Numerical Implementation of an Eulerian Description of Finite

Home , Objective stress rate

Numerical implementation of an Eulerian description of ﬁnite elastoplasticity

Dissertation

zur

Erlangung des Grades Doktor-Ingenieurin (Dr.-Ing.)

der

FakultätfürBau- und Umweltingenieurwissenschaften der Ruhr-UniversitätBochum

von

M. Sc. Marina Trajkovi´c- Milenkovi´c

aus Leskovac, Serbien

Bochum 2016 Dissertation eingereicht am: 10.08.2016

Tag der m¨undlichen Pr¨ufung: 21.10.2016

Erster Referent: Prof. em. Dr.-Ing. Otto T. Bruhns Zweiter Referent: Vertr.-Prof. Dr.-Ing. Ralf J¨anicke To my little M. Summary

The numerical implementation and validation of a self-consistent Eulerian constitutive theory of ﬁnite elastoplasticity, based on the logarithmic rate and additive decomposition of the stretching tensor, via commercial ﬁnite element software has been the objective of this thesis.

The elastic material behaviour has been modelled by a hypo-elastic relation while for the plastic behaviour an associated ﬂow rule has been adopted. In accordance with the objectivity requirement, in the constitutive relations the objective time derivatives have been used; the logarithmic rate, the Jaumann and Green-Naghdi rates as corotational rates and the Truesdell rate, the Ol- droyd rate and the Cotter-Rivlin rate as non-corotational rates.

It has been shown that for the large elastic deformation analysis the reliability of the Jaumann and Green-Naghdi rates decreases with the increased inﬂuence of ﬁnite rotations while the non-corotational rates have to be excluded from the constitutive relations. It has been proved that the implementation of the Log- rate successfully solves the so far existing integrability problem and the residual stress occurrence at the end of closed elastic strain cycles.

The constitutive relation for ﬁnite elastoplasticity based on the INTERATOM model has been developed and implemented. The obtained results have been compared with the experimental records. The proposed constitutive model has been proved as reliable for monotonic deformations while for cyclic deformations some improvements remain to be implemented. The directions for improvements have been given as well. Zusammenfassung

Das Ziel dieser Arbeit sind die numerische Implementierung und die Validierung einer selbstkonsistenten Eulerschen Beschreibung der ﬁniten Elastoplastizit¨at, die auf der logarithmischen Rate und additiven Zerlegung des Dehnungstensors basiert. Implementierung und Validierung erfolgen in eine kommerzielle Finite Elemente Software.

Das elastische Materialverhalten wird durch eine hypo-elastisch Beziehung mod- elliert, währendfür das plastische Verhalten eine assoziative Fließregel angenom- men worden ist. Entsprechend der Forderung nach Objektivitätder konstitutiven Gleichungen sind objektive Zeitableitungen verwendet worden; und zwar die logarithmische Rate, die Jaumann Rate und die Green-Naghdi Rate als mitrotierende und die Truesdell Rate, die Oldroyd Rate und die Cotter-Rivlin Rate als nicht-mitrotierende Raten Zeitableitungen.

Es wird gezeigt, dass fürgroße elastische Verformungen die Genauigkeit der Zeitableitungen nach Jaumann und Green-Naghdi mit zunehmendem Einfluss größererRotationen abnimmt, währenddie nicht-mitrotierenden Zeitableitun- gen als ungeeignet aus konstitutiven Beziehungen auszuschließen sind. Mit Hilfe der logarithmischen Rate konnte das bisher ungelösteIntegrabilitätsproblem gelöstund das Auftreten von Eigenspannungen am Ende geschlossener elastis- cher Beanspruchungszyklen vermieden werden.

Das verwendete Materialgesetz fürdie finite Elastoplastizitätbaut auf dem IN- TERATOM Modell auf und wurde hier an ein vereinfachtes Materialverhalten angepasst und implementiert. Die erhaltenen Ergebnisse wurden mit experi- mentellen Ergebnissen verglichen. Das vorgeschlagene Materialgesetz kann für monotone Verformungen als zuverlässigangesehen werden. Fürzyklische Verfor- mungen dagegen werden einige Verbesserungen als notwendig erachtet. Mögliche Verbesserungsvorschlägewerden gegeben. Acknowledgments

I would like to express my sincere gratitude to Prof. em. Dr.-Ing. Otto T. Bruhns for giving me the privilege to work in this area and for his guidance throughout my work on this thesis. I really appreciate his warm hospitality during my research stays at the Institute of Mechanics of the Ruhr University in Bochum.

I want to thank to Vertr.-Prof. Dr.-Ing. Ralf J¨anicke for his acceptance to be the co-reviewer of my thesis.

I also want to thank to Prof. Dr.-Ing. RüdigerHöfferfor beeing the faculty reviewer of my thesis.

My sincere gratitude to Prof. Dr Dragoslav Sumarac,ˇ my co-advisor, for his support and fruitful cooperation all these years.

I also want to thank to Prof. Dr.-Ing. Günther Schmid, Prof. Dr.-Ing. Rüdiger Höfferand Prof. Dr Stanko Brˇcićfor their advice and motivation.

Being the guest researcher at the Institute of Mechanics of the Ruhr University in Bochum was the extremely valuable experience for me. For co-ﬁnancing my stays in Bochum I deeply appreciate the DAAD grants obtained through the SEEFORM programme. I want to thank to Prof. Dr. Heng Xiao and Dr.-Ing. Albert Meyers for very useful discussions while I was doing the ﬁrst steps toward this goal. I want to thank to all members of the (former) Chair of Technical Me- chanics for friendly and pleasantly working atmosphere. My special thanks to our small Serbian community in Bochum for their help.

I really appreciate the support of my friends and colleagues from the Faculty of Civil Engineering and Architecture in Niˇs.Many thanks for my colleagues from the Faculty of Mechanical Engineering in Niˇsfor fruitful discussions.

Finally, I want to thank to my parents, my sister and her family, and mostly to my son and husband for their constant support, love and encouragement.

Bochum, 2016 Marina Trajkovi´cMilenkovi´c Contents

List of Figures iv

List of Tables vii

Conventions and Notations viii

1 Introduction1 1.1 Motivation...... 2 1.2 Aim of the thesis...... 4 1.3 Outline...... 4

2 Deformation and motion5 2.1 Introduction...... 5 2.2 Kinematics of deformable body...... 5 2.2.1 Configurations and body motion...... 5 2.2.2 Lagrangian and Eulerian description...... 7 2.2.3 Material and spatial time derivative...... 9 2.3 Analysis of deformation...... 10 2.3.1 Deformation gradient...... 10 2.3.2 Polar decomposition...... 11 2.4 Analysis of strain...... 13 2.4.1 Generalized strain measures...... 13 2.5 Analysis of motion...... 15 2.5.1 Velocity gradient...... 15 2.5.2 Rate of deformation...... 17 2.5.3 Vorticity tensor...... 17 2.6 Objectivity of a tensor field...... 19 2.6.1 Lagrangian and Eulerian objectivity...... 19 2.6.2 Objective time derivatives...... 20 2.6.3 Corotational and convective frame...... 21 2.6.4 Non-corotational rates...... 23 2.6.5 Corotational rates...... 25 2.7 Decomposition of finite deformation...... 29

3 Conservation equations and stress measures 32 3.1 Introduction...... 32 3.2 Conservation laws...... 32

i ii

3.2.1 Conservation of mass...... 32 3.2.2 Conservation of linear momentum and Cauchy’s ﬁrst law of motion...... 33 3.2.3 Conservation of angular momentum and Cauchy’s second law of motion...... 35 3.3 Stress measures and stress rates...... 36 3.3.1 Alternative stress measures...... 36 3.3.2 Lagrangian ﬁeld equations...... 38 3.3.3 Work rate and conservation of mechanical energy..... 39 3.3.4 Stress rates...... 41 3.4 Conjugate stress analysis...... 44 3.5 Weak form of balance of momentum...... 48 3.5.1 Principle of virtual work...... 48 3.5.2 Rate of the weak form of balance of momentum...... 50

4 Constitutive relations 52 4.1 Introduction...... 52 4.2 Continuum thermodynamics...... 53 4.2.1 First law of thermodynamics...... 54 4.2.2 Second law of thermodynamics and principle of maximum dissipation...... 57 4.3 Decomposition of the stress power...... 60 4.4 Hypo-elasticity...... 61 4.4.1 Hypo-elastic relation for De - From small to ﬁnite strains 61 4.4.2 Hyper-elastic potential...... 63 4.5 Plasticity...... 65 4.5.1 Yield surface and loading condition...... 67 4.5.2 Priniple of maximum plastic dissipation and its consequences 69 4.5.3 Eulerian description of ﬁnite plastic deformations..... 71 4.5.3.1 Classical models from IA model...... 75

5 Numerical results 77 5.1 Introduction...... 77 5.2 Hypo-elasticity...... 77 5.2.1 Finite simple shear...... 78 5.2.2 Plate with a hole in ﬁnite tension...... 83 5.2.3 Closed elastic strain path - Hypo-elastic cyclic deformation 85 5.2.3.1 Small rotations (ξ = 0.1)...... 86 5.2.3.2 Moderate rotations (ξ = 1)...... 89 5.2.3.3 Large rotations (ξ > 1)...... 93 5.3 Elastoplasticity...... 101 5.3.1 Small cyclic elastoplastic deformations...... 101 5.3.2 Finite elastoplastic deformations...... 107 5.3.2.1 Elastic - perfectly plastic...... 108 iii

5.3.2.2 Elastic - linear kinematic hardening...... 111

6 Concluding remarks 116 6.1 Summary and conclusions...... 116 6.2 Possible extensions of the present research...... 117

Bibliography 119 List of Figures

2.1 Undeformed and deformed conﬁguration of the material body..6 2.2 Material and spatial coordinate systems...... 8 2.3 Polar decomposition...... 12 2.4 Velocity gradient...... 16 2.5 Angular velocity vector...... 19

4.1 Initial and succeeding yield surfaces and material stability.... 68 4.2 Loading condition, normality rule and convexity of the yield surface 68

5.1 Finite simple shear problem...... 79 5.2 Shear stress vs shear strain with Log-rate: Hypo-elastic yield point 80 5.3 Normal stress vs shear strain at simple shear for various rates.. 82 5.4 Shear stress vs shear strain at simple shear for various rates.. 82 5.5 Plate with a hole in tension: Model...... 83 5.6 Plate with a hole in tension: Undeformed and deformed configuration and normal stress distribution for large elastic deformation 84 5.7 Plate with a hole in tension: Normalized normal stress distribution for large elastic deformation in the node with the maximum normal stress...... 84 5.8 Plate with a hole in tension: Normalized normal stress distribution for large elastic deformation in the node with maximum shear stress...... 85 5.9 Deformation cycle...... 86 5.10 τ 11, τ 12 and τ 22 single cycle development, ξ = 0.1, η = 0.02... 86 5.11 Enlarged representation of τ 12 at first cycle end, ξ = 0.1, η = 0.02 87 5.12 τ 11, τ 12 and τ 22 single cycle development, ξ = 0.1, η = 0.1.... 88 5.13 τ 12 single cycle development, ξ = 0.1, η = 0.1...... 88 5.14 τ 11, τ 12 and τ 22 single cycle development, ξ=1, η=0.1...... 89 5.15 τ 12 single cycle development, ξ=1, η=0.1...... 90 5.16 Enlarged representation of τ 12 at first cycle end, ξ=1, η=0.1.. 91 5.17 Enlarged representation of τ 11 at first cycle end, ξ=1, η=0.1.. 91 5.18 Hundred cycle stress development for corotational rates, ξ=1, η=0.1 92 5.19 τ 12 development for Jaumann rate, ξ=1, η=0.1...... 92 5.20 Normal stress τ 11 and τ 22 single cycle development, ξ=5, η=0.1 93 5.21 Shear stress τ 12 single cycle development, ξ=5, η=0.1...... 94

iv v

5.22 Enlarged representation of τ 11 and τ 12 single cycle development, ξ=5, η=0.1...... 94 5.23 Ten and hundred cycle stress development for the Logarithmic rate, ξ=5, η=0.1...... 95 5.24 Ten cycle stress development for the Jaumann rate UMAT, ξ=5, η=0.1...... 96 5.25 Ten cycle stress development for the Jaumann rate ABAQUS, ξ=5, η=0.1...... 96 5.26 Hundred cycle stress development for the Jaumann rate, ξ=5, η=0.1 97 5.27 Ten cycle stress development for the Green-Naghdi rate, ξ=5, η=0.1 98 5.28 Hundred cycle stress development for the Green-Naghdi rate, ξ=5, η=0.1...... 98 5.29 Hundred cycle stress development for the Trusdell rate, ξ=5, η=0.1 99 5.30 Hundred cycle stress development for the Oldroyd rate, ξ=5, η=0.1 99 5.31 Hundred cycle stress development for the Cotter/Rivlin rate, ξ=5, η=0.1...... 100 5.32 Residual normal and shear stresses for the given rates, ξ=5, η=0.1 100 5.33 Monotonic tensile test, source: Westerhoff(1995)...... 102 5.34 Uniaxial cyclic experiments with strain amplitudes ∆ε = 0.005, 0.015 and 0.02, source: Westerhoff(1995)...... 102 5.35 Numerical results vs monotonic tensile test...... 103 5.36 Comparison between computation and experiment for the loading- unloading-reloading case...... 104 5.37 First two cycles of the stress-strain curve for various strain ranges, ∆ε = 0.005, 0.015 and 0.02, respectively...... 104 5.38 Saturated cycles: numerical results (solid line) vs monotonic tensile test (dashed line), source: Westerhoff(1995)...... 105 5.39 IA-model uniaxial cyclic response for ∆ε = 0.005, 0.015 and 0.02 106 5.40 Comparison of IA-model, linear and nonlinear kinematic hardening, with experimental record for strain range ∆ε = 0.02.... 107 5.41 Normalised normal stress vs shear strain for elastic-perfectly plastic behaviour...... 109 5.42 Normalized shear stress vs shear strain for elastic-perfectly plastic behaviour and its enlarged representation...... 109 5.43 Elastic-perfect plasticity: Normalized normal stress development and enlarged representation of the normalized shear stress development for real material parameters...... 110 5.44 Normalized normal stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.6...... 111 5.45 Normalized shear stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.6...... 112 5.46 Normalized shear stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.022...... 113 vi

5.47 Normalized normal stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.022...... 113 5.48 Normalized normal and shear stress vs shear strain for the material parameters according to Westerhoﬀ(1995)...... 114 5.49 Plate with a hole in tension: Undeformed and deformed conﬁgu- ration and normal stress distribution for large elastoplastic deformation with linear kinematic hardening...... 115 5.50 Plate with a hole in tension: Normalized normal stress distribution for large elastoplastic deformation with linear kinematic hardening...... 115 List of Tables

2.1 Transformation and objectivity of kinematical quantities..... 21

5.1 Residual shear stresses τ 12/2G for ξ = 0.1, η = 0.02...... 87 5.2 Residual stresses after ﬁrst cycle for ξ = 1, η = 0.1...... 90 5.3 Residual τ 12 compared to τ 12max in % for ξ = 1, η = 0.1..... 90 5.4 Residual stresses after 10 cycles for ξ = 10, η = 0.2...... 101

vii Conventions and Notations

Scalars (italics) B Body bi Distinct eigenvalue of the Cauchy-Green tensors Dp Plastic dissipation dv Volume element in the current configuration dV Volume element in the reference configuration dx Length of the spatial vector dx dX Length of the material vector dX E Young’s modulus Et Tangent modulus F Loading function G Shear modulus J Determinant of the deformation gradient (Jacobian determinant) m Mass P Particle P0 Initial position of the particle p Current position of the particle q Heat flux vector Q Heat r Internal heat source s Specific entropy S Entropy t Time u Specific internal energy U Internal energy w˙ Stress power per unit volume wp Accumulated plastic work W Elastic potential W¯ Complementary elastic potential xi Component of the position vector in the current configuration Xα Component of the position vector in the reference configuration Y0 Tensile yield strength

Scalars (Greek characters) γ Shear strain

viii ix

p εeqv Equivalent plastic strain κ Isotropic hardening variable Λ Plastic multiplier λ First Laméconstant µ Second Laméconstant ν Poisson’s ratio θ Absolute temperature ρ Mass density in the current configuration ρ0 Mass density in the reference configuration σij Component of the stress tensor σy Yield stress Σ Elastic potential ψ Specific Helmholtz free energy

Vectors (boldface roman) b Current body force per unit mass b0 Referential body force per unit mass c Rigid body translation of the observer da Area element in the current configuration dA Area element in the reference configuration dl Load vector in the current configuration dL Load vector in the reference configuration dx Line element in the current configuration dX Line element in the reference configuration n Unit normal vector in the current configuration N Unit normal vector in the reference configuration o Origin in the current configuration O Origin in the reference configuration ei Unit vector in the current configuration Eα Unit vector in the reference configuration t Traction ¯t Traction boundary conditions u Displacement δu Virtual displacement u¯ Displacement boundary conditions v Velocity δv Virtual velocity x Position vector in the current configuration X Position vector in the reference configuration

Vectors (Greek characters) ω Angular velocity vector x

Second order tensors (boldface roman) 1 Unit tensor A Eulerian tensor A0 Lagrangian tensor a Finger tensor A Piola tensor B Left Cauchy-Green tensor Bi Eigenprojection of the left Cauchy-Green tensor C Right Cauchy-Green tensor Ci Eigenprojection of the right Cauchy-Green tensor D Stretching tensor e Almansi-Eulerian strain tensor E Green-Lagrangian strain tensor e(m) Eulerian strain tensor E(m) Lagrangian strain tensor F Deformation gradient h Hencky strain L Velocity gradient P Nominal stress tensor Q Rotation tensor R Rotation tensor (polar decomposition) RLog Logarithmic rotation tensor S Second Piola Kirchhoﬀ stress tensor T First Piola Kirchhoﬀ stress tensor U Right stretch tensor V Left stretch tensor W Vorticity tensor

Second order tensors (Greek characters) α Back stress ε Linearized (engineering) strain π Eulerian stress measure Π Lagrangian stress measure σ Cauchy stress tensor σ∗ Corotational Cauchy stress tensor σ˜ Finger tensor τ Kirchhoﬀ stress tensor τ¯ Shifted Kirchhoﬀ stress tensor Ω¯ Angular velocity tensor Ω Spin tensor ΩJ Zaremba-Jaumann spin tensor ΩR Polar spin tensor xi

ΩLog Logarithmic spin tensor

Fourth order tensors (boldface roman) C Elastic stiﬀness tensor H Hypo-elasticity tensor I Identity tensor K Elastic compliance tensor

Other symbols B Configuration Bt Current configuration B0 Reference configuration B¯t Intermediate configuration ∂Bt Boundary of the body in Bt ∂B0 Boundary of the body in B0 ∂uB0 Displacement boundary of B in B0 ∂σB0 Traction boundary of B in B0 ∂uBt Displacement boundary of B in Bt ∂σBt Traction boundary of B in Bt ∂vBt Boundary of B in Bt with prescribed velocity E Euclidean point space O Observer

Special symbols & functions det Determinant div Divergence with respect to x g(·) Scale function h(·) Spin function grad Gradient with respect to x Grad Gradient with respect to X ∇ Gradient with respect to x ∇0 Gradient with respect to X sym(·) Symmetric part of a tensor tr(·) Trace of a tensor (·)T Transpose of (·) (·)−1 Inverse of (·) (·˙) Material time derivative of (·) ◦ (·) L Lie derivative of (·) (·) Arbitrary objective time derivative of (·) 5 (·) Arbitrary objective non-corotational rate of (·) xii

5 (·) Ol (Upper) Oldroyd rate of (·) 5 (·) CR Cotter-Rivlin (lower Oldroyd) rate of (·) 5 (·) Tr Truesdell rate of (·) ◦ (·) Arbitrary objective corotational rate of (·) ◦ (·) J Zaremba-Jaumann rate of (·) ◦ (·) GN Polar or Green-Naghdi rate of (·) ◦ (·) Log Logarithmic rate of (·) · Scalar product : Double contraction × Vector product ⊗ Dyadic product ? Rayleigh product LHS Left-hand side of the equation RHS Right-hand side of the equation

Superscripts (·)e Elastic (·)p Plastic (·)? Transformed (·)0 Deviator of tensorial quantity 1 Introduction

Contemporary theories, established to describe the behaviour of materials that undergo elastoplastic deformations, usually belong to the group of phenomenological theories which rather try to mathematically formulate the behaviour of materials observed during the experiments than to really explain a complex phenomenon of plasticity. The latter is the topic of the physical theories of plasticity that are beyond the scope of this treatise. Application of the phenomenological theories is extremely valuable in engineering practice such as in structural design or metal forming.

Engineering structure elements during their serviceability period are usually exposed to small strains with rare exceptions. But during their manufacturing materials could undergo very large strains usually accompanied with very large rotations, such as metals during metal forming. Even though the phenomenological models have been established and very well elaborated for the case of small deformations, especially for metals, ﬁnite elastoplastic deformations represent a wide and yet not fully discovered ﬁeld of research.

Nowadays powerful tools of numerical simulations help engineers in resolving the existing problems. Wide application of numerical simulations significantly shortened the time required to obtain the final product and therefore made the manufacturing process less expensive. For example, until recently the design of metal forming was based on the knowledge earned trough a long practical experience or expensive experimental try-outs. Nowadays, finite element simulations are involved from the early stage of the production process. Each stage in this process can be simulated and the errors can be corrected and avoided prior to the actual manufacturing. The implemented theories can be tested, improved and finally accepted for wide practical applications.

New achievements in the ﬁeld of material sciences result with the introduction of new materials, such as shape memory alloys and composites, in the wide range of applications from medical purpose to numerous engineering structures. In civil engineering structures shape memory alloys can be used for devices like dampers and base isolators. On the other hand, some known materials, such as the rubber-like materials, see their use expanded beyond their traditional purpose, for example in civil engineering structures as a part in building supports to protect the structure during the dynamic excitations. Having in mind that one of the main tasks of engineers is to provide the reliable lifetime assessment procedures of engineering structures, the establishment of the constitutive the-

1 2 Chapter 1. Introduction ory seems to be a complex, demanding and challenging process especially if the materials are exposed to large deformations.

The more reliable and accurate constitutive model we want to establish, the more material parameters have to be introduced and defined. Since the experimental tests concerning large deformations are very difficult to perform and can be followed by different unpredictable instabilities, there is a lack of a sufficient number of experiments necessary for the material parameter identification. For example, the large simple shear problem is used very often in numerical calculations since it involves as much rotation as stretching. From this example we can conclude the validity range of the proposed constitutive models. But, ”the experimental execution of a simple glide test, that allows one, in principle, to discriminate between models, is confronted with great difficulties as deformation becomes heterogeneous rather quickly, so that there remains a lot of mysteries to be elucidated concerning very large strains” (Besson et al., 2009).

1.1 Motivation

Concerning the finite strains, in order to fulfil the material frame indifference in the constitutive model instead of a material time derivative objective rates have to be implemented. The occurrence of the large rotations, such as in the aforementioned large simple shear example, can cause totally different material stress responses for different stress rates introduced in the same constitutive model. Some of these results can be inconsistent with the experimental or theoretical predictions.

Looking back to history, Hencky was the ﬁrst who has realized that for the ﬁ- nite deformations analysis in an Eulerian description the time derivative must be independent of the respective rigid-body rotation, i.e. in the present notion must be objective. He replaced the material time derivative with an expression similar to that presently known as a Jaumann rate.

Unfortunately, this idea of Hencky has to wait for two decades to be reconsidered. Oldroyd was Hencky’s successor in developing the idea of the material time derivative replacement with, in his approach, ”convected differentiation with respect to time” in the finite deformation problems and he formulated four different relations of the Cauchy stress derivatives (for more details and references the reader is referred to Bruhns, 2014).

The Prandtl-Reuss theory of elastoplastic material behaviour, adopted here, implies the additive decomposition of an increment (rate) of a total deformation into its elastic and plastic parts. The deﬁciency of the constitutive law formulated in rates of stresses and deformation for purely elastic processes, later denoted as hypo-elastic by Truesdell, has been ﬁrstly observed by the same author. It 1.1. Motivation 3 can be demonstrated by the occurrence of the residual stresses at the end of a closed elastic cyclic process. Much later this was the topic of investigation of various researchers, some of them are Kojic & Bathe(1987), Lin et al.(2003), Xiao et al. (2006b).

When a shear oscillatory phenomenon has been discovered and proved by Lehmann (1972), Dienes(1979) and Nagtegaal & de Jong(1982), i.e. it was shown that the hypo-elasticity model based on the Jaumann rate gives the unstable response at simple shear, it has been concluded that for the case of large rotation the Jau- mann rate cannot be an appropriate choice. In the work of Simo and Pister (cf. Simo & Pister, 1984) it has been shown also that for the case of pure elastic deformation, where the natural deformation rate is equal to its elastic part, none of the constitutive hypo-elastic relations, based on at that time known objective rates, fulﬁl the requirement that is exactly integrable to give an elastic relation.

In Besson et al.(2009) it has been shown that for the simple shear problem the oscillatory stress response occurs for the Jaumann rate if the elastic behaviour of materials is taken into account. For pure plastic behaviour with isotropic hardening the shear oscillation vanishes while for the linear kinematic hardening it still exists. Only for adopted nonlinear kinematic hardening for pure plasticity the oscillation in the stress response disappears. For elastoplastic behaviour of materials even for isotropic hardening the stress response has oscillatory character while its occurrence for adopted elastic-nonlinear kinematic hardening behaviour is sensitive on the hardening modulus and shear modulus ratio. For usual values of this ratio the oscillation in stress response exists.

These problems have been solved not long ago with the introduction of the so- called logarithmic rate. In Bruhns et al.(1999), Xiao et al. (1997a, 1997b, 1999) the authors proved the correctness of Log-rate implementations in the constitutive models for ﬁnite elastoplasticity.

But even though Jaumann himself pointed out the lacks of his rate, i.e. he ”states that the behaviour of this rate material is indeed not elastic and only in the limit of infinitesimal deformations changes to that of an elastic body” (Bruhns, 2014), this time derivative has been widely accepted. The Jaumann rate accompanied with the Green-Naghdi rate is incorporated in all commercial finite element codes for structural analysis especially for the case of finite deformations which caused a lot of debates among researches in last decade (see for example Baˇzant et al., 2012, and Baˇzant & Vorel, 2014).

