it reaches stability. Note that if the matrix work hardening rate is reduced, this simply causes the model to elongate further in order to achieve stability. In the limit of perfect matrix plasticity, the particulate models extend indefinitely. The continuous fibre model extends only enough for the fibre to bear elastically the load over and above the matrix yield stress. The bcc model is less effective in constraining matrix flow than the simple cubic model, exhibiting a smaller maximum stress drop. This result is consistent with the work of Daymond and Withers [75], noted in 2.7.3. Their models of short fibre composites demonstrated the relative ease of plastic flow along§ shear bands between fibres arranged in a body-centred array, in comparison to an aligned fibre array. The existence of channels of relatively unconstrained flow limits load transfer to the reinforcing inclusions. It is clear from the diffraction data that the arrangement of cementite inclusions in the HC steel is highly effective in constraining matrix flow and facilitating load transfer. Therefore, in the following section, the simple cubic model is selected for comparison of the model output with the experimental data.

4.11.4 Comparison of FE Model Predictions With Experimental Data The success of the FE model in capturing the development of internal stress in HC steel is evaluated using a similar strategy to that used to evaluate the EPSC model in 4.9.2. That is, the constituent properties are varied until a good fit is achieved with the macroscopic§ flow curve. Only then is the internal state observed. While in the EPSC case, the variable parameters were the single crystal yield and hardening parameters, in the FE model, it is the matrix yield stress, softening and hardening which are to be varied. These parameters are optimised for the HC unswaged sample in the simple matrix flow curve shown in Fig. 4.27. The flow curve consists of three stages: initial softening, linear work hardening, and then a drop in the rate of linear work Chapter 4. Interphase and Intergranular Stress In Carbon Steels 108

applied stress

reinforcement matrix

Figure 4.25: Geometry of continuous ﬁbre model with reinforcement volume fraction of 20%.

300

continuous fibre 250 simple cubic bcc

200

150

100 Matrix stress drop [MPa]

0 0 50 100 150 200 250 300 Matrix softening [MPa]

Figure 4.26: Diﬀerence between axial matrix stress component just after and just before yield (at the same applied stress), as a function of the yield point softening. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 109

hardening. Although this curve is clearly over-simplified, it maintains a low number of variable parameters yet remains adequate to give an excellent fit of the model’s macroscopic response to the experimental flow curve of the unswaged HC sample. The red dotted curve in Fig. 4.27 shows the macroscopic response of the model to a monotonically increasing uniaxial stress. This provides an excellent fit to the experimental flow curve (continuous line). An important feature is the Luders¨ band region. Through the imposition of matrix softening of 150MPa, the model is able to capture this feature successfully. It may seem surprising that a matrix flow curve with discontinuous gradient changes can give rise to a smooth macroscopic response. The explanation for this is that different regions of matrix experience different stress states, and therefore reach the gradient discontinuities at different stages during the deformation. Although this is not a fully realistic representation of the true matrix behaviour, it is felt that refining the matrix flow curve will add little to the information provided by the model.

800

Model matrix Model macroscopic 600 Experimental macroscopic

400 Stress [MPa] 200

0 0 2 4 6 8 10 Strain [%]

Figure 4.27: Flow curves showing matrix ﬂow properties (blue dashed) chosen such that the model macroscopic response (red dotted) closely resembles the experimental ﬂow curve (continuous black).

The ABAQUS output is post-processed to calculate the average stress and elastic strain in each phase. The axial and transverse elastic strains are plotted against applied stress in Fig. 4.28, alongside the experimental lattice strains. The elastic responses of the phases are well matched to the experimental data (within a few GPa difference in slope). When yielding occurs, the model predicts a large drop in the average axial ferrite strain, and correspondingly an even larger tensile jump in the average axial cementite strain. However, in the experimental data, these jumps are even greater than those predicted. In the transverse direction, the lattice strain jumps are again captured qualitatively, but underestimated. In both directions, the predicted strain gradients after yielding are consistent with the experimental data. Inflections can be seen in the slopes of the curves. These may be attributed to the change in linear work hardening which is incorporated into the matrix flow curve. When the matrix work hardening rate drops, the matrix bears a lower proportion of an increment in applied stress, while the inclusions bear greater stress. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 110

800

700

600

500

400

300 Applied stress [MPa] Fe (expt) Fe (model) 200 Cementite (expt) Cementite (model) 100

0 −4000 −2000 0 2000 4000 6000 8000 Lattice strain [microstrain]

Figure 4.28: Experimental data and 3D simple cubic model predictions for volume averaged lattice strain development in both phases of high carbon steel.

4.11.5 Summary of Findings from FE Model of Interphase Stress

The FE model was constructed in order to emulate computationally the internal stress and strain state which arises in HC steel due to plastic deformation. Important features in the elastic strain evolution are the large jumps in the phase strains during yielding. Through the incorporation of matrix softening, the model has qualitatively captured these features. The HC steel flow curve has been successfully emulated; in particular the Luders¨ band region which is associated with matrix softening. Quantitatively, the model is unable to emulate the magnitudes of the yield point elastic strain shifts. This highlights the efficacy of a random dispersion of inclusions in constraining matrix flow. A simple cubic array of inclusions is more effective in constraining matrix flow than a body centred cubic array, but both are significantly less constraining than the experimentally-observed random dispersion. This demonstrates that such a dispersion is highly effective in facilitating load transfer from a plastically deforming matrix to elastically-deforming inclusions.

4.12 Combined Model of High Carbon Steel

In comparison to LC steel, the presence of a reinforcing phase adds a further level of diﬃculty to the prediction of ferrite intergranular strains in HC steel. A method of broaching this problem is to use the FE model presented in the previous section to predict the phase stresses, and then apply the predicted average matrix stress state as the boundary condition within the EPSC model employed in 4.9. Alternatively, the intergranular strains may be predicted directly in the EPSC model through§ the incorporation of a population of non-yielding grains. This approach has been presented for iron-cementite in the paper by Daymond and Priesmeyer [133]. In this section, the former approach is used to predict intergranular strains in the HC steel tested experimentally in this chapter. The combined FE/EPSC model is then tailored to the steel studied by Daymond and Priesmeyer, in order to compare lattice strain predictions from the two methods. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 111

4.12.1 Combined Modelling Strategy In modelling intergranular strains in LC steel, the strategy employed was to vary the single crystal yield and hardening parameters within the EPSC model, in order to fit the single phase macroscopic flow curve. In order to predict interphase strains in HC steel, the bulk matrix properties were varied within a finite element model in order to fit the composite macroscopic flow curve. In this section, these strategies are combined, with the aim of producing a consistent matrix flow curve in both the EPSC and FE models. The bulk matrix flow properties are again varied within the FE model in order to fit the unswaged HC steel flow curve. By varying the single crystal plasticity parameters, an attempt is then made to achieve a matrix flow curve within the EPSC model which is as similar as possible to that specified in the FE model. Once the single crystal parameters are tailored in this way, the EPSC model is used for the prediction of intergranular strains. In 4.12.2, application of the combined model to the HC steel studied in this chapter is describ§ed. However, full agreement between the FE and EPSC matrix flow curves cannot be achieved. While the introduction of matrix softening is crucial to the success of the FE model in fitting the macroscopic flow curve, as discussed in 4.9.1, it is not possible to incorporate softening within the present EPSC model implementation.§ However, application of the model to the steel studied by Daymond and Priesmeyer does not present this problem because this steel does not exhibit matrix softening. In 4.12.3, data from the paper of Daymond and Priesmeyer [133] is reproduced with the kind permission§ of the authors, and used to test the combined modelling strategy. Although matrix softening cannot be incorporated within the EPSC implementation, it is still instructive to proceed with this modelling strategy for the HC steel. It may be argued that the matrix exhibits two distinct states, pre- and post-softening. Softening occurs immediately at the onset of plasticity. Thereafter, throughout plastic flow, the matrix adopts the latter state. Therefore, the build up of internal stress depends mainly on the matrix properties in this state, not the initial state. The EPSC model can emulate the matrix properties after the softening process and is thus still valid for the prediction of internal stress. In this work, the EPSC model is run using a computer generated population of 1000 randomly generated grain orientations. This is a reasonable representation of the ferrite phase in the unswaged HC steel, which was shown in 4.7.2 to be untextured. Isotropic hardening between slip systems is assumed and the initial critical§ resolved shear stress is set equal for all systems.

4.12.2 Combined Model Results Fig. 4.29 shows the fitted matrix and composite flow curves. The matrix flow curve specified in the FE model is shown in blue. This is similar to that shown in Fig. 4.27, comprising an elastic region, followed by yield point softening and then linear work hardening. However, the curve specified in Fig. 4.27 includes a drop in work hardening rate. This is excluded from the current curve in favour of a constant work hardening rate, which may be more easily captured in the EPSC model. The simpler curve results in a poorer fit between the HC unswaged experimental flow curve (black continuous) and the FE-calculated composite response (green dotted) at high plastic strain, but it remains very good up to 6% strain. In order to gain maximum consistency between the EPSC and FE ferrite flow curves, single crystal linear work hardening is specified in the EPSC model, giving rise to macroscopic response which is linear in the limit of high strain. The macroscopic response in uniaxial tension of the fitted EPSC model is shown in red in Fig. 4.29. Beyond 1% strain, the EPSC flow curve coincides with that specified in the FE model. At smaller strains, the curves do not coincide, owing to the lack of yield point softening and the fact that yielding is necessarily gradual in the EPSC model (due to differently-oriented grains yielding at different applied stresses). However, since the lattice strains measured in the plastic regime all correspond to strains much larger Chapter 4. Interphase and Intergranular Stress In Carbon Steels 112

than 1%, the discrepancies below this strain are not critically important.

800

experimental FE matrix behaviour 600 FE output EPSC matrix behaviour

400 Stress [MPa] 200

0 0 2 4 6 8 10 Strain [%]

Figure 4.29: Flow curves illustrating strategy for combining two phase FE and single phase EPSC deformation models. In the FE model, the matrix is assigned yield point softening followed by linear hardening. The stress drop and hardening parameters are fixed such that the predicted macroscopic response matches the experimental flow curve as closely as possible. Then the parameters of the linear hardening EPSC model are chosen to achieve maximum consistency with the FE matrix flow curve.

Having characterised a consistent bulk matrix response, the single crystal plasticity parameters may be used in a separate EPSC simulation. Firstly, the FE model is used to calculate the average (triaxial) matrix stress as a function of applied uniaxial stress. This stress state then forms the boundary condition in the EPSC simulation, in order to determine the evolution of the average grain family stress and strain states with applied stress. A difficulty arises in that at the yield point, the mapping of applied stress to matrix stress is not one-to-one. Just prior to yield, the matrix experiences the uniaxial yield stress; after Luders¨ band propagation, the applied stress is the same, but the matrix experiences a lower axial stress, as well as a transverse stress component. This coincides with the change in matrix properties pre- and post-softening. In order to reflect the two yield point stress states and the change in matrix properties, two separate EPSC runs are performed. In the first run, the critical resolved shear stress is increased such that the matrix deforms purely elastically up to the macroscopic yield stress. In the second run, the matrix is assigned the single crystal plasticity parameters corresponding to the EPSC flow curve of Fig. 4.29, and the triaxial stress state is imposed. The grain family lattice strain versus applied stress curves which result from this process are shown in Fig. 4.30. Also plotted are the measured lattice strains. The elastic strains are well captured, both in the axial and transverse direction. Since the cementite is assumed to have the same bulk elastic constant as ferrite, these strains are calculated solely using the EPSC model, and so serve as additional confirmation to that given in 4.9 that the self-consistent method correctly predicts grain family strains in the elastic regime§ in single phase ferrite. Considering that the elastic and plastic regimes correspond to different EPSC runs, it is wholly unsurprising that lattice strain shifts are observed at the yield point. The relative magnitudes of the shifts are of interest, however. In the transverse direction, the shifts are in very good agreement with the measured lattice strains. The 200 grain family once more Chapter 4. Interphase and Intergranular Stress In Carbon Steels 113

exhibits the greatest tensile shift. The volume averaged response, determined solely from the FE model, lies very close to the 110 family response. This concurs with the experimentally observed proximity of the volume averaged and 110 transverse responses. The level of agreement achieved between measured and calculated strains in the transverse direction must, however, be viewed cautiously, since it was shown in 4.11.4 that the shift in volume averaged elastic strain is underestimated in the FE model. In§the axial direction, the shifts in the 110 and 310 lattice strains are well captured, but the 200 shift is underestimated. The volume averaged curve deviates further from the 110 curve than observed experimentally. In summary, the combined model approach achieves a degree of success in predicting the lattice strain development in HC steel. However, due to the complexity introduced by matrix softening, the model must be substantially tailored in order to eﬀectively simulate the real material. This reduces the extent to which the approach may be objectively judged. Sim- pler material behaviour is required for a better test of the ability of this method to predict intergranular strains.

Transverse Axial 700

600 yield point

500

400

300 110 expt 200 expt Applied stress [MPa] 200 310 expt 110 model 200 model 100 310 model vol. average 0 −1000 0 1000 2000 3000 Lattice strain [microstrain]

Figure 4.30: Individual grain family lattice strain predictions from EPSC model, applying triaxial stress state boundary conditions from 3D simple cubic FE model. The lines are composed from output from two EPSC runs: for pre- and post-softening respectively. The yield criteria are altered between these runs. Datapoints give experimental results.

4.12.3 Comparison of Combined Model Approach With Two Phase EPSC Model The carbon steel studied by Daymond and Priesmeyer [133] does not exhibit matrix softening, and the dataset presented in their paper may thus be used for a more objective evaluation Chapter 4. Interphase and Intergranular Stress In Carbon Steels 114

of the success of the combined FE/EPSC modelling approach. Moreover, these authors have presented lattice strain calculations using a two phase EPSC model, the results of which may be compared to those from the combined model. Such a comparison is performed in this section, with the aim of assessing the performance of both modelling strategies. Results from [133] are reproduced with the kind permission of the authors. The microstructure of this alternative steel consists of pearlite and ferrite. This is not so well approximated by a matrix containing spherical inclusions as the spheroidised microstructure of the HC steel studied in this chapter. However, both the combined FE/EPSC and two phase EPSC model assume spherical inclusions, and thus the difference from the real microstructure does not affect comparison between the models. The cementite volume fraction, determined by Rietveld refinement of neutron diffraction spectra, is quoted in [133] as 5%. However, at such small volume fractions, this technique is not necessarily reliable, and the authors found a value of 8% best matched the observed load partitioning between phases, in agreement with the nominal carbon content of the steel. Therefore, this value is used in the combined model. A new simple cubic FE mesh incorporating this reduced volume fraction was constructed for this purpose. In order to concur with the hardening law used in [133], in this section, single crystal hardening employing an extended Voce law [157] is specified in the single phase EPSC model, rather than the linear hardening of previous sections. This more highly parameterised law enables a better fit to the experimental flow curve. The law has the form:

dτî θi Γ dτ i = hij dγj where τî = τ i + τ i + θi Γ 1 exp 0 (4.27) dΓ 0 1 1 − − τ i j 1 X j i and Γ is the accumulated shear strain: Γ = i γ . The parameters τ0 (the initial critical i i i resolved shear stress), τ1, θ0 and θ1 are assumed equal for all systems i. Isotropic hardening between slip systems is again assumed, so that Phij = 1 (note that hij here differs from the definition used previously). The randomly oriented grain population employed in the simulations of the previous section is used again here. Following the strategy of the previous section, an EPSC simulation of uniaxial tension is performed, and the resulting macroscopic response in uniaxial tension is used to define the bulk matrix response in the FE model. The single crystal plasticity parameters are varied in the EPSC model, until the macroscopic composite response of the FE model matches the experimental flow curve. The resulting matrix (blue dashed) and composite (continuous black) flow curves are shown in Fig. 4.31, together with the experimental flow curve (round datapoints). While in the previous section, complete consistency of the matrix response could not be achieved between the EPSC and FE models, in this case the bulk macroscopic response output from the EPSC model is used exactly as input in the FE model. The resulting composite flow curve matches the experimental flow curve almost exactly. The average elastic phase strains as functions of applied uniaxial stress, calculated using the FE model, are shown in Fig. 4.32 as continuous lines. Also shown are the strains calculated from the two phase EPSC model (dot-dashed lines), and the experimental data published in [133] (triangular datapoints). Ferrite strains are shown in red, cementite strains in green. Upon yielding, the axial ferrite strain deviates from the linear response towards lower lattice strain, but this is more pronounced in the two phase EPSC response than in the FE response. Also, in the transverse direction, the tensile deviation of the ferrite strain is greater in the two phase EPSC response. However, the corresponding deviations of the cementite strains are similar in both models in both directions. This is surprising. Recall equations (4.2) and (4.3), which follow from stress balance and the assumption that the composite is elastically isotropic and homogeneous:

σA f el + (1 f) el = (4.28) k I − k M E D E D E Chapter 4. Interphase and Intergranular Stress In Carbon Steels 115

800

700

600

500

400 Stress [MPa] 300 Matrix flow curve Composite flow curve 200 Experimental flow curve

100

0 0 0.5 1 1.5 2 2.5 3 3.5 4 Strain [%]

Figure 4.31: Flow curves from combined FE/EPSC model of the iron-cementite system studied by Daymond and Priesmeyer [133]. An EPSC run was performed, and the calculated macroscopic ferrite flow curve was used as input in the FE model. This procedure was repeated successively, varying the single crystal slip and hardening parameters in the EPSC model, until consistency was achieved between the composite flow curve predicted by the FE model, and the experimental flow curve. The figure shows the resulting macroscopic ferrite response (dashed blue line) and corresponding composite response (continuous black line), together with the experimental flow curve (round datapoints). Chapter 4. Interphase and Intergranular Stress In Carbon Steels 116

