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LSU Historical Dissertations and Theses Graduate School
1996 Crystal Plasticity Model With Back Stress Evolution. Wei Huang Louisiana State University and Agricultural & Mechanical College
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Recommended Citation Huang, Wei, "Crystal Plasticity Model With Back Stress Evolution." (1996). LSU Historical Dissertations and Theses. 6154. https://digitalcommons.lsu.edu/gradschool_disstheses/6154
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CRYSTAL PLASTICITY MODEL WITH BACK STRESS EVOLUTION
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in
The Interdepartmental Program in Engineering Science
by Wei Huang B.S., Shanghai Jiao Tong University, 1982 M.S., Shanghai Jiao Tong University, 1985 M.S., Louisiana State University, 1991 May 1996
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS
I would like to express my thanks to my beloved advisor Professor George
Z. Voyiadjis, who not only guided, supported and encouraged me during my
doctoral study, but also made a tremendous effort to make this dissertation become
possible. I would also like to thank him for the financial aid that he provided me
during this study.
I would like to thank other members of my doctoral advisory committee,
Professor M. Sabbaghian, Dr. S. S. Peng, Professor J. R. Dorroh and Professor
R. Gambrell for reviewing the dissertation and serving in the committee.
I wish to thank my colleagues at LSU, Dr. A. R. Venson, Dr. P. U.
Kurup, Dr. P. I. Kattan, Mr. R. Echle, Mr. G. Thiagarajan, and Dr. T. Park for
many useful discussions and numerous help in the aspect of numerical
computation. They also made my stay at LSU a pleasant one.
My beloved wife, Yuxian, with whom I have been married for ten years
and have two wonderful daughters, Linda and Jennifer, deserves special worlds of
thanks. My wife helped and supported me during this study in so many ways that
cannot be listed. My parents have been the key roles in my career in pouring me
with love and emphasized the importance of education in my life. My father, now
a retired electrical engineer, was the first one in my early childhood to introduce
many wonderful fields of science and engineering. In recent years, my mother
made two trips to the U.S., helping me and my wife take care of our children
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. while we were working and studying. It has been a dream for her that I complete
this study. I would like to dedicate this dissertation to her.
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS
ACKNOWLEDGMENTS ...... ii
LIST OF TABLES ...... vi
LIST OF FIGURES...... vii
ABSTRACT ...... ix
CHAPTER 1 INTRODUCTION...... 1
2 THEORETICAL BACKGROUND...... 5 2.1 Phenomenological Approach ...... 5 2.1.1 Isotropic Strain Hardening ...... 6 2.1.2 Kinematic Hardening ...... 6 2.1.3 Anisotropic Hardening ...... 7 2.2 Crystal Plasticity Approach ...... 8 2.2.1 Single Crystal Plasticity ...... 8 2.2.2 Concepts of Dislocation and Backstress...... 10 2.2.3 Polycrystal Plasticity ...... 16 2.2.4 Current Crystal Plasticity Approaches by Asaro and Needleman and Rashid and Nemat-Nasser ...... 17 2.3 Objectives and Method of Approach ...... 21
3 THEORETICAL FORMULATION OF CONSTITUTIVE EQUATION ...... 22 3.1 Fundamentals ...... 22 3.2 The Hardening Rule and the Evolution of the Backstress X ...... 25
4 NUMERICAL IMPLEMENTATION ...... 32 4.1 Application to the Simple Shear ...... 32 4.2 Numerical Results...... 35 4.2.1 Comparison Between the Two Models ...... 36 4.2.2 Effect of Initial Orientation ...... 40 4.2.3 Evaluation of Coefficients A, B, C for Model 2 ...... 43 4.2.4 Evaluation of the Interacting Factors qj ...... 49
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 DISCUSSION OF RESULTS ...... 52
6 CONTINUUM PLASTICITY THEORY FOR POLYCRYSTALS WITH MICROSTRUCTURAL CHARACTERIZATION...... 56 6.1 Introduction ...... 56 6.2 Continuum Mechanics and Dislocation T h eo ry ...... 56 6.3 Macroscopic Mechanical Behavior of Polycrystalline Materials ...... 59
7 PROPOSED FUTURE WORK FOR THE THREE- DIMENSIONAL CASE ...... 65 7.1 "Pan-Cake" Type Polycrystal ...... 66 7.2 "Hexagonal Column" Type Polycrystal ...... 68
8 SUMMARY AND CONCLUSION ...... 72
REFERENCES...... 74
APPENDIX ...... 79
VITA ...... 100
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES
Table 2.1 Designation of slip systems in FCC crystal ...... 9
Table 4.1 List of different numerical problems conducted ...... 37
vi