The aim of this thesis is to provide a contribution in this ﬁeld of ﬁnite elastoplastic deformation analysis. 4 Chapter 1. Introduction

1.2 Aim of the thesis

The main goal of this treatise is to implement the various widely used stress rates as well as the recently introduced logarithmic rate into the proposed constitutive models of ﬁnite elasticity and elastoplasticity, based on the Prandtl-Reuss theory, via the commercial ﬁnite element software in order to examine the range of validity of the tested rates separately.

The second goal of this thesis is to test the new constitutive model for ﬁnite elastoplastic deformations based on the INTERATOM model originally established for small deformation case (see Bruhns et al., 1988).

1.3 Outline

This thesis has been organized in six Chapters.

After this introduction, the basic relations of the non-linear continuum mechanics necessary for ﬁnite deformation description have been presented in the second Chapter. The objectivity principle has taken a prominent place in this Chapter because of its crucial role in the Eulerian description of physical processes. The objective time derivatives have been introduced and elaborated in detail.

The third Chapter, after a short review of the conservation laws, presents various Lagrangian and Eulerian stress measures and deﬁnes several mostly used objective corotational and non-corotational rates. The special attention has been assigned to the work conjugacy analysis from which the direct relation between the rate of deformation tensor and the Eulerian Hencky strain has been obtained. A brief description of the weak form of the balance of momentum has been given as well.

In the fourth Chapter the constitutive relations, based on the consistent Eulerian theory of ﬁnite elastoplasticity, have been presented and the advantage of the logarithmic rate implementation in the proposed constitutive models has been clariﬁed.

The ﬁfth Chapter shows the results obtained from the numerical implementation of the various objective stress rates, through the proposed constitutive models, in the commercial ﬁnite element code ABAQUS/Standard using the UMAT subroutine.

The main conclusions and some possible extensions of the present research work have been given in the last, sixth Chapter. 2 Deformation and motion

2.1 Introduction

The aim of this Chapter is to provide a brief review on kinematics of non-linear continuum mechanics necessary for the subsequent Chapters. For an elaborated overview of continuum mechanics, the reader is referred to Malvern(1969), Mars- den & Hughes(1983), Ogden(1984), Haupt(2000), Ba¸sar& Weichert(2000).

2.2 Kinematics of deformable body

2.2.1 Conﬁgurations and body motion Even though in physical reality material body is a discontinuous system which consists of molecules and atoms, from the engineering point of view, material body can be considered as continuum where each particle of the body retains all physical properties of the parent body.

The continuum model for material bodies is acceptable to engineers for two very good reasons. The characteristic dimensions of the scale in which we usually consider bodies of steel, aluminium, concrete, etc., are extremely large in comparison with molecular distances so the continuum model provides a very useful and reliable representation. Additionally, our knowledge of the mechanical behaviour of materials is based almost entirely upon experimental data gathered by tests on relatively large specimens (Mase & Mase, 1999).

Therefore, the deformable body of interest B, depicted in Fig. 2.1, may be considered as a set of continuously distributed material points or particles P ∈ B occupying the region B of the Euclidian point space E. This one-to-one mapping is termed a conﬁguration of B. As the body moves, the region B changes accordingly. The set of conﬁgurations depending on a parameter t, time interval, represents a motion of the body B.

A domain of the body at initial time, t = 0, is termed an initial conﬁguration and denoted B0. A place P0 ∈ E , occupied at initial time by the material point

P ∈ B, is deﬁned with the aforementioned mapping denoted by χ0 according to the relation

P0 = χ0(P ), (2.1) as it is presented in Fig. 2.1.

5 6 Chapter 2. Deformation and motion

A region of E occupied by B at present time is termed a current or Eulerian conﬁguration and denoted Bt. The current position p of the material point P is deﬁned by a mapping χ

p = χ(P, t), (2.2) where the function χ as well as its inverse function χ−1 must be continuous and at least twice continuously diﬀerentiable.

The motion of the body is defined with respect to the reference or Lagrangian configuration. The reference configuration does not need to be the initial configuration or any configuration that was occupied by the material body during the process of deformation. But having in mind that all variables are defined with respect to the reference configuration it is always convenient that the initial and reference configurations are identical, as it has been assumed here.

The process of deformation from the reference to the current conﬁguration in-

Figure 2.1: Undeformed and deformed configuration of the material body cludes changing in shape and position of the body of interest. While the former leads to a varying distance between the arbitrary pairs of particles of the body, the latter reflects a rigid body motion. Both of these phenomena can be observed by one or several physical observers and, therefore, can be described in different ways. But, unlike their kinematical description, physical phenomena do not depend on the choice of the observer. That reflects on the mathematical formulation of physical laws as constitutive models, developed later. 2.2. Kinematics of deformable body 7

An observer O monitors the deformation process from his position o ∈ E. If in the origin o we assume the Cartesian coordinate system with basis ei, the position of the point p, occupied by the material point P in Bt, can be described with a position vector x by the relation

x = xiei, (2.3) as it is depicted in Fig. 2.2. In the last relation the Einstein’s summation convention has been used.

The vector x designates a deformed position of the material point P . The pair (x, t), recorded by O, is termed an event. Components of the vector x, xi in equation (2.3), are called current, spatial or Eulerian coordinates of the point p. Italic indices in the further text will be used for the presentation of vector or tensor components in the current conﬁguration.

The position vector of point P0, X, in the reference conﬁguration, described by

X = XαEα, (2.4) represents an undeformed position of the material point P . In equation (2.4) basis of the Cartesian coordinate system in the origin O are designated with Eα, while Xα are called referential, material or Lagrangian coordinates of the point P0. Greek character indices in the further text will be used for the presentation of vector or tensor components in the reference conﬁguration.

The motion of the body can now be described by

x = ϕ(X, t), (2.5) where the function ϕ(X, t) maps the reference configuration into the current configuration at time t and it is a one-to-one continuously differentiable and invertible function, i.e.

X = ϕ−1(x, t). (2.6)

2.2.2 Lagrangian and Eulerian description A scalar, vector or tensor ﬁeld can be deﬁned by either material or spatial coordinates as independent variables. For example, a displacement u of the material point P from its position in B0 to its position in Bt, depicted in Fig. 2.2, can be expressed in terms of X by

u(X, t) = c + x − X = c(X, t) + ϕ(X, t) − X, (2.7) or in terms of x

u(x, t) = c + x − X = c(x, t) + x − ϕ−1(x, t), (2.8) 8 Chapter 2. Deformation and motion

Figure 2.2: Material and spatial coordinate systems where Eqs. (2.5) and (2.6) have been used.

For the sake of simplicity and clarity, a vector c, representing the distance between the origins O and o, will take the zero value. The displacement u in the material and spatial description is then respectively given by relations

u(X, t) = x − X = ϕ(X, t) − X, (2.9)

u(x, t) = x − X = x − ϕ−1(x, t). (2.10)

Equations (2.9) and (2.10) have diﬀerent physical meanings. While the former expresses the displacement of the particle with the position vector X in B0 at current time t, the letter is the displacement of any particle which is currently at the position x. These two approaches in describing physical ﬁelds are known as the material or Lagrangian description and the spatial or Eulerian description, respectively. The Lagrangian description refers to the behaviour of a material particle, whereas the Eulerian description refers to the behaviour at a spatial position. The choice of the description depends on the nature of the problem under consideration. In this treatise the Eulerian description has been adopted.

If a tensor is completely defined in the Lagrangian configuration it is labelled 2.2. Kinematics of deformable body 9 a Lagrangian tensor, while an Eulerian tensor is completely defined in the current configuration. A second-order tensor field, defined in both, the reference and current configuration, is termed a two-point or mixed Eulerian-Lagrangian tensor.

2.2.3 Material and spatial time derivative In the material description a velocity vector is the rate of change of the position vector for a particle of interest, i.e. the time derivative with X held constant. Time derivatives with X held constant are termed material time derivatives, Lagrangian, total derivatives or shortened material derivatives.

dϕ(X, t) du(X, t) v(X, t) = = ≡ u˙ . (2.11) dt dt An acceleration of the material point in the Lagrangian description is then de- ﬁned as the rate of change of the velocity, or the material time derivative of the velocity, and can be written as

d2u(X, t) dv(X, t) u¨(X, t) = = ≡ v˙ . (2.12) dt2 dt A superposed dot denotes the material time derivative or ordinary time derivative when the variable is only a function of time.

When a variable is given in an Eulerian description, i.e. it is expressed in terms of spatial coordinates and time, the material time derivative, designated as d(•)/dt at ﬁxed X, is obtained by the chain rule for partial derivatives

d(•) ∂(•) (•˙) = | = | + v · grad(•), (2.13) dt X ∂t x where v is the velocity given in the spatial description and grad(•) represents the gradient with respect to the spatial coordinates. In the RHS of Eq.(2.13) the ﬁrst term is a spatial time derivative or Eulerian time derivative, and the second is a convective or transport term. For details see Belytschko et al.(2000).

As an example, the material time derivatives of the velocity v(x, t) and tensor function σ(x, t) will be given by relations

∂v(x, t) ∂v(x, t) u¨(x, t) = v˙ (x, t) = + v(x, t) · , (2.14) ∂t ∂x

∂σ(x, t) ∂σ(x, t) σ˙ (x, t) = + v(x, t) · . (2.15) ∂t ∂x 10 Chapter 2. Deformation and motion

2.3 Analysis of deformation

2.3.1 Deformation gradient

The distance between two close material points P0 and Q0 in the reference con- ﬁguration is determined by the material vector dX (see Fig. 2.2). During the process of deformation this elemental material vector is transforming in a corresponding space vector dx according to the law

dx = F · dX. (2.16)

A tensor F that describes such a kind of transformation is called a deformation gradient. It is a two-point tensor that lives partially in the reference conﬁgura- tion and partially in the current conﬁguration and it can be represented by the relation

∂ϕ(X, t) ∂xi T F = = ei ⊗ Eα = (Grad x) , (2.17) ∂X ∂Xα where the operator Grad(•) represents a gradient with respect to the material coordinates.

Since one spot in space can be occupied by not more than one material point, meaning ϕ is a one-to-one mapping between X and x, the deformation gradient F is always a revertible tensor, i.e.

dX = F−1 · dx. (2.18)

Therefore, and from the reason to avoid physically unrealistic case where the deformation reduces the length of a line element to zero, a Jacobian determinant of the tensor F, or shortly Jacobian, J has to have a non-zero value, i.e.

J = det(F) 6= 0. (2.19)

In most of research literature, relation (2.16) is called a push forward transformation and the space vector dx is the push forward equivalent of the material vector dX. Inversely, relation (2.18) is called a pull back transformation and the material vector dX is the pull back equivalent of the space vector dx.

Similarly to the previously explained line element transformation, according to the Nanson’s formula (cf. Ogden, 1984), a referential surface element dA can be transformed to its current representation da, i.e.

da = JF−T · dA. (2.20)

An elemental volume dV in the reference and its corresponding volume dv in the current conﬁguration are related to each other by the Jacobian as

dv = JdV. (2.21) 2.3. Analysis of deformation 11

From the last equation one additional mathematical requirement for J is that it must be non-negative. Regarding the last statement and Eq. (2.19), we can conclude that

J > 0. (2.22)

The deformation at which J = 1 is called isochoric or volume preserving at X, as in a simple shear case.

2.3.2 Polar decomposition Rotation, especially combined with deformation, is fundamental to nonlinear continuum mechanics. The most important theorem which elucidates the role of rotation in large deformation problems is a polar decomposition theorem (see for example Malvern, 1969).

Following this theorem, the deformation gradient F, as a non-singular second- order tensor, can be uniquely multiplicatively decomposed into a positive deﬁnite symmetric second-order tensor V or U, and an orthogonal second-order tensor R such that

F = V · R = R · U, (2.23) where the rotation tensor R is proper orthogonal, i.e.

RT = R−1 , RT · R = R · RT = 1 with det(R) = 1. (2.24)

From (2.24) it follows that

det(F) = det(V) = det(U). (2.25)

Tensors V and U are called left and right stretch tensors, respectively. V is an Eulerian, and U is a Lagrangian tensor, while R is, as F, a two-point tensor.

In the case of a rigid body motion F = R and U = V = 1, where 1 represents a symmetric unit tensor of the second order. During the deformation where U = V = F, the rotation tensor is a unit two-point tensor and that is the case of a pure strain. In general, deformation consists of the rigid body motion, represented by R, and the stretching, represented by V, that is following the rigid body motion. This decomposition of the deformation is termed a left polar decomposition. In a right polar decomposition the stretching, represented by U, is followed by the rigid body motion, as shown in Fig. 2.3 for one dimensional element dX.

The left stretch tensor V can be obtained from the right stretch tensor U by forward rotation with R and vice versa

V = R ? U = R · U · RT , U = RT ? V = RT · V · R. (2.26) 12 Chapter 2. Deformation and motion

Figure 2.3: Polar decomposition

A star product in the last relation is called a Rayleigh product and it describes a transformation between the Lagrangian and Eulerian conﬁguration. There- fore, V is the Eulerian counterpart of the Lagrangian tensor U and U is the Lagrangian counterpart of the Eulerian tensor V.

Even though V and U live in diﬀerent conﬁgurations, from (2.26) it is obtained that their eigenvalues have the same values and, additionally, in accordance with (2.25) are equal to those of F, because all eigenvalues of R are equal to 1 (cf. Bruhns, 2005).

Sometimes it is more convenient to use the squares of V and U, termed as left and right Cauchy-Green tensors and deﬁned as

B = V2 = F · FT , C = U2 = FT · F. (2.27)

The relation between the Eulerian tensor B and its Lagrangian counterpart C can be deﬁned as

B = R ? C , C = RT ? B. (2.28) 2.4. Analysis of strain 13

2.4 Analysis of strain

2.4.1 Generalized strain measures If the distance dX, between any close material points P and Q (see Fig. 2.2), remains unchanged during the process of deformation, the deformable body B is exposed to a rigid body displacement. Otherwise, the body B is deformed and the length dX becomes dx in the current configuration (cf. Lubliner, 2008). The difference between squares of distances between particles of interest in the reference and current configuration can be expressed in terms of material and spatial coordinates in the following way

|dx|2 − |dX|2 = dX · (FT · F − 1) · dX = 2dX · E · dX (2.29) and

−1 |dx|2 − |dX|2 = dx · (1 − (F · FT ) ) · dx = 2dx · e · dx, (2.30) where E and e are Lagrangian and Eulerian strain measures, respectively.

Variables that give us an insight into the deformation in the vicinity of any material point of the body are elongations (stretches) and shear strains. The set of all elongations and shear strains for all possible directions from one material point deﬁnes a state of deformation at that point and can be described by different either material or spatial strain tensors.

Based on the right and left stretch tensors U and V, Seth, 1964 and Hill (1968, 1978) introduced a general class of Lagrangian and Eulerian strain tensors designated by E(m) and e(m), respectively. Since the application of Cauchy-Green tensors C and B is more comfortable for numerical purposes due to their quadratic form, a modiﬁed deﬁnition of the general class of strain measures is given by

n (m) X E = g(C) = g(bi)Ci i=1 n (2.31) (m) X e = g(B) = g(bi)Bi, i=1 where bi represents n distinct eigenvalues of tensors C and B, while Ci and Bi are the corresponding eigenprojections (for details see Xiao et al., 1998b; Bruhns, 2005).

The scale function g(•) is a smooth and monotonously increasing function with initial conditions 1 g(1) = 0 and g0(1) = , (2.32) 2 14 Chapter 2. Deformation and motion where g0(•) is the ﬁrst derivative of g(•). The scale function is deﬁned as:

( 1 m 2m (bi − 1) for m 6= 0 g(bi) = 1 (2.33) 2 ln(bi) for m = 0, (see Doyle & Ericksen, 1956, and Seth, 1964).

In this way, for diﬀerent values of the parameter m, all commonly known strain measures can be obtained. For example, for m = 1, from Eq. (2.31)1 the material Green-Lagrangian strain tensor can be derived

n 1 X 1 1 2 E = (b − 1)C = (C − 1) = (U − 1), (2.34) 2 i i 2 2 i=1 and from Eq. (2.31)2 the spatial Finger tensor comes as

n 1 X 1 1 2 a = (b − 1)B = (B − 1) = (V − 1). (2.35) 2 i i 2 2 i=1

For m = −1, the material Piola tensor comes from (2.31)1

n 1 X −1 1 −1 1 2 A = (1 − b )C = (1 − C ) = (V − 1) (2.36) 2 i i 2 2 i=1 and the spatial Almansi-Eulerian strain tensor

n 1 X −1 1 −1 1 −2 e = (1 − b )B = (1 − B ) = (1 − V ) (2.37) 2 i i 2 2 i=1 can be obtained from (2.31)2.

In the special case, for m = 0, one can obtain a Hencky or Logarithmic strain tensor in the Lagrangian

n 1 X 1 H = lnb C = lnC = lnU (2.38) 2 i i 2 i=1 and Eulerian description

n 1 X 1 h = lnb B = lnB = lnV. (2.39) 2 i i 2 i=1 The spatial and material logarithmic strain tensors h and H are also known as true or natural strain measures and are of great importance in a ﬁnite elastoplastic deformation description. One of their advantages compared with other strain measures is the additivity property of successive strains. Logarithmic 2.5. Analysis of motion 15 strain measures were disregarded for a long period even though Hencky introduced them in theoretical mechanics in 1928 (cf. Hencky, 1928, 1929a, 1929b).

Having in mind that rotations, accompanied with distortions, in the case of large deformations have a prominent role, the strain measure appropriate for this kind of deformation calculation has to be dependent of the stretching but independent of the rigid body motion. All aforementioned strain measures satisfy this requirement.

As for any other corresponding pair of tensors in the referential and current conﬁguration, relation (2.28) can be applied for all strain tensors as well

e(m) = R ? E(m) and E(m) = RT ? e(m) (2.40)

2.5 Analysis of motion

The deformation gradient F, stretching tensors U and V, Cauchy-Green tensors C and B and strain measures introduced in the last Section as well are quantities independent of time. However, most of formulations of plastic behaviour of materials (viscoplasticity is an obvious example) are given as functions of rates of variables. Even a rate-independent model of plasticity is usually written in a rate form for a numerical implementation into ﬁnite element based programs. Therefore, it is necessary to reconsider how the aforementioned quantities can be expressed in rate form, i.e. as functions of time.

2.5.1 Velocity gradient The velocity of the material point, introduced in Section (2.2.3), is by nature an Eulerian quantity even though in Eq. (2.11) it is expressed in terms of material coordinates. The velocity can be deﬁned as a function of spatial coordinates as well dx v(t) = x˙ = . (2.41) dt A velocity increment dv, which emerges by virtue of a spatial position change dx, in the same deformed conﬁguration (cf. Fig. 2.4), can be written in the form

∂v(x, t) dv(x, t) = · dx. (2.42) ∂x An Eulerian second-order tensor defined as ∂v(x, t) L = = ∇v (2.43) ∂x 16 Chapter 2. Deformation and motion is termed a velocity gradient. It maps the material line element dx in its rate in the current configuration dv = dx˙ = L · dx. (2.44) Contrary to the deformation gradient that describes the local deformation state in the material point P , the velocity gradient L describes a rate of change of a local deformation state of the particle P , and, contrary to F, it is not related to the reference configuration. As it is depicted in Fig. 2.4, L is the spatial tensor defining the relative velocity of the material point at the current position q in comparison with the velocity of the material point occupying the current position p.

Figure 2.4: Velocity gradient

The material derivative of the deformation gradient, which is deﬁned by Eq. (2.16), can be written as d ∂ϕ(X, t) F˙ = = ∇ ⊗ v(X, t) = (∇ ⊗ v(x, t)) · F = L · F. (2.45) dt ∂X 0 From the last relation, another for a numerical implementation more convenient deﬁnition of the velocity gradient follows L = F˙ · F−1. (2.46) A decomposition of the velocity gradient tensor on its symmetrical part, related to stretching, and skew-symmetric part, related to rotations, can be performed on the following way L = D + W. (2.47) 2.5. Analysis of motion 17

The tensor D is termed as a rate of deformation tensor or a stretching tensor and it can be calculated using the following formula

1 1 D = (L + LT ) = (∇v + (∇v)T ), (2.48) 2 2 while the tensor W is named a vorticity tensor or a spin tensor, and can be obtained by the relation

1 1 W = (L − LT ) = (∇v − (∇v)T ), (2.49) 2 2 (see Malvern, 1969, and Mi´cunovi´c, 1990).

2.5.2 Rate of deformation The material rate of the Green-Lagrangian strain tensor, given by Eq. (2.34), and using relation (2.27)2, can be determined in the following way

1 1 T 1 E˙ = C˙ = (F˙ · F + FT · F˙ ) = FT · (L + LT ) · F = FT · D · F. (2.50) 2 2 2

The last relation demonstrates that the material rate of deformation E˙ represents a pull-back equivalent of the Eulerian rate of deformation tensor D. Both quantities define the rate of change of the scalar product of two elemental vectors in the current configuration (cf. Bonet & Wood, 2008). While the latter gives the rate of change in terms of elemental vectors in the current configuration, the former gives the rate of change in terms of corresponding elemental vectors in the reference configuration. We can conclude from here that E˙ and D are the Lagrangian and, respectively, Eulerian measure of the rate of change of the material line length as well as the rate of change of the angle between two material line elements of interest, during the process of deformation.

Even though the aforementioned conclusions are generally accepted in nonlinear continuum mechanics, until recently it was considered that the stretching tensor D cannot be deﬁned either as a Lagrangian or as an Eulerian strain rate tensor and therefore it was not considered as a rate of deformation (see Ogden, 1984). However, Xiao et al.( 1997b, 1998b) proved that the stretching tensor D can be integrated to give the Hencky strain tensor h, deﬁned in the Eulerian description by relation (2.39).

2.5.3 Vorticity tensor In order to understand better the nature of the vorticity tensor, which deﬁnes the rotation of the body during deformation, further transformations will be performed. 18 Chapter 2. Deformation and motion

The material rate of the deformation gradient, deﬁned by polar decomposition (2.23)1, is given by

F˙ = R˙ · U + R · U˙ . (2.51)

Relation (2.46) can be now rewritten as

L = F˙ · F−1 = (R˙ · U + R · U˙ ) · U−1 · R−1 = R˙ · RT + R · U˙ · U−1 · RT . (2.52)

The stretching and vorticity tensors are then given by relations

1 D = R · (U˙ · U−1 + U−1 · U˙ ) · RT (2.53) 2 and

1 W = R˙ · RT + R · (U˙ · U−1 − U−1 · U˙ ) · RT . (2.54) 2

The ﬁrst term on the RHS of the last equation represents an angular velocity tensor and it depends on the rigid body rotation and its rate but not on the stretching of the body

Ω¯ = R˙ · RT . (2.55)

As it can be seen from relations (2.53) and (2.54), decomposition of the velocity gradient on the vorticity and stretching tensor is evidently more complicated than decomposition of the deformation gradient on pure rotation and pure stretching (see (2.23)). The reason for that lies in the fact that the vorticity tensor, apart from the rigid body rotation, depends on the elongation and elongation rate.

In the absence of deformation, i.e. in the case of a rigid body motion, D = 0, resulting in W = L = Ω¯ . The increment of the relative velocity of the material point Q, occupying the position q, in comparison with the velocity of the particle P , that occupies the position p, as it is depicted in Fig. 2.5, can then be deﬁned as

dv = W · dx = ω × dx, (2.56) where ω is the angular velocity vector obeying the rule

Ω¯ · r = ω × r, (2.57) for each vector r. 2.6. Objectivity of a tensor ﬁeld 19

Figure 2.5: Angular velocity vector

2.6 Objectivity of a tensor ﬁeld

As it was stated in Section 2.2.1, the same process of deformation can be moni- tored by two different observers O and O∗ that occupy different positions in the space, o and o∗ respectively, and can move relative to one another. Moreover, the observers can monitor the same process with time difference, but they must agree on the space and time distance recorded by them while they are following the movement of a same material point. Using the language of mathematics, there exists a one-to-one mapping between the event (x, t), recorded by O, and the event (x∗, t∗) recorded by O∗. This mapping is termed a change of the observer or observer transformation (cf. Ogden, 1984) that is represented by the relation

x∗ = Q(t) · x + c(t) and t∗ = t − a, (2.58) where Q is the proper orthogonal tensor of relative rotation and c is the vector of relative translation of one observer relatively to another, while with the scalar quantity a time distance in records has been designated. This scalar quantity will be disregarded in the further text, i.e. it will be considered that a = 0.

Physical processes are independent of an observer. Therefore a demand is that tensor ﬁelds (scalars, vectors or tensors of higher rank) that qualitatively and quantitatively describe those physical processes stay independent of the change of the observer i.e. must stay objective.

2.6.1 Lagrangian and Eulerian objectivity According to Ogden(1984), the objectivity feature depends on the conﬁguration in which the quantity has been deﬁned.

Scalar quantity α0 , vector α0 and second-order tensor A0, deﬁned in the La- 20 Chapter 2. Deformation and motion grangian conﬁguration, are objective if they stay unchanged after the change of the observer, i.e.

∗ ∗ α0(X, t ) = α0(X, t) ∗ ∗ α0(X, t ) = α0(X, t) (2.59) ∗ ∗ A0(X, t ) = A0(X, t). Concerning Eulerian quantities, scalar α , vector α and second-order tensor A, are objective if they obey the following rules of transformation by virtue of the change of the observer

α∗(x, t∗) = α(x, t) α∗(x, t∗) = Q(t) · α(x, t) = Q(t) ? α(x, t) (2.60) A∗(x, t∗) = Q(t) · A(x, t) · Q(t)T = Q(t) ? A(x, t).

For a Lagrange-Eulerian, or Euler-Lagrangian two-point tensor, deﬁned as

Aˆ = α0 ⊗ α, or A˘ = α ⊗ α0 (2.61) the objectivity criteria is deﬁned as

∗ ∗ Aˆ = Aˆ · QT , and A˘ = Q · A˘ , (2.62) respectively.