A el el σ f I + (1 f) M = ν . (4.29) ⊥ − ⊥ − E

That is, the volume fraction-weighted average of the elastic strains equals the elastic strain which would exist in both phases had yielding not occurred. Using these strain balance equations and f = 8%, cementite strains are calculated from the ferrite strains calculated in the FE and two phase EPSC models. These are shown respectively on Fig. 4.32 as green circles and green squares. For the FE model, the strains calculated from strain balance match the averaged strains calculated in the model exactly. This is indeed a requirement of the model, since the same isotropic elastic constants are specified in both phases. However, the model-calculated and strain balanced cementite strains differ greatly in the case of the EPSC model. Since bulk isotropic elasticity is not specified in this model, this does not violate the stress balance principle. However, given that the simulations were run using a randomly oriented grain population, the bulk elastic properties are expected to be nearly isotropic, and therefore the magnitude of the discrepancy is surprising. The experimental deviations from linearity of the ferrite lattice strains agree best with the predictions of the two phase EPSC model. Again, the FE model suffers from the simple cubic geometry which, as discussed in 4.11.3, restricts the extent to which the inclusions restrict matrix flow. Since no specific geometry§ is specified in the EPSC model, it is not affected by this problem. The experimental cementite strains are rather scattered (for error bars, see [133]), and the drops in magnitude at high stress may indicate yielding of the cementite. At lower stresses, the predictions of both models are consistent with the data, given that the error bars published in [133] are quite large. As in 4.12.2, the fitted FE model is used to calculate the average matrix stress state as a function of§ applied stress. This state is specified as the boundary condition within the single phase EPSC model. The ferrite grain family strains calculated in this simulation are plotted against applied uniaxial stress in Fig. 4.33. Alongside the combined model strains (continuous lines) are those calculated using the two phase EPSC model, and also lattice strains determined experimentally. Surprisingly, the axial strains calculated by both methods are very similar, even though the volume averaged strains have been shown to differ significantly. The calculated axial strains also show good agreement with the measured strains. In the transverse direction, the responses differ to a greater extent, absolutely as well as relatively. Both modelling strategies predict tensile shifts in the 200 and 310 lattice strains, but the two phase EPSC model captures the magnitudes more closely. Neither model fully captures the curves. However, as discussed earlier, transverse strain calculations are sensitive to the exact grain population specified, and therefore it is not surprising that there are differences between modelling strategies and between the calculated and measured strains. Note the discrepancy between the predicted linear elastic responses of the transverse 110 response. As discussed in 4.10, this response is particularly sensitive to texture. § Since the reinforcement volume fraction is small, the trends in the predicted lattice strain curves do not differ greatly from those of single phase ferrite. A composite with a greater reinforcement volume fraction would provide a better system of study. However, this section has demonstrated the viability of combining a two phase FE model with a single phase EPSC model, in order to predict matrix intergranular stresses. The present FE model underestimates the constraint imposed by inclusions on matrix flow, but the predicted intergranular strains have shown reasonable agreement with measured elastic strains. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 117

Transverse Axial

700

600

500

400

300 Applied stress [MPa]

200

100

0 −5000 0 5000 10000 Lattice strain [microstrain]

Figure 4.32: Comparison between FE model and 2 phase EPSC model predictions of axial and transverse average elastic phase strains, as a function of applied stress. Ferrite strains are shown in red, cementite strains in green. Continuous lines: FE model strains; dot- dashed lines: 2 phase EPSC model strains. Green circles represent the cementite strains calculated from the FE model ferrite strains using the principle of stress balance. Green squares represent the cementite strains calculated in the same manner from the EPSC model ferrite strains. Triangular datapoints represent experimental data from reference [133]. The dashed black lines extrapolate the elastic responses. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 118

800 800 110 200 211 310

600 600

400 400

200 200 Macroscopic Stress [MPa] Macroscopic Stress [MPa]

0 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 Lattice strain [10−6] Lattice strain [10−6] (a) (b)

800 800 110 200 211 310

600 600

400 400

200 200 Macroscopic Stress [MPa] Macroscopic Stress [MPa]

0 0 −1000 −800 −600 −400 −200 0 −1000 −800 −600 −400 −200 0 Lattice strain [10−6] Lattice strain [10−6] (c) (d)

Figure 4.33: Comparison of grain family average lattice strain predictions from combined FE/EPSC model (continuous lines) and two phase EPSC model (dashed lines), plotted alongside experimental datapoints [133]. (a) and (b): axial direction; (c) and (d): transverse direction. Chapter 4. Interphase and Intergranular Stress In Carbon Steels 119

4.13 Summary of Chapter

In this chapter, neutron diffraction and a range of modelling strategies have been used to measure and interpret the development of internal stress in carbon steels during uniaxial tensile loading. The subjects of intergranular and interphase stress development have both been addressed, through the study of a single phase low carbon steel as well as a two phase high carbon steel. In both materials, the majority phase is ferrite, which has a bcc crystal structure. The development of intergranular strain in this material has been followed by measuring shifts in single diffraction peaks during and after tensile loading. The elastic strain determined from a single diffraction peak relates to the average in a family of grains all of which have the same crystallographic direction aligned along the scattering vector. The grain family strains have been interpreted in light of the knowledge of operable slip systems in bcc crystals. Since these modes differ from those which operate in fcc materials, the observed trends in intergranular stress evolution differ from those reported by other authors in austenitic steels. The grain family residual strains measured parallel to the tensile axis are in reasonable agreement with the predictions of an elastoplastic self-consistent model. The model has been used to demonstrate that elastic anisotropy tends to counteract plastic anisotropy in ferrite, reducing the magnitude of the axial strains. Poorer agreement is seen between the strains measured in the transverse direction and the corresponding self-consistent calculations. This has been reported in previous studies of ferrite and other materials. It has been noted in this chapter that in comparison to the axial strains, the average transverse strains are more sensitive to the exact distribution of grain orientations. This helps to explain why prediction of the transverse strains is more difficult. The transverse grain family residual strains are of much greater magnitude than those measured parallel to the tensile axis. This has also been reported previously, but not adequately explained. It has been emphasised in this chapter that the set of grains which contribute to a particular reflection in the axial spectrum differs from the set which contributes to the corresponding reflection in the transverse spectrum. It is important to recognise this point in order to properly understand the large transverse residual strains. A simple argument based on the pencil glide model of slip in bcc crystals has been used to demonstrate that on average the grains in some transverse families are better oriented for slip than in other families. In particular, grains with 200 lying in the transverse plane are well-oriented, and this helps to explain the remarkablehtensilei shift seen in the transverse 200 elastic strain response during yielding. A further point to arise from the elastoplastic self-consistent analysis is that the gradients of the linear elastic responses of some transverse families are far more sensitive to texture than others. Symmetry-based arguments and consideration of the single crystal elastic stiffness tensor have demonstrated why this is the case. In ferrite the linear elastic response of the 110 transverse family is particularly sensitive to texture. This arises from the fact that in a single crystal of ferrite, the Poisson strain developed along [110] varies greatly with the direction perpendicular to this along which tensile stress is applied: it is negative for some orientations of the tensile axis, but positive for others. In high carbon steel, much larger internal stresses develop than in the low carbon steel, due to the presence of spheroidal cementite inclusions. Upon yielding, load is rapidly transferred from the ferrite matrix, indicating that the cementite inclusions continue to deform elastically while the matrix deforms plastically. This leads to the generation of misfit between the phases, and hence the generation of interphase stress. By comparison to an unrelaxed model, it is seen that the misfit begins to saturate after a few percent plastic strain. Comparison of elastic strain at load and after unloading also indicates some relaxation of the internal stresses during unloading. The load transfer is particularly dramatic because of yield point softening in the matrix. Directly after yielding the matrix bears significantly less stress than directly before. This effect Chapter 4. Interphase and Intergranular Stress In Carbon Steels 120

has been emulated in a finite element model. However, using simple cubic and body-centred cubic arrays of inclusions, it was not possible to fully capture the extent of load transfer due to softening. This demonstrates that back stress hardening is very effective in a composite containing a random dispersion of reinforcing inclusions. Despite the large interphase stresses, intergranular stresses are also evident in the ferrite phase of the high carbon steel. The axial grain family residual strains vary more greatly than in the low carbon steel. This has been explained on the basis that elastic anisotropy causes grain families to respond differently to the overall axially compressive residual stress existing in the matrix. In the final sections of this chapter, the viability of combining the finite element and self-consistent models for the full prediction of interphase and intergranular stresses in composites has been demonstrated. Chapter 5

Internal Stress Development During Stress-Induced Martensitic Transformation

The previous chapter concerned the generation of internal stress in a material containing a fixed volume fraction of reinforcing particles. This chapter moves further by studying internal stress generation in a material in which there is dynamic nucleation and growth of the reinforcing phase. The material chosen for this study is a Fe-Ni-C steel which exhibits transformation- induced plasticity. Again, the main experimental method employed is neutron diffraction. This provides an excellent means of studying stress-induced transformation. Not only can elastic strain be measured in both phases, but the diffraction spectra also provide a measure of the increasing volume fraction of reinforcement, as well as information on the evolution of crystallographic texture, revealing which grain families of the parent phase are preferentially oriented for transformation. Internal stress is a key consideration in explaining the observed behaviour. The chapter is structured as follows. A review of transformation-induced plasticity is given, followed by characterisation of the microstructure, mechanical properties and texture of the alloy selected for study. The neutron diffraction study is then presented. Results concerning the issue of preferential transformation are reported and interpreted using the Eshelby-based theory of martensitic transformation presented in 2.8.2. The other major issue addressed is the development of interphase stress and the role§ that this plays in the phenomenon of transformation-induced plasticity. Finally, intergranular stress development in austenite is considered.

5.1 Review of Transformation-Induced Plasticity (TRIP)

This section reviews the phenomenon of transformation-induced plasticity (TRIP), in order to provide the necessary background for the following investigation.

5.1.1 Overview Transformation-Induced Plasticity (TRIP) is a phenomenon in which deformation-induced martensitic transformation promotes plastic deformation and gives rise to improved ductility. The term was coined in 1967 by Zackay, et al. [43], who studied Fe-Ni-Cr alloys of varying composition. They noted that strength and ductility tend to be inversely related, and proposed transformation-induced plasticity as a mechanism to increase the strain hardening rate and prevent early necking. By subjecting the alloys to various heat and deformation treatments,

121 Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 122

they achieved a combination of strength and ductility superior to that of existing maraging alloys and ausform steels. Early efforts at exploiting the TRIP effect focused on modified stainless steels and Fe-Ni-C alloys with fully austenitic initial microstructures [43, 158, 159]. Although these alloys demonstrated promise for use in dynamic applications, this was hindered by strain rate sensitivity of the transformation [160]. At high strain rates, adiabatic heating may inhibit transformation and reduce ductility [161]. Moreover, the high alloying and demands on metallurgical control render the manufacture of these steels expensive. For these reasons, they have not found widespread application. A new generation of ‘simple’ TRIP steels developed by Japanese companies have, however, found widespread use in automotive applications [162]. Their excellent ductility makes them ideal materials to absorb energy in a motor accident: a quality known as crashworthiness [163]. These steels are based on a composition containing around 0.2%C-1.5%Mn-1.5%Si, and have a mixed microstructure containing ferrite, bainite, martensite and austenite. Only the austenite undergoes TRIP, and for this reason the alloys are sometimes termed TRIP-assisted steels [164]. An Fe-Ni-C alloy is studied in this chapter, since transformation is achieved throughout the entire microstructure, providing a good system for the study of internal stress evolution during the growth of a second phase. Therefore, this review focuses on this type of alloy.

5.1.2 Observed Features of the TRIP Phenomenon The critical stress for martensite formation in a TRIP steel is shown schematically as a function of temperature in Fig. 5.1. At the Ms temperature, martensite forms without the application of stress. As the temperature is increased, the critical stress is observed to increase linearly with temperature. This is consistent with a linear decrease in the chemical driving force, compensated by a linear increase in mechanical driving force, as suggested by Patel and Cohen [165]. In this regime, transformation occurs without slip of the parent austenite phase. However, σ at the temperature labelled Ms , the critical stress coincides with the austenite yield stress. At higher temperatures, the stress required to augment the chemical driving force cannot initially be sustained within the austenite. However, as straining progresses and the austenite work hardens, transformation is induced at a stress below that predicted by extrapolation of the σ linear curve (shown by the dashed line). It is in this regime, bounded by the temperatures Ms and Md (the maximum temperature at which applied stress causes transformation) that the TRIP effect is observed. It is clear from the above discussion that the martensitic transformation which forms the origin of the TRIP phenomenon is dependent upon the strain state of the material. Some σ authors choose to reflect this by describing the transformation above Ms as strain-induced, σ while describing that below Ms as stress-assisted. Olson and Cohen [166] used this terminology σ to specifically describe the formation of martensite nuclei, noting that below Ms , martensite embryos nucleate on the same sites as those which operate when the transformation is induced σ purely by cooling, while above Ms nucleation occurs on new sites which are produced by straining. However, in other papers (e.g. [167]) the terminology is used in a less rigorously defined manner. Some authors use the term stress-induced to refer to the transformation whether or not it accompanies plastic flow (e.g. [168]). The view taken in this dissertation is that the transformation is best described as stress-induced, since it is the stress, which unlike plastic strain is a state variable, which contributes to the thermodynamic driving force for transformation. Of course, plastic strain plays a role in influencing the internal stress state, as well as promoting the transformation kinetics.

Tensile Properties of a TRIP Steel Stress-strain curves of a typical Fe-Ni-C alloy (Fe-29wt.%Ni-0.26wt.%C) are shown in Fig. 5.2, reproduced from the paper of Tamura et al. [169]. The alloy has Ms = 60◦C. The − Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 123 Applied Stress → − Austenite yield stress

σ Ms Ms Temperature

Figure 5.1: Schematic illustration of the critical stress required for martensitic transformation in a TRIP steel, as a function of temperature. Reproduced from [161].

maximum temperature at which stress-induced martensite is formed is denoted Md. In this case, Md = +25◦C. Therefore, the test at +58◦C gives an example of the deformation of a fully austenitic alloy. In contrast, at -150◦C the material is fully martensitic. Although yield stress increases as temperature is reduced, the yield stress at -150◦C is substantially greater than would be predicted by extrapolation of the austenite yield stress to this temperature. This indicates that martensite may act as a reinforcing phase within an austenite matrix; this is fundamentally related to the origin of the TRIP eﬀect. The reason for the greater yield strength is that carbon atoms in interstitial sites cause greater distortion of the bct lattice, more eﬀectively inhibiting dislocation motion [41]. At intermediate temperatures, martensitic transformation is induced during the tensile tests. This is indicated in the constant strain-rate curves by characteristic serrations. These arise because the transformation strain which develops when a martensite plate forms increases the σ natural length of the sample, abruptly relieving the applied stress. Since the Ms temperature is greater than -50◦C, serrations are observed before austenite yielding in the tests at -70◦C and -50◦C. However, in the tests at -30◦C and -10◦C, transformation is only evident after austenite yielding. It is in this temperature range that TRIP occurs, as seen from the greatly increased elongations which are achieved. Zackay et al. [43] demonstrated that the maximum elongation developed during tensile testing at a particular temperature can be enhanced by prior deformation (by rolling) at a temperature at which austenite is stable with respect to transformation. Processing in this way work hardens the austenite, increasing the yield stress. This enables the applied stress driving force to be increased, further promoting transformation. It also increases the density of potential martensite nucleation sites. However, although the austenite yield stress continues to rise with the extent of pre-deformation, there is an optimal level for maximum subsequent tensile elongation. Beyond this level, the loss of further austenite ductility limits the total elongation. The tests of Zackay et al. also demonstrate that pre-deformation causes the alloy to develop Luders¨ bands during tensile testing. The Luders¨ strain increases with the level of pre-deformation. This type of heteregeneous deformation has been discussed in 4.1. § Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 124

Serrations − → − ←

Figure 5.2: Stress-strain curves of Fe-29wt.%Ni-0.26wt.%C alloy (Ms = 60◦C, − 4 1 Md = +25◦C), tensile tested at various temperatures at a strain rate of 5.5 10− s− . Reproduced from [169]. ×

Origin of Increased Elongation

Fig. 5.3, reproduced from the paper of Tsuchida and Tomota [170], shows the strains which develop when a Fe-27wt.%Ni-0.4wt.%C alloy is cooled below Ms (=239K). When no stress is applied, positive strains εx and εy of equal magnitude develop in two orthogonal directions, indicating that the strain is purely dilatational, arising due to the volume expansion which occurs during transformation. However, when a constant tensile stress is applied (equal to half the austenite yield stress at 300K, so that the austenite deforms elastically over the whole temperature range), the strain which develops parallel to the tensile axis (εy) becomes larger, while a much smaller strain (εx) develops in the transverse direction. This shows that the effect of applied stress is to promote the production of those martensite variants which cause the greatest elongation along the tensile axis. In this way, the mechanical potential energy is minimised. However, the maximum elongation attained by transformation under applied stress in Fig. 5.3 is 1.1%, which is much smaller than the increased elongations in the TRIP temperature range ≈seen in Fig. 5.2. Moreover, if the increased elongation were due to preferential selection alone, then the increased elongations would also be seen in the tensile tests at the temperatures at which transformation occurs below the austenite yield stress. An alternative explanation for the increased elongation is the additional work hardening which is introduced as the reinforcing martensite phase develops. This acts to maintain the stability of plastic flow. As discussed in 4.1, the condition for tensile plastic instability is that if a region begins to neck, then the stress§ concentration which develops is not sufficiently compensated by the additional work hardening of the region. Repeating equation (4.1), this is expressed as dσ y < σ . (5.1) d y

dσy Fig. 5.4 reproduces plots from [169] of true stress σy and d against true strain for some of the tensile tests of Fig. 5.2. In the tests in which no transformation occurs, the work hardening Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 125

12 ε (under load) y ε (under load) x ε (no load) y 10 ε (no load) x

) 8 −3

Nominal strain (/ 10 4

0 195 200 205 210 215 220 225 230 235 240 Temperature (K)

Figure 5.3: Strains measured by cooling below Ms = 239K with and without constant tensile stress in an Fe-27wt.%Ni-0.4wt.%C alloy. εx and εy refer to the strains measured transverse and parallel to the tensile axis respectively. The applied tensile stress is half the yield strength of austenite at 300K. Reproduced from [170]. rate falls monotonically until condition (5.1) is met and the material necks to failure. However, at -10◦C, when stress-induced transformation occurs during plastic ﬂow, the work hardening rate stops falling and begins to increase. This suppresses necking, resulting in greatly increased elongation. In the case of pre-deformed alloys which deform by Luders¨ band elongation, the initiation of the Luders¨ band may be viewed as identical to necking. Azrin et al. [160] demonstrated this by testing at elevated temperature, such that transformation is inhibited. In this case, the neck does not stabilise, leading to early failure. In the TRIP temperature regime, the onset of transformation provides the additional work hardening required to stabilise plastic ﬂow, leading to Luders¨ band propagation. It is clear from the above discussion of the reinforcing role of martensite that the development of internal stress plays an important role in the origin of the TRIP phenomenon. However, there is a lack of internal stress measurements on TRIP steel in the literature. This is addressed by the work presented in this chapter.