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Figure 2.1 A schematic illustration of a dislocation line BC created by shearing on plane ABC along the direction AC in a simple cubic crystal ...... 11
Figure 2.2 Microphoto and schematic illustration showing two groups of slip lines along two different {111} planes in a plastically deformed Cu3Au single crystal ...... 13
Figure 2.3 Dislocation configuration in a tensile tested alpha brass single crystal showing dislocations in several slip systems ...... 14
Figure 2.4 Dislocation configuration in a tensile tested pure copper sample showing high density of dislocations in the dislocations cells ...... 15
Figure 2.5 Schematic illustration of configurations and the decomposition of deformation gradient F ...... 19
Figure 4.1 Schematic illustration of the planar deformation model. Two slip systems denoted by their direction and normal to slip planes (s1, n 1), (s2, n2) ...... 33
Figure 4.2a Comparison of stress changes during shear deformation for two different backstress models and the ideal plastic crystal cases ...... 38
Figure 4.2b Comparison of the backstress evolution for the two backstress m odels ...... 39
Figure 4.3a Effect of initial orientation on the change of stresses during shear deformation ...... 41
Figure 4.3b Effect of initial orientation on the change of orientation angle,
Figure 4.4a Effect of parameter B0 on the stress a during shear deformation ...... 44
Figure 4.4b Effect of parameter B0 on the backstress X during shear deformation ...... 45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.5a Effect of parameter C0 on the stress a during shear deformation ...... 47
Figure 4.5b Effect of parameter C0 on the backstress X during shear deformation ...... 48
Figure 4.6a Effect of interactive factors qj on the stress a during shear deformation ...... 50
Figure 4.6b Effect of interactive factors q, on the backstress X during shear deformation ...... 51
Figure 7.1 Schematic illustration of a "pan-cake" type polycrystal m odel ...... 67
Figure 7.2 Schematic illustration of a "hexagonal column" type polycrystal m odel ...... 69
Figure 7.3 Microphotos showing flattened grains in a cold rolled drawing quality SAE 1008 steel (top) resemble the "pan-cake" polycrystal model and equiaxial grains in a hot rolled SAE 1008 steel (bottom) resemble the hexagonal polycrystal model (400X) ...... 71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT
In this work, two backstress models and a modified yield criterion for
crystalline materials are proposed. The proposed work is for the identification of
texture development in metals when subjected to large deformation processes such
as rolling, extrusion, stamping and deep drawing, etc. These models are incor
porated into a numerical procedure similar to the approaches by Asaro (1983),
Peirce et al. (1983), Nemat-Nasser (1983), and Rashid and Nemat-Nasser (1992).
Also, similar to their approaches, a two-dimensional crystal structure is used for a
feasibility study of the model by solving a simple shear problem. More impor
tantly, the proposed formulations are strain rate independent expressions used to
evaluate the plastic behavior and texture development in rate independent materials.
This modification to the numerical procedure avoids the inconsistencies of mathe
matical manipulations by other approaches which use a strain rate dependent
materials model to solve the problems of strain rate independent materials.
Furthermore, the parameters in the proposed models are studied to evaluate their
effects on the plastic behavior of the modeled materials. Numerical results showed
that both backstress models are physically sound and feasible. The method in this
work is quite flexible and can be used to model complex anisotropic behavior of
the materials. This approach is also compared with the works by Zbib and
Aifantis (1987) and Lamar (1989). It is noticed that the proposed method in this
work has more advantages in physical process than the other compared approaches.
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, a general form of yield surface in polycrystalline materials is proposed that
incorporates the Tresca and von Mises and other models.
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION
The elasto-plastic behavior of polycrystalline metal material is of great
importance to industrial manufacturing and structural design. Generally, modeling
the plastic behavior of crystal materials has been attempted through two kinds of
approaches: the phenomenological and physical approaches.