Based on the aforementioned transformation rules, in Table 2.6.1 the kinematical quantities introduced in this Chapter and their transformed forms have been presented and their objectivity examined. It must be pointed out that if the Eulerian tensor is objective its corresponding Lagrangian tensor is objective as well, and vice versa. (see Xiao et al., 1998c; Bruhns, 2005).

2.6.2 Objective time derivatives If two coordinate systems are associated with the positions of the observers O and O∗, i.e. with o and o∗, respectively, transformation between O and O∗ can be also determined as a change of frame. As it can be seen from Eq. (2.59)3, the Lagrangian strain tensors are not aﬀected by the change of the observer, i.e. by the change of frame. Thus, the material time derivative of the transformed Lagrangian second-order tensor, given by the relation ˙ ∗ ˙ A0 = A0, (2.63) satisﬁes the objectivity requirement as well.

Contrary to the Lagrangian tensor, the objective Eulerian second-order tensor 2.6. Objectivity of a tensor ﬁeld 21

Transformed quantity Quantity Objective Lagrangian Eulerian Two-point FF∗ = Q · F yes JJ ∗ = J yes RR∗ = Q · R yes VV∗ = Q ? V yes UU∗ = U yes BB∗ = Q ? B yes CC∗ = C yes LL∗ = Q ? L + Q˙ · QT no DD∗ = Q ? D yes WW∗ = Q ? W + Q˙ · QT no

Table 2.1: Transformation and objectivity of kinematical quantities

transforms according to (2.60)3 and therefore it is dependent of the change of frame. From that follows that the material time derivative of the objective Eulerian second-order tensor will be transformed according to the relation

∗ ˙ A˙ = Q ?˙ A = Q · A · QT = Q˙ · A · QT + Q · A˙ · QT + Q · A · Q˙ T. (2.64)

From the last relation it turns out

∗ A˙ 6= Q · A˙ · QT, (2.65) and thus it can be concluded that the material time derivative of the objective Eulerian second-order tensor is not an objective quantity. To preserve the objectivity requirement in the Eulerian description one has to use instead of the material time derivative an objective time derivative that satisﬁes the relation

∗ A˙ = Q ? A, (2.66)

where A is the objective time derivative of the Eulerian second-order tensor A.

2.6.3 Corotational and convective frame Let us consider that the observer O is located at the fixed point of space o, while the observer O∗ is situated on the moving body at point o∗ and it moves and rotates together with the deformable body. The sitting point of the observer O, o, is the origin of a so-called fixed background frame, while o∗ is then the origin of a so-called co-deforming frame. In that case relation (2.58) represents the transformation between the background and co-deforming frame. That means that the pair (x,t) represents the point in the Galilean space-time, occupied by 22 Chapter 2. Deformation and motion the particle P , observed by O from the background frame, while the pair (x∗,t) represents the same point in the space observed by O∗ from the transformed moving frame. Both observers have recorded the position of the particle at the same time, i.e. the time difference vanishes.

We will now rewrite relation (2.58) and assume transformation between the frames as

x∗ = K(t) · x + c(t) and t∗ = t, (2.67) where the time dependent tensor K, is not a proper orthogonal but general asymmetric second-order tensor determined by the following ﬁrst-order diﬀeren- tial system with a prescribed initial value

K˙ = Ψ · K, K|t=0 = 1. (2.68)

In the previous relation Ψ is the asymmetric second-order tensor given by

Ψ = Ω + Γ, (2.69) where Ω is the antisymmetric part and Γ is the symmetric part of Ψ. The skew-symmetric tensor Ω is called a spin and has the properties of the angular velocity Ω¯ introduced in Section 2.5.3. Transformation of an objective Eulerian tensorial quantity A, deﬁned in the background frame, to A∗ in the co-deforming frame can be determined by the following transformation rule

A∗ = K · A · KT = K ? A. (2.70)

Accordingly, the material time derivative of the transformed quantity A∗ can be deﬁned as

∗ A˙ = K ?˙ A = K˙ · A · KT + K · A˙ · KT + K · A · K˙ T = K ? A, (2.71) where the objective time derivative, introduced in the previous Section, is determined by the tensor Ψ as

A= A˙ + A · Ψ + ΨT · A. (2.72)

The kinematical property of relation (2.71) can be understood in such a way that a material time derivative of a counter part of an Eulerian quantity A in a ∗ co-deforming frame, that is A˙ , is a co-deforming counter part of an objective time derivative of the same quantity A in a background frame.

The objective rates of the symmetric Eulerian second-order ﬁeld have been so far given in a general form A. In modelling of various material behaviours, using the formulation with Eulerian rates, objective rates play an essential role. 2.6. Objectivity of a tensor ﬁeld 23

Therefore, the type of the objective rate should be carefully chosen. Depending on the choice of the tensor Ψ the objective rates can be generally classiﬁed in two categories of corotational and non-corotational objective rates ( cf. Bruhns et al., 2004).

Whenever the symmetric part of Ψ, given by relation (2.69), vanishes, i.e. Ψ becomes equal to the spin Ω, the frame deﬁned by Eqs. (2.67) and (2.68) is a spinning or corotating frame and K is then equal to the rotation tensor. Oth- erwise, Ψ determines a convective frame. While the former experiences only constant rotation the latter can deform and rotate continuously during the deformation process. In the case of a convective frame the coordinate system in o∗ is no longer a Cartesian coordinate system and under the change of frame a physical or kinematical quantity can lose some important features. For example, the eigenvalues of the quantity of interest can be modiﬁed during the process of deformation. If we want to preserve the physical or kinematical features of a physical or a kinematical tensor, the tensor Ψ must be a skew-symmetric tensor, meaning Ψ = Ω. That leads to K = Q, i.e. K is the proper orthogonal rotation tensor. For details see Bruhns et al.(2004) and Xiao et al.(2005). Integration of (2.71) leads to the generalised objective time integration of the objective rate

Z A = K−1 · K · A· KTdt · K−T, (2.73) t and it is applied in the co-deforming frame. This relation will be very useful for the numerical application which results are presented in Chapter5.

2.6.4 Non-corotational rates The objective non-corotational rate of the objective Eulerian quantity can be generally deﬁned as

5 A≡ A˙ + A · Ψ + ΨT · A. (2.74)

Two classes of objective non-corotational rates were introduced by Hill(1968, 1970, 1978).

5 A≡ A˙ + A · W − W · A − m (A · D + D · A), (2.75)

5 A≡ A˙ + A · W − W · A + tr (D) · A − m (A · D + D · A), (2.76) where m is any given real number. With the particular values for m diﬀerent particular objective rates can be deﬁned from Eqs. (2.75) and (2.76). If one introduces m = 0, −1, 1 the former class yields the Jaumann rate, the lower 24 Chapter 2. Deformation and motion

Oldroyd, known as Cotter-Rivlin rate as well, and the upper Oldroyd rate, respectively. The latter class produces the Truesdell rate and Durban-Baruch rate, respectively, for m = 1, 0.5. With the speciﬁc choice of Ψ

Ψ = W + mD + c tr (D)1 (2.77) in (2.74), even a broader class of objective non-corotational rates can be deﬁned

5 A≡ A˙ + A · (W + mD + c tr (D)1) + (W + mD + c tr (D)1)T · A, (2.78) where m and c are real numbers. With a speciﬁc choice for m and c in the last relation the well known rates can be obtained: the (upper) Oldroyd rate (cf. Oldroyd, 1950)

5 A Ol ≡ A˙ − L · A − A · LT for m = −1 and c = 0, (2.79) the Cotter-Rivlin rate (cf. Cotter & Rivlin, 1955)

5 A CR ≡ A˙ + LT · A + A · L for m = 1 and c = 0, (2.80) the Truesdell rate (cf. Truesdell, 1953)

5 A Tr ≡ A˙ − L · A − A · LT + tr (D) · A for m = −1 and c = 0.5. (2.81)

The Oldroyd and Cotter-Rivlin objective rates have a speciﬁc property that the Oldroyd rate of the Finger tensor and the Cotter-Rivlin rate of the Almansi strain tensor are exactly the stretching, i.e.

5 a Ol = a˙ − L · a − a · LT = D, (2.82)

5 e CR = e˙ + LT · e + e · L = D, (2.83) but this conclusion cannot be generalised for all possible rates obtained from classes (2.75) and (2.76) (or (2.78)) for any m ( and c).

For the derivation of a general class of non-corotational rates that are satisfying the demand that the stretching D can be expressed as a Hill type non- corotational rate of any given Seth-Hill strain, refer to Bruhns et al.(2004). 2.6. Objectivity of a tensor ﬁeld 25

2.6.5 Corotational rates As it was pointed out in Section 2.6.3, the symmetric part of tensor Ψ may vanish. In that case Ψ = Ω. The rotating, or co-deforming, frame is then determined by the skew-symmetric second-order Eulerian tensor Q instead of a general asymmetric second-order tensor K, introduced in (2.67). The skew- symmetric spin tensor Ω, determining the rotating frame, is deﬁned by:

Ω = Q˙ T · Q = −QT · Q˙ = −ΩT. (2.84)

The rotating frame becomes the corotating frame and the general objective time ◦ derivative A becomes the corotational rate A, deﬁned as

◦ A = A˙ + A · Ω − Ω · A. (2.85)

Transformation of the objective Eulerian second-order tensor quantity from the background to the corotating frame is described by the relation

A∗ = Q · A · QT = Q ? A. (2.86)

Similarly to (2.71), the material time derivative of the tensor A∗ in the corotating frame is deﬁned by

∗ ◦ A˙ = Q ? A, (2.87) and it can be concluded that the corotational rate of the objective Eulerian tensor A corresponds to the material rate of A in the corotating frame (cf. Xiao et al., 1998a). The last relation does not hold for the tensors that are not objective.

For different choices of the spin tensor Q different corotational rates can be determined. But, even though, for the chosen spin, relation (2.87) satisfies transformation rule (2.60)3, the corresponding corotational rate need not be an objective quantity. That means that the crucial demand of objectivity of corotational rate (cf. Truesdell et al., 2004) may be violated. Having in mind that the objective corotational rate has the fundamental importance in a material behaviour description, especially of an inelastic behaviour, the significance of the proper choice of the objective rate and their defining spin tensors, is again pointed out.

Inspired by the introduction of a general class of strain measures through a single scale function (cf. Section 2.4.1), Xiao et al. in (1998a) and (1998b) deﬁned a general class of spin tensors and corresponding general class of objective corotational rates by introducing a single antisymmetric real function called spin function. The general classes deﬁned in this way include all known spin tensors and corresponding rates in a natural way.

Any spin tensor Ω has to fulﬁl the certain necessary requirements (cf. Xiao et al. 26 Chapter 2. Deformation and motion

1998b) and then it becomes a kinematical quantity of the same kind as the vorticity tensor W. The general class of spin tensors, for which the corotational rate of an Eulerian tensor A is an objective quantity, can be deﬁned as

Ω = W + Υ(B, D), (2.88) where Υ denotes an isotropic skew-symmetric tensor-valued function depending on the left Cauchy-Green tensor B and the stretching D.

The general class of spin tensors can be written as a function of the spin function h as n X bi bj Ω = W + h , Bi · D · Bj, (2.89) I1 I1 i6=j where

I1 = tr (B) = tr (C). (2.90)

The remaining quantities are already introduced in Section 2.4.1.

A subclass of general class (2.89), given by the relation n n X bi X Ω = W + h Bi · D · Bj = W + h(z) Bi · D · Bj, (2.91) bj i6=j i6=j is broad enough to include all known spin tensors, and therefore make their numerical implementation easier.

In (2.91) the simpliﬁed spin function h(z) has the property

h(z−1) = −h(z), ∀z > 0. (2.92)

The choice of the spin function as

h(z) = hJ(z) = 0 (2.93) yields the spin tensor

ΩJ = W, (2.94) which implemented in (2.85) deﬁnes the well-known Zaremba-Jaumann rate

◦ A J = A˙ + A · W − W · A (2.95)

(cf. Zaremba, 1903, and Jaumann, 1911).

The Jaumann rate was the first introduced in the rate formulation of inelastic material behaviour. It is widely used since it can be relatively easy implemented 2.6. Objectivity of a tensor field 27 in numerical calculations. It is also incorporated in several commercial finite element codes. But, even though the use of the Jaumann rate in constitutive theories gives appropriate results for the case of small deformations it is proved that this rate is not an adequate choice for the case of finite deformations (cf. Lehmann, 1972; Dienes, 1979; Simo & Pister, 1984; Szabó& Balla, 1989; Khan & Huang, 1995; Baˇzant & Vorel, 2014). This topic will be examined in more details in Chapters4 and5.

In order to overcome the deﬁciencies encountered during the Jaumann rate implementation in ﬁnite deformation formulation, numerous alternative corotational rates have been developed (cf. Xiao et al.2000a).

One of them is the polar or Green-Naghdi rate. If the spin function takes the form √ 1 − z h(z) = hR(z) = √ , (2.96) 1 + z the polar spin ΩR will be obtained

n pb pb R ˙ T X j − i Ω = R · R = W + p p Bi · D · Bj, (2.97) i6=j bj + bi which substituted in (2.85) deﬁnes the Green-Naghdi rate

◦ A GN = A˙ + A · ΩR − ΩR · A, (2.98)

(see Green & Naghdi, 1965, and Green & McInnis, 1967).

In this treatise, the special attention will be given to the choice of the so-called logarithmic spin function √ 1 + z 2 h(z) = hLog(z) = √ + , (2.99) 1 − z ln(z) that leads to the logarithmic spin tensor

n Log Log Log T X 1 + (bi/bj) 2 Ω = R˙ · (R ) = W + + Bi · D · Bj. 1 − (bi/bj) ln (bi/bj) i6=j (2.100)

The implementation of the logarithmic spin in (2.85) yields the logarithmic corotational rate or Log-rate of A

◦ A Log = A˙ + A · ΩLog − ΩLog · A. (2.101) 28 Chapter 2. Deformation and motion

For more details the reader is referred to Lehmann et al.(1991), Reinhardt & Dubey(1996) and Xiao et al. (1997b).

Since the symmetric stretching tensor D is a natural characterization of the rate of change of the local deformation state, we want to present it as a direct ﬂux of a strain measure. Therefore we are interested which Eulerian strain measure and which corotational time derivative satisfy the following relation

◦ e(m) = e˙ (m) + e(m) · Ω − Ω · e(m) = D. (2.102)

In Xiao et al. (1997b), (1998a), (1998b) the authors proved that among all strain measures and among all corotational rates the spatial logarithmic strain measure h and the logarithmic rate are an unique choice that satisﬁes the above demand, i.e.

◦ h Log = h˙ + h · ΩLog − ΩLog · h = D. (2.103)

As it can be seen from (2.100), the logarithmic spin is determined by the proper orthogonal logarithmic rotation tensor RLog that is deﬁned by the linear tensorial diﬀerential equation

Log Log Log Log R˙ = −R · Ω , R |t=0 = 1. (2.104)

The corotating frame obtained from the background frame by the rotation RLog is named a logarithmic corotating frame. In the logarithmic corotating frame the material time derivative of an objective Eulerian quantity A is exactly the logarithmic rate of the same quantity, i.e.

˙ ◦ RLog ? A = RLog ? A Log. (2.105)

Applying the last assertion to h and the stretching D, the following relation will be obtained ˙ RLog ? h = RLog ? D, (2.106) that means, in the logarithmic corotating frame stretching D is a true time rate of h.

Integration of (2.105) leads to the corotational integration

Z ◦ A = (RLog)T ? RLog ? A Logdt , (2.107) t and it is applied in the logarithmic corotating frame. The last relation is a particular form of the generalised objective time integration given by (2.73). This relation will be of great importance in further numerical calculations. 2.7. Decomposition of ﬁnite deformation 29

2.7 Decomposition of ﬁnite deformation

Elastoplasticity represents a combination of two completely different types of material behaviour, namely elasticity and plasticity. Most of the modern theories of elastoplasticity are confined to the description of a rate-independent behaviour of elastoplastic materials; that means that viscous effects are ignored. Before 1960, most contributions in the rate-independent elastoplasticity theory were confined to the field of small deformations. Some of the basic ideas of the theories for small deformations can be fully or partially applied for the case where finite deformations are occurring (cf. Naghdi, 1990, and Xiao et al., 2006a).

One of those ideas is the composite structure of elastoplasticity. It means that a total deformation, or total deformation rate, of elastoplastic material can be decomposed into its elastic, or reversible, and plastic, or irreversible, part and then a separate constitutive relation for each part has to be established. Having in mind the incremental essence of elastoplastic behaviour of material, we are more interested in the strain rate than the strain itself. The rate of inﬁnitesimal strain ε can be additively decomposed in the following form:

ε˙ = ε˙ e + ε˙ p. (2.108)

Even though the researchers agree with the aforementioned statement for the case of small deformations, the decomposition of ﬁnite deformation on its reversible and irreversible part causes the disagreement within the members of the plasticity community dividing them into several various schools of plasticity.

The ﬁrst group are the followers of the idea that the classical Prandtl and Reuss formulation (2.108) can be extended to the ﬁnite deformation description using the additive decomposition of the natural deformation rate to its elastic and plastic part, i.e.

D = De + Dp. (2.109)

It was thought that the above decomposition of the stretching could hold only for certain restrictive cases of deformation and materials, such as small elastic and ﬁnite plastic deformations in metals (Simo & Hughes, 1998). The reason for that was comming from the fact that in order to fulﬁll the objectivity requirement the rate type model must involve an objective rate, instead of the material time derivative, and the use of Jaumann and some other well-known objective rates in the context of decomposition (2.109) produced irregular results. Therefore, this decomposition was refused as inappropriate for a general purpose. This held true until recently when an implementation of the newly discovered logarithmic rate solved the existing problems (Xiao et al., 1997b; Bruhns et al., 1999). This topic will be discussed in detail in Chapter4.

The second approach is the most common in the ﬁnite deformation theories; that 30 Chapter 2. Deformation and motion is the multiplicative decomposition of the deformation gradient, given by F = Fe · Fp, (2.110) where Fe and Fp represent elastic and plastic part of the deformation gradient, respectively.

This formulation introduces an intermediate stress-free configuration B¯t achieved by an elastic unloading from the current configuration Bt. Therefore, while the mapping from the reference to the current configuration is described by the deformation gradient F, the mapping from the reference to the intermediate configuration can be described by its plastic part Fp and from the intermediate to the current configuration by its elastic part Fe.

Since an arbitrary rigid body rotation Q superimposed on the intermediate configuration has no effect on the decomposition (2.110), the determination of the elastic and plastic part of deformation gradient is not unique, i.e. F = Fe · Fp = F¯ e · F¯ p, (2.111) where F¯ e = Fe · Q (2.112) and F¯ p = QT · Fp. (2.113) Additionally, according to Naghdi (cf. Naghdi, 1990), the multiplicative decomposition (2.110) has several shortcomings. The first ”lies in the fact that the stress at a point in an elastic-plastic material can be reduced to zero without changing plastic strain only if the origin in stress space remains in the region en- closed by the yield surface.” This is usually not the case, except of some special cases such as isotropic hardening, because the yield surface moves in the stress space due to deformation. Another shortcoming is that ”even if the stress can be reduced to zero at each material point, the resulting configuration will not, in general, form a configuration for the body as a whole, but only a collection of local configurations.”

The third physically admissible decomposition was established by Green & Naghdi (1965) postulating the additive decomposition of the Green strain, the Lagrangian strain measure. The authors introduced the strain-like variable of a Lagrangian type, called a plastic strain Ep, as a primitive variable. The total Green strain can then be decomposed using the following form E = Ee + Ep, (2.114) where only for the case of small deformations Ee can be denoted as elastic strain. Having well understood the limited applicability of the additive separation of E 2.7. Decomposition of ﬁnite deformation 31 for large deformations, the authors did not interpret the diﬀerence E − Ep as an elastic strain or part, but as an alternative convenient variable used for well motivated purposes.

Since the Eulerian description is the topic of this treatise, the pure Eulerian ﬁnite elastoplasticity theory based on the additive decomposition of the stretching, summarised in Xiao et al. (2006a), will be adopted here.

As a recommendation, the reader is refer to Xiao et al. (2000b), where the authors developed the approach that consistently combining the additive and multiplicative decompositions of the stretching and deformation gradient. The authors proved that the latter combination results in a unique determination of the elastic and plastic deformation Fe and Fp and all their related kinematical quantities, and that the new theory, based on the use of the logarithmic rate in constitutive relations, overcomes the discrepancies that were appearing with application of the other presented decompositions.

Once the elastic deformation Fe is available, the plastic deformation Fp can be obtained from (2.110) as Fp = Fe−1 · F. (2.115) The material time derivative of the deformation gradient can be derived from (2.110) as F˙ = F˙ e · Fp + Fe · F˙ p. (2.116) If we recall relation (2.46), that is deﬁning the velocity gradient, L = F˙ · F−1, (2.117) and apply it to the elastic and plastic part of the velocity gradient, we will obtain

Le = F˙ e · Fe−1, Lp = F˙ p · Fp−1. (2.118) Using (2.110), (2.115), (2.116) and the last relations, (2.117) can be rewritten as L = F˙ e · Fe−1 + Fe · F˙ p · Fp−1 · Fe−1 = Le + Fe · Lp · Fe−1. (2.119) Substituting the last equation in (2.48) D = sym(L) = sym(Le + Fe · Lp · Fe−1), (2.120) and assuming decomposition (2.109), the ﬁnal forms for the elastic and plastic part of the deformation rate can be obtained as De = sym(Le) = sym(F˙ e · Fe−1), (2.121)

Dp = sym(Fe · Lp · Fe−1). (2.122) 3 Conservation equations and stress measures

3.1 Introduction

The main interest of engineers is always a determination of a structural response to applied loads.

The ﬁrst group of laws known as general principles, that determine a relation between the applied forces and the motion of the body of interest, are independent of the material properties. The brief overview of these principles will be presented in this Chapter.

For the correct deﬁnition of constitutive equations, the examination of appropriate stress measures and stress rates is necessary and will be done in this Chapter, as well.

3.2 Conservation laws

3.2.1 Conservation of mass A deformable body B is assumed as a part of space continuously ﬁlled with matter and during the process of deformation it changes its shape and dimensions but not the amount of material it consists of.

A mass m of the body B is a positive scalar independent of the body motion. That means that its change during the time must vanish, i.e.

m˙ (B) = 0. (3.1)

The last relation is known as a continuity equation and it represents the principle of conservation of mass.

The mass m can be deﬁned as Z Z m = ρ0 dV = ρ dv, (3.2)

B0 Bt where with ρ0 and ρ scalar ﬁelds of a mass density in the reference and the current conﬁguration, respectively, are designated. dV and dv are the elemental volume in B0 and Bt respectively, as it was pointed out in Section 2.3.1.

32 3.2. Conservation laws 33

Relation (3.2) can be rewritten as Z Z Z ρ0 dV − ρ dv = (ρ0 − Jρ) dV = 0, (3.3)

B0 Bt B0 where Eq. (2.21) has been used. From the last relation the following conclusion can be derived

−1 ρ = J ρ0. (3.4)

Combining Eqs. (3.1) and (3.2) we will obtain d Z m˙ = ρ dv = 0. (3.5) dt Bt The application of the Reynold’s transport theorem

d Z Z df Z f(x, t) dv = + fdiv(v) dv = f˙ + f∇ · v dv dt dt Bt Bt Bt in Eq. (3.5) leads to the global form of continuity equation Z (˙ρ + ρ∇ · v) dv = 0. (3.6)

The above relation is valid for any subdomain of Bt. Therefore, the integrand must vanish as well, and the local form of continuity equation can be written as

ρ˙ + ρ ∇ · v = 0. (3.7)

3.2.2 Conservation of linear momentum and Cauchy’s first law of motion A linear momentum I of the material body in the Eulerian description represents a vector defined by the velocity field and the mass of the body as Z Z I(x, t) = v(x, t) dm = ρ(x, t)v(x, t) dv. (3.8)

Bt Bt Its rate of change is equal to the vectorial sum of all forces acting on the body of interest d d Z Z dv I˙ = I = ρv dv = ρ dv = Fall, (3.9) dt dt dt Bt Bt 34 Chapter 3. Conservation equations and stress measures where the second form of the Reynold’s transport theorem for a density-weighted integrand d Z Z ρf dv = ρf˙ dv (3.10) dt Bt Bt has been applied.

all A resulting force F is the sum of the volume or body-force acting over Bt and the surface or contact force distributed over ∂Bt. Z Z Fall = ρb dv + t da, (3.11)

Bt ∂Bt where with b a body-force density is denoted while the integrand of the contact force is termed as a contact-force density, traction or Cauchy stress vector t. Both densities, b and t, are objective Eulerian quantities.

Combining (3.8)-(3.11), a global form of the principle of conservation of linear momentum can be written as Z dv Z Z ρ dv = ρb dv + t da. (3.12) dt Bt Bt ∂Bt According to the Cauchy’s fundamental principle (cf. Ogden, 1984) the stress vector t at point x, dependent of the unit normal n on the surface in x, can be transformed into the second-order tensor ﬁeld, independent of n, following the transformation relation known as Cauchy’s theorem

t(x, n, t) = σ(x, t) · n. (3.13)

The Eulerian second-order tensor ﬁeld σ is an objective quantity called a Cauchy stress tensor or true stress tensor. Given in the component form, the Cauchy’s theorem is given as

ti = σijnj . (3.14)

Using (3.13) and the divergence theorem, the balance of linear momentum (3.12) can be rewritten in the form Z dv Z Z Z ρ dv − ρb dv − σ · n da = (ρv˙ − ρb − ∇ · σT) dv = 0. (3.15) dt Bt Bt ∂Bt Bt Since the RHS of the last equation is deﬁned for an arbitrary domain, the integrand must vanish as well, i.e.

∇ · σT + ρb = ρv˙ . (3.16) 3.2. Conservation laws 35

The last relation represents the local form of the balance of linear momentum known as Cauchy’s first law of motion and it is the key equation in the nonlinear finite element procedures (Belytschko et al., 2000). In a component form the Cauchy’s first law of motion can be written as

∂σij + ρbi = ρv˙i. (3.17) ∂xj

3.2.3 Conservation of angular momentum and Cauchy’s second law of motion For the arbitrarily chosen ﬁxed point in the Euclidean space, whose position vector is designated with x0, a rotational or angular momentum or moment of momentum of the body B can be deﬁned as Z ρ(x − x0) × v dv, (3.18)

Bt where with x a position vector of an arbitrary particle of the body B has been designated.