5.1.3 Models of Stress-Induced Transformation in TRIP Steels A range of diﬀerent approaches have been presented for the modelling of the mechanical properties of TRIP steels. Early models proposed expressions for martensite volume fraction as a function of plastic strain [171, 172]. Olson and Cohen [166] presented a model of this type, relating the evolution of martensite to the probability of martensite nucleation at shear band intersections. The model produces reasonable ﬁts to the experimental, sigmoidal-shaped curves of martensite volume fraction versus plastic strain. However, it takes no account of the crystallography of the transformation, nor of the applied stress. Stringfellow et al. [173] generalised the Olsen and Cohen model by incorporating stress state dependence into the expression for the thermodynamic driving force. However, their model uses the rather simplistic parameter of the ratio of volumetric and deviatoric stress invariants, in Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 126

Figure 5.4: Variation of work hardening rate dσ/d and true stress σ in Fe-29wt.%Ni- 0.26wt.%C alloy, deformed at: (a) +58◦C; (b) -10◦C; (c) -150◦C.

order to reflect the dependence of the transformation on the triaxiality of the stress state. Using this approach, they demonstrated that transformation is particularly favoured where the triaxiality is high, such as around crack tips, helping to inhibit fracture. However, the model again makes no reference to the crystallographic theory of martensitic transformation. It therefore does not realistically describe the mechanical potential energy, since this requires the transformation strain to be explicitly specified. Stringfellow et al. incorporated their constitutive equation within a self-consistent model in order to predict the polycrystal flow curve. They used the Eshelby solution for a spherical inclusion, rather than for a plate, which is more realistic of the true morphology. Nevertheless, their predictions are in reasonable agreement with experimental flow curves, albeit through the fitting of a rather large number of parameters. Iwamoto and Tsuta have further modified the Olson and Cohen model to account for the dependence of TRIP on austenite grain size [174]. Tsuchida and Tomota have emphasised the partitioning of stress between the austenite and martensite phases [170]. They attempted to predict the composite flow curve on the basis of the constituent single phase material flow curves, by applying the mean field method, and using the concept of instantaneous stiffness, as discussed in the context of the elastoplastic self-consistent method in 2.6.4. That is, at any stage in the deformation, a plastically deforming phase is conceptually§ replaced by an elastic, but compliant, phase, in order that the Eshelby theory may be applied to determine the subsequent partitioning of stress. Tsuchida and Tomota relied, however, on the experimentally-determined variation of martensite volume fraction with strain, and thus their work does not constitute a complete model of the TRIP phenomenon. A more fundamental approach was taken by Bhattacharyya and Weng [168]. They developed a criterion for transformation by consideration of the Gibbs free energy of the system, accounting for the mechanical potential energy and chemical energy of the two phase system. They neglected surface energy between phases, quoting data reported by Patel and Cohen [165] and Cohen and Wayman [175] indicating that this term is small in comparison to the chemical energy term, except when the surface area to volume ratio of the inclusions is very high, i.e. during nucleation. Therefore their model neglects the kinetics of nucleation, unlike the Olson and Cohen model which is based upon consideration of the nucleation rate. The mechanical potential energy depends on the average internal stress. Bhattacharyya and Weng calculated this using the Eshelby and mean field theories, and the prediction from finite deformation theory of the martensite transformation strain. Matrix plasticity was accounted for using the Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 127

concept of instantaneous stiffness. They assumed martensite inclusions to be fully elastic, randomly oriented thin plates. This inclusion shape is certainly more realistic than the spheres assumed by Stringfellow et al. However,the assumption of a random orientation distribution is not supported by optical microscopy shown later in this chapter (see Fig. 5.7). The flow curve predicted by the model qualitatively resembles the experimental curve, exhibiting an increase in work hardening as the martensite volume fraction grows. Cherkaoui, Berveiller and Sabar formulated a constitutive equation for an austenitic single crystal undergoing TRIP [176]. The basis of their model is similar to that of Bhattacharyya and Weng, although the formulation is very different. They took due account of plasticity in both phases, internal stress – calculated using the Eshelby and mean field theories – and the transformation strain predicted by the finite deformation theory, to formulate an expression for the rate of energy dissipation under an applied stress, in terms of the present values and rates of change of thermodynamic internal variables and the external variables of applied stress and temperature. The internal variables are the average plastic strain and volume fraction of individual martensite variants and the austenite matrix. The expression may be interpreted in terms of a thermodynamic driving force for the evolution of each internal variable. The consistent evolution of all internal variables may be solved numerically under an imposed variation of the external variables. Cherkaoui et al. have furthermore incorporated their single crystal constitutive equation into an elastoplastic self-consistent polycrystal model [167]. In similar fashion to other groups, they validated this model by comparison to experimental macroscopic stress-strain curves, obtaining qualitatively good agreement with the variation in shape of the curves with temperature. However, in principle their model also predicts which austenite grains are oriented preferentially for transformation to martensite, and furthermore which of the possible martensite variants are preferentially formed under an imposed external stress. It also predicts the development of intergranular and interphase stress. However, the authors did not consider experimental validation of any of these predictions. Reisner, Werner and Fischer have incorporated a transformation criterion within a finite element model of a TRIP-assisted steel, in order to assess the dependence of transformation on austenite orientation [177]. By simulating 12 different grain orientations, they demonstrated that the extent of transformation is indeed strongly dependent on parent grain orientation. Curiously, however, they did not elaborate as to which orientations were favourable for transformation, nor as to the martensite variants which are preferentially selected. To summarise, the most recent models of the TRIP phenomenon have emphasised the roles played by internal stress and the preferential transformation of favourably oriented parent grains into selected martensite variants. However, these issues have rarely been investigated experimentally. These are the subjects of primary interest in this chapter.

5.2 Materials

A melt of TRIP steel was specially prepared for this investigation. Its composition was chosen to match that of alloy B in the paper by Tomota et al. [178]. This composition is given in Table 5.1. The alloy was kindly produced by Dr Michio Okabe and Mr Shigeki Ueta of the Research and Development Department of Daido Steel Company Ltd, Nagoya, Japan. After forging and hot rolling, the material was solution treated at 1150◦C for three hours. It was supplied in this condition as 12mm diameter rods.

C Si Mn P S Ni Fe 0.38 0.32 0.30 0.019 0.016 25.44 Balance

Table 5.1: Composition of the TRIP steel in wt.%. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 128

5.3 Characterisation of Microstructure and Mechanical Behaviour

In alloys such as the TRIP steel studied in this chapter, the temperature at which transformation begins on cooling, Ms, (and hence the range of temperatures over which stress-induced transformation occurs) is very sensitive to composition [41]. Although the composition was selected to exhibit TRIP at room temperature, slight inaccuracies both in the quoted composition and the alloy produced following this composition may alter the temperature at which TRIP is observed. Therefore, prior to studying the alloy with neutron diﬀraction, characterisation of the stress and temperature dependence of the transformation was vital. This is described in this section.

5.3.1 As-Received Microstructure An optical micrograph of the as-received material is shown in Fig. 5.5. In this condition, the material is fully austenitic, with a large grain size of 100µm. Unless stated otherwise, optical microscopy specimens were prepared by electropolishing∼ in 10% perchloric acid, 90% acetic acid, and etching in 10% nital solution.

100 µ m

Figure 5.5: Optical micrograph of as-received material.

5.3.2 Ms Temperature

Tomota reported a martensite starting temperature in alloy B of Ms = 37◦C. As mentioned, − however, Ms is sensitive to the exact composition [41], and for this reason a short investigation was performed to determine Ms for the as-received alloy. Samples were held in methanol cooled by liquid nitrogen, then inspected metallographically. This was repeated for a range of temperatures. The development of martensite on cooling is shown in the optical micrographs of Fig. 5.6. As seen, the Ms temperature was 55◦C, which is lower than that reported by Tomota. ≈ − 5.3.3 Mechanical Behaviour

The reduced Ms temperature may have implications for the temperature at which TRIP is induced. To determine whether the as-received material exhibited TRIP at room temperature, Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 129

100 µ m 100 µ m 100 µ m

(a) -52◦C (b) -57◦C (c) -93◦C

Figure 5.6: Development with cooling of martensite in as-received material.

a specimen was tensile tested to failure and observed metallographically. The specimen developed a very prominent neck prior to failure. This was manifested in a large engineering stress drop, as seen in the room temperature stress–strain curve shown in Fig. 5.8, to be discussed shortly. Fig. 5.7 shows the variation of microstructure behind the fracture surface. In the necked region, martensite laths can be seen in many of the original austenite grains. The aligned nature of the laths is characteristic of a stress-induced transformation. In contrast, martensite laths formed by cooling (Fig. 5.6) are randomly oriented. Further from the fracture surface, the volume fraction of martensite falls oﬀ. Thus it appears that transformation occurs during necking, but this does not provide suﬃcient work hardening to stabilise the neck and produce Luders¨ deformation. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 130 Figure solution surface Fracture 5.7: Microstructural Figure Necked va region 5.8: riation

Dep Engineering stress [MPa] 1000 1200 b 200 400 600 800 endence ehind 0 fracture of 50% tensile RT surface b ehaviour after +1 C o Engineering strain ro on om test temp −22 C temp erature o erature tensile fo −40 C r ST. o test Strain on −50 C as-received rate: o 0.067min Un−necked material. − region 1 . 0.5mm Etched in 10% nital Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 131

As the temperature is lowered, martensite is stabilized relative to austenite, promoting the stress-induced transformation. To investigate this, constant strain rate tensile tests were performed at a series of temperatures. The tensile specimens were loaded in a temperature- controlled box cooled by nitrogen gas. Temperature stability of +/-2◦C was achieved. The resulting stress-strain curves are shown in Fig. 5.8.

The change in behaviour on cooling is clear. The neck which forms at 1◦C is less prominent than that at room temperature, and this is evidenced in the lack of an engineering stress drop prior to failure. However, the strain achieved before failure is 50% at both temperatures. ≈ At -22◦C however, rather than fail at this strain, the sample continues to deform, entering a new phase of work hardening. Prominent serrations are visible in the stress–strain curve. As noted in 5.1.2, these are characteristic of deformations which occur via rapid shape changes, such as martensitic§ transformation or twinning. The ﬁnal strain achieved at failure is in excess of 70%. Metallographic inspection of the post-test microstructure (Fig. 5.9) reveals a great deal of martensite has formed, and thus supports the assertion that the increased ductility is associated with transformation.

Below -22◦C, the ductility falls but the ultimate tensile strength rises. It is evident from the prominent serrations and increases in work hardening rate seen in the ﬂow curves at -40◦C and - 50◦C that transformation is induced at these lower temperatures. Further incidental evidence of the formation of martensite was gained by qualitative testing with a small magnet. Martensite is ferromagnetic and attracts a magnet more strongly than the paramagnetic austenite phase [41]. Strong attraction was observed along the entire gauge length of the low temperature samples, but only in the necked region of the samples deformed at room temperature and 1◦C.

100 µ m

Figure 5.9: Transverse section of solution-treated TRIP steel after loading to failure at 22◦C. −

5.3.4 Martensite Volume Fraction

As seen in the above subsection, tensile deformation of the as-received material at approximately -20◦C leads to the development of a large martensite volume fraction, while deformation at room temperature does not. For this reason, the material was cooled to -20◦C during the neutron diffraction tests described later in this chapter. In this section, the martensite volume fraction developed through tensile straining at this temperature is quantified metallographically. Inde- pendent quantification may also be achieved by Rietveld refinement of the neutron diffraction spectra. Results using these two methods are compared in 5.10. § Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 132

Method

Hounsfield tensile specimens of the unswaged material were tensile tested at 20◦C. This temperature was maintained by conducting the tests within a temperature controlled− box cooled 1 by nitrogen gas. The Hounsfield specimens were deformed at a strain rate of 0.067min− , to a range of tensile strains between 14% and 67%. After testing, the specimens were cut into both transverse and longitudinal sections ready for metallographic preparation. Sections were also taken from the unswaged sample tested in the neutron diffraction experiment. The sections were ground and polished, and the martensite phase was revealed by etching in nital solution. Micrographs were recorded using an optical microscope and digital camera. The Kontron KS400 image analysis software was used to enhance and threshold the images, and then to measure the area fraction of martensite, which was assumed to provide a good estimate of volume fraction. The thresholding process is illustrated in Fig. 5.10. Between 5 and 18 micrographs were analysed for each measurement, corresponding to a sampled area of several square millimetres per measurement. By assuming the area fraction determined in each micrograph to be sampled from a normally distributed population, the uncertainty in the measurement was estimated to be σ2/n, where σ2 is the estimated variance of the population and n is the number of micrographs analysed [179]. p

(a) (b)

Figure 5.10: Illustration of thresholding, to estimate martensite volume fraction from optical micrographs: (a) the original micrograph, consisting of light (austenite) and dark (martensite) regions; (b) the micrograph after thresholding – the black area fraction provides an estimate of the martensite volume fraction.

Results The variation of martensite volume fraction with plastic strain is plotted in Fig. 5.11. Good agreement is seen between the measurements made from longitudinal sections and those made from transverse sections. No martensite is visible through optical microscopy in the sample deformed to 14% strain, although some serrations are observed in the flow curve at approximately this strain (Fig. 5.8). At 30% strain, the material contains approximately 5% martensite; note that this strain corresponds approximately to the inflection point in the tensile flow curve, when the rate of work hardening increases. Beyond 30% strain, the volume fraction increases more rapidly with strain, to over 50% in the final measurement at 67% strain.

5.3.5 Inﬂuence of Hot Swaging As noted in 5.1.2, a method to encourage transformation during tensile testing other than by cooling is to§pre-deform the material at elevated temperature, such that austenite is stabilised relative to martensite. For this purpose, rods of the as-received material were heated in a Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 133

Longitudinal section 0.5 Transverse section

0.4

0.3

0.2 Martensite volume fraction 0.1

0 0 10 20 30 40 50 60 70 Plastic strain [%]

Figure 5.11: Variation of martensite volume fraction with plastic strain (tensile deformation at -20◦ C), determined using image analysis of optical micrographs.

furnace to 200◦C, then withdrawn and immediately swaged (in several passes) to a reduction in cross sectional area of 52%. The microstructure after hot swaging is shown in Fig. 5.12. The austenite grains are heavily elongated by the process, as seen in the longitudinal section, Fig. 5.12a. The heavy deformation is also evident from the prominent strain markings throughout the structure. These are most clearly seen in the transverse section, Fig. 5.12b. In the higher magnification micrograph, Fig. 5.12c, markings can be seen lying parallel to a twin boundary. This indicates that the markings are etch pits along stacking faults, arising due to the dissociation of partial dislocations. Martensite is not present in the material after hot swaging. This is clear from the microscopy, testing with a magnet, and diffraction spectra (see 5.7). The room temperature tensile flow curve after§hot swaging is shown in Fig. 5.13. For comparison, the flow curve of the unswaged material tested at -22◦C is also shown. The yield stress of the hot swaged material is slightly greater than the ultimate tensile strength of the unswaged material; thus the process is indeed effective in raising the driving force for transformation. Clear upper and lower yield points are visible, and plastic flow initially occurs by Luders¨ band propagation (seen easily by visual inspection). After the Luders¨ band has propagated, homogeneous deformation follows with a small degree of further work hardening. Transformation occurs during Luders¨ band propagation. This is seen in Fig. 5.14, which shows a longitudinal section through a Luders¨ band front. Dark martensite laths are seen in the propagated region (left) but not in the region through which the band has yet to pass, where very little plastic straining has occurred. Testing with a magnet confirms these observations: the magnet is more strongly attracted by the propagated region. The serrated nature of the flow curve after the band has fully propagated indicates that further transformation occurs during the homogeneous straining stage. The final microstructure is shown in Fig. 5.15. It is clear from inspection that martensite is the majority phase. In summary, the influence of hot swaging is to work harden the material, to cause heterogeneous plastic straining, and to promote stress-induced transformation at room temperature. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 134

swaging axis ←− −→

100 µ m

(a)

100 µ m

(b)

50 µ m

(c)

Figure 5.12: Microstructure of hot swaged material: (a) longitudinal section; (b) transverse section; (c) transverse section showing twin boundaries and etch pits at stacking faults. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 135

1100 1000 Hot swaged, RT 900 800 Unswaged, -22◦C 700 Luders¨ −propagation→ 600 500 400 300 200

Engineering stress [MPa] 100 0

50% Engineering strain [%]

Figure 5.13: Tensile behaviour before and after hot swaging. Unswaged: strain rate 1 1 0.067min− . Hot swaged: strain rate 0.017min− . Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 136 Figure 5.14: Figure Longitudinal 5.15: T ransverse section section through of L hot uders ¨ sw aged band tensile front of sp hot ecimen sw aged 50 m after µ tensile failure. sp ecimen. Etched in Etched 10% nital in 10% solution. nital solution. 0.5mm Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 137

5.4 Evolution of Crystallographic Texture

The martensitic phase transformation should be manifested in the diffraction spectrum of the TRIP steel in the weakening of intensity of certain austenite reflections, and the appearance of new martensite reflections. However, the grain rotation associated with plastic deformation (see 2.5.1) may also cause intensity changes. It is clear from the heavily elongated grain microstructure§ seen in Fig.5.12a that this effect is potentially significant. It is important to isolate the effect of grain rotation from that of preferential transformation. To achieve this, a sample of unswaged material was deformed in uniaxial tension to 41% plastic strain. The tensile test was performed at room temperature, thus inhibiting TRIP. It will be demonstrated in 5.7.1 that the unswaged material is initially untextured. With TRIP inhibited, the texture whic§ h evolves during tensile testing may thus be attributed to grain rotation alone. In order to measure texture in the tested sample, the method of electron back scattered diffraction (EBSD) was employed. A sample was prepared by sectioning and electropolishing in the same solution of acetic and perchloric acid used for optical microscopy samples. As stated, prior to tensile testing, the unswaged material is untextured. However, as shown in the pole figures of Fig. 5.16, texture develops during straining. A strong 111 component grows parallel to the tensile axis. The 100 orientation parallel to the axis alsoh remainsi significant, but the 110 component completelyh idisappears in this direction. This is a characteristic drawing or extrusionh i texture for a fcc metal with low stacking fault energy [110].