The phenomenological approach is the one based on the description of the
experimental data obtained from the stress strain curves. A yield surface function,
its motion as well as the change of its shape and size are the major ingredients of
this approach. It is quite simplistic and straight forward compared to the physical
approach. However, it lacks the microstructural interpretation of the physical
process in the metals. Also, it cannot correctly model the anisotropic behavior
caused by the microstructure re-orientation during deformation such as the texture
evolution. On the other hand, the physical modeling approach (micromechanical
approach) attempts to describe more accurately the behavior of metals through the
microstructural process. One of the big advantages of this is the capability to
predict the texture evolution of the material in a realistic way. When a polycrystal
material undergoes large plastic deformation, each grain will rotate toward a more
stable orientation with a different speed. This rotation process depends on the
original orientation of the grain and the loading direction. Therefore, the initially
uniform orientation distribution will change to a non-uniform one with
1
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concentrations of several specific orientations. However, due to the fact that many
factors can influence the elasto-plastic behavior such as the size, shape, orientation
of grains, secondary phase distribution and the stacking fault energy of the metal,
etc., the physical modeling approach usually lacks simplicity and needs much more
intensive computational techniques as well as various simplified assumptions.
There has been a constant need to predict the elasto-plastic behavior of
polycrystalline metal materials in order to assist in the proper processing of metal
products. Taylor and Elam (1925), Taylor (1934) and Sachs (1928) are among the
first to lay down some of the framework of crystal plasticity. In Sachs’ model,
stresses are assumed homogeneously distributed in the grains. Therefore, strains in
neighboring grains may be different due to the lattice orientation. However, no
restrains are applied to this difference between different neighboring grains.
Whereas in Taylor’s model (1934), the elastic strain is neglected compared to the
large value of the plastic strain and more importantly a homogeneous plastic
deformation in each individual grain is assumed. Individual crystals deform by
slipping over preferred crystallographic planes and along preferred crystallographic
directions under a critical shear stress.
Taylor’s model was further developed and elaborated in more details by
Bishop and Hill (1951a, 1951b). More advanced models that incorporate elasto-
plastic behavior were later proposed by Lin (1964), Lin and Ito (1966), Kroner and
Teodosiu (1972), Hill and Rice (1972), Hutchinson (1976), Asaro (1983), Peirce et
al. (1982, 1983), Nemat-Nasser (1983), Rashid and Nemat-Nasser (1992), Havner
I I | }
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(1992), and many other people. The development of the theory makes us more
closely understand the real physical processes that occur inside the materials. With
the advent of computer technology, the crystal plasticity theory has made
tremendous progress. This is reflected in the works of van Houtte (1976), Asaro
and Needleman (1985), Needleman et al. (1985) and Rashid and Nemat-Nasser
(1992).
On the other hand, phenomelogical approaches without considering the
crystal behavior of the material are also used for the convenience of numerical
calculation. Mroz (1967), Tseng and Lee et al. (1983), Zbib and Aifantis (1987,
1988), Lamar (1989) are some representatives in this group. In these models, the
shape, size and center of yield surfaces are predicted based on the load increment
and yield functions in order to describe elasto-plastic behavior of materials.
It is beyond the scope of this research to document all the details of the
developments in crystal plasticity theory in the last six decades. Of interest are the
approaches by Peirce, Asaro and Needleman (1982), Nemat-Nasser (1983) and
Rashid and Nemat-Nasser (1992). They have been very successful in predicting
the elasto-plastic behavior as well as the texture evolution of crystalline materials.
Unfortunately, the backstress evolution has been neglected in these models.
Backstress is a residual stress embedded in a polycrystalline or a single crystal
material at the crystal-lattice level due to plastic deformation. In the stress space,
it is expressed by the stress tensor from the origin to the center of the yield sur
face. Since it affects the evolution of the yield surface, it plays a very important
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role in modeling the plastic deformation and describing the material anisotropy.
Consequently, there have been numerous attempts recently in the crystal plasticity
field in order to model the plasticity behavior of materials with consideration of the
backstress evolution.
In this work, the physical concept of dislocation slips in crystalline
materials is used in order to model the backstress evolution. Hence, two types of
backstress evolution models are proposed based on the crystallographic slips in the
crystal. These backstress models are then incorporated into the constitutive
equations so that kinematic hardening can be included in the models used by Asaro
(1983), Peirce et al. (1983), Nemat-Nasser (1983) and Rashid and Nemat-Nasser
(1992). To study the physical feasibility of the proposed models, the simple shear
problem is used in this investigation. The main purpose of this study is focused on
the feasibility of the use of backstress in crystal plasticity. Therefore, a fictitious
single crystal with two slip systems is used here for simplicity and for a better
representation of the physical process of the proposed model.