The same as for the linear momentum, the rotational momentum is inﬂuenced by the forces to which the material body B is subjected. The distribution of the surface and volume force densities b and t forms a system of forces that determines a resultant torque Z Z M0 = ρ(x − x0) × b dv + (x − x0) × t da. (3.19)

Bt ∂Bt

The resultant torque M0 is strongly dependent of the position vector x0 of the chosen point to which the rotational momentum refers and therefore it is not an objective quantity.

The rate of change of the angular momentum is equal to the resultant torque and hence the global form of the principle of balance of rotational momentum can be deﬁned as d Z Z Z ρ(x − x ) × v dv = ρ(x − x ) × b dv + (x − x ) × t da. (3.20) dt 0 0 0 Bt Bt ∂Bt Introducing the Cauchy’s ﬁrst law of motion (3.16) and the Cauchy’s theorem (3.13) into the balance of rotational momentum the following equation will be obtained Z Z T (x − x0) × ∇ · σ dv = (x − x0) × (σ · n) da. (3.21)

Bt ∂Bt 36 Chapter 3. Conservation equations and stress measures

Using the properties of a cross product the last relation can be rewritten as Z T T [(x − x0) ⊗ (∇ · σ ) − (∇ · σ ) ⊗ (x − x0)] dv

Bt Z = [(x − x0) ⊗ (σ · n) − (σ · n) ⊗ (x − x0)] da . (3.22)

∂Bt Applying the divergence theorem to the surface integral and the following property

T T ∇ · (σ ⊗ (x − x0)) = (∇ · σ ) ⊗ (x − x0) + σ (3.23) the balance of the angular momentum in the global form leads to Z (σ − σT) dv = 0. (3.24)

Bt Since the integral holds for an arbitrary domain

σT − σ = 0, i.e σT = σ. (3.25)

The last relation is known as the Cauchy’s second law of motion and can be written in the component form as

σij = σji. (3.26)

The balance of mass and the Cauchy’s first and second law of motion are defined in the current configuration and make a set of Eulerian field equations

Conservation of massρ ˙ + ρ ∇ · v = 0 Cauchy’s ﬁrst law of motion ∇ · σT + ρb = ρv˙ Cauchy’s second law of motion σT = σ.

3.3 Stress measures and stress rates

3.3.1 Alternative stress measures In nonlinear continuum mechanics, besides already introduced true stress σ, several alternative stress measures are in use and will be introduced here.

Stress measures, as their corresponding strain measures presented in the preceding Chapter, can be defined in the current or reference configuration. Some measures are connected with both configurations and are defined by two-point tensors. 3.3. Stress measures and stress rates 37

The load vector dl will be introduced as an Eulerian measure deﬁned as a traction t acting on the deformed inﬁnitesimal area da. It can be represented as a function of true stress σ using the Cauchy’s postulate (3.13) as

dl = t da = σ · n da = σ · da, (3.27) where n is the outward unit normal vector to the current elemental surface.

The Eulerian load vector can be expressed in terms of Lagrangian quantities as

dl = Jσ · F−T · dA = Jσ · F−T · N dA = T · N dA = T · dA, (3.28) where T is the first Piola-Kirchhoff stress defined as

T = Jσ · F−T. (3.29)

In relation (3.28) Nanson’s formula (2.20) for transformation of the current to the referential surface element has been used. With N the unit normal vector to the surface element in the reference conﬁguration has been designated.

The resultant contact force on the boundary ∂Bt of the current conﬁguration Bt in terms of the force on the referential boundary ∂B0 can be deﬁned as Z Z σ · n da = T · N dA. (3.30)

∂Bt ∂B0 The ﬁrst Piola-Kirchhoﬀ tensor is an objective but antisymmetric two-point tensor as well as its transpose nominal stress tensor P

P = TT = JF−1 · σ. (3.31)

To overcome the disadvantage of unsymmetry of the first Piola-Kirchhoff tensor the symmetric and objective Lagrangian stress tensor named second Piola- Kirchhoff tensor is usually used. For the derivation of this second Piola-Kirchhoff tensor a Lagrangian load vector dL as a referential counterpart of an Eulerian measure dl will be introduced first1

dL = F−1 · dl. (3.32)

Since

dl = σ · da = T · dA, (3.33) the Lagrangian load vector can be deﬁned as

dL = F−1 · T · dA = S · dA. (3.34)

1The load vector dL should be distinguished from the velocity gradient tensor introduced in Eq. (2.43). 38 Chapter 3. Conservation equations and stress measures

In the last relation with S the second Piola-Kirchhoﬀ tensor has been designated and its relation with the other stress measures can be presented as

S = JF−1 · σ · F−T = F−1 · T = P · F−T. (3.35)

To complete the set of stress measures related to the load vectors, the forth tensor, named Finger tensor, designated by σ˜, is introduced. The Lagrangian load vector can be expressed in terms of Eulerian quantities as

dL = F−1 · σ · da = σ˜ · da. (3.36)

The Finger tensor and its relation to the Cauchy stress and nominal stress is given by

σ˜ = F−1 · σ = J −1P. (3.37)

In finite elastoplasticity the weighted Cauchy or Kirchhoff stress tensor τ plays an important role and thus is defined as

τ = Jσ. (3.38)

The second Piola-Kirchhoﬀ stress and the Kirchhof stress can be treated as corresponding Lagrangian and Eulerian quantities. Their correlation can be expressed as

S = F−1 · τ · F−T τ = F · S · FT. (3.39)

If the deformation is observed in the corotational frame, as it was explained in Section 2.6.3, any Eulerian tensor can be expressed in terms of components in this coordinate system that rotates with the material. The Cauchy stress in the rotating frame becomes a corotational Cauchy stress σ∗, deﬁned as

σ∗ = Q ? σ, (3.40) and determined by the introduced spin tensor Ω (see Eq. (2.84)).

3.3.2 Lagrangian ﬁeld equations Conservation laws derived in Section 3.2 are given in the Eulerian description. They can also be expressed in terms of the Lagrangian quantities.

From Eq. (3.2) the Lagrangian form of the mass conservation principle can be deﬁned as ρ ρ(X, t)J(X, t) = ρ (X) or 0 = J. (3.41) 0 ρ For numerical purpose, it must be pointed out that the last equation holds only for material points and it is used as an algebraic form of mass conservation in 3.3. Stress measures and stress rates 39

Lagrangian meshes. In Eulerian meshes, continuity equation (3.7) must be used (cf. Belytschko et al., 2000).

Recalling (2.5), (2.20) and (2.21), the Eulerian principle of balance of linear momentum in global form, deﬁned by (3.12), can be given in Lagrangian description as Z Z Z T ρ0ϕ¨ dV = ρ0b0 dV + P · N dA, (3.42)

B0 B0 ∂B0 where the referential body-force density is given by

b0(X, t) = b(x, t) = b(ϕ(X, t), t).

Applying the divergence theorem, the local form of balance of linear momentum in terms of material coordinates, known as a Lagrangian equation of motion is obtained from (3.42) as

∇0 · P + ρ0b0 = ρ0ϕ¨. (3.43)

The Cauchy’s second law of motion (3.25) in Lagrangian description gives

F · P = PT · FT and S = ST. (3.44)

The set of Lagrangian ﬁeld equations can be summarized as

Conservation of mass ρ0 = Jρ

Cauchy’s ﬁrst law of motion ∇0 · P + ρ0b0 = ρ0ϕ¨ Cauchy’s second law of motion F · P = PT · FT

3.3.3 Work rate and conservation of mechanical energy The balance of mechanical energy and balance of virtual work are the important consequences of the conservation laws derived earlier. A physical background of the balance equations can be understood better within these laws, i.e. the mechanical process implies the process of the mechanical power transformation into energy (cf. Malvern, 1969; Haupt, 2000). The variational principles and approximate solutions as the basis for numerical calculations are directly derived from the work balances.

The mechanical work done by the external forces (traction and body forces) supplies the body with additional energy and it’s rate can be written as

ext Z Z W˙ = ρb · v dv + t · v da. (3.45)

Bt ∂Bt 40 Chapter 3. Conservation equations and stress measures

Applying the Cauchy’s and divergence theorem and (2.43), the last equation can be rewritten as

ext Z W˙ = [(ρb + ∇ · σ) · v + tr (σ · L)] dv. (3.46)

Bt Since the Cauchy tensor is a symmetric tensor according to (3.25), the following relation holds

tr (σ · L) = tr (σ · D). (3.47) 1 Using (3.16), (3.10) and the property v˙ · v = (v · v)˙, Eq. (3.46) can be written 2 as

ext d Z 1 Z W˙ = ρv · v dv + tr (σ · D) dv. (3.48) dt 2 Bt Bt The ﬁrst term on the RHS of the last equation represents the rate of change of the kinetic energy of the body B, while the second term, referred to as the stress power, is the rate of work of the stresses on the body.

Therefore, the rate of change of the total energy in the body can be given in the form

ext kin int W˙ = W˙ + W˙ . (3.49)

Finally, the mechanical energy balance equation in Eulerian description can be deﬁned as Z Z d Z 1 Z ρb · v dv + t · v da = ρv · v dv + tr (σ · D) dv. (3.50) dt 2 Bt ∂Bt Bt Bt

int The rate of internal work W˙ is deﬁned as

int Z Z W˙ = tr (σ · D) dv = w˙ dv, (3.51)

Bt Bt where the stress power per current unit volume w˙ is

w˙ = tr (σ · D) = J −1tr (τ · D). (3.52)

Since J, D and τ are all objective quantities, the stress power per unit volume w˙ is an objective quantity as well. It can be deﬁned per referential unit volume as w˙ = tr P · F˙ = tr (P · (L · F)) = Jtr (σ · D) = tr (τ · D) = τ : D. (3.53) 3.3. Stress measures and stress rates 41

From the last relation for the rate of deformation D is said that it is conjugate in power to the weighted Cauchy stress τ . The stress power can be represented as a combination of different stress and strain rate measures defined in the Eulerian as well as in Lagrangian configuration that form the work-conjugated pair. Some of these quantities may not be objective but they must form an objective combination. This very important topic will be discussed in more detail in Section 3.4.

The conservation of mechanical energy law can be deﬁned in Lagrangian description as Z Z T ρ0b0 · ϕ˙ dV + (P · N) · ϕ˙ dA

B0 ∂B0 d Z 1 Z = ρ ϕ˙ · ϕ˙ dV + tr P · F˙ dV , (3.54) dt 2 0 B0 B0 where the relations between stress measures derived in Section 3.3.1 were used.

3.3.4 Stress rates In the rate-independent elastoplasticity theory constitutive equations are given in the rate form. As it has already been shown, the material time derivative of an objective Eulerian second-order tensor is not an objective quantity. For example, the Cauchy stress tensor transforms according to rule (2.60)3, i.e. σ∗(x∗, t∗) = Q(t) ? σ(x, t), (3.55) but its material time derivative, given by ∂σ σ˙ = + v · (∇ ⊗ σ), (3.56) ∂t is not an objective quantity. It can be proved by the following consideration

σ˙ ∗(x∗, t∗) = Q(t) ? σ˙ + (Q˙ (t) · QT(t)) · σ∗ − σ∗ · (Q˙ (t) · QT(t)) 6= Q(t) ? σ˙ (x, t). (3.57) Since the mechanical behaviour of the body is independent of the choice of observer, even though its kinematical description may be, in order to preserve the objectivity in constitutive relations the modiﬁed time derivatives, known as the objective rates, must be applied.

The corotational and non-corotational objective rates have been introduced in Sections 2.6.5 and 2.6.4. Their diﬀerences and the advantages or disadvantages of their implementation were also discussed therein. The most important rep- resentatives of the corotational and non-corotational stress rates that have been 42 Chapter 3. Conservation equations and stress measures used in the numerical calculations in this treatise will be given in this Section.

Any kind of the known objective stress rates is a particular case of Lie derivatives. The Lie derivative of an objective Eulerian stress tensor implies the push-forward time derivative of the pulled-back Eulerian stress tensor (see Marsden & Hughes, 1983). Starting from the simplest example, the Lie derivative of the Kirchhoﬀ stress tensor is given by

◦ d τ L = F · S˙ · FT = F · [ (F−1 · τ · F−T)] · FT, (3.58) dt where the correlation between the second Piola-Kirchhoﬀ tensor and the Kirch- hoﬀ stress tensor (3.39) has been used. Applied d F−1 = −F−1 · L (3.59) dt to the RHS of the last relation, the well known Oldroyd stress rate will be obtained as

◦ 5 τ L = τ Ol = τ˙ − L · τ − τ · LT. (3.60)

The Lie derivative of the Cauchy stress tensor leads to the objective Truesdell stress rate

◦ 5 σ L = σ Tr = σ˙ − L · σ − σ · LT + tr (D)σ. (3.61)

According to the previously given deﬁnition of the Lie derivative, if the pull-back and after that the push-forward transformations are performed on the Kirchhoﬀ stress tensor with FT and F−T respectively, as a result the Cotter-Rivlin stress rate is obtained

5 d τ CR = F−T ? [ (FT ? τ )] = τ˙ + LT · τ + τ · L. (3.62) dt All these rates belong to the group of non-corotational rates. The implementation of the second group of rates, named corotational rates, has some advantages coming from the following properties of the corotational rates. First, the principal invariants of the stress remain stationary iﬀ an objective corotational rate of stress vanishes. The second property is that the objective corotational rate of an objective Eulerian tensor, such as the Cauchy stress tensor, in the rotating frame is a true time derivative. Third, as presented in the succeeding Section a work-conjugacy relation between the Eulerian stress and strain measure is established by virtue of the objective corotational rate. Forth, it is found that the stretching tensor D, as the fundamental quantity describing the change of a deforming state, is expressible as an objective corotational rate of the Eulerian Hencky strain h (cf. Xiao et al., 1998b).

The ﬁrst from the group of corotational stress rates, named the Green-Naghdi 3.3. Stress measures and stress rates 43 stress rate, will be obtained if the above transformation is assumed due to the rotation tensor R only, i.e. the stretching component of deformation gradient is ignored. Then the Green-Naghdi rate of the Kirchhoﬀ stress is given as

◦ d τ GN = R · [ (RT · τ · R)] · RT = τ˙ + τ · ΩR − ΩR · τ , (3.63) dt where

R · R˙ T = −R˙ · RT (3.64) has been adopted.

The most used corotational stress rate is the Jaumann rate, defined by the spin Ω = ΩJ = W. It was the first objective corotational rate incorporated in the formulation of finite elastoplastic constitutive relations. Applied to the Kirchhoff stress tensor it is given by

◦ τ J = τ˙ + τ · W − W · τ . (3.65)

The relationship between the Lie derivative and the Jaumann rate of the Kirch- hoﬀ stress is, according to Simo & Pister(1984), given by

◦ τ L = τ˙ − L · τ − τ · LT = τ˙ − (W + D) · τ − τ · (W + D)T (3.66) ◦ = τ J − D · τ − τ · D.

Both corotational stress rates presented above can be obtained as well from the general class of spin tensors introduced by (2.88) (cf. Xiao et al., 1998b). Recall Eq. (2.89)

n X bi bj Ω = W + h , Bi · D · Bj. (3.67) I1 I1 i6=j

Introducing spin function (2.93) and (2.96), respectively, in the last relation the Jaumann and Green-Naghdi corotational rate of any objective Eulerian second- order tensor will be obtained. For the chosen Kirchhoﬀ stress tensor the obtained relations for corotational rates will be the same as (3.65) and (3.63), respectively.

For the logarithmic spin function (2.99) the logarithmic spin ΩLog will be obtained as

ΩLog = W + ΥLog, (3.68) 44 Chapter 3. Conservation equations and stress measures where the isotropic skew-symmetric tensor-valued function ΥLog can be obtained by the following relations  0, n = 1,  ΥLog = ν[BD] n = 2, (3.69)  2 2 ν1[BD] + ν2[B D] + ν3[B DB] n = 3,

1 1 + (b /b ) 2 ν = 1 2 + (3.70) b1 − b2 1 − (b1/b2) ln (b1/b2)

3 1 X 3−k 1 + εi 2 νk = (−bi) + , k = 1, 2, 3, ∆ 1 − εi lnεi i=1 (3.71) ε1 = (b2/b3) , ε2 = (b3/b1) , ε3 = (b1/b2)

∆ =(b1 − b2)(b2 − b3)(b3 − b1).

The logarithmic rate of the Kirchhof stress is written as

◦ τ Log = τ˙ + τ · ΩLog − ΩLog · τ . (3.72)

Then the relationship between the logarithmic and Jaumann stress rate as well as the logarithmic and Lie derivative is given by

◦ ◦ τ Log = τ J − ΥLog · τ + τ · ΥLog, (3.73) that is to say

◦ ◦ τ L = τ Log + ΥLog · τ − τ · ΥLog − D · τ − τ · D. (3.74)

3.4 Conjugate stress analysis

One of the main tasks in the formulation of elastoplastic behaviour of materials is a proper choice of stress and strain measures, among inﬁnitely many, that are forming the constitutive relations.

The chosen stress and strain measures, either they are Eulerian or Lagrangian quantities, have to fulﬁl the requirement that the inner product of the stress and (proper) strain rate furnishes the stress power per unit reference volumew ˙

w˙ = τ : D, (3.75) introduced in Section 3.3.3. 3.4. Conjugate stress analysis 45

The original formulation of the work-conjugacy principle was deﬁned by Hill in (Hill, 1968), covering only the Lagrangian measures, and is given by the following relation

w˙ = T(m) : E˙ (m). (3.76)

T(m) and E(m) are the Lagrangian stress and strain measure, respectively. They are forming a work-conjugate pair such as the second Piola-Kirchhoﬀ stress tensor S and Green strain tensor E.

This relation is generally not applicable to Eulerian quantities. It is not as general as it is necessary even for the Lagrangian quantities, i.e. some stress measures do not have their appropriate partners among the strain measures. One example is the Lagrangian counterpart of the Kirchhoff stress tensor τ known as T the rotated Kirchhoff stress or Lagrangian Kirchhoff stress τ 0 = R ? τ , useful in formulation of the elastic and elastoplastic Lagrangian description of material behaviour.

Xiao et al. in (Xiao et al.,1998c) proposed an extension of Hill’s work-conjugacy relation (3.76) that can be applied to the objective Eulerian stress and strain measures as well.

The stretching tensor D and its counterpart in the reference conﬁguration D0, named Lagrangian stretching tensor, according to (2.40) are in the following relation

T D0 = R ? D. (3.77) If the Eulerian quantities form a work-conjugated (cf. (3.75)), their Lagrangian counterparts are work-conjugated as well and vice versa.

Therefore the following relation and (3.75) are the standard formulas for the stress powerw ˙ per unit reference volume

w˙ = τ 0 : D0. (3.78) The same conclusion can be made for the Eulerian stress-strain pair defined in the fixed backgroud frame and their counterparts that refer to the corotational frame (see Section 2.6). If the symmetric and objective Eulerian stress tensor t and strain tensor e2 are defined in the fixed background frame, their counterparts in the corotating frame, defined by the spin Ω, are equal to Q ? t and Q ? e, respectively. The scalar product of the rotated stress tensor and strain rate tensor in the corotating frame yields the following relation

˙ ◦ (Q ? t): (Q ? e) = (Q ? t):(Q ? e). (3.79)

2Eulerian stress tensor t and strain tensor e should be distinguished from the Cauchy stress vector and Almansi strain tensor introduced in Eqs. (3.11) and (2.37), respectively. 46 Chapter 3. Conservation equations and stress measures

In the last equation (2.87) was used and the corotational rate of the strain tensor e deﬁned by the spin Ω was given by

◦ e = e˙ + e · Ω − Ω · e. (3.80)

Observed from the corotational frame, the pair (t, e) is a Ω-work-conjugate pair ˙ if the inner product of Q?t and (Q ? e) furnishes the stress powerw ˙ as in (3.75), i.e. ˙ w˙ = (Q ? t): (Q ? e), (3.81) or, equivalently (3.78), if the relation

◦ w˙ = t : e (3.82) is satisﬁed.

Relation (3.81), i.e (3.82), represents the extended work-conjugacy relation de- ﬁned in the corotating frame and it holds true only for the objective corotational ◦ rates e (for details see Xiao et al., 1998c).

◦ The Lagrangian counterparts of the pair (t, e), obtained by the pull-back trans- ◦ ◦ formation rule (2.40)2 are introduced and labelled as t0 = T and e0 = E.

The Lagrangian quantities (T, E) are a (RT ? Ω)-work-conjugate pair if the following relation is satisﬁed

◦ ◦ w˙ = (RT ? t):(RT ? e) = T : E. (3.83)

Eqs. (3.83) and (3.82) represent the uniﬁed work-conjugacy relation between the Lagrangian and Eulerian stress and strain measures, respectively, where Hill’s work-conjugacy relation (3.76) is incorporated as a particular case of a ΩR-work- conjugated relation.

As it has already been stated in Section 2.4.1, any Eulerian strain measure e can be represented using the spectral decomposition

n X e = g(B) = g(bα)Bα. (3.84) α=1 Choosing the special case, when the scale function g equals 1 g(b ) = lnb , (3.85) α 2 α (3.84) leads to the Eulerian Hencky strain h (cf. (2.39)). 3.4. Conjugate stress analysis 47

Its ΩR-conjugated stress is denoted by π and equals

n q X −1 −1 bα − bβ π = bα bβ Bα · τ · Bβ . (3.86) lnbα − lnbβ α,β=1

From (3.82) and (3.83), the (RT ? Ω)-work-conjugate relation can be written as

◦ ◦ w˙ = π : h GN = (RT ? π):(RT ? h GN) = Π : H˙ , (3.87) where

Π = RT ? π and π = R ? Π, (3.88) hold. The Lagrangian counterparts (Π, H) of an Eulerian pair (π, h) form a work-conjugated pair as well.

If, instead of the polar spin ΩR, in (3.80) the logarithmic spin ΩLog has been introduced, another stress-strain pair can be obtained. The ΩLog-work-conjugate stress to any Eulerian strain tensor can be obtained as

n X 1 lnbα − lnbβ t = Bα · τ · Bβ . (3.89) 2 g(bα) − g(bβ α,β=1

Speciﬁcally, if the strain tensor is the Hencky strain h, its ΩLog-work conjugate stress is exactly the Kirchhoﬀ stress

n X t = Bα · τ · Bβ = τ . (3.90) α,β=1

From Eq. (3.82) it follows

◦ ◦ w˙ = t : e = τ : h Log, (3.91) and it must be equal to (3.75). From the last statement the most important conclusion comes out

◦ τ : h Log = τ : D, (3.92) i.e.

◦ h Log = D. (3.93) 48 Chapter 3. Conservation equations and stress measures

3.5 Weak form of balance of momentum

There are a few practical problems in elastoplasticity that can be solved analytically whereas for solving more complex problems the numerical procedure has to be used. The ﬁnite element method gives the approximate solution for solid mechanics problems satisfying the equilibrium requirement for the body B as a whole, even though the equilibrium is not satisﬁed in the strong sense, i.e. at every point in the continuum.

Therefore, instead of solving the strong form of the balance of linear momentum along with the boundary conditions, the finite element method solves its integral over the domain, known as a weak form of the balance of momentum, where the geometric boundary conditions are strongly satisfied while the static boundary conditions must be satisfied only in an integral form.

The weak formulation of the equation of motion can be obtained considering the principle of virtual forces, the Hamilton’s principle of continuum or the principle of virtual work. The last one is universally applicable to arbitrary materials and excellently programmable, therefore will be used here.

3.5.1 Principle of virtual work

In the initial state B0, the boundary of the body ∂B0 may consists of two parts, ∂uB0 named the displacement or Dirichlet boundary, where the displacement u is prescribed, and ∂σB0 named the traction or Neumann boundary, on which the traction t is prescribed, in such a way that

∂uB0 ∩ ∂σB0 = ∅ and ∂B0 = ∂uB0 ∪ ∂σB0 (3.94) is satisﬁed.

The displacement boundary conditions are deﬁned on ∂uB0 and given by u = u¯, (3.95) while the traction boundary conditions are deﬁned on ∂σB0 by

T P · N = ¯t0. (3.96) If we conﬁne ourselves to quasi-static problems, i.e. dynamic problems are to be excluded, the Lagrangian equation of motion (3.43) may be replaced by the equilibrium equation

∇0 · P + ρ0b0 = 0. (3.97) For the displacement field u(X, t) is said that it is kinematically admissible if it satisfies the displacement boundary conditions on ∂uB0. Then a virtual displacement field, represented by the test function δu(X, t), can be introduced as 3.5. Weak form of balance of momentum 49 a difference between neighbouring kinematically admissible displacement fields, i.e.

δu(X, t) = u∗(X, t) − u(X, t), (3.98) where the test function has to fulﬁl the requirement

δu(X, t) = 0 on ∂uB0. (3.99)

Multiplying equilibrium equation (3.97) with the virtual displacement δu and integrating over the reference conﬁguration, the following relation will be obtained

Z Z ∇0 · P · δu dV + ρ0b0 · δu dV = 0. (3.100)

B0 B0 Applying the divergence theorem, the last relation can be rewritten as Z Z Z T T P :(∇0 ⊗ δu) dV = P · N · δu dA + ρ0b0 · δu dV , (3.101)

B0 ∂B0 B0 and further Z Z Z T P : δF0 dV = ¯t0 · δu dA + ρ0b0 · δu dV , (3.102)

B0 ∂B0 B0 where with δF0 the virtual displacement gradient (∇0 ⊗ δu) is designated.

The first term on the RHS can be reduced to the integral over the surface ∂σB0 since the virtual displacement field satisfies requirement (3.99). The weak or variational form of the principle of virtual work in the Lagrangian description can be given as Z Z Z T P : δF0 dV = ¯t0 · δu dA + ρ0b0 · δu dV . (3.103)

B0 ∂σ B0 B0

int ext If with δW 0 and δW 0 the work of internal and external forces on the virtual displacement δu, respectively, are denoted, i.e. Z int T δW 0 = P : δF0 dV , (3.104)

Z Z ext δW 0 = ¯t0 · δu dA + ρ0b0 · δu dV , (3.105)

∂σ B0 B0 50 Chapter 3. Conservation equations and stress measures the principle of virtual work, deﬁned in the reference conﬁguration, can be written in short form as

int ext δW 0 = δW 0 . (3.106)

The same procedure can be applied to obtain the weak formulation of the principle of virtual work in an Eulerian description.