Figure 5.16: Pole ﬁgures of unswaged material tensile tested to 41% plastic strain at room temperature, measured by EBSD. Contours are integer multiples above random. The central z-axis corresponds to the tensile axis.

5.5 Neutron Diﬀraction Method

The neutron diﬀraction study was completed using the ENGIN instrument, described in 3.3. Cylindrical tensile specimens were machined from both the as-received (hereafter denoted§ unswaged) and hot swaged materials, with diameters 6.5mm and 5mm respectively. Diﬀrac- tion spectra were recorded in situ during uniaxial tensile loading using the ENGIN loading rig. Measurements were performed both under load (in load control), and after unloading. Count times were approximately 45 minutes, but longer when a small volume fraction of martensite was detected. An extensometer was not used, due to the large strains which would develop and because it would interfere with the cooling of the unswaged samples (see 3.3.5). Instead, macroscopic strain was monitored using crosshead movement. In order to induce TRIP in the unswaged material, the cooling grips described in 3.3.5 were § used to grip the sample. Using this design, cooling to 20◦C was achieved. A temperature − Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 138

gradient of several degrees Celsius from the shoulder to sample centre was observed, but the temperature was stable to within 1◦C. The sample temperature increased systematically with 6 elongation. The maximum change was 6◦C, corresponding to thermal strain of about 100 10− , which is small relative to the measured lattice strains. The increase does however act to sligh× tly discourage transformation.

5.6 Macroscopic Response During Neutron Diﬀraction Tests

The flow curves acquired during the neutron diffraction tests are shown in Fig. 5.17. Although similar to those acquired under constant strain rate conditions (Fig. 5.13), the flow curves are modified by the different loading regime, i.e. maintaining constant engineering stress for long periods of time. This causes the characteristic steps, as also observed in the carbon steel flow curves of Chapter 4. As noted for the carbon steels, after holding at load, the stress has to be increased significantly for further plastic flow to occur. As explained in 4.4, this suggests some ageing occurs due to the pinning of dislocations by the diffusion of carb§on atoms towards dislocation cores. During the experiment, it was observed that after increasing the load beyond a certain level the plastic strain increased rapidly (within a few seconds); this is consistent with a degree of softening due to the unpinning of dislocations. Upon subsequent holding at constant load, only a very small amount of creep was observed to develop. Note that ageing due to pinning tends to be more prevalent in ferritic rather than austenitic steels, because the bcc lattice is more greatly distorted by the occupancy of an interstitial site by a carbon atom, owing to the smaller size of these sites compared to those in the fcc lattice [143] (this also explains the greater solubility of carbon in austenite). Nevertheless, as given in Table 5.1, there is a substantial amount of carbon in the TRIP steel, and thus the density of pinning sites may be high, causing this mechanism to give rise to significant ageing. The yield stress of the hot swaged material is greater than that measured in the constant strain rate test. This may also possibly be attributable to ageing, since the constant strain rate test was conducted within a day of hot swaging, while the neutron tests were conducted several weeks later. This time scale is significantly longer than that required to cause ageing during testing, however. The ultimate tensile strength and failure strain in the unswaged material are reduced compared to those attained at 22◦C under the condition of constant strain rate. The material fails at less than 30% strain,−whereas in the constant strain rate test a strain of over 80% was achieved. This may be due to the systematic rise in temperature with strain during the neutron diffraction test. Since increasing temperature stabilises austenite relative to martensite, this encourages the sample to fail before sufficient transformation occurs to adequately increase the rate of work hardening. In fact, it is demonstrated in subsequent sections that the volume fraction of martensite developed in the unswaged material was indeed limited. The hot swaged material was not tested to failure, but as shown in 5.7, a large volume fraction of martensite was formed in this material. § A further difference between the flow curves acquired during testing with neutrons and those acquired under constant strain rate conditions is that no serrations are seen in the former. This is simply due to the tests being conducted in load control. In a constant strain rate test, the elongation which occurs when a small region transforms causes the stress required to continue straining to momentarily drop. However, in load control, it is the strain which jumps suddenly, while the stress remains constant. In the test on the hot swaged material, the Luders¨ band was propagated under strain control, and the applied stress does indeed fluctuate in this region. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 139

900 1400

800 1200 Luders band propagation 700

1000 600

500 800

400 600

True Stress [MPa] 300

Engineering Stress [MPa] 400 200

200 100

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Macroscopic Strain [%] Macroscopic Strain [%] (a) (b)

Figure 5.17: Macroscopic flow curves acquired during neutron diffraction tests: (a) unswaged; (b) hot swaged sample. Red datapoints indicate where diffraction measurements were made. In the hot swaged case, strain is plotted against engineering stress (force/(original cross sectional area)) rather than true stress, because flow is inhomogeneous during Luders¨ band propagation, causing the true stress to vary along the sample. A separate sample was used to measure residual strain in the unswaged material.

5.7 Evolution of Diﬀraction Spectra and Observation of Pref- erential Transformation

Diﬀraction spectra recorded before and after tensile testing are shown for both the unswaged and hot swaged materials in Fig. 5.18. In this section, the range of information contained within the spectra is discussed. It should be noted that at a given plastic strain, the spectra recorded under applied load and after unloading diﬀer only in peak shifts due to changes in lattice strain. That is, there is no evidence of reverse transformation during unloading.

5.7.1 Texture Prior to Testing The initial axial and transverse diffraction patterns of the unswaged material closely resemble one another, and that expected of a powder sample of fcc material. This provides evidence that there is no preferred orientation in the unswaged material. However, the relative intensities of reflections in the initial spectra of the hot swaged material differ greatly from those expected from a powder sample. This indicates that the hot swaged material is strongly textured. In the axial spectrum, some reflections are particularly weak or completely absent. Only the 111 and 200 reflections remain strong, indicating that a large majority of grains are aligned with one of these directions lying close to the tensile axis. In 5.4, a similar texture was shown to develop during tensile testing of the unswaged material §(at a temperature at which stress-induced transformation is inhibited). It is very reasonable that similar texture develops through swaging and uniaxial tension, for the following reason. The stress state imposed during swaging is very comparable to strong hydrostatic compression, superposed with uniaxial tension. Hydrostatic stress does not cause plastic deformation; thus it is the tensile component which leads to grain rotation. In the initial transverse spectrum of the hot swaged material, reflections are seen which are weak or absent in the axial direction. Since most grains lie with either a 111 or 200 direction parallel to the tensile axis, the intense transverse reflections correspond toh setsi of planesh i in either a 111 or 200 zone. Notably, this includes the 220 reflection, which is correspondingly strong h i h i Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 140 transverse; Figure { ma 101

rtensite Intensity [neutrons per second]

10 20 30 40 50 60 70 80 Intensity [neutrons per second] 10 12 14 16 18 20 22 } 0 0 2 4 6 8

{422} − − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − − − − − − − − 0 5.18: ).

{420}

0.8 −− − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − −− − − −

0.8 − γ

−− − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − − − − − − p {331} (d) γ

eaks

{400} −− − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − −− − − − − − − − −

γ −− − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − {220}

{310} Diﬀraction hot γ {222}

a −− − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − α

α 1

' −− − − − − − − − − − − − − − − − − − − − − − − − re − ' γ

sw −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − lab {311}

aged

{211}

−− − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − −− − − − − − − − − − − − elled − γ

1.2 1.2 α

sp '

−− − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − transverse. − {220} ectra d−spacing [10 d−spacing [10 with γ

1.4 1.4 just b efo one re −10 −10 Austenite 1.6 1.6 m] m] (red) set of

and

{200} − − − − − − − − − − − − − − − − − 1.8 1.8 indices, ( γ γ

after ) and {110}

2 2

although −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − α (green) '

ma − − − {111} rtensite γ

(b) (a) 2.2 2.2 Before After Before tensile After in realit ( α Intensity [neutrons per second] {422} 0 10 15 20 25 30 35 40 Intensity [neutrons per second] testing: 10 12 14 16 18 {420} ) 0 5 0 2 4 6 8 y γ

{422} −− − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − −− − − − − − reﬂections − {331} each γ

{420}

− − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − −− − − − − − 0.8 −

0.8

− − − − − − − − − − − − − − − − − −− − − − − − − − − − − −− − − − − − − γ

{331} γ

{400} consists − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −

(a)

γ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − {220}

{310} γ {222}

α −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −

unsw − 1 α

a '

1 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − '

re γ

−− − − − − − − − − − − − − − − − − − − − − − − − − − − − of {311} lab aged,

{211} − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − at − γ 1.2 elled

1.2 α

least

− −− − − {220} d−spacing [10 axial; d−spacing [10 in γ

t 1.4

w 1.4 black

{200}

− − − − − − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − − − − − − − − − − − − − o − (b) merged α ' −10 −10 and hot 1.6 m] 1.6 m] sw blue reﬂections aged,

{200}

− − − − − − − − − − − − − − − 1.8 − 1.8 resp γ

axial; ectively {110} (e.g.

2 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 2 − α

{111}

− − − − − − − − − − − − − − − − − − − − − − − − − (c) − . γ {

110 F unsw o 2.2 2.2 r } After (d) (c) Before Before After cla α aged, 0 rit and y , Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 141

in the transverse direction, while completely absent in the axial direction. However, it excludes the 111 reﬂection, which is correspondingly weak in the transverse direction.

5.7.2 Development of Martensite As the applied stress and plastic strain increase, new peaks emerge in the spectra, indicating transformation to martensite. The martensite peaks are labelled in blue in Fig. 5.18. As noted in 2.8.1, the martensite phase is body-centred tetragonal, not cubic. Therefore, each martensite§ peak in fact consists of at least two merged reflections, which cannot be resolved. For example, 110 planes have a slightly different d-spacing to 101 planes, but the separation is too small relativ{ }e to the peak width for the reflections to be{resolv} ed. This point should be noted, but for brevity the peaks are hereupon denoted with one set of Miller indices only, while non-equivalent permutations of the indices are implied. In the axial spectra of both the unswaged and hot swaged materials, the only prominent martensite peak to emerge is 110. The reflections 211 and 310 are also visible, but are very weak. This provides evidence of preferential transformation. Since there is a direct correspondence between the orientations of the parent and transformed phases, more than one prominent reflection would be anticipated if austenite grains of all orientations transformed to an equal extent. The orientation dependence of the transformation is analysed in 5.8. § In the transverse spectra, other reflections are prominent in addition to 110, in particular the 200 and 211 reflections. Again, these transversely reflecting plane families lie in the zones of the crystallographic directions which lie parallel to the tensile axis in the majority of grains, i.e. 110 . h i The growth of the martensite peaks is quantified in Fig. 5.19 in terms of the variation of integrated peak intensity with applied stress. Figs. 5.19 (a) to (d) show the variation in absolute intensities. While the intensities grow gradually with applied stress in the case of the unswaged material, prominent martensite peaks emerge suddenly upon yielding in the hot swaged material. This is consistent with the micrograph of Fig. 5.14, which shows that a large amount of martensite is formed during Luders¨ band propagation. Despite this difference in the extent of transformation after yielding, the relative intensities of the individual martensite peaks are similar in both materials, even though the textures of the parent phase differ markedly. As described above, in both materials, the 110 reflection is by far the most prominent in the axial spectra. A weak axial 211 reflection is also present in both materials, although relative to 110, it is more intense in the unswaged material. This suggests that few parent grains in the hot swaged material have the required orientation to produce such martensite grains. Since in both the materials the vast majority of martensite grains have a 110 direction aligned along the tensile axis, and this is also an axis of cylindrical symmetryh , iit may be asserted that the texture of the martensite phase is similar in both materials, despite the parent phase textures differing greatly (strictly, this argument neglects the peak splitting due to tetragonality, but since this cannot be resolved, it has no influence on the arguments which will now be advanced with respect to peak intensities). This is of course the result of preferential transformation, as discussed above. Further, it implies that the relative intensities of reflections in the transverse spectra should be similar in both materials. This is indeed the case. In the transverse spectra of both materials, the 200 reflection is slightly less intense than the 211 reflection, which has approximately half the intensity of 110. The integrated intensities are also plotted in Figs. 5.19 (e) to (h) as a fraction of their value at the highest applied stress. Continuing the above argument, the failure to emerge of any strong axial reflections other than 110 indicates that the martensite texture does not change markedly with applied stress. Hence the relative transverse intensities should also not vary markedly with applied stress. Support for this argument is seen in Figs. 5.19 (f) and (h); the fractional intensities for the unswaged and hot swaged transverse spectra respectively. In the hot swaged case, the curves of all reflections follow one another very closely. This confirms that the proportions of the various martensite grain orientations do not vary significantly, and thus Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 142

Plastic strain [%]: Plastic strain [%]: 0.2 1.4 4.6 8.8 14.2 18.7 25.2 0.2 1.4 4.6 8.8 14.2 18.7 25.2 160 60 110 (a) Unswaged, axial (b) Unswaged, transverse 140 110 50

120 0.2% yield stress 0.2% yield stress 40 100

80 30

60 20 211 40 200

Integrated Intensity [neutrons per second] 10 20 211 Integrated Intensity [neutrons per second]

0 0 100 200 300 400 500 600 700 800 900 0 0 100 200 300 400 500 600 700 800 900 Applied Stress [MPa] Applied Stress [MPa]

Plastic strain [%]: 0 12.3 17.8 21.8 26.1 Plastic strain [%]: 0 12.3 17.8 21.8 26.1 1200 300 (c) Hot swaged, axial 110 (d) Hot swaged, transverse 110 1000 250

800 200

upper yield stress upper yield stress 211 600 150 200

400 100

200 50 Integrated intensity [neutrons per second] Integrated Intensity [neutrons per second]

211 0 0 800 900 1000 1100 1200 1300 1400 1500 1600 1700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 Applied Stress [MPa] Applied Stress [MPa]

Unswaged axial Unswaged transverse Hot swaged axial Hot swaged transverse 1 1 1 1 110 110 (e) (f) 211 (g) (h) 0.8 0.8 0.8 0.8 211 200 final 0.6 0.6 0.6 0.6 211 211 0.4 0.4 110 0.4 0.4

0.2 110 0.2 0.2 0.2 200 Intensity/Intensity

0 0 0 0 200 400 600 800 200 400 600 800 1000 1500 1000 1500

Applied Stress [MPa]

Figure 5.19: Variation of martensite peak integrated intensities with applied stress. (a) to (d): absolute intensities (plastic strain is also indicated on a non-linear scale); (e) to (h): relative to final intensity, using the same colour coding. Background is removed by subtracting the integrated intensity prior to tensile testing, when no martensite is present. The tail of the 110 peak is truncated to remove the overlapping austenite 111 peak. Although the peaks{ }are indexed with Miller indices hkl, each actually comprises{ }merged reflections of all permutations of hkl, for which the d-spacings differ slightly. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 143

supports the conclusion that the martensite texture does not change. The unswaged data is more noisy, but the curves again follow one another reasonably closely. If full transformation was achieved then the texture would be expected to change, because the transformation of unfavourably oriented austenite grains would give rise to martensite of diﬀerent orientations. It appears that the extent of transformation is not great enough for this to occur. The axial plots of fractional intensities, Figs. 5.19 (e) and (g), may be used to assess whether the proportion of the minor 211 orientation remains constant with applied stress. The 211 curves do indeed follow the 110 curves closely (although again, the unswaged data is more noisy). It may be expected that the emergence of the 211 peak would lag behind that of the 110 peak, since the latter is clearly the more preferred orientation. However, it appears that up to the strains achieved, the proportions of the two martensite grain families remain approximately constant.

5.7.3 Changes in Austenite Texture Accompanying the stress-induced transformation, the austenite texture is expected to change. As noted previously, two factors influence this process: the transformation itself, which selec- tively removes grains from the austenite population, and plastic deformation, which gradually causes grains to rotate relative to the external reference frame. Through comparison of the unswaged and hot swaged spectra and with reference to the texture measurement presented in 5.4, it is however possible to separate the influence of the two contributions and hence determine§ which orientations of austenite grain are favoured for transformation. The integrated intensities of austenite peaks are plotted as functions of applied stress in Fig. 5.20. The plastic strain is also indicated, in the non-linear scale at the top of each plot. For comparison between different grain families, the intensities are scaled relative to their values in the initial, unloaded spectra.

Unswaged Material

Consider first the variation of intensities in the unswaged axial spectra (Fig. 5.20a). The intensities remain essentially unchanged until macroscopic yielding occurs. Immediately after yielding, the intensities begin to change. The 111 reflection increases strongly throughout deformation, until at the highest applied stress its intensity has grown almost by a factor of 3. Since martensitic transformation removes grains from the austenite population, it cannot account for such an increase in intensity. There- fore the growth may be attributed fully to the effect of grain rotation. This agrees with the texture measurement of Fig. 5.16, which shows that a strong component of 111 fibre texture develops when transformation is inhibited. As discussed earlier, the initial hot swaged spectra also demonstrate this. The 220 reflection weakens considerably after yielding. The Bain correspondence discussed in 2.8 does not, however, predict that austenite grains with 220 parallel to the tensile axis transform§ into martensite grains with 110 parallel. Since hthisiis the commonly observed martensite orientation, the weakening ofh austenitei 220 is not attributed to transformation. However, Fig. 5.16 and the initial hot swaged spectra again confirm that the change in intensity may be attributed to grain rotation alone: during plastic deformation, the 110 texture component is seen to disappear parallel to the tensile axis. This corresponds to the alignment of all grains such that either 111 or 100 lies parallel to the axis. h i h i The 200 reflection shows the most interesting response. After yielding, the intensity initially increases. This is expected from grain rotation. However, at about 400MPa, the intensity stops growing, and at higher applied stress begins to fall. It is also at about 400MPa that the martensite peak intensities become significant, as seen in Fig. 5.19. It thus appears that two competing mechanisms are active. The growth of the 200 intensity due to grain rotation is counteracted Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 144

Plastic strain [%]: 0.2 1.4 4.6 8.8 14.2 18.7 25.2 Plastic strain [%]: 0.2 1.4 4.6 8.8 14.2 18.7 25.2 3 1.2 111 220 1.1 2.5 1

0.2% yield stress 0.9 2 initial initial 0.8

1.5 0.7 331 200 0.2% yield stress 311 0.6 420 422 422 Intensity/Intensity 1 Intensity/Intensity 311 0.5 200

0.4 0.5 111 331 0.3 420 220 0 0.2 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 Applied Stress [MPa] Applied Stress [MPa] (a) Unswaged, axial (b) Unswaged, transverse

Plastic strain [%]: 0 12.3 17.8 21.8 26.1 Plastic strain [%]: 0 12.3 17.8 21.8 26.1 1.4 1.4

1.2 1.2

1 1 initial initial 0.8 111 0.8

331 111 0.6 upper yield stress 0.6 upper yield stress 420 220 Intensity/Intensity Intensity/Intensity 311 0.4 311 0.4

331 420 0.2 0.2 200 200

0 0 0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 1800 Applied Stress [MPa] Applied Stress [MPa] (c) Hot swaged, axial (d) Hot swaged, transverse

Figure 5.20: Fractional change in austenite integrated peak intensities with stress (plastic strain is also indicated on a non-linear scale). Background is removed by linear interpolation between the tails of each peak, except in the case of the 111 peak in the hot swaged spectra. Due to the strong overlap of these peaks with martensite{ } 110 , they are truncated and no background correction is made. However, 111 is an intense{ }reﬂection for which the peak-to-background ratio is high. { } Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 145

by the removal of austenite grains from the family, due to stress-induced transformation. As discussed in 5.8, this is consistent with the crystallography of the transformation. § The other reflections which are plotted show responses within the extremes described above. The 331 and 420 reflections weaken to a similar extent to the 220, while the integrated intensities of the 311 and 422 reflections remain approximately constant with applied stress / plastic strain. The axial hot swaged spectrum before testing (Fig. 5.18b) reveals a significant difference between these two pairs of families. While the 311 and 422 reflections are clearly seen in the spectrum, 331 and 420 are absent. Thus the intensities of the latter pair are reduced to a greater extent through grain rotation, explaining the greater weakening observed during tensile loading. In the unswaged transverse spectra (Fig. 5.20b), the intensities of all reflections drop by a similar proportion, with the exception of 220, which grows slightly. The lack of diversity between reflections arises because grains with equivalent orientations relative to the tensile axis may contribute to many different transverse reflections: e.g. a grain with 110 parallel to the tensile axis may contribute to any one of the 111, 200, 220, 311, 331 or 422hreflections.i The 220 reflection is exceptional because it is the only transverse reflection to which grains having 111 parallel to the tensile axis contribute; since this orientation becomes increasingly commonh asi deformation progresses, the 220 reflection grows in intensity, while all the others weaken.

Hot Swaged Material Consider now the intensity changes in the hot swaged material. Since this material is highly plastically deformed prior to testing, further grain rotation is inhibited. Therefore, the intensity changes relate more directly to the stress-induced transformation, allowing an unambiguous interpretation. There is, however, a smaller range of original austenite orientations than in the unswaged material. As in the case of the unswaged material, there are no significant changes in intensity until macroscopic yielding. After Luders¨ band propagation, the intensities of all the austenite peaks fall sharply, both in the axial and transverse directions. This indicates the loss of a substantial volume fraction of austenite, and the corresponding growth of a substantial amount of martensite. The actual volume fraction is quantified by Rietveld refinement in 5.10. § In the axial spectra, the greatest reduction in intensity is that of the 200 reflection, which drops by 85% between the initial and final measurements. The reflection which changes least is 111. It is difficult to obtain a reliable value of the integrated intensity of the 111 peak, because it overlaps strongly with the martensite 110 reflection, but it is clear even from visual inspection that the intensity is not reduced by any great extent. The reductions in intensity of the 311, 331 and 420 reflections are intermediate between the extremes set by 200 and 111. These observations agree well with the conclusions drawn from the unswaged material. In that material, grain rotation plays an important role, but the onset of preferential transformation is seen in the 200 grain family. In the hot swaged material, this is seen unambiguously. There is no evidence that the intensity of the 111 reflection is diminished by transformation in the unswaged material, and in the hot swaged material it is seen that this grain family is least diminished by transformation. Again, the transverse data simply help to corroborate the conclusions drawn from the axial data. The 200 and 420 grain families are diminished to the greatest extent. These families include grains with 200 parallel to the tensile axis, but not grains with 111 parallel. Although 220 includes grainsh withi 200 parallel to the tensile axis, it also includesh i grains with 111 parallel, and hence is diminishedh i to a lesser extent. h i

5.7.4 Summary In this section, the diﬀraction spectra obtained during tensile testing of TRIP steel have been discussed and quantitatively analysed for the purpose of understanding texture evolution in the Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 146

parent austenite and stress-induced martensite phases. While grain rotation plays a major role in the development of texture in the unswaged material, evidence is also seen of transformation of preferentially oriented austenite grains. This mechanism is dominant in the hot swaged material. The observations from both materials indicate that austenite grains oriented with 100 parallel to the tensile axis are preferentially transformed to martensite, while grains having h111i parallel to the tensile axis are the least optimally oriented for transformation. Martensite grainsh i with 110 lying parallel to the tensile axis are formed preferentially. In the following section, it ishshowni that this is consistent with the crystallography of the transformation.

5.8 Orientation Dependence of Transformation

In this section, an explanation is forwarded for the observed trend of preferential transformation to martensite of certain well-oriented austenite grains, following the inﬁnitesimal deformation approach described in 2.8.2. As observed in the previous section, austenite grains with 100 parallel to the tensile axis§ transform preferentially, while those having 111 parallel tendh toi remain untransformed. h i As discussed in 2.8.2, a twinned martensite plate may form without internal stress generation. The eigenstrain§ of the combined plate is solved as

0 0 0 = 0 a + c 0 (5.2) ij∗   0 0 a   where the Bain strain components are written in terms of the stress-free lattice parameters of austenite, aγ, and martensite, aα0 , cα0 , as

√2aα0 aγ cα0 aγ a = − ; c = − . (5.3) aγ aγ

In 5.9, estimates of the transformation strains determined by Rietveld reﬁnement of the neutron diﬀraction§ spectra are given as

a = 0.13 ; c = 0.20 . (5.4) −

Note that a+c < 0 and a > 0. For this reason, the following shorthand notation is introduced, in which (5.2) is written as = diag (0, , ) . (5.5) ij∗ ⊕ The diag notation is used to represent a diagonalised tensor, the diagonal elements being given inside the brackets, and the symbols

= a + c and = a (5.6) ⊕ are used to indicate that the elements have negative and positive signs respectively. The eigenstrains of other stress-free combined martensite variants are obtained by permutation of the diagonal elements in (5.5). For tensile directions li which lie in the standard crystallographic triangle, i.e. l3 l1 l2 0, the transformation strain resolved along the tensile axis is maximised by the varian≥ t≥whose≥ eigenstrain is written explicitly in (5.5). Resolving along the tensile axis, the elongation strain is given by

elong 2 2 = lilj = l + l . (5.7) ij∗ 2 3 ⊕ Within the standard triangle, this is maximised by l = [001], and minimised by l = 1 [111]. i i √3 For an applied uniaxial tensile stress of σA, assuming the material is elastically isotropic, the Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 147

change in mechanical potential energy per unit volume of plate is given by

P = σAelong . (5.8) − This is the only term contributing to the Gibbs free energy which varies with the orientation of the tensile axis. Hence the Gibbs free energy is minimised when the elongation is longest. That is, transformation is energetically most favoured when the tensile axis is parallel to 100 , and least favoured when it is parallel to 111 , as observed. It should be noted, however,hthatisince > , there always exists a martensiteh i variant whose free energy is lowered by uniaxial |tensile⊕ | stress,| | whatever the austenite orientation. The analysis also explains the emergence of the 110 martensite peak. An austenite grain having [001] parallel to the tensile axis transforms into a martensite plate consisting of twinned variants with eigenstrains diag (c, a, a) and diag (a, c, a) respectively (such that both cause elongation a along the tensile axis). According to the Bain correspondence, both variants have a 110 martensite plane normal aligned with the tensile axis, explaining the observed diﬀraction{ p}eak. This is illustrated in Fig. 5.21.

Stress axis

BCV (1)

2 3

BCV (2)

Figure 5.21: Illustration of the changes in crystal structure to form the two Bain Corre- spondence Variants (BCVs) favoured by tensile stress along austenite [001] (the 3-axis). BCV(1) is formed by contraction along the 1-axis and uniform dilation perpendicular to this axis; similarly, BCV(2) is formed by contraction along and dilation perpendicular to the 2-axis. In both cases, neglecting rotation, a martensite 110 plane normal lies parallel to the tensile axis.

Strictly, this argument neglects the lattice rotation required for compatibility. As is now shown, this rotation is small. When twins form, they must rotate relative to one another in order that they share a common interface plane. However, the axis of rotation is austenite [001]. This simply causes rotation of the martensite 110 plane about its normal, and does not alter its orientation with respect to the tensile axis.{ This} is illustrated in Fig. 5.22. Another rotation of the entire twinned plate is also required, such that the habit plane remains unrotated. Recalling equation (2.104), the habit plane normal is

n = 0, 1 + a/c, a/c . (5.9) − h p p i This is perpendicular to austenite [100], which therefore lies within the habit plane. This is to be expected, since the eigenstrain (5.2) has no component along the 1-axis, which is therefore an undistorted direction. Moreover, the eigenstrain causes no rotation of this direction. This Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 148

BCV(1)

θ 2

3 1 θ

BCV(2)

Figure 5.22: Illustration of rotation of BCV twins to share a common interface plane. A cube of austenite is shown, looking down the 3-axis (tensile axis). The white region transforms to BCV(1), the grey region to BCV(2). The twins must rotate about the 3-axis in opposite senses in order to share a common interface plane. This leaves unchanged the crystallographic direction lying parallel to the 3-axis.

implies that h(1) = [100] is the axis about which the plate is rotated. The amount of rotation can be found from the change in direction of the habit plane vector h(2) lying perpendicular to h(1), i.e. (2) (1) h = h n = 0, a/c, 1 + a/c . (5.10) × − − h p p i The eigenstrain ij∗ changes this vector to

(2) (2) (2) (2) h h 0 = h + h (5.11) i −→ i i ij∗ j (2) (note that h is rotated but unchanged in length). Using the values of a and c given in (5.4), (2) (2) the angle between h and h 0 is evaluated as 5.4◦. Since austenite [001] is also perpendicular to the rotation axis h(1), this is also the angle through which the martensite 110 plane normal is rotated away from the tensile axis. Since it is smaller than the scattering{ v}ector range of the detectors used in the diffraction experiment, the martensite 110 diffraction peak remains strong. Furthermore, the rotation brings the 110 plane normal closer to the tensile axis in some of the grains which are oriented close to {the optimal} orientation. In the above analysis, elastic isotropy is assumed, such that all grains experience the same uniaxial stress. In fact, as discussed many times in Chapter 4, due to elastic anisotropy the stress in a grain is dependent upon its orientation. Since 100 and 111 are respectively the elastically most compliant and stiffest directions in austenite,h thei effecth ofi elastic anisotropy is to counteract the above effect of preferential transformation. Grains with 111 parallel to the tensile axis require the greatest stress for transformation, but in the elastich regimei they also bear the greatest stress. This may have some significance in the deformation of superelastic materials. However, in TRIP steel, the transformation accompanies plastic deformation by slip. Using EPSC modelling, it is demonstrated in 5.13 that slip tends to increase the stress in grains aligned with 100 parallel to the tensile axis,§ since these are among the last to yield. Thus it is seen that theh effecti of plastic slip is to encourage further the preferential transformation of such grains. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 149

Right Left Unswaged austenite 3.588968 0.000046 3.587884 0.00006 Unswaged martensite 2.860291 0.00224 2.859887 0.002113 Hot swaged austenite 3.590433 0.000058 3.588252 0.00007 Hot swaged martensite 2.871937 0.000596 2.853234 0.001533 Table 5.2: Lattice parameters in unswaged and hot swaged materials, as determined by Rietveld refinement of the spectra recorded in the right and left detector banks. The austenite lattice parameters are determined from the spectra recorded prior to loading. The martensite lattice parameters (refined using the constraint c = a, due to the difficulty of resolving martensite tetragonality) are determined from spectra recorded in the absence of load after plastic straining.

5.9 Rietveld Reﬁnement

The following sections return to the analysis of the neutron diffraction spectra. Rietveld refinement of the spectra enables the determination of not only volume averaged phase strains, but also the volume fraction of the evolving martensite phase. However, certain issues arise in the Rietveld analysis. Firstly, the material evolves from a single phase to a two phase material. Since no martensite exists in the initial, unstressed state, stress-free lattice parameters cannot be determined for this phase. Moreover, since the evolution is gradual in the case of the unswaged material, some measurements are made when the martensite volume fraction is very small, limiting the accuracy of lattice parameter determination. This is also affected by the strong texture of the evolving phase. As discussed earlier, the only strong martensite reflection in the axial spectra is 110, which strongly overlaps with austenite 111. Rietveld refinement can often be used to reliably determine phase strains from diffraction spectra with overlapping peaks, since the fitting method takes account of all reflections. However, if a phase produces only one strong reflection, a systematic error may occur. The austenite and transverse martensite lattice parameters should not be strongly affected by this, since they are determined from multiple reflections. The texture of each phase is fitted in the Rietveld refinement, using a spherical harmonic model. A further issue arises because the peak splitting due to martensite tetragonality is not resolvable. This means it is not possible to distinguish, and thus reliably determine, both the a and c lattice parameters of the martensite phase. Therefore, the constraint c = a is imposed, effectively neglecting tetragonality and treating the martensite as body-centred cubic. Using this refinement strategy, lattice parameters determined for austenite and martensite from both the unswaged and hot swaged spectra are shown in Table 5.2. These lattice parameters give estimates of the transformation strain components (equations (5.3)) of

a = 0.13 0.01 ; c = 0.20 0.01 . (5.12) − where the uncertainties correspond to the variation in values determined from diﬀerent spectra (unswaged and swaged, axial and transverse). These are only approximate estimates, owing to the points made above, i.e. tetragonality is neglected and the martensite lattice parameter is not determined under stress-free conditions. Nevertheless, the values are certainly reasonable: Nishiyama [41] quotes values of a = 0.12 and c = 0.17 for a similar steel, of composition Fe- 22%Ni-0.8%C. The values given in (5.12) are only used− explicitly in 5.8 for the determination of the habit plane rotation; slight discrepancies in the values do not§alter the conclusion that this rotation is small. The following two sections present the data determined from Rietveld reﬁnement regarding martensite volume fraction and elastic phase strains. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 150

5.10 Martensite Volume Fraction

The development of martensite has already been discussed in 5.7.2. In this section, the phase volume fraction is quantiﬁed using Rietveld reﬁnement. Firstly, this method is validated by comparison to measurements made metallographically. The Rietveld-determined volume fractions are used in later sections for stress balance calculations.

Validation of Rietveld-Determined Volume Fractions Rietveld refinement provides an alternative measure of volume fraction to the metallographic method described in 5.3.4. By comparing results made using the two methods on the same sample, the reliability§ of the Rietveld method may be assessed. For this purpose, metallographic measurements were made on the neutron-tested unswaged sample, according to the method described in 5.3.4. The metallographic measurements were made using transverse and longitudinal sections§taken from the gauge length of the sample, away from the neck which developed just prior to failure. In Table 5.3, the results are compared to the values determined by Rietveld refinement of the final spectrum recorded for the sample (note that although the word “transverse” is used to describe both sections and spectra, there is not expected to be any more correlation between the values determined from transverse sections and the transverse spectrum than that between the values determined from transverse sections and the axial spectrum). The values determined from the two methods are in reasonable agreement. The metallographically-determined values are slightly lower than those determined by Rietveld refinement; this may be due to the fact that the micrographs were taken in the centre of the sample sections, and the volume fraction was observed to be greater in the vicinity of the sample surface. However, the agreement provides validation that good estimates of the phase volume fractions may be determined using Rietveld refinement. It is also encouraging that the values determined by refinement of the axial and transverse spectra agree very closely.

Metallographic measurements: Longitudinal sections 12.2 +/- 0.4 Transverse sections 14.4 +/- 2.1 Rietveld measurements: Axial spectrum 15.2 +/- 0.7 Transverse spectrum 15.5 +/- 1.4

Table 5.3: Measurements of martensite volume fraction (in %) in unswaged sample, made metallographically and by Rietveld reﬁnement.