The Eulerian gradient of the virtual displacement is deﬁned as

−1 −1 δF = δF0 · F or ∇ ⊗ δu = (∇0 ⊗ δu) · F . (3.107)

Applying the last relation, (3.31), (3.29), (2.20) and (2.21) into Lagrangian form (3.103) the Eulerian form of the principle of virtual work will be obtained as Z Z Z σ :(∇ ⊗ δu) dv = ¯t · δu da + ρb · δu dv, (3.108)

Bt ∂σ Bt Bt with the traction boundary conditions deﬁned on ∂σBt

σ · n = ¯t (3.109) and displacement boundary conditions deﬁned on ∂uBt

u = u¯. (3.110)

3.5.2 Rate of the weak form of balance of momentum Many numerical solution schemes for quasi-static elastoplastic problems perform the incremental solution based on the rate formulation of the weak form of the balance of momentum. In the ﬁnite element formulation it leads to the continuum tangent stiﬀness matrix (cf. Simo & Hughes, 1998). Therefore, the rate of the weak form of equilibrium equation will be introduced here.

Instead of the virtual displacement field δu a kinematically admissible spatial virtual velocity field δv may be introduced. The test function δv in any configuration Bt satisfies the requirement to vanish on ∂vBt and is tangent to the deformation ϕ. Its corresponding quantity in the Lagrangian configuration is denoted by δv0(X, t).

The material time derivative of equilibrium equation (3.97) deﬁned in terms of the Lagrangian quantities is

∇0 · P˙ + ρ0b˙ 0 = 0. (3.111)

Multiplying the last equation by the virtual velocity δv0, integrating over the reference conﬁguration and applying the divergence theorem, the rate-incremental 3.5. Weak form of balance of momentum 51 version of the Lagrangian weak form of the balance of momentum will be obtained as Z Z Z T ˙ P˙ :(∇0 ⊗ δv0) dV = ¯t0 · δv0 dA + ρ0b˙ 0 · δv0 dV . (3.112)

B0 ∂σ B0 B0 Similarly as in (3.107) the spatial and referential gradient of virtual velocity are related as

−1 ∇ ⊗ δv = (∇0 ⊗ δv0) · F . (3.113)

Using (3.35), (2.45), (3.39), (3.58) and the last relation the virtual power density of the LHS of equation (3.112) can be rewritten as

T ◦ L −1 P˙ :(∇0 ⊗ δv0) = (L · τ + τ ):(∇0 ⊗ δv0) · F (3.114) ◦ = (L · τ + τ L):(∇ ⊗ δv).

Applying the last relation, (2.20) and (2.21) to the Lagrangian rate form the rate of the Eulerian weak form of the balance of momentum can be written as

Z ◦ Z Z J −1(L · τ + τ L):(∇ ⊗ δv) dv = ¯t˙ · δv da + ρb˙ · δv dv, (3.115)

Bt ∂σ Bt Bt or, inserting the logarithmic objective rate instead of the Lie derivative of the Kirchhoﬀ stress according to (3.74),

Z ◦ J −1(L · τ + τ Log + ΥLog · τ − τ · ΥLog − D · τ − τ · D):(∇ ⊗ δv) dv =

Bt Z Z ¯t˙ · δv da + ρb˙ · δv dv. (3.116)

∂σ Bt Bt 4 Constitutive relations

4.1 Introduction

In Chapter3 the general principles, that determine the relation between applied forces and the motion of the body of interest, were derived independently from the properties of the material the body is made of. On the other side, the stresses inside the body are the reactions of the deformation of material under consideration induced by the external forces, thus, they are dependent of the material properties. Therefore it is necessary to express the stresses in terms of deformation measures, such as strain is. The group of laws that describe the relationship between the deformation measure and the stress measure, or between their rates, in continuum mechanics are termed constitutive equations. They obviously depend on the type of material and have to stay invariant with respect to the change of frame since the material properties are independent of the frame of reference.

The aim of this Chapter is to provide the constitutive relations that can model and predict the complex behaviour of materials under elastoplastic deformations, especially if they are large. The phenomenological aspect in developing the constitutive relations rather than the physically motivated approach of elastoplasticity is adopted here.

Elastoplasticity in general is a single entity that represents a complicated combination of two totally diﬀerent kinds of idealized material behaviour; pure elasticity and pure plasticity. The pure elastic process implies that during deformation the whole work done is stored in the body as the internal energy and will be completely recovered during the unloading process. That is why the pure elastic process is assumed as reversible and dissipationless. On the other side, the plastic deformation of material stands for irreversible and dissipative process which implies the permanent change in the shape and usually in the volume of the body of interest that cannot be recovered after the removal of the external agencies.

Elastoplastic processes are dependent of the loading history and deformation and stress states in the body will be described by the time-dependent tensor ﬁelds. Therefore the best way to model the elastoplastic behaviour of materials would be by an incremental or rate type formulation, i.e. to give the relationship between instantaneous elastic and plastic deformation increments (rates) and the instantaneous stress increment (rate). The incremental essence of elastoplastic deformation is a common base for both elastoplastic rate formulation and its

52 4.2. Continuum thermodynamics 53

ﬁnite element implementation.

The inﬂuences of the micro-mechanical processes in the material in the course of elastoplastic deformations to the behaviour of the body on the macroscopic level, such as hardening eﬀect occurrence, in phenomenological model are represented by the set of internal variables, for which the appropriate governing equations have been developed individually.

In order to provide physically consistent prediction of the material behaviour the thermodynamic laws are implemented. The introduced constitutive models are restricted to the isothermal, quasi-static and rate-independent processes.

4.2 Continuum thermodynamics

Elastoplastic deformations, whether they are small or ﬁnite, are by their nature partially reversible and partially irreversible processes. During the irreversible process, usually addressed to plasticity, the dissipation of mechanical work occurs. The last conclusion is valid even for isothermal processes, which will be assumed here. During elastoplastic deformation only mechanical work and heat are assumed to be the sources of energy, i.e. any other inﬂuence on the deformable body, such as chemical, electrical or magnetic, has been neglected. Thus, the elastoplastic deformations of the solid bodies can be treated as thermo- mechanical processes.

Our goal is to develop physically consistent material laws of ﬁnite elastoplasticity. Therefore, the kinematics and kinetics principles of a deformable body, presented in preceding Chapters2 and3, have to be strongly coupled with thermodynamic principles (see Lehmann, 1989). Later on, when we restrict our attention to the purely mechanical theory of elastoplasticity, any temperature eﬀects will be ignored.

Within the concept of thermodynamics, the body of interest B, introduced in Section 2.2.1, geometrically occupies the region in space known as a control volume. The region outside the control volume represents the environment with which the body B thermodynamically correlates. The control volume will be considered as a closed thermodynamic system for which the energy exchange with the surrounding will be allowed while no matter exchange between the body and the environment is permitted.

In Lehmann(1989), and the same will be adopted here, the basic assumptions are the following: ”the body can be considered as a classical continuum”, i.e. the kinematics of the body of interest is deﬁned according to the description given in Chapter2, and ” the thermodynamical state of each material element is determined uniquely by the actual values of a set of external and internal state variables even if the body is not in thermodynamical equilibrium.” The mechan- 54 Chapter 4. Constitutive relations ical state variables can describe the mechanical features of the body, for example its position in space or velocity, while the thermodynamical state variables, such as density, temperature or stress, have to describe the thermodynamical state of the control volume. The change of the thermodynamic system state from its initial state B0 to the actual state Bt is determined by the interaction between the thermodynamic system and its environment. This change represents a thermodynamic process which implies the change of the magnitudes of the state variables as well. If the initial state of the system can be recovered by some additional processes we can say that thermodynamic process is reversible. Otherwise, the thermodynamic process is assumed to be irreversible and it is characterised by the signiﬁcant change in the environment. In nature the irreversible processes are predominant.

Another idealisation adopted here will be that the process of deformation is very slow, i.e. the rate of deformation is very low, in such a way that the subsequent internal equilibrium state in the body can be reached faster than the external process is processed. Therefore, the external process can be assumed as quasi- static. The reversible processes must be quasi-static while the opposite is not necessary to be the case.

Also, we will assume that the temperature changes in the system occur slowly enough that the system can continually adjust to the constant temperature through the heat exchange with the environment. Likewise, the body B will not be exposed to any exterior thermal inﬂuence. That means that only isothermal processes are to be considered here which means that the temperature of the body is held constant, and independent of the particle spatial position and time, but the thermal exchange between the thermodynamic system and its environment still exists.

The state variables evolutions, whether these variables are tensors or scalars, have to be governed by the appropriate diﬀerential equations. Knowing the current values of the state variables provides that the current stress state of the body of interest is determined without knowledge of the history of the process that precede the current state of the deformable body. Regarding the implementation of the developed phenomenological model in ﬁnite element procedure a limited number of internal variables, accompanied with their appropriate evolution equations, have been adopted here.

4.2.1 First law of thermodynamics In a thermo-mechanical process, where only mechanical work and heat are assumed to be the sources of energy, the ﬁrst law of thermodynamics plays a crucial role. That means that the thermodynamic process in material body implies the strong coupling between deformation and heat, i.e. that is the process in which the mechanical work and the heat are constantly converted into each other. 4.2. Continuum thermodynamics 55

Developing the first law of thermodynamics we will consider arbitrary thermodynamic processes firstly and later on we will confine ourselves to the restricted thermo-mechanical processes. For arbitrary thermodynamic process, the sum of the work rate of the applied forces and the rate of the heat supplied to the body equals the rate of the total energy of the body, i.e. the rate of the kinetic and the internal energy of the body. Shortly, the rate form of the balance of energy can be written as ext kin W˙ + Q˙ = W˙ + U˙ . (4.1) In the last relation, the rate of the heat supplied to the body is determined by Z Z Z Q˙ = −q˙ · n da + rρ˙ dv = [−∇ · q˙ +rρ ˙ ] dv, (4.2)

∂Bt Bt Bt where the heat transfer across the boundaries of the body is determined by heat flux vector q(x, t) while the heat generated within the body is defined by heat source r(x, t). Since the positive flux is directed out of the body the heat flux term in the last relation has to be of the negative sign if the heat is added to the body, and the opposite. In the last relation the divergence theorem has been applied.

We can now recall Eq. (3.45)

ext Z Z W˙ = ρb · v dv + t · v da, (4.3)

Bt ∂Bt and deﬁne the kinetic energy rate in the same manner as in Section 3.3.3 as

kin d Z 1 W˙ = ρv · v dv. (4.4) dt 2 Bt If the internal energy U of the body B will be represented by a speciﬁc internal energy u, the rate of internal energy can be deﬁned as Z U˙ = u˙ ρ dv. (4.5)

Bt Introducing Eqs. (4.3), (4.2), (4.4) and (4.5), respectively, into (4.1), the following relation will be obtained Z Z Z ρb · v dv + t · v da + [−∇ · q˙ +rρ ˙ ] dv

Bt ∂Bt Bt d Z 1 Z = ρv · v dv + u˙ ρ dv, (4.6) dt 2 Bt Bt 56 Chapter 4. Constitutive relations which represents a global form of the total energy balance for an arbitrary thermodynamic process.

Since the rate of work of the applied forces can be presented as (3.48), i.e. the ﬁrst two terms in the last equation can be replaced with the RHS of Eq. (3.48), we can conclude that the rate of internal energy is given as Z Z Z u˙ ρ dv = tr (σ · D) dv + [−∇ · q˙ +rρ ˙ ] dv, (4.7)

Bt Bt Bt that is Z Z u˙ ρ dv = [tr (σ · D) − ∇ · q˙ +rρ ˙ ] dv. (4.8)

Bt Bt The last relation represents the global form of the balance of internal energy of the body. From relation (4.7) comes that the rate of internal energy is equal to the sum of the rate of internal work, deﬁned by Eq. (3.51), and the rate of the heat supplied to the body, i.e.

int U˙ = W˙ + Q˙ . (4.9)

Since relation (4.8) is valid for any subdomain of Bt, the corresponding local form of the balance of internal energy can be written as u˙ ρ = tr (σ · D) − ∇ · q˙ +rρ ˙ . (4.10) Eq. (4.10) states that the local form of the first law of thermodynamics, for arbitrary thermodynamic process, represents the sum of the internal stress power per current unit volume (defined earlier by Eq. (3.52)) and the internal non- mechanical power per current unit volume due to the heat flux and heat sources, i.e. u˙ ρ =w ˙ − ∇ · q˙ +rρ ˙ . (4.11) As it has already been mentioned, we are focused only on the pure mechanical isothermal processes where no additional heat has been given to the body or removed from it neither through the boundary nor through the volume of the system. The only heat producer is the body itself due to the micro-mechanical changes in the structure of the body during the irreversible plastic deformation, which causes internal dissipative processes. The source of this part of conducted heat is hidden in the stress power. According to Lehmann(1982), the stress power can be decomposed into three parts:w ˙ r that is reversible in strict sense, w˙ d that is dissipated immediately, and the third,w ˙ h, which represents the stored mechanical energy due to the changes of the internal structure of the material. The local form of the balance of internal energy is then given as u˙ ρ =w ˙ =w ˙ r +w ˙ d +w ˙ h. (4.12) 4.2. Continuum thermodynamics 57

That means that during the isothermal and quasi-static process the speciﬁc internal energy will be changed only due to the change of the stress power and the latter depends on the internal processes in the body. Globally, it means that the rate of internal energy of the body will be equal to the rate of internal work or, according to (3.49), the global form of the ﬁrst law of thermodynamics for isothermal and quasi-static processes yields to

int ext U˙ = W˙ = W˙ . (4.13)

That means that, neglecting all external thermal eﬀects, the whole work produced by the external forces will be used for increasing the internal energy of the deformable body, which later can be partially dissipated and partially not.

4.2.2 Second law of thermodynamics and principle of maximum dissipation The first law of thermodynamics defines the energy balance of the deformable body, but it cannot describe macroscopic irreversible effects of thermodynamic processes due to some microscopic occurrences, such as the local rearrangements of microstructure caused by elastoplastic deformations. It is also well known that heat can be transferred only from wormer to colder body. Likewise, mechanical work can be converted into heat but the opposite process is possible only under special circumstances and with not so high coefficient of utilization. For the description of these phenomena the second law of thermodynamics has to be introduced. The latter defines the possible directions of the energy exchange and some restrictions of the observed thermodynamic process as well.

We will again start from an arbitrary thermodynamic process. As a measure of a microstructural disorder of the system and a total local dissipation of the thermodynamic process a scalar quantity named entropy S has been introduced. Its rate of change can be deﬁned as Z −q˙ · n Z r˙ S˙ ≥ da + ρ dv, (4.14) θ θ ∂Bt Bt which represents the global form of the Clausius-Duhem inequality, where with θ the absolute (local) temperature has been designated.

For pure reversible processes the following relation holds

rev Z −q˙ · n Z r˙ S˙ = da + ρ dv, (4.15) θ θ ∂Bt Bt which demands the existence of the irreversible part of the rate of entropy, too, for arbitrary reversible-irreversible process such as elastoplastic deformation, i.e. 58 Chapter 4. Constitutive relations

irr rev S˙ = S˙ − S˙ ≥ 0. (4.16)

rev The reversible part of the rate of entropy S˙ can be exchanged with the enviro- irr ment through the boundaries while the other part S˙ represents the production of the entropy inside of the body due to the irreversible process. Therefore, the global form of the balance of entropy, for an arbitrary thermodynamic process is given as

rev irr S˙ = S˙ + S˙ , (4.17) i.e. Z Z q˙ r˙ sρ˙ dv = −∇ · + ρ +s ˙irrρ dv, (4.18) θ θ Bt Bt where with s and sirr speciﬁc entropy and speciﬁc irreversible entropy per current unit volume have been designated, respectively. In the local form, the balance of entropy can be presented as

q˙ r˙ sρ˙ = −∇ · + ρ +s ˙irrρ. (4.19) θ θ

Expanding the term ∇ · (q˙ /θ) and applying the ﬁrst law of thermodynamics for arbitrary thermodynamic process (4.11), the following equation will be obtained

1 θsρ˙ = q˙ · ∇θ +uρ ˙ − w˙ + θs˙irrρ. (4.20) θ Since sirr physically represents the local dissipation, and therefore it is always greater than or equal to zero, the dissipation function D will be introduced as

D = θs˙irrρ ≥ 0. (4.21)

In the last relation a zeroth law of thermodynamics has been applied, too, i.e. it is assumed the absolute temperature is θ ≥ 0. Thus, comparing the last two relations, the following conclusion can be made 1 D =w ˙ − uρ˙ + θsρ˙ − q˙ · ∇θ ≥ 0. (4.22) θ According to Truesdell & Noll(2004), the dissipation function can be split into mechanical and thermal part as

D = Dmech + Dth ≥ 0, (4.23) 4.2. Continuum thermodynamics 59 where

Dmech =w ˙ − uρ˙ + θsρ˙ ≥ 0, (4.24) and 1 D = q˙ · ∇θ ≥ 0. (4.25) th θ To represent a thermodynamic potential that measures the work that can be used from a closed thermodynamic system, the speciﬁc Helmholtz free energy ψ per unit reference volume, deﬁned as

ψ = u − θ s, (4.26) will be introduced now.

The rate of the speciﬁc Helmholtz free energy will be deﬁned as

ψ˙ =u ˙ − θ˙ s − θs˙, (4.27) and for the arbitrary thermodynamic process, the rate of the speciﬁc Helmholtz free energy per unit current volume is

ψρ˙ =uρ ˙ − θ˙ sρ − θsρ˙ . (4.28)

The last relation can be rewritten as

θsρ˙ =uρ ˙ − θ˙ sρ − ψρ˙ , (4.29)

If the last relation is introduced in Eq. (4.24), the deﬁnition of mechanical dissipation in terms of Helmholtz free energy of the arbitrary thermodynamic process is obtained as

Dmech =w ˙ − θ˙ sρ − ψρ˙ . (4.30)

For adopted isothermal processes, where θ˙ = 0, the last relation yields

Dmech =w ˙ − ψρ˙ = σ : D − ψρ˙ ≥ 0, (4.31) per unit current volume, or per unit reference volume

Dmech = τ : D − ψ˙ ≥ 0. (4.32)

The classical rate-independent plasticity model can be obtained by postulating the last relation that stands for the local principle of maximum dissipation (cf. Simo & Hughes, 1998). Our goal would be to deﬁne the actual state of the system, i.e. to determine the actual values of the stress and the adopted internal variables among all admissible states of stress and internal variables for which the plastic dissipation, or Dmech, attains a maximum. 60 Chapter 4. Constitutive relations

Since our concern is the purely mechanical theory of elastoplasticity we will be interested in the ﬁrst term of the dissipation inequality (4.32), i.e. in the stress power, only. The further elaboration of this topic from a general point of view will be excluded here. For our further research the main interest is in the consequences of the aforestated principle of maximum dissipation, and they will be given in Section 4.5, while the exact derivation of the constitutive relation with the principle of maximum dissipation as a starting point would be omitted here. For more details the reader is referred to the following references that are dealing with this problematic Lehmann(1982), Simo(1988), Lehmann(1989), Poh´e& Bruhns(1993), Lubarda(2001), Xiao et al.(2007), Luig(2008), Bertram (2012).

In this treatise, in developing the constitutive relations as a starting point will be used the work postulate and this topic will be elaborated in Section 4.5.

4.3 Decomposition of the stress power

Even though the balance of mechanical energy developed in Section 3.3.3 does not exist in reality, because each process has some thermal dissipation, according to the assumptions specified in Section 4.2, the first law of thermodynamics for adopted isothermal and quasi-static process (4.12) should be assumed as valid. It states that the increment of the specific internal energy is equal to the stress power, i.e.

u˙ ρ =w ˙ =w ˙ r +w ˙ d +w ˙ h. (4.33)

For a purely elastic deformation, the energy produced by external forces will cause the increase of internal energy of the body that will be stored as the elastic internal energy and is fully recoverable during the process of unloading. In that case the total stress powerw ˙ will be identical to its reversible or recoverable partw ˙ r. On the other side, during the elastoplastic deformation the internal energy will be partially recovered and partially dissipated. According to Lehmann(1989), in general, alongside the recoverable part of the stress power w˙ r addressed to elastic deformation, the inelastic or irrecoverable part of the stress power, addressed here to plastic deformation, will occur and it will not dissipated totally. One partw ˙ d dissipates immediately while the second part w˙ h ”is stored in the micro-stress ﬁelds generated by lattice defects (leading to a hardening of the material)”. That leads us to the conclusion that the total stress powerw ˙ is composed of a recoverable part and an irrecoverable part during every process of elastoplastic deformation. If withw ˙ e =w ˙ r andw ˙ p =w ˙ d +w ˙ h the recoverable (elastic) and irrecoverable (plastic) part of the total stress power per unit reference volume are designated, respectively, the work conjugacy relation (3.75) can be rewritten as

w˙ = τ : D =w ˙ e +w ˙ p. (4.34) 4.4. Hypo-elasticity 61

For large elastoplastic deformations, applying the additive decomposition of an Eulerian rate of deformation tensor D given by (2.109), the stress power parts can be deﬁned as

w˙ e = τ : De andw ˙ p = τ : Dp. (4.35)

To ensure that this separation is consistent with the physical background of it, the constitutive formulations for De and Dp should be established in such a way that De is indeed elastic-like (recoverable), while Dp is plastic-like (dissipative).

4.4 Hypo-elasticity

4.4.1 Hypo-elastic relation for De - From small to ﬁnite strains The elastic behaviour at small deformations may be described by the generalized Hooke’s law. Its rate form may be given by 1 + ν ν ε˙ e = σ˙ − (tr (σ˙ ))1, (4.36) E E where with ν and E the material constants Poisson’s ratio and Young’s modulus were designated while 1 represents the unit second-order tensor.

The last relation can be applied to the case of ﬁnite deformations as well by simply replacing the elastic strain rate ε˙ e with the elastic stretching De. As it has already been shown, the material time derivative of an Eulerian stress tensor is not an objective quantity. Therefore the material time derivative in Eulerian description has to be replaced by an objective rate. Constitutive equations have to be consistent with thermodynamics, i.e. the stress and strain measures have to be the work-conjugated pairs. Thus, we will choose, instead of the Cauchy stress, the Kirchhoﬀ stress in conjunction with an Eulerian Hencky strain.

Finally, the objective extension of Eq. (4.36) for the case of ﬁnite deformation may be given as

1 + ν ◦ ν ◦ De = τ − (tr τ )1. (4.37) E E The last relations represents a hypo-elasticity model of grade zero. It is the simplest form of the general hypo-elasticity model

e ◦ D = H(τ ): τ , (4.38) at first introduced by Truesdell (cf. Truesdell, 1955a, 1955b, 1956). Here, H(τ ) is the fourth-order stress-dependent moduli named hypo-elasticity tensor, which in order to fulfil the requirement of material frame-indifference has to be isotropic with respect to τ ( cf. Xiao et al., 1997a). 62 Chapter 4. Constitutive relations

In developing the hypo-elastic constitutive relation the most important step is a proper choice of the corotational rate type implemented in the previous relations.

The ﬁrst proposed and widely used objective rate was the corotational Jau- mann rate. It is also implemented in all commercial codes for ﬁnite deformation calculations. However, in the studies of Lehmann(1972), Dienes(1979) and Nagtegaal & de Jong(1982) it was shown that the hypo-elasticity model based on the Jaumann rate gives an unstable response at simple shear, known as shear oscillatory phenomenon, and that for the case of large rotation the Jaumann rate is an inappropriate choice.

The aforementioned facts motivated the developing and application of several diﬀerent objective rates, corotational and non-corotational, such as the Trues- dell rate, the Green-Naghdi rate, the Cotter-Rivlin rate, the Durban-Baruch rate, etc. (see Sections 2.6.4 and 2.6.5 and references therein for details).

Simo and Pister (cf. Simo & Pister, 1984) have elucidated that for the case of pure elastic deformation, where D = De and when the recoverable elastic-like behaviour is expected, none of the constitutive hypo-elastic relations, based on the at that time known objective rates, san fulﬁl the integrability condition to give an elastic relation, i.e. the path-dependent and dissipative processes have been detected for all rates.

Kojic & Bathe(1987) have also shown that application of the Jaumann stress rate produces residual stresses at the end of an elastic closed strain path. This topic will be discussed in detail in Section 5.2.

All above stated were the reasons why the additive decomposition of the stretching was treated as inappropriate starting point for finite elastic-plastic deformation descriptions except for the case of metal plasticity where finite total but small elastic deformations occur. It was taught that the error in modelling the elastic deformation part by a hypo-elastic relation can be ignored since that deformation is small in comparison with the plastic deformation (cf. Simo & Hughes, 1998). Even though the intention of Truesdell was that hypo-elastic rate equation (4.38) described finite elastic deformation behaviour, the reliability of this relation stayed at that time at the level of small elastic deformations.

When the logarithmic rate was discovered by several researchers independently (cf. Lehmann et al., 1991; Reinhardt & Dubey, 1996; Xiao et al., 1997b, 1998a), the above stated problems have been solved by introducing the logarithmic rate in the hypo-elastic constitutive relation.

If in the general hypo-elastic relation (4.38) and in its simplest form of grade zero (4.37) the logarithmic rate is implemented, the following relations will be 4.4. Hypo-elasticity 63 obtained

e ◦ Log D = H(τ ): τ , (4.39) and

1 + ν ◦ ν ◦ De = τ Log − (tr τ Log )1, (4.40) E E respectively.