Comparison of Constant Strain Rate and Neutron Tests In Fig. 5.23, the values of martensite volume fraction determined by Rietveld refinement of the axial and transverse diffraction spectra are plotted against plastic strain. For comparison, the metallographically-determined fractions developed under constant strain rate conditions at -20◦C are reproduced from Fig. 5.11. The values determined from the axial and transverse spectra are in excellent agreement. There are, however, substantial differences between the constant strain-rate and neutron-test curves. While martensite is detectable in the diffraction spectra directly after the onset of plasticity, none is visible in the optical micrographs up to 14% plastic strain. This may indicate strain rate dependence, but it is difficult to make a definite judgement about this because the sample tested under neutrons was subjected to a 7 1 variable strain rate: very slow during data collection ( 10− s− ), and relatively fast between 3 1 ∼ measurements (up to 5 10− s− ). The curves may also differ due to differences in sample temperature. The accuracy∼ × of the sample temperature during the constant strain rate deformation is questionable, because in the set-up used, the thermocouple does not make thermal contact with the sample. Adiabatic heating may increase the sample temperature above that Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 151

of the surroundings; this would act to inhibit transformation and reduce the martensite volume fraction. Note that the neutron-tested sample failed at a much lower plastic strain than the sample deformed at constant strain rate, even though the results indicate that the former developed martensite more rapidly with plastic strain. This would be expected to enhance work hardening. However, other factors associated with holding at load (such as ageing) may contribute to early failure.

Constant strain rate, longitudinal section Constant strain rate, transverse section 0.5 Rietveld, axial spectrum Rietveld, transverse spectrum

0.4

0.3

0.2 Martensite volume fraction

0.1

0 0 10 20 30 40 50 60 70 Plastic strain [%]

Figure 5.23: Variation of martensite volume fraction with plastic strain in unswaged material (tensile deformation at -20◦ C). The constant strain rate curves were determined using the metallographic method described in 5.3.4; the neutron test curves are determined from Rietveld reﬁnement of neutron diﬀraction§ spectra.

Rietveld-Determined Volume Fractions The Rietveld-determined martensite volume fractions determined both from the axial and transverse spectra are plotted in Fig. 5.24, for both the unswaged and hot swaged materials. Since the intensities of particular martensite reflections differ greatly in the axial and transverse directions, comparison of the curves in the two directions gives a further indication of the reliability of the Rietveld-determined values. As noted above, the curves agree well in the unswaged case. However, there is some discrepancy in the hot swaged case. This is possibly due to the failure of the refinements to fully capture the martensite peak shape, and thus intensity. Nevertheless, some very clear observations can be made from Fig. 5.24. The maximum martensite volume fraction achieved in the unswaged material is only 15%, but a much greater volume fraction is produced in the hot swaged material, exceeding 50%. In the unswaged material, the volume fraction grows approximately linearly with plastic strain. This appears also to be the case in the hot swaged material, although no measurements are made in the region of heterogeneous deformation up to 12% plastic strain.

5.11 Elastic Phase Strains

The difficulties in extracting reliable martensite strains by Rietveld refinement have been discussed in 5.9. Due to these problems, the main discussion of interphase stress development in this section§ focuses on data acquired for the austenite phase. Therefore, this data is presented first. The martensite data is then discussed on the basis that it provides supplementary information. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 152

0.7

Axial 0.6 Transverse Hot swaged

0.5

0.4

0.3

Martensite volume fraction 0.2 Unswaged

0.1

0 0 5 10 15 20 25 30 Plastic strain [%]

Figure 5.24: Martensite volume fraction as a function of plastic strain, determined by Rietveld reﬁnement of both axial and transverse diﬀraction spectra, for both the unswaged and hot swaged materials.

5.11.1 Elastic Strain in Austenite The Rietveld-determined austenite lattice strains for the unswaged and hot swaged materials are presented in Figs. 5.25a and 5.25c respectively, as functions of applied stress. Since martensite development has an important inﬂuence on the evolution of lattice strain, plots of martensite volume fractions are also presented alongside the corresponding lattice strain plots (Figs. 5.25b and 5.25d), but as functions of applied stress rather than plastic strain (as was plotted in Fig. 5.24).

Unswaged Material In the unswaged material, the axial elastic strain response remains very nearly linear up to and beyond the yield point. The gradient corresponds to a Young’s modulus of 190GPa. The lack of any deviation at yield is reasonable, since the material remains essentially a single phase up to stresses much greater than the yield stress, as seen from the plot of martensite content in Fig. 5.25b. While the material remains in a single phase, this must bear the entire load; hence the linear response. However, as the martensite volume fraction becomes significant, the austenite strain deviates towards lower lattice strain. This indicates the development of axially compressive internal stress in austenite, balanced by tensile stress in martensite. A clear, linear transverse response is also seen in the elastic regime. The gradient corresponds to a Poisson’s ratio of 0.25, which is slightly low for steel. Upon yielding, there does appear to be a slight shift in the transverse response. As mentioned in 4.5, such shifts have previously been reported in single phase materials, and attributed to regions§ of non-diffracting material which bear increased load. However, since the shift is not detected in the axial data, it is difficult to ascertain whether this is the actual cause. Certainly, however, the deviation from the linear elastic response increases as martensite is formed, as also observed in the axial data.

Hot Swaged Material The trends in austenite lattice strain data collected from the hot swaged material are very clear. The elastic responses are well characterised. The slope of the axial data corresponds to a Young’s modulus of 190GPa. This is the same value determined for the unswaged material; Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 153

Transverse Axial 900 900

800 800

700 700

600 600

500 500

400 400

Applied Stress [MPa] 300 300 0.2% yield stress Applied Stress [MPa] 0.2% yield stress

200 200 Axial 100 100 Transverse

0 −1000 0 1000 2000 3000 4000 5000 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 6 Austenite lattice strain × 10 Martensite volume fraction (a) (b)

Transverse Axial 1600 1600

1400 1400

1200 Upper yield stress 1200 Upper yield stress 1000 1000

800 800

600 600 Applied Stress [MPa] Applied Stress [MPa] 400 400 Axial 200 Transverse 200

0 −2000 0 2000 4000 6000 8000 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 6 Austenite lattice strain × 10 Martensite volume fraction (c) (d)

Figure 5.25: Evolution of Rietveld-determined austenite elastic phase strain and martensite volume fraction with applied stress. (a) and (b): unswaged sample; (c) and (d): hot swaged sample. The dashed lines passing through datapoints are ﬁts through the initial linear, elastic responses. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 154

thus the development of strong texture has no detectable influence on the bulk elastic stiffness. This is reasonable, since although the majority of grains are now oriented with 111 , the stiffest crystallographic direction, aligned along the tensile axis, there is also a significanh ti minority of grains oriented with 100 , the most compliant direction, aligned along the axis. The gradient of theh transvi erse linear elastic response corresponds to a Poisson’s ratio of 0.28. This is very reasonable for steel, and is slightly greater than the value determined for the unswaged material. Beyond the yield point, the change in the responses is dramatic. Both the axial and transverse curves become almost vertical. After yielding and Luders¨ band propagation, an additional 500MPa is applied, but the austenite barely carries any more load at the maximum stress than at yield. In order for the austenite to bear less than the applied stress, another phase must bear more. The volume fraction plot shows that between the measurements taken just before and just after Luders¨ band propagation, the martensite content increases from complete absence to a volume fraction of approximately 30-40%. The martensite content continues to grow as deformation proceeds. This provides clear evidence that martensite bears the extra load, exerting back stress on the austenite and thus acting as a reinforcing phase.

Residual Strains Further evidence of the reinforcing role of martensite is seen in the austenite residual strains. These are plotted against macroscopic plastic strain for the unswaged and hot swaged materials in Figs. 5.26a and 5.26b respectively. In the case of the unswaged material, the residual strain measurements were made on a diﬀerent sample to the at-load measurements, but in the case of the hot swaged material, both sets of measurements were made on the same sample, by performing a series of unloads during tensile testing. The variation of martensite volume fraction is also represented on the graphs. The residual strains which develop in the hot swaged material are of much greater magnitudes than in the unswaged material, but the trends in both are similar. As the martensite volume fraction grows, the austenite develops a compressive residual strain in the axial direction, and tensile residual strain in the transverse direction. This is in agreement with the at-load results, showing that the martensite exerts a component of back stress on the austenite, thereby reinforcing it. Since the martensite volume fraction is greater in the hot swaged than unswaged material, so too is the back stress exerted on the austenite. Hence the greater residual strain magnitudes. In both materials, the back stress continues to increase with plastic strain.

1000 300 0.18 Axial Axial Transverse Transverse Martensite vol. frac. 0.16 Martensite vol. frac. 200 0.5 500 0.14 6 6 100 10 10

× 0.4

× 0.12 0 0 0.1 0.3

−100 0.08 −500

0.06 0.2 Martensite volume fraction

−200 Residual lattice strain Martensite volume fraction Residual lattice strain

0.04 −1000 0.1 −300 0.02

−400 0 −1500 0 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 Plastic strain [%] Plastic strain [%] (a) Unswaged (b) Hot swaged

Figure 5.26: Evolution of Rietveld-determined austenite residual lattice strain and martensite volume fraction with plastic strain. The volume fraction datapoints show the means of the values determine from the axial and transverse spectra. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 155

5.11.2 Elastic Strain in Martensite

As noted, the reliability of martensite axial lattice strains determined by Rietveld refinement is questionable, since the one prominent martensite peak in the axial diffraction spectrum strongly overlaps an austenite reflection. Furthermore, no stress-free lattice parameter is available. To assess reliability and overcome the lack of a reference lattice parameter, martensite strains (both axial and transverse) are calculated using the principle of stress balance. It is assumed that the phases have the same isotropic elastic properties, determined from the austenite response in the elastic regime. The strains are calculated using equations (4.2) and (4.3), taking the martensite volume fraction as the mean of the values determined from the axial and transverse spectra. Strains calculated in this way are plotted versus applied stress in Fig. 5.27. The Rietveld- determined martensite strains are also shown. To compensate for the lack of a reference lattice parameter, these strains are calculated in the following manner. At the lowest stress at which martensite is detected, the strain is assumed to be equal to that calculated by stress balance. Subsequent strains are calculated relative to this from the shifts in the lattice parameter, as determined by Rietveld refinement. The reliability of these experimental strains can then be assessed from the extent of divergence of the experimental and calculated curves.

Hot Swaged Material

The hot swaged data (Fig. 5.27b) is addressed first, since the greater martensite volume allows more accurate strain determination. It is seen that the calculated and Rietveld-determined axial curves do not agree well. The Rietveld-determined strain grows more rapidly with applied stress than the calculated strain. However, the transverse strains determined by the two methods agree very well. This suggests that the strains calculated by stress balance are realistic, and that Rietveld refinement of the axial data gives rise to systematic error for the reason of peak overlap explained earlier. Refinement of transverse spectra does not suffer from this problem because there is a greater number of strong martensite reflections. It is interesting to note that both the inferred axial and transverse curves run parallel to the respective initial austenite linear, elastic responses. This indicates that the average residual stress in the martensite phase does not increase with applied stress. Load transfer away from austenite can of course continue, because the martensite volume fraction continues to grow.

Unswaged Material

In the unswaged case, the first few calculated values of axial strain are very unreliable, since they are extremely sensitive to the deviation of the austenite response from linearity, and to the martensite volume fraction, both of which are very small. Since the first part of the calculated curve is not reliable, it is difficult to assess from it the reliability of the overall experimental curve. There is, however, little similarity between the axial calculated and experimental curves, once more suggesting that the Rietveld refinement suffers from systematic error. As in the hot swaged case, the agreement of the transverse strain curves is much better (the curves begin at higher stress, because a stable refinement of the martensite lattice parameter could not be achieved at smaller volume fractions). There is an apparent positive shift in transverse strain as the applied stress increases, suggesting the average martensite residual stress falls. This is possible, due to the growing volume fraction of martensite, but does not seem consistent with the calculated axial strain, which increases, implying increasing residual stress. In summary, firm conclusions about the martensite internal stress state are difficult to draw from the unswaged dataset, due to the small martensite volume fraction. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 156

Transverse Axial Transverse Axial 900 1600

800 1400 700 1200 600

1000 500

800 400

600

Applied Stress [MPa] 300 Applied Stress [MPa]

200 400 Austenite Austenite Martensite calculated Martensite calculated 100 Martensite fitted Martensite fitted 200

0 −4000 −2000 0 2000 4000 6000 8000 10000 0 −2000 0 2000 4000 6000 8000 10000 12000 Lattice strain × 106 Lattice strain × 106 (a) Unswaged (b) Hot swaged

Figure 5.27: Evolution of austenite and martensite lattice strains with applied stress. The experimental strains are determined from Rietveld reﬁnement. The calculated martensite strains are determined from the experimental austenite strains, using the principle of stress balance and the Rietveld-determined martensite volume fraction (the mean of the axial and transverse values is taken). Since no stress free lattice parameter is determined for martensite, the experimental martensite strain at lowest applied stress is chosen to coincide with the calculated value, and subsequent strains are calculated relative to this value.

Residual Strains

Corroboratory information is once more available from plots of residual strain versus macroscopic plastic strain. The strategy adopted above is again employed, i.e. calculating the martensite strains using stress balance, and choosing the first experimental strain to coincide with the first calculated value. The results are shown in Fig. 5.28. Again, the hot swaged dataset (Fig. 5.28b) is considered first. As seen in the at-load strains, the axial fitted strains do not match the calculated values, while the transverse curves are in excellent agreement. This supports the argument that the axial fitted strains are subject to a systematic error. The inferred axial and transverse strains hardly change at all with increasing plastic strain. This is in good agreement with the conclusion drawn from the at-load data, that the average martensite residual stress remains approximately constant. The unswaged transverse curves (Fig. 5.28a) are also in good agreement. These indicate that the average residual stress in the martensite phase also remains almost constant in the unswaged material. However, the magnitude of the residual strain is somewhat larger than in the hot swaged material: approximately 2000µe in comparison to 1000µe. Also, a slight fall is evident, in agreement with the observation made from the at-load data. Although noisy, the axial calculated curve is reasonably consistent with constancy of residual stress. Once more, the axial experimental data is not in agreement and appears unreliable.

5.11.3 Discussion and Comparison with High Carbon Steel

The results presented in the above sections provide experimental validation that the development of stress-induced martensite in TRIP steel contributes an additional source of work hardening, helping to stabilise plastic ﬂow and thus contributing to the TRIP eﬀect. Dur- ing uniaxial tension, the evolving martensite phase develops on average axially tensile residual stress, balanced by axially compressive residual stress in the austenite matrix. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 157

10000 6000 Calculated Fitted Calculated 5000 8000 Fitted

4000 6 6000 6 10 10 × × 3000 4000

2000

2000 Axial

1000 Axial

0 Residual lattice strain Residual lattice strain 0

−2000 −1000 Transverse −4000 −2000 Transverse 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Plastic strain [%] Plastic strain [%] (a) Unswaged (b) Hot swaged

Figure 5.28: Evolution of martensite residual lattice strains with macroscopic plastic strain. The calculated martensite strains are determined from the experimental austenite strains, using the principle of stress balance and the Rietveld-determined martensite volume fraction (taking the mean of the axial and transverse values). The experimental strains are determined from Rietveld reﬁnement. Since no stress free lattice parameter is available for martensite, the experimental strain at lowest plastic strain is chosen to coincide with the calculated value, and subsequent strains are calculated relative to this value.

Sources of Misfit Assuming the phases to have similar elastic constants, there are two possible sources of misfit which may lead to internal stress development. The transformation strain associated with the phase change is one such source. However, as discussed in 2.8.2 and 5.8, the martensite plate morphology develops so as to accommodate the phase transformation§ § strain in a manner that minimises internal stress. In fact, it has been demonstrated for an elastically isotropic material that a plate can form without the generation of internal stress. The search for stress-free plate morphologies forms the basis of the infinitesimal deformation theory approach to martensitic transformation. As discussed in 2.8.2, the phenomenological theory of Weschler, Lieberman and Read [82] and Bowles and Mac§ kenzie [83, 84] instead seeks to find crystallographic planes which remain invariant under transformation. However, this amounts to the same assumption, as seen from the convergence of the solutions provided by both theories when the transformation strains become small. It is reasonable therefore to attribute the internal stress generation primarily to the second source of misfit: that due to inhomogeneous plastic flow. As noted in 5.1.2, the yield stress of martensite is substantially higher than that of austenite. Therefore§greater plastic flow is expected in austenite than within martensite, generating back stress in the matrix. Thus the mechanism of reinforcement provided by martensite is similar to that provided by cementite in high carbon (HC) steel. There are of course differences, such as different inclusion shape and variable reinforcement volume fraction.

Eﬀect of Changing Reinforcement Content In the HC steel, the average residual stress in each phase was observed to saturate after a few percent plastic strain. In the TRIP steel, the back stress in austenite continues to increase approximately linearly beyond plastic strains of 25%. However, the evidence suggests that the average martensite residual stress remains approximately constant. Consistency with stress balance is possible because the martensite volume fraction increases. The changing reinforcement content adds considerable complexity to the system. Whereas it may be assumed that Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 158

the misfit is similar for all inclusions in HC steel, this may not be assumed of the martensite plates in TRIP steel, because these are formed at different stages throughout the deformation. The misfit depends not on the total plastic strain developed but that which has developed since a plate was formed. If transformation indeed causes minimal internal stress, then on average a plate forms with the average residual stress of the matrix, i.e. axially compressive. Thus newly formed plates will tend to lower the average residual stress in the phase. As plastic straining continues, these plates will begin to develop axially tensile residual stress. Thus there is a dynamically changing martensite plate population, in which the axial tensile residual stress in each plate tends to increase, but the phase average does not necessarily increase because of the introduction of new plates. There is also the possibility that plates grow in size after initial formation; since the stress state in the surrounding matrix varies greatly with position around the plate, it is not straightforward to predict how this affects the average phase stress, but it further demonstrates the complexity of the problem. Considering this complexity, it seems somewhat coincidental that the average martensite residual stress appears to remain approximately constant with plastic strain, and it is fair to speculate whether there is an underlying reason for this. A possible explanation is that the system has some built-in stability. For example, this might be the case if there is a limiting internal stress within a plate, above which the plastic misfit begins to relax. This would limit the extent of work hardening for a given martensite volume fraction. This may lead to plastic instability (discussed in 4.1 and 5.1.2), in turn leading to the formation of a neck. However, the increased stress and plastic§ strain§ in the necked region would induce further transformation; if great enough, this would restore plastic stability. If the internal stress in the majority of plates is close to the saturation level, then the average phase residual stress would indeed remain approximately constant, with the required work hardening being provided by the newly formed martensite.

Origin of Lüders Bands Restoration of plastic stability is certainly seen in the hot swaged material since, as discussed in 4.1 and 5.1.2, this is the origin of Luders¨ band propagation. This phenomenon is also seen in the§ HC steel.§ However, it is instructive to compare the development of internal stress during and after Luders¨ band propagation in the two materials, since there are significant differences. During Luders¨ band propagation in HC steel, the average stress in the matrix phase drops. After this initial drop, however, the ferrite matrix continues to bear additional increments of load. Conversely, in the hot swaged TRIP steel there is no evidence of a stress drop in the austenite matrix during Luders¨ band propagation. However, after yielding the austenite bears hardly any additional increments of load. The matrix stress drop in the HC steel was explained in Chapter 4 in terms of yield point softening. That is, due to softening, the ferrite matrix cannot bear directly after yielding the stress that is was able to bear directly before yielding. In order to sustain a constant applied load, the composite as a whole must work harden. This is partly achieved through hardening of the matrix itself, but partly through load transfer to the cementite inclusions. Therefore, at fixed applied stress, the matrix bears less of the applied stress after yielding than before; hence the observed matrix stress drop. The formation of Luders¨ bands in carbon steels is also associated with the softening. Recall from 4.1 and 5.1.2 that in general terms Luders¨ band propagation occurs because the developmen§t of a nec§k is arrested due to an increase in work hardening rate. At the onset of yielding in the HC steel, the rapid drop in the stress which the matrix can bear causes the overall composite to have a negative rate of strain hardening. From equation (5.1), this implies tensile plastic instability and thus leads to necking. However, the hardening which follows restores plastic stability, leading to Luders¨ band propagation. The fact that such a stress drop is not observed in the hot swaged TRIP steel suggests that matrix softening is not the basis of Luders¨ band propagation in this material. If matrix softening was partially compensated by load transfer to the reinforcing phase, then such a stress drop Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 159

would be expected. As detailed above, load transfer to the evolving martensite phase certainly does occur, but the lack of a matrix stress drop indicates that this compensates for the failure of the matrix to bear significant increments of applied stress over and above the initial yield stress, rather than compensating for matrix softening. The absence of Luders¨ bands during tensile deformation of the unswaged TRIP steel supports the assertion that matrix softening is a minor effect. Although evidence has been presented in 5.6 that suggests that some ageing and hence softening does occur in the TRIP steel, the lack§ of inhomogeneous plasticity during tensile deformation of the unswaged steel suggests that the softening is indeed minor, and that another mechanism operates to give rise to Luders¨ band propagation in the hot swaged steel. Clearly the onset of transformation provides such a mechanism. As detailed above, the formation of martensite plates leads to back stress hardening; this may arrest necking and lead to the formation of a Luders¨ band. It remains to explain why necking initiates in the hot swaged material but not in the unswaged material. A major difference between the materials is the capacity for further austenite work hardening. In the unswaged material, this capacity is great enough that the condition of plastic stability is satisfied at the beginning of plastic straining; thus the material deforms homogeneously. However, the hot swaged material has been strongly work hardened by the swaging process; hence its much greater yield stress. Thus the capacity for further work hardening is severely limited. Conventional steels processed in the same way would neck to failure directly after yielding. However, in the hot swaged TRIP steel, the onset of transformation restores stability. The lack of capacity for further austenite work hardening is evidenced in the lack of a major increase in the stress borne by the phase after initial yielding. This indicates that the hardening in the hot swaged material is almost entirely attributable to the generation of back stress. To summarise, the evidence demonstrates that the origin of Luders¨ bands in carbon steels is associated with matrix softening, but in the TRIP steel it is associated with a lack of capacity for austenite hardening, and the introduction of a new source of hardening when transformation begins. The work of Azrin et al. [160] supports the latter assertion. They measured true stress during inhomogeneous deformation of an Fe-Cr-Ni-C TRIP steel, and showed that at no stage does the bulk material exhibit a true stress drop, as it would if softening was significant. Using the phase selectivity of the neutron diffraction method, their conclusion may be extended: neither the bulk material nor the austenite phase undergoes a true stress drop.

5.12 Intergranular Strain in Austenite

The main subjects in this chapter of preferential transformation and interphase stress development in TRIP steel have now been addressed. However, the neutron diffraction experiment also provides excellent information on the development of intergranular strain in the austenite phase, which is worthwhile to present. Therefore, to complete the chapter, this subject is addressed in the final two sections before the chapter summary. Fig. 5.29 shows the development with applied stress of lattice strain in individual austenite grain families, determined by single peak fitting.

Unswaged Material Before the martensite content becomes substantial, the axial lattice strains which develop in the unswaged material (Fig. 5.29a) are typical of those which develop in a single phase austenitic steel; compare, for example, the results of Clausen et al, shown in Fig. 3.4 [104]. The 111 and 200 grain families are respectively the stiﬀest and most compliant in the elastic regime, concurring with single crystal elastic anisotropy. When plasticity begins, the 200 family develops a large tensile strain, indicating that grains in the family tend to bear increased stress as other grains begin to slip. The strains of the 220 and 331 families deviate towards lower values, indicating that these families are amongst the ﬁrst to slip, and thus shed load. As described Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 160

900 900

800 800

700 700

600 600 Rietveld 500 111 500 200 220 400 400 311 331

Applied Stress [MPa] 300

Applied Stress [MPa] 300 Rietveld 200 200 111 200 220 100 100 311 331 0 0 0 1000 2000 3000 4000 5000 6000 7000 −1500 −1000 −500 0 Lattice strain × 106 Lattice strain × 106 (a) (b)

1600 1600

1400 1400

1200 1200 Upper yield stress Upper yield stress

1000 1000

800 800

600 600 Applied Stress [MPa] Applied Stress [MPa]

400 400 Rietveld 111 Rietveld 200 200 111 200 220 200 311 0 0 2000 4000 6000 8000 10000 12000 0 −3500 −3000 −2500 −2000 −1500 −1000 −500 0 Lattice strain × 106 Lattice strain × 106 (c) (d)

Figure 5.29: Evolution of austenite grain family lattice strains with applied stress: (a) unswaged, axial; (b) unswaged, transverse; (c) hot swaged, axial; (d) hot swaged, transverse. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 161

earlier, the Rietveld response deviates away from linearity when the martensite volume fraction becomes significant. The extent to which this happens in individual grain families varies. The most noticeable deviation is that of the 200 family. As discussed in 5.7, this is also the family which is most favourably oriented for transformation. Since the §most highly stressed grains within the family will tend to transform earliest, the large deviation is perhaps partly accounted for by the selective removal of these grains, reducing the average stress of the family. The transverse strains which develop in the unswaged material (Fig. 5.29b) are also typical of those reported in austenitic steel [104, 107]. As discussed at length in Chapter 4, a transversely reflecting family contains grains of many orientations with respect to the tensile axis, and thus the lattice strain distribution within the family is relatively greater than that in an axially reflecting family. Moreover, the transverse responses are far more sensitive to crystallographic texture. As noted in 3.4.1, in fcc materials the 200 transverse response tends to be most sensitive. In this instance,§ a clear kink towards the stress axis is seen. This is typical, but Daymond et al. [107] have reported deviations of varying magnitude, depending on sample texture. The transverse data is not sufficiently accurate to draw conclusions about the influence of the stress-induced transformation.

Hot Swaged Material Single peak fits are performed only on the 111 and 200 reflections in the hot swaged axial spectra (Fig. 5.29c), since these are the only first order reflections of sufficient intensity. Although these reflections correspond respectively to the stiffest and most compliant grain families, the slopes of the elastic responses are very similar: the diffraction elastic constants are 202GPa and 170GPa for the 111 and 200 reflections respectively. In comparison, in the unswaged material the diffraction elastic constants are respectively 226GPa and 125GPa. It seems the swaging process causes the grain families to impose additional constraint on one another, with 111-type grains forced to strain more, and 200-type grains less than in the unswaged material. An explanation for this is revealed by looking once again at the microstructure of the swaged material (Fig. 5.12). The grains are highly elongated along the tensile axis. This does indeed tend to constrain the strain to be similar in all grains. To understand this note that the structure is comparable to that of an aligned fibre composite. In the extreme case of a continuous fibre composite in which the fibres are aligned along the tensile axis, the axial strains in the fibre and matrix are constrained to be equal. The same is not true of the transverse strains, however (Fig. 5.29d). Consider again the simple slab composite of Fig. 2.1, constituting slabs of two materials with different stiffnesses. The slabs are constrained to have the same axial strain, but there is no such constraint on the transverse strains. Correspondingly, the hot swaged transverse gradients differ by a much greater factor than the axial gradients. The hot swaged axial responses begin to deviate from linearity at an applied stress of approximately 800MPa. This occurs even though there is no evidence of macroscopic yielding until an applied stress of 1000MPa. This suggests that micro-yielding occurs before macroscopic yielding is detected. As in the unswaged material, during yielding the 200 grain family develops greater tensile strain, indicating redistribution of tensile stress towards this family. The 200 grain population is, of course, also greatly diminished by preferential transformation, as observed from the diffraction peak intensities in 5.7 and explained in 5.8. As noted in that section, the driving force for transformation of grains§ in the 200 family§ is increased by the stress redistribution towards this family during yielding. However, this effect alone certainly can not adequately explain the preferential transformation, since the reason that 200-type grains yield later than 111-type grains is that they bear substantially less stress in the elastic regime, due to elastic anisotropy. As noted in 3.4.1, calculations show that in fcc materials grains oriented with 200 parallel to the tensile§ axis are actually better oriented for slip than those oriented with h111i parallel, and so would yield first if the grains were elastically isotropic. As discussed in h i Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 162

4.9.3, elastic anisotropy also tends to counteract plastic anisotropy in bcc materials. § After Luders¨ band propagation, the gradients of both the 200 and 111 axial responses increase sharply, corresponding to the transfer of load to martensite, as discussed in 5.11. The Rietveld response moves closer to the 111 curve, reflecting the diminishing 200 grain§ family population. In the hot swaged material, there is a clearer correspondence between axial and transverse grain families than in the unswaged material. This is because most grains are oriented with either 111 or 200 aligned along the tensile axis. Neglecting less populated orientations, the corresphondencei hbetwieen transverse and axial grain families may be deduced. For example, if a 200 direction lies in the transverse plane, it is impossible for 111 to lie parallel to the tensile haxis;ihence 200 must lie parallel. Therefore there is a direct corresph i ondence between the axial and transverseh 200i responses. This is indeed seen in the experimental data: the transverse 200 response develops Poisson strain which is consistent with the 200 tensile strain. However, there is not such a one-to-one relationship between all axial and transverse families. Grains with 220 in the transverse plane may have either 111 or 200 lying along the tensile axis. Moreohver,ias discussed in 4.10, the linear elastic resph onsei ofh thisi particular transverse family is highly sensitive to the distribution§ of orientations relative to the tensile axis. The 111 and 311 transverse reflections do not correspond to either of the major axial grain families; for this reason they are much weaker, as evidenced in the large peak fitting uncertainties, indicated by error bars. It is the 111 response which shows the most dramatic gradient change upon yielding. However, after yielding this weak reflection overlaps the strong martensite 110 reflection, and is therefore liable to systematic error.

5.13 Elastoplastic Self-Consistent Simulation of Deformation in Austenite

In this final section prior to the chapter summary, the intergranular strain measurements presented in 5.12 are compared to predictions made using the same single phase EPSC model employed in§ Chapter 4. Up to 10% plastic strain, the austenite internal stress development in the unswaged material may be safely attributed to intergranular effects, because the martensite volume fraction is negligible. The experimental lattice strains may therefore be directly compared to the predictions of the EPSC model. In the hot swaged material, all lattice strains measured in the plastic regime are strongly influenced by the interaction between phases. How- ever, comparison of measured and predicted strains is of some interest in the elastic regime. The computer code employed in Chapter 4 is again applied here. A modification was written to improve the statistical average of the transverse response, which was shown in Chapter 4 to be very sensitive to the exact population of contributory grains. Rather than considering all grains with hkl lying within a certain angle of a specific transverse direction to contribute to the hkl reflectionh ialong that direction, the modified code considers all grains with hkl lying within a certain angle of the transverse plane to contribute to a generic transverse reflection.h i The elastic strain is resolved along this hkl direction in each reflecting grain. This dramatically increases the number of contributory hgrains,i improving the statistical average.

5.13.1 Simulation of Unswaged Material For simulation of the elastoplastic deformation of the unswaged material, the modelling strategy described in Chapter 4 is again employed. That is, the single crystal plasticity parameters (initial critical resolved shear stress and hardening parameters) are used as fitting parameters to fit the macroscopic response to the experimental flow curve. The intergranular strain predictions are then compared to the experimental data. The 12 fcc slip systems, 111 110 , are specified. Isotropic hardening of slip systems is assumed. The extended Voce hardening{ } law employed in 4.12.3 (equation (4.27)) is specified. The single crystal elastic stiffness comp onen ts are taken § Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 163

L11 L12 L44 205.0 138.0 126.0

Table 5.4: Single crystal elastic stiﬀness tensor components of stainless steel in GPa [180], used in EPSC simulation of TRIP steel.

as published values for stainless steel [180]. They are given in Table 5.4. The polycrystal is represented by a set of 1000 randomly oriented grains, which reﬂects the lack of texture in the unswaged material.

500 500

400 400

300 300

200 200 111 Applied stress [MPa] 200 Applied stress [MPa] Model 220 Experimental 311 100 100 331

0 0 0 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Macroscopic strain [%] Lattice strain [microstrain] (a) Macroscopic ﬂow curves (b) Axial lattice strains

500 500

400 400

300 300

200 200 Applied stress [MPa] Applied stress [MPa] 220 311 100 111 331 200 100

0 0 −1000 −900 −800 −700 −600 −500 −400 −300 −200 −100 0 −1000 −900 −800 −700 −600 −500 −400 −300 −200 −100 0 Lattice strain [microstrain] Lattice strain [microstrain] (c) Transverse lattice strains (d) Transverse lattice strains

Figure 5.30: Comparison of experimental data with EPSC model calculations for unswaged material.

The results of the simulation of the unswaged material are shown in Fig. 5.30. The fitted macroscopic response is shown Fig. 5.30a. Using the Voce hardening law, a very good fit is achieved. Thus we may proceed to inspection of the grain family strains. The predicted axial strains versus applied stress are shown in Fig. 5.30b together with the measured strains. The agreement is very good. The elastic responses of all grain families are well predicted. There are not many experimental datapoints in the elastic-plastic transition region, since the main motivation of the experiment was to study the stress-induced transformation, which occurs at much higher plastic strains. Nevertheless, certain features are identifiable both in the experimental Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 164

and calculated strains. Of particular note is the crossover of the 111 response with those of the 220 and 331 families. This feature is predicted in the model and seen in the experimental data. The slight deflection predicted in the 311 response is also seen in the experimental response. The details of the predicted 200 response are not seen clearly in the experimental response, but the two curves remain close both before and after yielding. The transverse lattice strain predictions are significantly poorer. Fig. 5.30c shows the most extreme transverse responses: 200 and 111. The deflection during yielding of the 200 response seen in the experimental data is repeated in the calculated response, but the final gradient of the predicted response is too great. The deflection of the 111 response is of an opposite sense in the calculated and measured curves. The predicted and measured responses coincide more in the other grain families (Fig. 5.30d), perhaps simply because these show intermediate responses which do not tend to deflect strongly in either direction. In summary, the diffraction measurements made on unswaged TRIP steel prior to the onset of stress-induced transformation may be used for the validation of the EPSC model of intergranular stress development. As reported by other research on fcc steels [104, 107], and discussed for the alternative case of a bcc material in Chapter 4, axial lattice strains are predicted to far greater accuracy than transverse lattice strains. An explanation put forward for this in Chapter 4 is that a transversely reflecting family contain grains of many orientations with respect to the tensile axis, which therefore exhibit a much wider range of behaviour than the grains in an axially reflecting family, all of which are oriented similarly with respect to the axis. This makes the average strain measured (or calculated) in a transverse family very sensitive to the exact range of grain orientations within the family. Since it is difficult to exactly simulate the true reflecting grain population, measured and predicted transverse strains may not show good agreement.