According to Bruhns et al.(1999) and Xiao et al.(1999, 2000a), in order to be the real constitutive equation of finite elasticity, Eq. (4.39) has to fulfil the so- called elastic integrability criterion, i.e. ”for every process of elastic deformations with De = D, the rate equation (4.39) should be exactly integrable to deliver a dissipationless elastic relation and hence really characterize recoverable elastic behaviour”. This criterion will be fulfilled only if the objective rate in the same hypo-elastic relation is assumed to be the logarithmic rate. As a consequence, the simplified form of the hypo-elastic relation (4.40) will be exactly integrated to give a purely elastic relation; Cauchy elastic or Green elastic, if an appropriate potential exists, as it will be shown in the next Section. For integrability conditions relating hypo-elasticity to elasticity in Cauchy’s sense and in Green’s sense the reader is reffered to Xiao et al. (1997a).

Regardless to the strong inﬂuence of rotations, which usually has caused a lot of problems with other rates, the implementation of the Log-rate in the numerical calculations gives the stable solution for ﬁnite simple shear and the dissipationless responses for the elastic cyclic deformations. This statement will be proved with the numerical results which will be shown in Section 5.2.3.

4.4.2 Hyper-elastic potential For the group of hyper-elastic or Green-elastic materials holds that there exists a scalar elastic potential Σ as function of Lagrangian strain E and its derivative with respect to the strain E is simply related with the Lagrangian stress T as

∂Σ T = . (4.41) ∂E

A similar relation can be written for the Eulerian quantities as well. If we want the true stress to be derived from the elastic potential then an elastic potential W as a function of the Eulerian Hencky strain h has to be introduced. For isotropic elastic materials the last relation can be written as

∂W τ = Jσ = , (4.42) ∂h 64 Chapter 4. Constitutive relations where the elastic potential W is an isotropic function of h, i.e. W = W (h). There exists a complementary elastic potential W¯ as function of the Kirchhoﬀ stress, i.e. W¯ = W¯ (τ ), such that the Hencky strain is derived from

∂W¯ h = . (4.43) ∂τ These two elastic potentials satisfy the relation

W + W¯ = τ : h. (4.44)

The second gradients of the elastic potential, ∂2W /∂h2, and complementary elastic potential, ∂2W¯ /∂τ 2, represent the elastic tangent stiﬀness and elastic tangent compliance tensor, respectively. Replacing the hypo-elasticity tensor H by the elastic tangent compliance tensor, the hypo-elastic relation (4.38) for hyper-elastic materials will be given as

2 ∂ W¯ ◦ De = : τ . (4.45) ∂τ 2 With the logarithmic rate implementation in the last relation, the following hypo-elastic relation will be obtained

2 ∂ W¯ ◦ De = : τ Log, (4.46) ∂τ 2 which can be exactly integrated to give the hyper-elastic relation

∂W¯ h = . (4.47) ∂τ For the simplest form of the complementary potential W¯ given by an isotropic quadratic function, the second gradient of the complementary elastic potential results in a constant and isotropic instantaneous elastic compliance tensor K 1 + ν ν = −1 = − 1 ⊗ 1, (4.48) K C E I E where with C the instantaneous elasticity tensor has been designated. Tensor C is also constant and isotropic tensor which represents the second gradient of the elastic potential W . With I and 1 the symmetric fourth-order and the second- order unit tensors, respectively, have been introduced.

The hypo-elastic relation (4.45) than will be given in the form

◦ 1 + ν ◦ ν ◦ De = : τ = τ − (tr τ )1. (4.49) K E E That means that over the inﬁnitesimal time interval (t, t + dt), from Eq. (4.49) the relation between the instantaneous elastic strain increment De dt and the 4.5. Plasticity 65

◦ instantaneous stress increment τ dt can be established via the instantaneous elastic compliance tensor K, where the latter will be constant over the whole integration elastic range. The last statement is very important for further numerical implementation of the constitutive law of elasticity using the hypo-elastic formulation.

The last relation can be written in its inverse form as

◦ e τ = C : D , (4.50) or in terms of Lam´eelastic constants λ and µ,

◦ τ = λtr De1 + 2µDe, (4.51) where, for the given Young’s modulus and Poisson’s ratio, the first Laméconstant λ and the second Laméconstant µ can be obtained by the following relations Eν E λ = and µ = . (4.52) (1 + ν)(1 − 2ν) 2(1 + ν)

With a use of the logarithmic stress rate, Eqs. (4.50) and (4.51) are given, respectively, as

◦ Log e ◦ Log e e τ = C : D and τ = λtr D 1 + 2µD . (4.53)

By an integration of equation (4.53)2 the isotropic elastic relation will be obtained as

τ = λln(detV)1 + 2µlnV, (4.54) where V is the left stretch tensor deﬁned in Section 2.3.2 and, according to (2.39), lnV = h.

The above given isotropic elastic relation (4.54) is the natural generalization of the small deformation Hooke’s law where the inﬁnitesimal strain is replaced by the Hencky strain. In a study by Anand(1979, 1986) it has been shown that this elastic relation is in a good agreement with experiments for a wide class of materials for moderately large deformations.

For comprehensive derivations and explanations see Xiao et al. (2006a) and references cited above.

4.5 Plasticity

If the deformation of the body of interest is increased beyond a certain limit so that it cannot be fully recovered during the unloading process, the residual deformations will stay locked in the body and the material starts to plastify. The 66 Chapter 4. Constitutive relations constitutive relations, derived in the last Section, are no longer valid in the areas where the plastification occurs and a new relationship between stress and strain has to be defined. Defining the constitutive relation in the field of plastic deformations is a more complex process because plastic deformations do not depend only on the stress but also on the whole history of the process. Regarding the last statement, constitutive relations of plasticity are generally developed in terms of strain increment (rate) and stress increment (rate). Additionally, according to the adopted approach presented in Section 4.2, the set of internal variables have to be introduced and their evolution equations have to be governed to ensure the correct determination of the current deformation state of the body.

Two approaches in developing the constitutive relations of rate-independent elastoplasticity have been considered in this treatise. The first is based on the principle of maximum plastic dissipation derived from the second law of thermodynamics, which is introduced in Section 4.2.2, while the second relies on the work postulate, which is also expressed in terms of the universal stress work concept in thermodynamics. The first one is more general and more physically consistent, while the second is more practical and simplified. In the second approach Drucker’s postulate and Ilyushin’s postulate play a fundamental role.

Our main achievement would be to obtain the results that are consistent with the experimental results, and that would be possible by any of aforestated approaches if a special care is taken about the physical nature of the examined deformation process. Even though the ﬁrst approach is more general, at some level some assumptions or idealisations have to be introduced. That demands a deep understanding of the problem in order to develop the correct theoretical base whose further task will be to produce the accurate results that will stay in the frame of theoretically and experimentally expected. On the other side, ”even it is not a universal principle in a general sense, the work postulate expresses a common physical essence only for a class of typical elastoplastic materials including metals and alloys etc. However, this class of typical materials is already broad enough and usually just the main concern or object in the development of elastoplastic theory. Thus, it might be desirable that a reasonable constitutive theory for this large class of typical materials should be endowed with a structure as simple as possible.” Also, ”here, our concern is with the purely mechanical theory of elastoplasticity without pronounced temperature eﬀects. In this case, the aforementioned work postulate may play a role as quasi-thermomechanical foundation for the class of typical elastoplastic materials. No attempt beyond this will be pursued” (Xiao et al., 2006a).

The main reason for adopting the work postulate based approach rather than the maximum plastic dissipation is in our intention to develop the constitutive relations that are as simple as possible but reliable in prediction of the ﬁnite elastoplastic material behaviour for the target group of materials, such as metals 4.5. Plasticity 67 and alloys. In numerical ﬁnite element calculations the integration of elastoplastic rate constitutive equations will be called for in every process of numerical iteration at each time step and for each element. That results into the fact that the numerical calculation could be time consuming and very cumbersome if the constitutive relations involved additional variables and their evolution equations as well, especially if the more demanding engineering model had to be solved.

Of course, if we want to introduce the more complex material model or if sig- niﬁcant thermal eﬀects have to be taken into account, where the temperature becomes one of the important state variables, the use of the work postulate has to be re-examined.

4.5.1 Yield surface and loading condition That behaviour of materials in the plastic region is observed as a history dependent process bounded by a yield condition, which for small deformations can be represented in the following form:

F (σ, εp, k(εp)) = 0, (4.55) if isothermal processes are taken into account. In the last relation, the scalar k, known as the hardening variable, takes into acount the history of deformation. According to Hill(1950), it can be represented by the accumulated plastic work

t Z p p k1 = w = σ : ε˙ dt, (4.56)

0 or the equivalent or eﬀective plastic strain

t Z p 2 1 k = ε = ( ε˙ p : ε˙ p) 2 dt. (4.57) 2 eqv 3 0 In this treatise the ﬁrst approach will be adopted.

In the stress space, the function F , given by (4.55), represents the surface that is surrounding the origin, where σ = 0, and the elastic range (see Fig. 4.1a). Usually, the function F is termed as a loading function and F0 is an initial yield function while their corresponding surfaces in the stress space are the loading surface and yield surface, respectively. For metals, due to the hardening, the yield surface will expand, translate and distort in stress space. Therefore, relation (4.55) represents a family of yield surfaces. In Figure 4.1a, with F = F1 and F = F2 the subsequent yield surfaces have been presented. For some materials, such as concrete and soil, the opposite behaviour has been observed, i.e. the yield surface contracts in the stress space due to softening phenomena. 68 Chapter 4. Constitutive relations

Figure 4.1: Initial and succeeding yield surfaces and material stability

Figure 4.2: Loading condition, normality rule and convexity of the yield surface

If the stress changes the loading function will change as well. From Eq. (4.55) it will be obtained ∂F ∂F ∂F ∂k dF = : dσ + + : dεp. (4.58) ∂σ ∂εp ∂k ∂εp As a result of the change of the loading function, three diﬀerent cases are possible.

If dF > 0, in the observed point of the deformable body the angle between the direction of the stress increment dσ1 and the direction of the gradient of the loading surface (or of possible direction of plastic deformation dεp as it will be shown later) is smaller than 90◦ (see Figure 4.2). This case corresponds to the loading process and plastic behaviour of material in the observed point.

The second case, when dF = 0 and the plastic deformation is not increasing 4.5. Plasticity 69 while the stress is changing, is characterised as a neutral loading. The direction ∂F of the stress increment dσ2 is orthogonal to . ∂σ The case when dF < 0 corresponds to the unloading process, dεp = 0 and material behaves elastically. The angle between the direction of the stress increment dσ3 and the direction of the gradient of the loading surface is now grater than 90◦.

Mathematically these cases can be described as

∂F : dσ1 > 0 (plastic behaviour) ∂σ ∂F : dσ2 = 0 (neutral process) (4.59) ∂σ ∂F : dσ3 < 0 (elastic behaviour). ∂σ

Thus, it can be concluded that for the materials which harden, two conditions have to be satisfied for the onset of plastic deformations. The first requirement is that the stress has to reach a threshold value on the yield surface specific for the observed material. The second requirement is that certain loading direction is required, described by the following loading condition

∂F LC = : dσ1 > 0. (4.60) ∂σ

4.5.2 Priniple of maximum plastic dissipation and its consequences

As it has been already mentioned, during the elastoplastic deformation the stored internal energy will be partially recovered and partially dissipated in the unloading process, i.e. the total stress power per unit volumew ˙ consists of a recoverable part and an irrecoverable part during every process of elastoplastic deformation. For small deformations the irreversible plastic mechanical work rate is deﬁned as

w˙ p = σ : ε˙ p. (4.61)

Generalising the relations valid for the one-dimensional tensile test for hardening materials, e.g. presented in Figure 4.1b, Drucker has defined the stable or workhardening materials. This definition is known as a Drucker’s postulate and states that stable materials are those in which additional stresses produce nonnegative work during a loading process and the work done by the additional stresses in the loading-unloading cycle is positive if plastic strains occur, or equal zero when the strains are purely elastic. These definitions can mathematically be presented 70 Chapter 4. Constitutive relations as dσ : dε ≥ 0, I (4.62) (σ − σ∗) : dε ≥ 0.

Let us consider the material has already been loaded above the initial yield ∗ limit. In the last relation, (4.62)2, σ is an arbitrary starting point in stress space which is in the current elastic range (point A in Figures4 .1a and4 .1b). Induced by the external agency the stress is increasing to the value σ that reaches the current yield surface F = F1 (point B in the Figures). The additional stress increment dσ will produce the additional plastic deformation during the loading process (point C in the Figures) and cause the expanding of the yield surface to the succeeding level F = F2, or stay on the current yield surface during a neutral process (point B0 in Figure4 .1a). The process of unloading from point C to the initial stress value σ∗ (point D in Figure4 .1b and A in4 .1a) closes the deformation cycle. It can be observed from Figure4 .1b that during this deformation the remained plastic strain dεp will occure at the end of the cycle.

Since the elastic strains are completely reversible and independent of the loading- unloading path, i.e. I (σ − σ∗) : dεe = 0, (4.63) for the arbitrary closed path ABCD it comes I (σ − σ∗) : dεp ≥ 0. (4.64)

Since the point A is arbitrarily chosen, inequality (4.62)2 will be satisﬁed if the integrand of the last integral is nonzero. Finally, the following inequalities are representing Drucker’s postulate (σ − σ∗) : dεp ≥ 0, (4.65) dσ : dεp ≥ 0. Drucker’s postulate consequently implies

(σ − σ∗): ε˙ p ≥ 0. (4.66)

The last inequality is known as principle of maximum of plastic dissipation.

The plastic dissipation is deﬁned as

p Dp = σ : ε˙ , (4.67) (cf. Lubliner, 2008). When the stress σ is found inside the yield surface, then the plastic strain rate and thus the dissipation vanish. If the stress during the 4.5. Plasticity 71 loading process is found on the yield surface, according to the second law of thermodynamics Dp ≥ 0. Relation (4.66) can now be rewritten as

∗ p Dp ≥ σ : ε˙ . (4.68)

As it was stated earlier in Section 4.2.2, the principle of maximum plastic dissipation characterises the actual stress state (accompained by the actual values of the state variables) as a state among all admissible states for which plastic dissipation attains a maximum (Simo & Hughes, 1998).

The consequences of this principle of maximum plastic dissipation are very important in plasticity theory.

The ﬁrst consequence is that the plastic strain increment in stress space is orthogonal to the yield surface in any point of it. This condition is often called the normality rule and is deﬁned as

∂F ε˙ p = Λ , (4.69) ∂σ where Λ is a plastic multiplier greater than zero that will be determined later.

From the last equation it can be observed that the yield function F defines the yield surface but it can be understood as a plastic potential as well. In this case normality rule (4.69) is called a flow rule that is associated with the yield criterion or associated flow rule in the stress space. For some materials the flow rule has to be defined from a plastic potential that is different from the yield function and it is then termed a non-associated flow rule.

The second consequence of the maximum plastic dissipation postulate is the convexity of the yield surface observed from the top of its outward normal vector.

The third consequence would be the loading conditions that are already elaborated in the previous Section.

For details the reader is reﬀered to more comprehensive texts such as Hill(1950), Lubliner(2008), Khan & Huang(1995), Simo & Hughes(1998), Lubarda(2001).

4.5.3 Eulerian description of ﬁnite plastic deformations In this Section the constitutive relations for ﬁnite plasticity will be derived, based on the approach proposed in the last Section for the small deformation case.

The general form of the yield condition in the Eulerian description can be presented as

F (σ, θ, qn) = 0, (4.70) 72 Chapter 4. Constitutive relations where σ is an appropriate Eulerian stress measure, θ is relative temperature and qn represents a set of internal variables. If we conﬁne ourselves to isothermal processes and take into account that for metals the plastic behaviour of material is usually independent of the hydrostatic stress, the yield condition can be given by the von Mises yield function F = (σ0 − α):(σ0 − α) − g(κ). (4.71) In the last relation, α and κ are a symmetric Eulerian second-order tensor and a scalar, respectively, representing internal variables. The former is named back stress and represents a midpoint coordinate of the yield surface while the latter can be interpreted as a reduced accumulating plastic work. The back stress corresponds to the kinematic hardening while the variable κ is characterizing the isotropic hardening.

The stress deviator, used in the last equation, is determined by the following relation 1 σ0 = σ − tr (σ)1. (4.72) 3 The reason for using the stress deviator instead of the stress itself can be clariﬁed with the introduction of the so-called π plane. In the three-dimensional space of principal stresses, the π plane is determined with a normal vector n inclined with equal angles to the coordinate axes. Since the yield surface does not depend on the hydrostatic stress the surface can be described by the same shape in planes with the given normal n. If the yield surface is forming the cilinder in the given coordinate system, its projection in the π plane will be a circle. The newly introduced isotropic hardening function g(κ) in Eq. (4.71) describes the growth of the yield surface in the π plane, and it depends on the internal parameter κ.

In the Eulerian finite elastoplasticity description the Eulerian Hencky strain h and Kirchhoff stress τ will be chosen as appropriate variables since they form a unique Eulerian work conjugate pair satisfying the necessary requirements, as it was pointed out in Section 3.4. Thus, relation (4.71) will become F = (τ 0 − α):(τ 0 − α) − g(κ). (4.73) If with τ¯ the shifted Kirchhoff stress has been designated and given as τ¯ = τ 0 − α, (4.74) the final shortened form of the yield condition is F = τ¯ : τ¯ − g(κ). (4.75) The normality rule (4.69) for large plastic deformations can be shown as ∂F Dp = Λ . (4.76) ∂τ 0 4.5. Plasticity 73

To fulﬁll the yield condition, during plastic deformations the stress must stay at the yield surface, i.e. the time derivative of the loading function must vanish

F˙ = 0. (4.77)

The last relation represents the so-called consistency condition. The material time derivative of yield function (4.73) is

◦ ∂F ∂F ◦ ∂F F˙ = : τ 0 Log + : α Log + κ˙ = 0. (4.78) ∂τ 0 ∂α ∂κ Here, the logarithmic rate has been introduced to fulﬁll the requirement of objectivity. Moreover, it has been shown in Xiao et al. (2000a), that once the objective rate has been ﬁxed this choice must be kept to avoid inconsistencies.

The internal variables α and κ may be governed by the following evolution equations

◦ α Log = c (κ) Dp, 1 ∂F (4.79) κ˙ = : Dp = τ¯ : Dp. 2 ∂τ 0 Based on the INTERATOM model (see Bruhns et al., 1988) the proposed relations for the scalar functions g(κ) and c(κ) are of the forms

g(κ) = g0 + gs [1 − exp (−c1κ)] , (4.80) c(κ) = cs [1 − exp (−c1κ)] .

For the known material parameters, Young’s modulus E and Poisson’s ratio ν as elastic parameters, and the yield stress σy and tangent modulus Et as plastic parameters, the unknown coeﬃcients in Equations (4.80) can be determined by 2 2 g = σ2, g = (σ2 − σ2), (4.81) 0 3 y s 3 s y where g0 and (g0 + gs) represent the squares of the initial and saturated radii of the yield surface, respectively, and σs is the saturated stress obtained from the experimental record of a uniaxial cyclic deformation process.

If a new material parameter B has been introduced and given as E B = t , Et (4.82) (1 − E ) then 2B c1 = and cs = B. (4.83) gs 74 Chapter 4. Constitutive relations

The proposed model describes a combined kinematic-isotropic hardening behaviour of material. As the cyclic process proceeds, at the beginning, the model shows a pronounced isotropic hardening behaviour of material, i.e. an expansion of the yield surface prevails. During the deformation process the kinematic hardening is starting to dominate, leading to a saturation of the stress-strain curve. This behaviour can be observed in cyclic tests for most metals and the proposed constitutive model predicts the material behaviour which is in accordance with that observed in experimental records. For the numerical results of the proposed model the reader is referred to Section 5.3.

The objective time derivative of the Kirchhoﬀ stress deviator can be determined as   ◦ 0 Log  τ¯ : C : D  τ = C : D − τ¯  , (4.84)  ∂g(κ)  g(κ) c(κ) + 1 + τ¯ : : τ¯ 2 ∂κ C where C is the instantaneous elastic stiﬀness tensor and D is the stretching tensor.

From the consistency condition (4.78), using (4.76) and (4.79), the plastic multiplier Λ can be established as

◦ τ¯ : τ 0 Log Λ = . (4.85) ∂g(κ) 2g(κ) c(κ) + 1 2 ∂κ

Introducing Λ in (4.76) will give the ﬁnal form of the normality rule for ﬁnite plastic deformations

◦ ∂F τ¯ : τ 0 Log Dp = Λ = ρ τ¯, (4.86) ∂τ 0 ∂g(κ) 2g(κ) c(κ) + 1 2 ∂κ where ρ is the loading-unloading factor, taking the values

 ∂F ◦ 0 if F < 0 or F = 0 and : τ 0 Log < 0,  ∂τ 0 ρ = ◦ (4.87) ∂F 0 Log 1 if F = 0 and : τ ≥ 0. ∂τ 0 The Eulerian constitutive relations for finite plasticity derived in this Section are the extension of the small deformation theory, known as INTERATOM model (briefly IA-model), proposed by Bruhns et al.(1984). For more details the reader is reffered to Bruhns et al.(1988), Bruhns(2014), Bruhns & Anding(1999), 4.5. Plasticity 75

Westerhoﬀ(1995), Anding(1997).

For the assumed additive decomposition of the deformation rate, the ﬁnal relation between the stress rate and strain rate of the ﬁnite deformation will be obtained combining equations (2.109), (4.46) and (4.76)

2 ∂ W¯ ◦ ∂F D = : τ Log + Λ . (4.88) ∂τ 2 ∂τ 0 The last relation represents the constitutive relation of the ﬁnite elastoplasticity in the Eulerian description.

To obtain the integral forms of rate equations (4.79) and (4.88), corotational integration (2.107) has to be adopted. That will give us the ﬁnal relations, suitable for the numerical implementation for the reduced plastic work, back stress and Eulerian Hencky strain Z κ = κ˙ dt , (4.89) t

Z ◦ α = (RLog)T ? RLog ? α Log dt , (4.90) t

∂W¯ Z h = + (RLog)T ? RLog ? Dpdt . (4.91) ∂τ t

4.5.3.1 Classical models from IA model The IA-model for the large elastoplastic deformations, presented in the preceding Subsection, can be speciﬁed to give the classical plasticity models. i) Purely linear isotropic hardening

If the hardening is described with the widening of the yield surface in the stress space the inﬂuence of the kinematic hardening has been ignored. The yield condition for isotropic hardening can be attained from the general case (4.73) by setting α = 0, i.e. F = τ 0 : τ 0 − g(κ). (4.92) A linear hardening material is characterized through a linear stress-strain relation in the uniaxial case. Tangent modulus Et has a constant value and therefore, the material parameter B is a constant as well. The scalar functions g(κ) and c(κ) are of the following forms

g(κ) = g0 + 2Bκ, (4.93) c(κ) = 0. 76 Chapter 4. Constitutive relations ii) Purely linear kinematic hardening

If the inﬂuence of the isotropic hardening has been abandoned, the yield condition for kinematic hardening case can be obtained from the general case (4.75) by setting κ = 0, i.e. g(κ) = g0 = const.

F = τ¯ : τ¯ − g0. (4.94)

The hardening is described with the movement of the yield surface with a constant radius in the π plane.

The relationship between stress and strain is again assumed to be the linear function, i.e. Et = const. Therefore, the coeﬃcients c1 and cs are constants, too. The scalar functions g(κ) and c(κ) can be shown to become

g = g0, (4.95) c = c0, where 2 c = B. (4.96) 0 3 iii) Ideal plasticity

To model the ideal plastic behaviour of material, the yield condition is of the form

0 0 F = τ : τ − g0. (4.97)

Once the stress in some point reaches the threshold value it stays constant while the deformation increases. That means that the material does not show hardening behaviour at all, i.e. Et = 0. The scalar functions g(κ) and c(κ) will in this case be presented as

g = g0, (4.98) c = 0.

All numerical models, examined in this Chapter, have been tested and the summarized results are given in Section 5.3. 5 Numerical results

5.1 Introduction

The aim of this Chapter is the presentation and comparison of the numerical results obtained by the implementation of the well known objective rates as well as the logarithmic rate in hypo-elastic and elastoplastic constitutive relations.

5.2 Hypo-elasticity

For the most of engineering materials subjected to finite elastoplastic deformations, the elastic part of deformation may be small in comparison to the plastic deformation part. However, since this elastic part can significantly influence the result of the total deformation, it must be properly formulated. This holds especially true for large deformation cyclic processes when small errors may accu- mulate. In the modern Eulerian formulation of finite elastoplasticity the elastic behaviour is usually described by a grade zero hypo-elastic law (4.50) that re- quires the implementation of an objective rate. As it can be seen from the literature, the objective Jaumann rate has been used by many researchers and found wide application in developing elastoplasticity theories. An unexpected discovery of the shear oscillation phenomenon by Lehmann(1972) and Dienes (1979) and later proved by Nagtegaal & de Jong(1982) has caused the doubts in the correctness of the Jaumann rate applied in the constitutive relations for finite deformations. The work of Kojic & Bathe(1987), where the authors have shown that the application of the Jaumann stress rate produces residual stresses at the end of an elastic closed strain path, have just confirmed the above conclusion for the inappropriateness of the Jaumann rate implementation in the constitutive relations even for the case of small but cyclic deformations.

These findings have caused the development of several different rates, corotational and non-corotational, such as the Truesdell rate, the Green-Naghdi rate, the Cotter-Rivlin rate, the Durban-Baruch rate, but their implementation has not completely solved the existing problems. Additionally, in the work of Simo & Pister(1984), it has been shown that for the case of pure elastic deformation, hypo-elastic rate equation (4.50), based on all then known objective rates, fails to be exactly integrable to really define an elastic behaviour of the material.

Even though the theoretical studies have proved that the classical rates produce unstable solutions for ﬁnite deformations, some of them are still incorporated in

77 78 Chapter 5. Numerical results widely used commercial ﬁnite element codes for structural analysis. For example, in ABAQUS, depending on the element type and constitutive model, the solver selects the Jaumann or Green-Naghdi rate (c.f. ABAQUS, 2013).

In Xiao et al. (1997a) and (1997b), Bruhns et al.(1999) and (2001b), Meyers et al.(2003) and (2006), the authors proved that the logarithmic rate implementation in hypo-elastic constitutive relations successfully solves the problems of shear oscillation and residual stresses for a closed strain path.

The aim of this Section is to prove through several numerical problems that the logarithmic rate implementation in the hypo-elastic constitutive relations provides an appropriate solution for the aforementioned problems.