5.13.2 Simulation of Hot Swaged Material The self-consistent model simulation of the hot swaged material is restricted to the elastic regime, since the strains in the plastic regime are strongly influenced by the interaction between phases. When plasticity does not occur, the model is equivalent to the purely elastic self- consistent model, described in 2.4.3. § The purpose of the elastic simulation is to support the assertion made earlier that the elongation of grains along the tensile axis reduces the divergence of axial strains in different grain families. In all previous simulations in this dissertation, grains have been represented in the EPSC model as spheres. However, grains may be represented by any ellipsoidal shape, since the Eshelby solution predicts uniform stress in any such inclusion. Therefore, for simulation of the hot swaged material, an ellipsoid of revolution is specified, with the axis of rotation parallel to the tensile axis. The aspect ratio is used as a fitting parameter. The divergence of the grain families is well matched when the axial to transverse aspect ratio is 4. This is a reasonable estimate of the average aspect ratio, lying between estimates gained by other means. Direct inspection of the micrograph of Fig. 5.12a suggests a value somewhat greater than 4. However, another estimate may be gained by assuming the average grain has aspect ratio of 1 prior to swaging, and that during swaging its axial and transverse dimensions change by the same factors as the macroscopic sample dimensions. For a reduction of cross sectional area of 50%, this corresponds to a final aspect ratio of 2√2 3. ≈ The strains calculated in the simulation, using the fitted aspect ratio of 4, are shown against applied stress in Fig. 5.31, alongside the experimental strains. The results of two simulations are presented, differing in the grain population which is specified. The first simulation (Figs. 5.31a and 5.31b) employs a randomly oriented grain population, as in the simulation of the unswaged material. The second simulation (Figs. 5.31c and 5.31d) uses a grain population which is representative of the texture in the hot swaged material. This is generated using the diffraction spectra, by the method explained in 4.9.1. The purpose of performing both simulations is to assess the influence of the strong texture§ on the elastic responses of the hot Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 165

swaged material. When the random grain population is used, the axial strains match the experimental data very well. The simulation is directly comparable to that of the elastic regime of the unswaged material, differing only in the specified shape of the ellipsoid. The elongation of the ellipsoid does indeed reduce the divergence of the axial grain family responses. The gradients vary more widely in the transverse direction, for the reason explained earlier. There is good agreement between the calculated and experimental responses in this direction, too. It was stated in 5.11 that the experimental results show no influence of the strong texture in the hot swaged material§ on the bulk elastic stiffness. However, the EPSC model does predict such an influence. The stiffnesses of the axial grain family responses increase when the random grain population is replaced by the textured population. Correspondingly, the axial responses do not match the experimental data as well as in the previous simulation, even though the texture is more realistically represented. The introduction of texture also increases the gradients of the transverse responses, so that they too exhibit poorer agreement with the experimental data. It is reasonable that the stiffness should increase, since the majority of grains are now oriented with the stiffest direction, 111 , lying close to the tensile axis. Quite why this is not observed experimentally remains unknoh wn.i In summary, use of the elastic self-consistent model predicts that the bulk stiffness of the hot swaged material is greater than that of the unswaged material, due to the influence of crystallographic texture. This is not observed experimentally. However, the use of an elongated grain shape in the model successfully accounts for the reduced divergence in the elastic regime of axial grain family responses in the hot swaged material, in comparison to the corresponding responses in the unswaged material. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 166

111 200 200 311 220 111 800 800

700 700

600 600

500 500

400 400

300 300 Applied stress [MPa] Applied stress [MPa] 200 200 111 111 200 200 100 100 220 311 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 −2000 −1500 −1000 −500 0 Lattice strain [microstrain] Lattice strain [microstrain] (a) (b)

111 200 200 311 220 111 800 800

700 700

600 600

500 500

400 400

300 300 Applied stress [MPa] 200 Applied stress [MPa] 200 111 111 200 200 100 100 220 311 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 −2000 −1500 −1000 −500 0 Lattice strain [microstrain] Lattice strain [microstrain] (c) (d)

Figure 5.31: Development of elastic grain family strains with applied stress during self- consistent simulation of elastic deformation of hot swaged material. In the simulations, grains are represented by ellipsoids of revolution, aligned parallel to the tensile axis. The plots above correspond to an axial to transverse aspect ratio of 4:1. (a) and (b): axial and transverse strains respectively, using random grain population; (c) and (d) axial and transverse strains, respectively, using experimental grain population. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 167

5.14 Summary of Chapter

In this chapter, the mechanical deformation of a Fe-Ni-C TRIP steel has been studied using neutron diffraction. The final two sections have considered intergranular stress development in austenite prior to the onset of stress-induced martensitic transformation. The results corroborate work by other authors: the evolution of lattice strain in differently oriented grain families is understood by taking account of the elastic and plastic anisotropy of the fcc crystal structure. The measured austenite intergranular strains are in good agreement with the predictions of an elastoplastic self-consistent model. In simulations of uniaxial tensile deformation, the model captures the observed deviations from linearity of axial lattice strain responses during yielding, and the influence of inclusion shape on strains in the elastic regime. As commonly reported by other authors, predictions of average grain family lattice strains are poorer for transverse than axial grain families. This is explained by the greater sensitivity of transverse responses to the exact grain orientation distribution. The majority of the chapter has focused on the subjects of the preferential stress-induced transformation of favourably oriented austenite grains, and the development of interphase stress as martensite is formed. Evidence of preferential transformation is seen from changes in diffraction peak intensities as tensile deformation progresses. Relative changes in austenite peak intensities reveal the evolution of crystallographic texture in the phase. By reference to texture measurements when transformation is inhibited, and by comparison of changes in the diffraction spectra of initially untextured and highly textured samples, the separate influences of grain rotation and preferential transformation on the texture evolution have been isolated. This has demonstrated that austenite grains with 100 pointing along the tensile axis are most favourably oriented for stress-induced transformation.h i Owing to the preferential transformation of such grains, the martensite which develops is highly textured with 110 -type directions aligned along the tensile axis. This correspondence and an explanation of theh observi ed orientation dependence of transformation has been given using a crystallographic theory of martensitic transformation. Rietveld refinement of the diffraction spectra has revealed the generation of back stress in the austenite matrix as the martensite phase evolves. This indicates that martensite acts as a reinforcement. Since martensite plates tend to form with a structure which minimises internal stress, the back stress has been primarily attributed to the generation of plastic misfit between the phases, arising due to the greater yield stress of martensite. The observed back stress hardening acts to stabilise plastic flow, thereby giving rise to improved ductility. This work has therefore provided direct experimental evidence of a phenomenon which is an important origin of the TRIP effect. Chapter 5. Stress-Induced Martensitic Transformation in TRIP Steel 168 Chapter 6

Conclusions and Suggestions for Further Work

The aim of this work has been to further the understanding of the generation of internal stresses in single and two phase engineering materials subjected to mechanical deformation. In order to achieve this aim, the time-of-flight neutron diffraction method of elastic strain measurement has been applied to case studies of carbon steels and stress-induced martensitic transformation in TRIP steel. A range of analytical and computational models have been applied in order to interpret the results of these studies. Detailed discussion and summaries of the specific results have been given in Chapters 4 and 5. The intention here is to give a more general overview of the work presented in the dissertation and suggest avenues for future research.

6.1 Summary and Conclusions

An introduction to the origins, theories and models of internal stress development in single and two phase materials has been given in Chapter 2. Models and concepts applied in later chapters were described in some detail. Of particular note is the Eshelby ellipsoidal inclusion theory. This underpins the mean field and self-consistent methods, and the infinitesimal deformation theory approach to martensitic transformation, all of which have been applied in Chapters 4 and 5. In Chapter 3, the neutron diffraction method of elastic strain measurement was introduced. The basis of the technique was summarised and a description given of the instrument used to acquire the data presented in the dissertation. In order to set the context for the subsequent experimental work, a review was presented of the application of the technique to the measurement of intergranular and interphase stress development, and the study of stress-induced martensitic transformation. The experimental studies of internal stress development have been presented in Chapters 4 and 5. One of the major subjects which has been addressed is that of intergranular stress generation. The understanding of intergranular stresses is important because stress concentrations on the scale of individual grains may ultimately be important in the initiation of processes such as fatigue or failure. Another reason is that diffraction measurements of residual stresses are typically made by determining the position of a single diffraction peak, which relates to the elastic strain in a particular family of grains. In order to correctly deduce bulk residual stresses, the elastic strain response of the family to applied stress (which, owing to plasticity, may be highly non-linear) must be characterised and understood. For this purpose, an elastoplastic self-consistent (EPSC) model has proved useful. The work presented on intergranular stresses in this dissertation has sought to further validate this model by comparison to measurements made upon materials with both body-centred and face-centred cubic crystal structures. In

169 Chapter 6. Conclusions and Suggestions for Further Work 170

addition, attention has been paid to the issue of the dramatic non-linearities which have been reported in average grain family elastic strains resolved transversely to applied uniaxial stress. In both Chapters 4 and 5, intergranular strains measured during uniaxial tensile deformation have been compared to predictions of the EPSC model. In each case, the predictions of elastic strains parallel to the tensile axis are better than those of transverse strains. It has been demonstrated that one reason for this is that average transverse strains are more sensitive to the exact distribution of grain orientations. A point which has been laboured is that there is not a simple correspondence between the grains which contribute to a diffraction peak in the axial spectrum and those which contribute to the corresponding reflection in the transverse spectrum. Although this point is simple to understand, it is emphasised because it is also simple to fall under the misconception that the axial and transverse responses corresponding to the same diffraction peak are directly related. The measurements on body-centred cubic low carbon steel show that the lattice strain responses of some transverse grain families exhibit large tensile shifts during yielding. As noted above, similarly dramatic non-linearities have previously been reported, but research has mainly focused on face-centred cubic materials. Two points raised in Chapter 4 help to explain the behaviour and why it is common to more than one crystal structure. Firstly, although grains in a transverse family have a range of orientations with respect to the tensile axis, it is still possible that the grains in one family are on average better oriented for slip compared to those in other families. This has been demonstrated for body-centred cubic materials. Secondly, for a given axial strain, a grain will tend to contract more in the transverse plane if the strain is developed plastically than if it is developed elastically. Therefore, in grains which tend to slip earlier than their surroundings, the greater transverse contraction due to plasticity is compensated by the development of relatively large tensile elastic strains. Thus such shifts can be expected in all materials in which some transverse families are on average better oriented for slip than others. This of course depends on the operable slip modes in the particular crystal structure. The subject of interphase stress is another of the major topics addressed in the experimental work. This subject is important because the generation of interphase stresses dominates the mechanical behaviour of composite materials. In this dissertation, this has been demonstrated in both high carbon steel and TRIP steel. In particular, in TRIP steel, measurements of back stress in the austenite matrix have demonstrated that load transfer to the evolving martensite phase plays a vital role in stabilising plastic flow and giving rise to the TRIP effect. Matrix back stresses also help to stabilise plastic flow in high carbon steel. In this material, redistribution of load between the phases is very dramatic during yielding, owing to softening of the ferrite matrix. A unit cell finite element (FE) model has been used to emulate this effect. However, the extent of load redistribution observed experimentally cannot be fully captured using simple cubic and body-centred cubic arrays of inclusions. This indicates the efficacy of a random dispersion of elastically-deforming inclusions in constraining matrix plastic flow. In both the TRIP and high carbon steels, the interphase stresses can be understood on the basis that plastic flow is greater in the matrix than in the inclusions, generating misfit between the phases. In high carbon steel, the interphase stresses saturate after a few percent plastic strain, indicating that the misfit does not increase indefinitely. Possible reasons for this are that the cementite inclusions begin to deform plastically, or that local plastic flow around the inclusions acts to relax the misfit. There is not direct evidence of such relaxation in TRIP steel, because the axially-compressive residual strain in the austenite phase continues to grow linearly with macroscopic plastic strain. However, the average residual strain in the martensite phase remains approximately constant, and it has been speculated that this may be achieved if the residual stress in a martensite plate saturates soon after its formation, but that load transfer from austenite continues due to the formation of new martensite plates. In addition to measurements of interphase strains in high carbon steel, intergranular strains have also been presented. The axial intergranular residual strains which remain in the matrix after unloading have been explained primarily on the basis that elastic anisotropy causes grain families to develop different elastic strains in response to the bulk residual stress in the phase. Chapter 6. Conclusions and Suggestions for Further Work 171

In order to simulate the overall development of intergranular and interphase stresses in high carbon steel, a combined analysis has been performed in which the average interphase stresses predicted by the FE model are applied as boundary conditions in the EPSC model. Comparison to experimental data has demonstrated this to be viable approach for the full prediction of internal stresses in composite materials. The third major issue which has been addressed is that of the preferential stress-induced transformation of favourably oriented austenite grains in TRIP steel. The experimentally- determined trend is consistent with the prediction of a crystallographic theory of martensitic transformation. The basis of this theory is that martensite plates are formed with a morphology which minimises internal stress development. Thus it has been demonstrated that consideration of internal stress is important for the understanding of stress-induced martensitic transformation. Although the mechanical properties diﬀer considerably, the analysis of preferential transformation applied to TRIP steel could also be applied to the same phenomenon in shape memory alloys. In conclusion, the experimental results and analysis presented in this dissertation have revealed and explained trends in the development of intergranular and interphase stresses, thereby contributing to the understanding of the generation of internal stresses in single and two phase engineering materials.

6.2 Suggestions for Further Work

There are numerous directions along which the work presented in this dissertation may be extended. The validation of models of internal stress development would benefit from the characterisation by neutron diffraction of internal stresses in many more materials. With regard to intergranular stresses, alloys with hexagonal close packed crystal structures are of particular interest. In comparison to cubic materials, hexagonal materials tend to exhibit greater plastic anisotropy, owing to the availability of fewer deformation modes. This may lead to greater intergranular strain magnitudes and sensitivity to crystallographic texture, providing distinct trends which may be compared to predictions from the EPSC model. With regard to interphase stresses, it would be beneficial to carry out further work to investigate the saturation of plastic misfit between phases. In particular, a simple extension of the TRIP steel study would be to induce a population of martensite plates by pre-straining, and then measure subsequent load partitioning at an elevated temperature in order to inhibit further transformation. This would confirm whether saturation of the misfit does occur, and may help to explain why the average martensite residual stress remains approximately constant. Concerning the issue of preferential transformation, it would be instructive to compare the observed trend in TRIP steel to the behaviour in other materials which exhibit stress-induced transformation, such as shape memory alloys. Since these materials deform mainly by phase or variant transformation rather than slip, the development of internal stress would be expected to differ to that observed in TRIP steel, but similarities would be expected in the selection of certain grain orientations for transformation. There are also avenues for the development of the modelling strategies which have been applied in this dissertation. There is considerable scope to integrate the FE and EPSC models more closely for the prediction of intergranular stresses in composites. Rather than take the average phase stress predicted by the FE model as the boundary condition within the EPSC model, the phase stress evaluated at a range of locations could be used for spatially-dependent calculations of the intergranular stress state. Since grain family responses are highly non-linear, the volume averaged intergranular strains calculated from this method might differ significantly from those determined using the simpler approach. Lastly, the present EPSC code could be modified to incorporate a criterion for martensitic transformation, providing a model of internal stress generation in TRIP steel. This would also require grain rotation to be accounted for, since the development of large plastic strains is an Chapter 6. Conclusions and Suggestions for Further Work 172

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Elastic Strain Response of a Cubic Crystal

This appendix shows that if uniaxial stress σ is applied to a cubic crystal along the unit vector [l, m, n] (speciﬁed in the crystal axes reference frame), then the elastic strain uvw developed along the direction of another unit vector [u, v, w] is given by uvw = AM + (1 A) M + BM (A.1) σ 11 − 12 44 where

A = l2u2 + m2v2 + n2w2 , (A.2) B = lmuv + lnuw + mnvw (A.3) and Mij is the single crystal compliance tensor in contracted matrix notation, following the convention explained in Nye [14]. Note that using this convention, the deﬁnitions of the 6 1 matrix representations diﬀer for the stress and strain tensors. The stress and strain tensor× components σij and ij respectively are related to the matrix representations σi and i according to: 1 1 σ1 σ6 σ5 1 2 6 2 5 1 1 σij = σ6 σ2 σ4 and ij = 2 6 2 2 4 . (A.4)    1 1  σ5 σ4 σ3 2 5 2 4 3    

Denote the crystal axes reference frame as X. The 3-axis of a second orthonormal reference frame X0 is chosen to lie parallel to [l, m, n]. The 1- and 2-axes are arbitrarily chosen. In this frame, the stress tensor is 0 0 0 σ0 = 0 0 0 . (A.5)  0 0 σ  In the X reference frame, the stress tensoris given by the transformation law:

T σ = a σ0a (A.6) where the transformation matrix a is given by

a = • • • . (A.7)  •l m• n•    183 Chapter A. Elastic Strain Response of a Cubic Crystal 184

The symbol is used to represent matrix elements which are not given explicitly (these are the • direction cosines of the arbitrarily chosen 1- and 2- axes of the X 0 frame). These elements have no bearing on the stress tensor when transformed into the X frame, using (A.6):

l2 lm ln 2 σ0 = σ lm m mn . (A.8)  ln mn n2    The strain matrix elements i are given by Hooke’s Law:

2 M11 M12 M12 0 0 0 l 2 M12 M11 M12 0 0 0 m    2  M12 M12 M11 0 0 0 n i = Mij σj = σ  0 0 0 M44 0 0   mn       0 0 0 0 M44 0   ln       0 0 0 0 0 M44   lm       2 2 2    l M11 + m + n M12 m2M + l2 + n2 M  11 12  n2M + l2 + m2 M = σ 11 12 . (A.9)  mnM   44   lnM   44   lmM   44    The strain resolved along the unit vector xi = [u, v, w] is given by

2 2 2 uvw = ij xixj = 11u + 22v + 33w + 223vw + 213uw + 212uv 2 2 2 = 1u + 2v + 3w + 4vw + 5uw + 6uv . (A.10)

Substituting in the values of i from (A.9), and noting that

u2 m2 + n2 + v2 l2 + n2 + w2 l2 + m2 = 1 l2u2 + m2v2 + n2w2 , (A.11) − (A.1) is giv en.

Noting that

A + 2B = l2u2 + m2v2 + n2w2 + 2 (lmuv + lnuw + mnvw) = (lu + mv + nw)2 (A.12)

and that lu + mv + nw is the dot product of the unit vectors [l, m, n] and [u, v, w], (A.1) is now examined for two diﬀerent relative orientations of [l, m, n] and [u, v, w].

(i) [u, v, w] is parallel to [l, m, n], i.e. [u, v, w] = [l, m, n]

In this case, the dot product of the unit vectors is equal to unity. Therefore

A + 2B = 1 . (A.13)

Substituting for A, (A.1) becomes

1 1 lmn = = M 2 M M M l2m2 + l2n2 + m2n2 , (A.14) σ E 11 − 11 − 12 − 2 44 lmn Chapter A. Elastic Strain Response of a Cubic Crystal 185

where Elmn is the directional Young’s modulus.

(ii) [u, v, w] is perpendicular to [l, m, n]

In this case, the dot product of the unit vectors vanishes. Therefore

A + 2B = 0 . (A.15)

Substituting for B, (A.1) becomes

1 uvw = M + M M M l2u2 + m2v2 + n2w2 . (A.16) σ 12 11 − 12 − 2 44