5.2.1 Finite simple shear

The analysis of simple shear deformation has become a popular benchmark for testing the appropriateness of elastic and elastoplastic finite deformation constitutive models. In the early 80’s of the last century the finite simple shear problem was the topic of research of many researchers for hypo-elasticity and elastoplasticity as well (Atluri, 1984; Johnson & Bammann, 1984; Moss, 1984). Using the simple shear model, Moss has shown that, besides the Jaumann rate, the application of the Green-Naghdi and Truesdell rates gives nonmonotonic solutions and consequently a loss of stability for large deformation analysis. The comprehensive study of Szabó& Balla(1989) has given similar conclusions.

This study will show that the use of the logarithmic rate in the ﬁnite simple shear analysis gives the correct results in accordance with the analytical solutions. The analytical solution for large simple shear has been derived by Xiao et al. (in 1997a and 1997b) and will be brieﬂy presented here.

As an example of the simple shear deformation, we can observe a deformation of a small element cut out from the thin-walled tube exposed to torsion (for details see Bruhns et al., 2001b). The deformation of the unit cube (see Fig. 5.1) in the e1 − e2 plane can be represented as

x = (X1 + γ X2)e1 + X2e2 + X3e3, (5.1) where γ is a shear strain. Then, the deformation gradient is

F = e1 ⊗ e1 + γ e1 ⊗ e2 + e2 ⊗ e2 + e3 ⊗ e3. (5.2)

The left Cauchy-Green tensor is of the form

2 B = (1 + γ ) e1 ⊗ e1 + γ (e1 ⊗ e2 + e2 ⊗ e1) + e2 ⊗ e2 + e3 ⊗ e3, (5.3) 5.2. Hypo-elasticity 79

Figure 5.1: Finite simple shear problem while its eigenvalues are as follows

2 p 2 b1 = (2 + γ + γ 4 + γ )/2,

2 p 2 −1 b2 = (2 + γ − γ 4 + γ )/2 = b1 , (5.4)

b3 = 1.

Introducing 1¯ = (1 − e3 ⊗ e3), the eigenprojections of B are 1 B1 = (B − b21¯ − e3 ⊗ e3), b1 − b2 1 (5.5) B2 = (B − b11¯ − e3 ⊗ e3), b2 − b1

B3 = e3 ⊗ e3.

Following (2.39), the Hencky strain tensor can be determined as

1 lnb1 − lnb2 b1lnb2 − b2lnb1 h = lnV = lnB = (B − e3 ⊗ e3) + 1¯. (5.6) 2 2(b1 − b2) 2(b1 − b2) √ Since J = b1b2b3 = 1, for simple shear σ = τ . From the last equation and (4.54), the expressions of the shear and normal stresses in terms of the shear strain γ for the simple shear case will be obatined as " r # 2µ γ2 γ2 τ12 = τ12(γ) = ln 1 + + γ 1 + , p 2 2 4 4 + γ (5.7) 1 τ = −τ = γτ . 11 22 2 12 80 Chapter 5. Numerical results

The same problem is solved numerically where a simple shear deformation of a single ﬁnite element has been considered.

In ABAQUS/Standard material behaviour can be defined in terms of a built-in or a user-defined material model. In the latter case the actual material model is defined in the user subroutine UMAT. Here, three objective corotational stress rates, namely the Jaumann rate, the Green-Naghdi rate and the logarithmic rate, and three objective non-corotational stress rates, namely the Truesdell rate, the Oldroyd rate and the Cotter-Rivlin rate have been incorporated in separate material models programmed in the user defined subroutine UMAT. The outputs have been compared mutually and with those obtained for ABAQUS built-in material model. For the chosen element type ABAQUS uses the Jaumann rate in the built-in material model and these results will be designated on the graphs with ABAQUS.

The numerical implementation of the Log-rate using the UMAT subroutine in ABAQUS/Standard has given the shear stress development as it is depicted in Fig. 5.2 and the results are in accordance with those obtained analytically (Xiao et al.(1997a)).

Τ12 2G 0.7 0.6 (3.015; 0.6646)

0.5

0.4 Analytical 0.3 FEA 0.2

0.1

Γ 1 2 3 4 5

Figure 5.2: Shear stress vs shear strain with Log-rate: Hypo-elastic yield point

In Truesdell(1956) the author has shown that, for the hypo-elasticity model, the curve of shear stress against shear strain in the case of simple shear at certain point ﬂattens and turns down. Even though it would not be expected during the experiment that for increasing strain the stress is starting to decrease, this phenomenon proves that hypo-elasticity gives the maximum shear stress that could be reached, i.e. ”the theory asserts that the load cannot increase, no 5.2. Hypo-elasticity 81 matter what happens.” That means ”... we have obtained an upper bound for the yield stress.” However, contrary to the theories of plasticity, in hypo-elasticity ”yield is not assumed as a postulate but rather is predicted.” This theoretical phenomenon is termed a hypo-elastic yield. In Xiao et al. (1997a) it has been proved that the grade-zero hypo-elasticity model (4.53) based on the logarithmic stress rate predicts the same stress-strain distribution as stated above. The hypo- elasticity yield point is reached for γm = 3.0177171 and τ m/2G = 0.6627434. In numerical calculation performed here, the hypo-elasticity yield point is reached at γm = 3.015 and τ m/2G = 0.6646 (see Fig. 5.2). Here, with G the shear modulus has been designated and, for the known elastic constants E and ν, obtained by relation

G = E/2(1 + ν). (5.8)

With all these results, the implementation of the hypo-elastic law through the proposed numerical model can be regarded as successful.

The following graphs, presented in Figures 5.3 and 5.4, show the dimensionless stress components τ 11/2G and τ 12/2G, respectively, as functions of the shear strain γ, for various stress rates.

As it can be seen from the plot in Fig. 5.4, and it was expected, the Jaumann rate implementation has given the oscillatory shear stress results, which is not possible from the physical consideration, while for the other stress rates the stress distribution monotonically increases. Only for the logarithmic rate the results are in accordance with analytical solutions.

The simple shear represents an isochoric deformation, i.e. det(F) = 1 (see Section 2.3.1). The stretching tensor is of the form

D = 1/2 (γ ˙ e1 ⊗ e2 +γ ˙ e2 ⊗ e1), (5.9) thus tr (D) = 0. The consequence of the latter is that the Oldroyd rate and the Truesdell rate, deﬁned by Eqs. (2.79) and (2.81), respectively, are of the same form for the examined deformation. Therefore, the stress responses for these two rates for simple shear are identical, as can be seen in the Figures 5.3 and 5.4.

It was proved that if the body is exposed to axial load (compression or tension) all rates can be used even for large deformations (see Section 5.2.2). But if the body is subjected to rotations, even to small, the inﬂuence of it is of great importance. From this simple example of simple shear it can be concluded that for the monotonically increasing deformation with small rotations the non- corotational rates, namely the Truesdell rate, the Oldroyd rate and the Cotter- Rivlin rate, can be used if the shear strain is up to 0.1 while the Jaumann rate is very accurate if the shear strains are up to 0.4 (see Fig. 5.4). The same conclusion for the Jaumann rate was given in the work of Dienes(1979). The 82 Chapter 5. Numerical results

Green-Naghdi rate is reliable if the shear strains are up to 0.6. However, beyond this limit only the logarithmic stress rate gives the accurate results. Therefore, we can conclude that for reliable results at large strains the logarithmic rate has to be implemented in the constitutive relations.

Τ11 2G 2.5 Oldroyd 2.0 Truesdell Logarithmic 1.5

1.0 Green/Naghdi ABAQUS 0.5 Jaumann

Γ 1 2 3 4 5 -0.5 Cotter/Rivlin -1.0

Figure 5.3: Normal stress vs shear strain at simple shear for various rates

Τ12 2G 2.5 Oldroyd 2.0 Truesdell

1.5 Cotter/Rivlin Green/Naghdi

1.0 Logarithmic 0.5

Γ 1 2 3 4 5 -0.5 Jaumann ABAQUS -1.0

Figure 5.4: Shear stress vs shear strain at simple shear for various rates 5.2. Hypo-elasticity 83

5.2.2 Plate with a hole in ﬁnite tension

Assuming the following material parameters, Young’s modulus E = 210 GPa and Poisson’s ratio ν = 0.3, the plate with a hole exposed to large axial elastic deformation has been considered. The assumed model has been presented in Figure 5.5.

Figure 5.5: Plate with a hole in tension: Model

In the numerical calculations, due to the symmetry, only a quarter of the plate has been considered. The deformed and undeformed conﬁgurations have been given in Figure 5.6 - left, while the ﬁnal normal stress distribution in the plate can be seen on the right side of the same Figure.

As it has already been pointed out, for the large uniaxial compression or tension case, such as it has been presented here, the stress responses are congruent for all examined corotational stress rates. This statement can be conﬁrmed by the graph, given in Figure 5.7, which presents the development of the normal stress, in the node exposed to the maximum stress, obtained for the Jaumann, Green- Naghdi and Log-rate.

But, even in this case, the slight diﬀerences among the stress responses for three examined rates can be observed in the node where the maximum shear stress has been determined. In Figure 5.8, on the left side the normal stress σ22 development for all rates has been shown while its enlarged representation is given on the right side of the Figure. 84 Chapter 5. Numerical results

Figure 5.6: Plate with a hole in tension: Undeformed and deformed conﬁguration and normal stress distribution for large elastic deformation

Σ22 2G 0.8

0.6

0.4

0.2

0.0 h22 0.0 0.2 0.4 0.6 0.8 1.0

Figure 5.7: Plate with a hole in tension: Normalized normal stress distribution for large elastic deformation in the node with the maximum normal stress 5.2. Hypo-elasticity 85

Σ22 2G

Σ22 2G 0.46 0.4 0.45

0.3 0.44 Jaumann

Green Naghdi 0.2 0.43 Logarithmic 0.1 0.42

0.41 h22 0.0 h22 0.35 0.36 0.37 0.38 0.39 0.40 0.0 0.1 0.2 0.3 0.4

Figure 5.8: Plate with a hole in tension: Normalized normal stress distribution for large elastic deformation in the node with maximum shear stress

5.2.3 Closed elastic strain path - Hypo-elastic cyclic deformation

In engineering practice strain cycles and cyclic loading as well can frequently occur if the agencies are repeated in large number of cycles. In Xiao et al. (2006b) it has been proved analytically that even for small but cyclic deformations the residual stress may be appreciable even after a single cycle and it may become very large with an increasing number of cycles.

Therefore, the analysis of cyclic deformation paths takes the important place in this treatise. For hypo-elastic law (4.50), the already mentioned objective stress rates, i.e. the Jaumann, Green-Naghdi and logarithmic rates as corotational rates and the Truesdell, Oldroyd and Cotter-Rivlin rates as non-corotational rates, have been compared in closed single parameter elastic deformation cycles.

In Kojic & Bathe(1987) and Lin et al.(2003), the authors have considered a four-phase plain strain cycle which consists of extension, shear, compression and return to original unstrained state.

Here the smooth strain cyclic deformation of the square element (see Fig. 5.9), has been considered. The square element of size H is subjected to a combined lengthening and shearing process in the e1−e2 plane, such that the upper corners are moving along the ellipse with radii a and b.

The deformation is described by

1 − cosϕ x = X + η ξ X , 1 1 1 + η sinϕ 2 (5.10) x2 = (1 + η sinϕ) X2,

x3 = X3, 86 Chapter 5. Numerical results where η and ξ are dimensionless parameters. η = b/H(0 < η < 1) represents the measure of tension or compression while ξ = a/b is the measure of rotation to which the element is subjected. The parameter ϕ in the last relation is the single parameter that describes the deformation of the square plate. The material of the plate has been considered as initially isotropic and stress free. The adopted values for Young’s modulus and Poisson’s ratio are E = 210 GPa and ν = 0.3.

Figure 5.9: Deformation cycle

5.2.3.1 Small rotations (ξ = 0.1) The square element, presented in Fig. 5.9, has been exposed to the small deformation of max 2% in axial direction 2-2 with the negligible rotation of max 0.2%, i.e. the parameters ξ and η are taking the values 0.1 and 0.02, respectively.

Τij 2G 0.04

0.02 Τ11 2 G

Τ12 2 G j 2Π 0.2 0.4 0.6 0.8 1.0 -0.02

Τ22 2 G -0.04

Figure 5.10: τ 11, τ 12 and τ 22 single cycle development, ξ = 0.1, η = 0.02 5.2. Hypo-elasticity 87

Development of the normal stresses τ 11 and τ 22 and the shear stress τ 12 versus the deformation angle ϕ for all rates have been presented in Fig. 5.10. As it can be seen from the plots the results are almost identical for all rates over the ﬁrst deformation cycle.

But, if we take a closer look at the end of the ﬁrst cycle (see Fig. 5.11), the deviation from the zero stress value can be noticed for the non-corotational rates. The residual shear stresses for the Oldroyd, Truesdell and Cotter-Rivlin rates after the ﬁrst cycle are 4.72%, 7.85% and 11% , respectively, in comparison with the maximum shear stress that occurs during the cycle and these values are accumulating for repeated cycles.

Τ12 2G Truesdell 0.0002

Oldroyd 0.0001 corotational rates j 2Π 0.95 1.00 1.05 1.10

-0.0001

Cotter/Rivlin -0.0002

Figure 5.11: Enlarged representation of τ 12 at ﬁrst cycle end, ξ = 0.1, η = 0.02

Log-rate Jaumann Green/ Oldroyd Truesdell Cotter/ Naghdi Rivlin 1st cycle 0.833e-8 0.299e-7 -0.252e-8 0.943e-4 0.157e-3 -0.220e-3 100 cycles -0.983e-8 0.208e-5 -0.110e-5 0.943e-3 0.157e-2 -0.220e-1

Table 5.1: Residual shear stresses τ 12/2G for ξ = 0.1, η = 0.02

The values of the residual stresses after 100 cycles are given in Table 5.1. As it can be seen from the table, the non-corotational rates give the unreliable results because the residual stresses are 47% - 110% of the maximum shear stress and these are obviously non negligible values.

If the deformation is increased in the axial direction up to 10% and the rotation is up to 1%, i.e. ξ = 0.1 and η = 0.1, the normal stress development shows greater diﬀerences for diﬀerent rates (cf. Fig. 5.12). 88 Chapter 5. Numerical results

Τij 2G Jaumann Green Naghdi Truesdell Oldroyd 0.1 Τ11 2 G Cotter Rivlin Logarithmic Τ12 2 G j 2Π 0.2 0.4 0.6 0.8 1.0

-0.1

Τ22 2 G

-0.2

Figure 5.12: τ 11, τ 12 and τ 22 single cycle development, ξ = 0.1, η = 0.1

Τ12 2G

0.010 Truesdell

0.005 Oldroyd corotational rates j 2Π 0.2 0.4 0.6 0.8 1.0 1.2

Cotter/Rivlin -0.005

Figure 5.13: τ 12 single cycle development, ξ = 0.1, η = 0.1

Concerning the shear stress, the development of it for different rates, presented in Fig. 5.13, shows the noticeable differences from the zero stress value at the end of the first cycle for the non-corotational rates.

Therefore, it can be concluded that for elastic deformations with even a small rotation the non-corotational rates, namely the Oldroyd, Truesdell and Cotter- Rivlin rates, give unreliable results and their implementation in constitutive relations should be avoided while the corotational rates, namely the Jaumann, Green-Naghdi and logarithmic rates, give more accurate results and they have 5.2. Hypo-elasticity 89 to be implemented in the hypo-elastic constitutive relations, especially if the deformations are cyclically repeated.

5.2.3.2 Moderate rotations (ξ = 1)

For the same elastic deformation in axial direction 2-2 up to 10% , i.e. η = 0.1, but with increased rotation up to the moderate value of 10%, i.e. ξ = 1.0, which mostly can be found in metals and thermoplastics, the stress developments in the square element have been examined for all aforementioned rates.

Τij 2G

Τ12 2 G 0.1 Τ11 2 G j 2Π 0.2 0.4 0.6 0.8 1.0

Jaumann Green Naghdi -0.1 Truesdell Oldroyd Τ 2 G Cotter Rivlin 22 -0.2 Logarithmic

Figure 5.14: τ 11, τ 12 and τ 22 single cycle development, ξ=1, η=0.1

It has been observed that the normal stress τ 22 development (see Fig. 5.14) diﬀers among diﬀerent rates as it was also noticed in the previous example (see Fig. 5.12). But the increased rotation caused that the normal stresses τ 11 for various rates are no longer congruent as it was for small rotations.

Concerning the shear stress development the diﬀerences among rates are now more evident, as it can be seen from Figure 5.15. The non-corotational rate implementation in the hypo-elastic constitutive relation caused the shear stress residuals at the end of strain cycle with much higher values than in the case of small rotation.

The normal and shear stress residuals for all rates at the end of the ﬁrst cycle are presented in Table 5.2 while their values compared with the maximum shear stress obtained in the cycle are given in Table 5.3. 90 Chapter 5. Numerical results

Τ12 2G

Truesdell 0.10

0.05 Oldroyd corotational rates j 2Π 0.2 0.4 0.6 0.8 1.0 1.2

Cotter/Rivlin -0.05

Figure 5.15: τ 12 single cycle development, ξ=1, η=0.1

Log-rate Jaumann Green/ Oldroyd Truesdell Cotter/ Naghdi Rivlin

τ 11/2G 0.440e-3 -0.118e-2 0.125e-2 -0.446e-2 -0.761e-2 -0.806e-2 τ 12/2G -0.628e-6 0.105e-4 -0.189e-4 0.242e-1 0.399e-1 -0.563e-1

Table 5.2: Residual stresses after ﬁrst cycle for ξ = 1, η = 0.1

Log-rate Jaumann Green/ Oldroyd Truesdell Cotter/ Naghdi Rivlin 1 cycle 0.628e-3 0.105e-1 0.189e-1 24.2 39.9 56.3 10 cycles 0.194e-2 2.40 0.109 - - - 100 cycles 0.433e-1 101.61 1.158 - - -

Table 5.3: Residual τ 12 compared to τ 12max in % for ξ = 1, η = 0.1

In the last table it can be seen that, when the moderate rotations occur, the residuals at the end of a single cycle, produced by implementation of the non- corotational rates, are 24.2% to 56.3% of the maximum shear stress. For the corotational rates, namely the Jaumann and Green-Naghdi rates, residual stresses of negligible values after the ﬁrst cycle (see Figures 5.16 and 5.17 and Table 5.3) have been observed but their values are no longer negligible for repeated number of cycles, especially for the Jaumann rate (see Fig. 5.18). Only for the logarithmic rate stresses are returning to the vanishing, initial state at the beginning of the cycle, thus conﬁrming the results found in Xiao et al.(1999). 5.2. Hypo-elasticity 91

Τ12 2G 0.00015 Jaumann 0.00010

0.00005 Log-rate

j 2Π 0.980 0.985 0.990 0.995 1.000 1.005 1.010 1.015

Green/Naghdi -0.00005

Figure 5.16: Enlarged representation of τ 12 at ﬁrst cycle end, ξ=1, η=0.1

Τ11 2G 0.015 ____ Green/Naghdi 0.010 ____ logarithmic ____ Jaumann 0.005

j 2Π 0.99 1.00 1.01 1.02 -0.005 ____ Oldroyd -0.010 ____ Truesdell ____ Cotter/Rivlin -0.015

-0.020

Figure 5.17: Enlarged representation of τ 11 at ﬁrst cycle end, ξ=1, η=0.1

As it can be seen from Figure 5.18, the shear stress obtained from the Jaumann rate implementation in the hypo-elastic constitutive relation shows the harmonic nature and residuals are very high while the number of cycles is increasing. The maximum values of the shear stress in the cycles are of constant value.

The Green-Naghdi rate gives better results concerning the residuals but the maximum value of the shear stress decreases monotonically for increased number of cycles. The logarithmic rate produces the negligible values of the residual stresses at the end of the cycles and the maximum shear stress is of constant value. 92 Chapter 5. Numerical results

Τ12 2G 0.10

0.05

j 2Π 20 40 60 80 100

Jaumann -0.05 Green Naghdi Logarithmic -0.10

Figure 5.18: Hundred cycle stress development for corotational rates, ξ=1, η=0.1

Thus it can be concluded that, when in the deformation process moderate rotations occur, the non-corotational rates should be excluded. The Jaumann rate can be used if the deformation is repeating in only a few cycles while the Green- Naghdi rate can be used for very low number of cycles, not more than 10. For the higher number of cycles their implementation is not adequate.

Τ12 2G 0.10 0.08

0.06

0.04

0.02

j 2Π 2 4 6 8 10

Figure 5.19: τ 12 development for Jaumann rate, ξ=1, η=0.1

In Figure 5.19 the shear stress developments for the Jaumann rate implementation have been presented, obtained from the UMAT subroutine (presented with 5.2. Hypo-elasticity 93 solid line) and from the ABAQUS built-in subroutine for hypo-elastic material model (presented with dashed line). As it can be seen, the stresses are completely congruent. Therefore, the further conclusions concerning the Jaumann rate based on the results obtained from the UMAT subroutine can be accepted as general conclusions for the Jaumann rate.

5.2.3.3 Large rotations (ξ > 1)

In this Section a stress response of the square element, presented in Fig. 5.9, subjected to the deformation with large rotation, has been examined. The extension in 2-2 direction will stay in the range of 0 − 10%, i.e. η = 0.1. The shear deformation is now predominant with relatively large values up to 50%, i.e. parameter ξ = 5.

The development of normal and shear stress components of the Kirchhoﬀ stress due to the above deﬁned deformation has been presented in Figures 5.20 and 5.21. In Figure 5.20, for the given rates, the normal stresses in 1-1 direction are marked with the dashed lines while the normal stresses τ 22 are presented by the solid lines.

The diagrams over a single cycle can generally be devided into three characteristic parts. The ﬁrst one is that where ϕ/2π is in the range of 0 − 0.1. These are the values that are usually met as elastic deformations in metals, for example in civil engineering structures and during metal forming. It can be seen that in this part for all rates the plots are almost congruent.

Τij 2G Jaumann 0.6 Green Naghdi Truesdell 0.4 Oldroyd Cotter Rivlin Τ 2 G Logarithmic 0.2 22 j 2Π 0.2 0.4 0.6 0.8 1.0

-0.2 Τ11 2 G

-0.4

Figure 5.20: Normal stress τ 11 and τ 22 single cycle development, ξ=5, η=0.1 94 Chapter 5. Numerical results

Τ12 2G

0.6 Truesdell 0.4

0.2 Oldroyd j 2Π 0.2 0.4 0.6 0.8 1.0 ____ Green/Naghdi -0.2 ____ Jaumann Cotter/Rivlin ____ Log-rate -0.4

Figure 5.21: Shear stress τ 12 single cycle development, ξ=5, η=0.1

For rubber-like and composite materials and shape memory alloys, which can be subjected to very large elastic deformations, the second and the third part of diagrams are of great importance. The second part, where ϕ/2π is in the range of 0.1 − 0.53, is characterized with an almost identical stress response for the corotational rates while the stress responses of non-corotational rates are drifting apart. Concerning the normal stress τ 11, from Fig. 5.20 it can be observed that the Oldroyd and Truesdell rates implementation in the hypo-elastic relation gives very high values while the Cotter-Rivlin rate gives a very low value for τ 11.

Τij 2G 0.06 Τ12 2 G 0.04 Log-rate 0.02 Green/Naghdi

j 2Π 0.90 0.95 1.00 1.05 1.10

-0.02 Jaumann

Τ 2 G -0.04 11

Figure 5.22: Enlarged representation of τ 11 and τ 12 single cycle development, ξ=5, η=0.1 5.2. Hypo-elasticity 95

The third part of the diagrams is beyond the limit of 0.53 for ϕ/2π where the plots of shear and normal stresses even for the corotational rates are starting to drift apart. At the end of the ﬁrst cycle some residual normal and shear stresses for the Green-Naghdi and Jaumann rates occur while the non-corotational rates produce extremely high values of residuals. Only the Log-rate gives zero stress values at the end of the elastic deformation cycle (see Fig. 5.22). Again, the normal stresses in 1-1 direction are marked with the dashed lines while the shear stresses are presented by the solid lines.

Τij 2G

0.4 Τ12 2 G

0.2 Τ11 2 G

j 2Π 2 4 6 8 10 12 Τ33 2 G -0.2 Τ22 2 G

- 0.4

Τij 2G

0.4 Τ12 2 G

0.2 Τ11 2 G

j 2Π 20 40 60 80 100 Τ33 2 G -0.2 Τ22 2 G -0.4 Figure 5.23: Ten and hundred cycle stress development for the Logarithmic rate, ξ=5, η=0.1

If the continuum square element is subjected to a deformation that repeats cyclically, the stress response error is accumulating for all rates except for the Log-rate. The development of normal and shear stresses during 10 and 100 cycles 96 Chapter 5. Numerical results for the logarithmic rate has been respectively presented up and down in Figure 5.23. It can be seen that the development of all Kirchhoﬀ stress components is regularly periodical with constant magnitudes and without any residuals at the end of each cycle.

Τij 2G

0.6 Τ12 2 G

0.4 Τ22 2 G

0.2

Τ33 2 G j 2Π 2 4 6 8 10 -0.2

Τ11 2 G -0.4

Figure 5.24: Ten cycle stress development for the Jaumann rate UMAT, ξ=5, η=0.1

Τij 2G

0.6 Τ12 2 G

Τ22 2 G 0.4 0.2

Τ33 2 G j 2Π 2 4 6 8 10 -0.2

Τ11 2 G -0.4 Figure 5.25: Ten cycle stress development for the Jaumann rate ABAQUS, ξ=5, η=0.1

The stress developments obtained using the Jaumann rate in the UMAT subroutine (see Fig. 5.24) and ABAQUS built-in subroutine for hypo-elastic con- 5.2. Hypo-elasticity 97 stitutive model (see Fig. 5.25) are completely congruent. It is obvious that the Jaumann formulation provides an oscillatory stress response for all stress components except τ 33 with variable magnitude of τ 11 and τ 12. The residual stresses at the end of cycles are of non-negligible values and show an oscillating character as well (see Fig. 5.32).

From Fig. 5.26, where the plots for 100 cycles have been presented, it can be concluded that the Jaumann rate gives results which are not in accordance with the physical behaviour of the materials. Therefore, this rate should not be implemented in the constitutive relations if large cyclic deformation occur.

Τij 2G

0.6

Τ11 2 G 0.4 Τ22 2 G

0.2 Τ 2 G 33 j 2Π 20 40 60 80 100 -0.2

-0.4 Τ12 2 G

-0.6

Figure 5.26: Hundred cycle stress development for the Jaumann rate, ξ=5, η=0.1

As in the case of moderate rotations, the stress responses of the Green-Naghdi rate formulation show the feature that the magnitude of the shear stress decreases as the number of cycles increases and after approximately 30 cycles its values are zero which is a totally unrealistic result. After that the maximum shear stress is starting to increase monotonically. The plots for normal stresses τ 11 and τ 22 are drifting away and changing their magnitude with the number of cycles (see Figures 5.27 and 5.28).

In the case of moderate rotations the Green-Naghdi rate was a more reliable choice than the Jaumann rate. Here, it can be seen that very high values of residuals occur even after a single cycle and their values are monotonically increasing as it is recorded in Figure 5.32. All above stated leads us to the conclusion that for the case of cyclic elastic deformations with large rotations the Green-Naghdi rate has to be excluded from the hypo-elastic formulations even for a low number of repeated cycles. 98 Chapter 5. Numerical results

Τij 2G

Τ 2 G 0.4 12 Τ 11 2 G 0.2

Τ33 2 G j 2Π 2 4 6 8 10 -0.2

Τ22 2 G -0.4 Figure 5.27: Ten cycle stress development for the Green-Naghdi rate, ξ=5, η=0.1

Τij 2G 2 Τ11 2 G

Τ33 2 G j 2Π 20 40 60 80 100 Τ 2 G 12 -1

Τ -2 22 2 G

Figure 5.28: Hundred cycle stress development for the Green-Naghdi rate, ξ=5, η=0.1

The Truesdell and the Oldroyd rates give similar stress responses as it is illus- trated in Figures 5.29 and 5.30 where the developments of the shear stress drift apart and increase the magnitude to the unrealistic high values of 30 times for the Oldroyd rate and 40 times for the Truesdell rate compared with the shear modulus. 5.2. Hypo-elasticity 99

Τij 2G

20 Τ12 2 G

Τ 2 G 33 j 2Π 20 40 60 80 100 Τ22 2 G -10

Τ11 2 G -20 Figure 5.29: Hundred cycle stress development for the Trusdell rate, ξ=5, η=0.1

Τij 2G 15 Τ12 2 G 10 5

Τ33 2 G j 2Π 20 40 60 80 100Τ22 2 G -5

-10 Τ11 2 G

Figure 5.30: Hundred cycle stress development for the Oldroyd rate, ξ=5, η=0.1

The results for the Cotter-Rivlin rate as well show great deviations from realistic values and therefore this rate must be excluded from the constitutive relation construction, too. 100 Chapter 5. Numerical results

Τij 2G

Τ22 2 G Τ33 2 G j 2Π 20 40 60 80 100 -10

-20

Τ 2 G -30 12

-40 Τ11 2 G -50 Figure 5.31: Hundred cycle stress development for the Cotter/Rivlin rate, ξ=5, η=0.1

Τij 2G 4 Τ11 2G

Τ12 2G 2

j 2Π 20 40 60 80 100

Jaumann Green Naghdi -2 Truesdell Oldroyd Cotter Rivlin -4

Figure 5.32: Residual normal and shear stresses for the given rates, ξ=5, η=0.1

In Table 5.4 the normalized residual stresses after 10 cycles have been given for all rates examined. The applied elastic deformation was with very large rotations, i.e. η = 0.2 and ξ = 10. It can be seen that except for the logarithmic rate the residual stresses are not negligible for the corotational rates and of extremely high values for the non-corotational rates. 5.3. Elastoplasticity 101

Logarithmic Jaumann Green/ Oldroyd Truesdell Cotter/ Naghdi Rivlin

τ 11/2G 0.45732e-4 0.035174 2.8971 -43.925 -69.584 -721.76 τ 12/2G -0.43250e-4 -0.52436 3.1558 10.437 16.700 -24.353

Table 5.4: Residual stresses after 10 cycles for ξ = 10, η = 0.2

Based on all presented results in this Section it can be concluded that for elastic deformations with large rotations the only choice among all objective rates would be the logarithmic rate since it is the only rate that gives reliable and physically admissible results for this kind of deformation, i.e. the only elasticity-consistent hypo-elastic constitutive relation would be the one based on the logarithmic rate.

5.3 Elastoplasticity

According to the modern industrial demands the engineers are faced with the task of finding and constantly improving reliable lifetime assessment procedures of engineering structures. One of the main problems in fulfilling that goal, for the structures exposed to elastoplastic deformations, is a proper definition of the constitutive model that is capable to predict the history dependent behaviour of the structural elements. On the other side, for practical applications, the engineers sometimes need quick answers which require simple but reliable methods.

Therefore, our intention was to test a simple constitutive model, with only a few necessary parameters, which describes the behaviour of metals during the process of small or ﬁnite monotonic as well as cyclic elastoplastic deformations.

5.3.1 Small cyclic elastoplastic deformations

In this Section the veriﬁcation of the numerical model for plastic material behaviour, proposed in Section 4.5.3, combined with the hypo-elastic model, given in Section 4.4.2, has been presented.

All necessary material parameters have been estimated from the experimental results taken from Westerhoﬀ(1995). The tested material a stainless steel SS 304, with elastic constants E = 205.73 GPa and ν = 0.278, has been exposed to uniaxial monotonic and cyclic deformations with various strain ranges, i.e. ∆ε = 0.005, 0.015 and 0.02, from which, for our numerical model, the results at room temperature with a constant strain rate of ε˙ = 10−5s−1 have been selected. The measurement data for monotonic and three cyclic tests have been shown in Figures 5.33 and 5.34, respectively. 102 Chapter 5. Numerical results

Figure 5.33: Monotonic tensile test, source: Westerhoﬀ(1995)

Figure 5.34: Uniaxial cyclic experiments with strain amplitudes ∆ε = 0.005, 0.015 and 0.02, source: Westerhoﬀ(1995)

Even though the original INTERATOM model, adopted here as a plasticity constitutive model, is constructed to anticipate the behaviour of the material at various temperature levels, in our model we have conﬁned ourselves to processes at constant room temperature. 5.3. Elastoplasticity 103

Since the test data themselves were no longer available and therefore an op- timization procedure could not be performed, the discussion of the numerical model has been done only according to the experimental plots, taken from the aforementioned reference.

First, the stress response due to the uniaxial monotonic tension has been determined using the IA-model for the assumed combined isotropic-kinematic hardening as well as for the linear kinematic and linear isotropic hardening indipen- dently. The results are presented in Figure 5.35. It can be observed that the proposed models give stress developments that are in a very good agreement with the measured values. Thus, it can be concluded that the suggested numerical models can be used for the computation of the material behaviour due to monotonically increasing loading.

Figure 5.35: Numerical results vs monotonic tensile test

The matching between the test and computational results, for the assumed combined hardening, can also be seen in the case of alternating loading, given in Figure 5.36.

Concerning the cyclic loading, during the strain-controlled experiment the examined material has shown the hardening behaviour which can be seen in Figure 5.34. Applying the same numerical model as tested before, with the hardening functions (4.80), the following results have been obtained for the three diﬀerent strain amplitudes, ∆ε = 0.005, 0.015 and 0.02. The ﬁrst two cycles for each strain range have been given in Figure 5.37. 104 Chapter 5. Numerical results

Σ22 MPa

300 @ D

200

100

0 h22 0.005 0.010 0.015 0.020 0.025

-100

-200

Figure 5.36: Comparison between computation and experiment for the loading- unloading-reloading case

Σ22 MPa Σ22 MPa 300 200 @ D @ D 200

100 100

h22 h22 -0.002 -0.001 0.001 0.002 -0.006 -0.004 -0.002 0.002 0.004 0.006

-100 -100 -200

-200 -300

Σ22 MPa

300@ D

200

100

h22 -0.010 -0.005 0.005 0.010 -100

-200

-300

Figure 5.37: First two cycles of the stress-strain curve for various strain ranges, ∆ε = 0.005, 0.015 and 0.02, respectively

Comparing the numerical with the experimental results, it has been detected that the stress distributions are qualitatively congruent while quantitatively the 5.3. Elastoplasticity 105 matching between the numerical and experimental plots has not been found. The reason for that is that the numerical model shows much stronger inﬂuence of the hardening than that which can be observed on the test stress-strain responses. As a consequence, numerically the stress saturation is achieved very soon, after several cycles for each strain amplitude. The saturation is noticed in the 15th cycle for ∆ε = 0.005 while for ∆ε = 0.015 and 0.02 it is reached in the 8th and 14th cycle, respectively.

The prediction of the cyclic saturated stress amplitudes for each strain range has been very accurate. It can be conﬁrmed with Figure 5.38 where the saturated experimental and numerical stress cycles have been presented.

Figure 5.38: Saturated cycles: numerical results (solid line) vs monotonic tensile test (dashed line), source: Westerhoﬀ(1995)

The computation of the chosen material behaviour using the linearized IA-model during the cyclic tests for the three examined strain ranges from the beginning to saturation has been given in Figure 5.39.

As a conclusion it can be said that, concerning the monotonic response, the numerical model shows a very good agreement with the experimental results and thus, it can be assumed capable to predict the material behaviour at small as well as ﬁnite strains. On the other side, this numerical calculation shows the necessity to modify the hardening functions (4.80) in order to obtain a more realistic model for the cyclic deformations.

Likewise, the numerical model at this stage has shown the lack in describing the cyclic behaviour at unknown strain ranges, since one of the constituents of the proposed model, coeﬃcient gs and c1 consequently, depends on the experimentally determined saturated stress. Therefore, in addition, the generalized 106 Chapter 5. Numerical results evolutionary constitutive model has to be developed in order to predict the behaviour of the material due to strain ranges that diﬀer from the experimentally considered, especially for the large deformation case.

But the model, in general, has proved itself as a very good initial step in prediction of the realistic behaviour of engineering materials.

Σ22 MPa

300@ D

200

100

h22 -0.010 -0.005 0.005 0.010 -100

-200

-300

Figure 5.39: IA-model uniaxial cyclic response for ∆ε = 0.005, 0.015 and 0.02

All things considered, some tips for future work might be: to find experimental records or carry out experiments in order to obtain data necessary for improving the existing linear model. Then, the modification of the hardening functions, or the coefficients therein, has to be performed in order to calibrate the numerical model which prediction would better fit to the experimental data. The evolution equation has to be developed as well to provide that the proposed numerical model is usable for unknown strain ranges, too.

Nevertheless, by introducing a nonlinear hardening behaviour of the material, much better results would be achieved and therefore, they would closely coincide with the experiments. For example, the proposed numerical model can be improved if the linear evolution equation (4.79)1 is replaced with the following nonlinear equation

◦ α Log = c (κ) Dp − ακ˙ . (5.11) In Figure 5.40, the computational and experimental results during the ﬁrst cycle of the cyclic deformation for the strain range ∆ε = 0.02 have been compared. The numerical results have been given for the previously suggested IA-model with combined isotropic-kinematic hardening and for the improved IA-model 5.3. Elastoplasticity 107 with combined isotropic-nonlinear kinematic hardening. It is obvious that the improved model give the more realistic stress-strain response.

Some further modiﬁcations of the hardening can also be performed and that will be the topic of future research work.

Figure 5.40: Comparison of IA-model, linear and nonlinear kinematic hardening, with experimental record for strain range ∆ε = 0.02

5.3.2 Finite elastoplastic deformations

The constitutive relations for finite elastoplasticity, derived in Section 4.5.3, have been implemented in the ABAQUS/Standard user subroutine UMAT in order to model the behaviour of the material exposed to large elastoplastic deformations. The large torsion problem of elastoplastic thin-walled cylindrical tubes with fixed ends, reduced to the large simple shear problem of a unit cube, defined in Section 5.2.1, has been considered. For the solution of the finite simple shear problem, the objective corotational rates, namely the logarithmic, Jaumann and Green- Naghdi rates, have been used in the constitutive relations and the results have been mutually compared.

In order to compare the results for the elastic-perfect plastic and linear kinematic hardening cases with the corresponding presented in Moss(1984), Szab´o & Balla(1989) and Bruhns et al.(2001b and 1999), the same material parameters used in the just mentioned references have been adopted for the numerical calculations. 108 Chapter 5. Numerical results

The real material parameters obtained from the experiments, conducted by Westerhoﬀ(1995), have also been used and the results are presented in this Section.

5.3.2.1 Elastic - perfectly plastic The scalar functions g(κ) and c(κ) that are introduced in the constitutive model, for the elastic-perfectly plastic case, following equation (4.98), will be

2 g = g0 = τ , 0 (5.12) c = 0, where the initial yield stress τ0 has been determined as r √ 2 6 τ = Y = µ. (5.13) 0 3 0 30

The tensile yield strength has been adopted as Y0 = 0.1µ, where µ is the second Lam´econstant, equal to the shear modulus deﬁned by relation 5.8. The elastic constants Young’s modulus and Poisson’s ratio have been chosen with the same values as those in the aforementioned references, i.e. E = 210GPa and ν = 0.3.

In the following graphs the normalized normal and shear stress distribution√ as function of the shear strain have been presented, where r = Y0/ 3 represents the initial yield stress associated with pure shear.

The results of the numerical calculation coincide with the analytical solutions for the Log-rate given in Bruhns et al.(2001a, 2001b and 1999), and are qualitatively in a very good agreement with analytical and numerical results for the Jaumann and Green-Naghdi rates, respectively, presented in Moss(1984) and Szab´o& Balla(1989).

In the enlarged representation of τ 12/r (see Figure 5.42) it can be seen that all three examined rates give a normalized shear stress development which elastic- plastic transition deviates from the constant unit value. For all rates, after the initial yielding the shear stress is starting to decrease up to its minimum at the shear strain γm. After that point the shear√ stress development shows a slight increase up to the rigid-plastic value Y0/ 3 for the logarithmic rate, while for the Green-Naghdi rate it is slightly above this value. For the Jaumann rate the stress distribution after the local minimum is almost constant. The minimum shear stress has been obtained for the shear strain γm = 0.319 which is similar to the theoretical value 0.3152, obtained in Bruhns et al.(2001b) (the double m value of ωpp therein).

According to Moss(1984), who has observed this deviation ﬁrst and called it ”instability”, the reason for this behaviour is not a consequence of the diﬀerent 5.3. Elastoplasticity 109 rates chosen. He suggests that the plastic rotation has to be properly taken into account to achieve the stable solution for an isotropic elastic-perfectly plastic material. For more details see the aforementioned reference.

Τ11 r 0.06 0.05 Logarithmic

0.04 Jaumann Green Naghdi 0.03 0.02

0.01

m Γ Γ 1 2 3 4 5

Figure 5.41: Normalised normal stress vs shear strain for elastic-perfectly plastic behaviour

Τ12 r

1.0

Τ12 r 0.8 1.0000 0.6 0.9995

0.4 0.9990

0.9985 0.2 m Γ Γ 1 2 3 4 5 Γ 1 2 3 4 5

Figure 5.42: Normalized shear stress vs shear strain for elastic-perfectly plastic behaviour and its enlarged representation 110 Chapter 5. Numerical results

In Bruhns et al.(2001a) the authors have concluded that the instability phenomenon occurs due to the absence of hardening. It can be observed from the consistency condition that in this case the rate of the shear stress and the rate of the normal stress have to be of opposite sign. This is the reason why the normal stress distribution shows a monotonic increase while the shear stress decreases m in a so-called boundary layer, i.e. after the yielding where γ = γ0 up to γ , and the opposite behaviour can be observed after this stationary point. The authors suggest that hardening has to be taken into accout to obtain a more realistic elastoplastic constitutive model.

Τ11 r

0.0012

0.0010

0.0008 Logarithmic Jaumann

Green Naghdi 0.0006

Γ 0.5 1.0 1.5 2.0 2.5

Τ12 r

1.01065

1.01060

1.01055

Γ 0.5 1.0 1.5 2.0 2.5

Figure 5.43: Elastic-perfect plasticity: Normalized normal stress development and enlarged representation of the normalized shear stress development for real material parameters 5.3. Elastoplasticity 111

Instead of a very high theoretical value of the initial tensile yield stress used in the previous example, the more realistic values of Y0 = 159.0MP a, E = 205.73GP a and ν = 0.278, taken from Anding(1997) and Westerhoﬀ(1995) for the experimentally tested material, have been used in the second numerical model. The boundary layer has not been observed for all examined rates (see Figure 5.43) but for the logarithmic rate and the Green-Naghdi rate the normal stress distribution monotonically decrease over the whole range of the shear strain while the shear stress values slowly increase monotonically. Since the time step used in the numerical integration was 2x10−5, the shear stress distribution in this case can be assumed as a constant over a whole range of γ.

5.3.2.2 Elastic - linear kinematic hardening If in the numerical model linear kinematic hardening has been assumed (see Sec- tion 4.5.3.1), the normal and shear stress developments due to large simple shear are strongly dependent on the choice of the hardening parameter c0, introduced in Eq. (4.95)2.

While the Jaumann rate shows oscillatory stress responses for both stresses, for the logarithmic and Green-Naghdi rates similar monotonically increasing results have been obtained for the normal stresses.

Τ11 Y0

0.8 Logarithmic

0.6 Jaumann Green Naghdi

0.4

0.2

Γ m 2 Γk 4 6 8 10 12

Figure 5.44: Normalized normal stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.6

If the hardening is more strong, for example c0/r = 0.6, for the logarithmic rate, after initial yielding the normal stress monotonically increases (see Figure 5.44), while the shear stress shows a monotonically growth up to its maximum 112 Chapter 5. Numerical results

m value, obtained at shear strain γk = 2.828 (which corresponds to the double m value of ωkp in Bruhns et al., 2001b) as it has been presented in Fig. 5.45. After that point√ the shear stress development monotonically decreases approaching the Y0/ 3 value as γ tends to inﬁnity.

For the Green-Naghdi rate both stress responses show a monotonic growth over the whole range of γ.

Τ12 r 3.5 3.0 Logarithmic 2.5 Jaumann

2.0 Green Naghdi

1.5

1.0

0.5

0.0 m Γ 2 Γk 4 6 8 10 12

Figure 5.45: Normalized shear stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.6

The afore-presented graphs show that the numerical model gives the stress responses which are in accordance with the analytical solutions given in Bruhns et al.(2001b). In the just cited reference the authors have shown that the elastoplasticity model based on the logarithmic rate gives the prediction of the stress-strain relationship that is in a good agreement with experimental results.

If the hardening is weak, here c0/r = 0.022 has been adopted, after initial yielding the shear stress development decreases to a value slightly smaller than the initial yield stress. After that the shear stress monotonically increases up to m its maximum value at γ1 . After having reached its√ maximum value, the shear stress monotonically falls approaching the value Y0/ 3 for γ tending to inﬁnity, as it has been shown in the graph given in Figure 5.46.

Concerning the normal stress, contrary to the case of strong hardening where the normal stress distribution is monotonically increasing, here two characteristic points have been observed (cf. Figure 5.46). After the yield point has been m reached, the normal stress is increasing up to the local maximum at γk1 from 5.3. Elastoplasticity 113

m where the stress drops down to its local minimum at γk2. After this turning point the τ11 again increases monotonicaly. The Jaumann rate in both Figures ex- hibits an unrealistic oscillating behaviour, whereas the Green-Naghdi rate leads to much too high shear stresses.

Τ12 r 1.02

1.01

1.00

0.99 Logarithmic

Jaumann 0.98 Green Naghdi

Γ 0.97 m 0 2 Γ1 4 6 8 10 12

Figure 5.46: Normalized shear stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.022

Τ11 Y0 0.05

0.04

0.03

0.02 Logarithmic

Jaumann 0.01 Green Naghdi

0.00 m m Γ 0 Γk1 2 Γk2 4 6 8 10 12

Figure 5.47: Normalized normal stress vs shear strain for linear kinematic hardening plasticity, c0/r = 0.022 114 Chapter 5. Numerical results

In a second example the material parameters obtained from the experiments by Westerhoﬀ(1995) have been used and the kinematic hardening modulus c0, according to Eq. (4.96), has been attained as

2 c = B = 1945.008MP a. (5.14) 0 3

The numerical calculations gave the following results if the logarithmic rate was used.

Τ11 Y0 Τ12 r

10 10

5 5

Γ 0 Γ 1 2 3 4 5 0 1 2 3 4 5

Figure 5.48: Normalized normal and shear stress vs shear strain for the material parameters according to Westerhoﬀ(1995)

It can be seen that the kinematic hardening is very strong, as it was already concluded in Section 5.3.1. From the numerical calculations the maximum shear m m stress has been achieved as τ12real = 1381.88 MP a for γ = 3.03 which is in m accordance with the analytical value τ12real = 1380.748 MP a , obatined using m m Eq. (6.19) from Bruhns et al.(2001b), for γ = 3.0176 (the double value of ωkp from the just mentioned reference).

Assuming the same material parameters as in the previous example, the plate with a hole that undergoes large axial elastoplastic deformation has been tested. The assumed model has been taken as previously introduced (see Figure 5.5). Here, the prescribed displacement in the axial direction has been 2.5 mm.

Due to the symmetry of the plate and loading, only a quarter of the plate has been considered in the numerical calculations. The deformed and undeformed conﬁgurations as well as the normal stress distribution in the plate at the end of deformation have been given in Figure 5.49. 5.3. Elastoplasticity 115

Figure 5.49: Plate with a hole in tension: Undeformed and deformed conﬁgura- tion and normal stress distribution for large elastoplastic deformation with linear kinematic hardening

With the Log-rate implemented in the constitutive relations, the development of the normalized normal stress for the node at the hole that undergoes the maximum stress has been presented in the following ﬁgure.

Σ22 2G 0.0025 0.0020

0.0015

0.0010

0.0005

0.0000 h22 0.00 0.01 0.02 0.03 0.04 0.05

Figure 5.50: Plate with a hole in tension: Normalized normal stress distribution for large elastoplastic deformation with linear kinematic hardening 6 Concluding remarks

6.1 Summary and conclusions

The numerical implementation of the self-consistent Eulerian ﬁnite elastoplasticity theory, based on the logarithmic rate and the additive decomposition of the natural deformation rate (stretching), in the commercial ﬁnite element software and its validation have been the main tasks of this treatise.

Alongside the Log-rate, the Jaumann and Green-Naghdi rates, as corotational, and the Truesdell rate, the Oldroyd rate and the Coter-Rivlin rate, as non- corotational rates, have been implemented in the constitutive models as well. The proposed numerical frameworks were veriﬁed on several benchmark exam- ples and the results were compared mutually and with the relevant literature. In order to preserve objectivity of the Eulerian rate model special attention is given to the incremental objectivity.

The stress-strain prediction due to large elastic deformations, usually met in a broad class of rubber-like materials, shape memory alloys, etc., has been modeled by the hypo-elastic constitutive relation, i.e. by the linear relationship between the objective rate (increment) of the Kirchhoff stress and the rate (increment) of its work-conjugate pair, the Eulerian Hancky strain, via a constant isotropic 4th-order tensor. The rate form of the hypo-elastic constitutive relation qualifies it to become a basic constituent in the structure of the adopted Eulerian rate theory of finite elastoplasticity, too, while the plasticity part has been modeled by the associated flow rule.

Concerning the pure elastic deformation, the numerical calculations have proved that, opposite to the other tested rates, the logarithmic rate predicts the stable solutions with no residuals in the closed elastic strain path, and thus conﬁrming that the hypo-elastic constitutive relation based on the Log-rate is self-consistent in the sense of elasticity.

The thesis has shown that the occurrence of the finite rotations in total deformation significantly influences stress responses. In the case of small rotations all examined objective corotational rates give the accurate results while the reliability of the Jaumann and Green-Naghdi rates is decreasing with the increase of rotations. For cyclic deformations with moderate and large rotations, the previously mentioned rates produce high values of residual stresses at the end of the cycle, especially if the deformation is repeated in large number of cycles,

116 6.2. Possible extensions of the present research 117 which is the usual case. The Log-rate has proved itself as the only one that gives responses in accordance with those theoretically expected. In this thesis it has been shown that the non-corotational rates have to be excluded from the constitutive relations even for moderately large rotations.

The numerical implementation is thereafter extended to elastoplastic deformations. Constitutive relations for ﬁnite plasticity, based on the INTERATOM model which originally has been established for the small deformation case, have been derived and numerically implemented. The behaviour of the material during small monotonic and cyclic elastoplastic deformations and ﬁnite monotonic elastoplastic deformations has been tested, too. The simple constitutive model concerning combined isotropic-kinematic hardening, established by introducing a few material constants, has been proposed and its stress-strain prediction has been compared to the experimental results. It has been concluded that the proposed model gives the results that are in a very good agreement with monotonic tests and in a qualitatively agreement with experiments for cyclic behaviour and therefore can be used as an initial step in the prediction of a realistic behaviour of engineering materials. Some improvements of the existing model have been proposed as well.

6.2 Possible extensions of the present research

The proposed Eulerian ﬁnite elastoplasticity model can be extended in several ways.

Although the current formulation of the given constitutive model focuses on rate- independent elastoplasticity, an extension to visco-plasticity is possible. More- over, the underlying IA model has been constructed to describe a visco-plastic behaviour (at higher temperatures). Here, isothermal processes have been the subject of research, but it can be broadened to adopt thermal eﬀects as well.

The proposed constitutive model of elastoplasticity relies on experimental results. In order to provide that the existing numerical model can be used for the strain ranges that have not been tested, the proper evolution constitutive equation has to be developed.

The ﬁrst improvement that can be made, and that was already indicated and begun in this work, is the introduction of the nonlinear hardening behaviour of materials.

Any step forward towards a generalization of the model would demand more material parameters and would make the analysis more complex. Some of the basic hypotheses than must be reconsidered.

In order to ensure more practical application of the proposed models, more com- 118 Chapter 6. Concluding remarks plex engineering structures with more realistic loadings have to be treated as well. One of these applications can be a modeling of metal forming processes where the spring eﬀect should be properly taken into account.

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