A proposed theoretical derivation is also discussed for the polycrystal
material with the intent to develop further the concept of dislocation tensor as
proposed by Mura (1963, 1965, 1968). Through a proper averaging technique, a
general yield criterion is derived which is applicable to the precipitate hardening
materials.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2
THEORETICAL BACKGROUND
There has been a constant need to predict the elasto-plastic behavior of
polycrystal materials in order to assist in the proper process of metal products.
The modeling approaches usually fall into two major categories: (1) the
phenomenological approach and (2) the micromechanical approach.
2.1 Phenomenological Approach
The phenomenological approach uses the continuum mechanics method to
depict the macroscopic elasto-plastic behavior of materials without looking into the
microstructural process inside the materials. To describe the plastic behavior, a
yield criterion is used which is a surface in the stress space. The plastic material
behavior is characterized through the isotropic hardening, kinematic hardening,
anisotropic hardening etc. These are expressed through the changes in the shape,
size and motion of the yield surface.
One of the first yield criteria was the Tresca criterion. It states that yield
occurs when the maximum shear stress reaches a critical value. Another very
important criterion is the von Mises yield criterion. The basic idea of this criterion
is that yield occurs when the elastic distortional strain energy stored in the con
tinuum reaches a critical value. It can be mathematically expressed as follows:
\ ra = k2 (2.1)
5
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where rkl is the deviatoric stress. rkl = crkl - amm 5kl/3 and k = aypl \JT. ayp is
the uniaxial tensile yield stress of the metal.
In general, the yield surface can be expressed as a function of stress such as
F (akl) = 0 (2.2)
The plastic strain rate is determined by the associated flow rule such that
where X is a Lagrange multiplier. As afore mentioned the yield and work
hardening behavior of the materials are modeled through the changes in the size,
shape and shifting of the yield surface. They can be listed as follows:
2.1.1 Isotropic Strain Hardening
For an isotropic hardening material, the yield surface expands in size during
plastic deformation which can be expressed as F(ffjj) = K(epeq), where K increases
in value with the increase of the equivalent plastic strain:
(2.4)
2.1.2 Kinematic Hardening
For this kind of materials, the yield surface has the same shape size and
orientation. However, the center of the surface changes position in the stress space
during the deformation due to the change of the backstress. This was first
introduced by Prager (1956) to interpret the Bauschinger effect in metals. The
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Bauschinger effect is the phenomenon of reducing the yield stress of a material
upon reverse loading that follows unloading.
2.1.3 Anisotropic Hardening
For this kind of materials, the yield surface changes its shape during plastic
deformation. This plastic deformation induced anisotropy is due to the develop
ment of texture in the material.
The evolution of backstress X is a very important part in modeling the
anisotropic hardening. In the stress space, the backstress is represented by the
stress tensor from the origin to the center of the yield surface. Prager proposed
that the center of the yield surface moves in the direction of the increment of
plastic strain which can be expressed as follows for the small strain theory,
dXy = C d cPy (2.5)
Later, Ziegler (1959) modified Prager’s hardening rule so that it is also valid in the
stress subspace. He proposed that the rate of translation takes place in the
direction of the reduced stress vector ay = ay - Xy, that is
dXy = dM (ay - Xy) (2.6)
where dyx is a positive proportionality factor which depends on the history of
deformation.
There are numerous other contemporary phenomenological approaches.
Among them, the works of Mroz (1967), Tseng and Lee (1983), Dafalias (1983,
1990, 1993), Zbib and Aifantis (1987, 1988), Voyiadjis and Kattan (1989, 1990,
1991) and Voyiadjis and Sivakumar (1991) are the major representative ones.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8
Their approaches try to depict the evolution of the backstress, plastic spin as well
as the proper definition of corotational stress rate. The physical concept of these
quantities are incorporated in their models. This has made the topic intensively
complex. Numerous researchers have tried to obtain rigorous formulations for the
elasto-plastic behavior of metals in the past several decades.
The advantages of the phenomenological model lie in its simplicity in the
calculation and its easy implementation in industry. However, it does not really
describe the real physical process in the material. One of the obvious disadvantage
is that it can not monitor the texture evolution during the plastic deformation.
2.2 Crystal Plasticity Approach
The crystal plasticity approach describes the elasto-plastic behavior of a
crystalline solid from its crystallographic point of view.
2.2.1 Single Crystal Plasticity
The plasticity behavior of a single crystal is described through the
occurrence of crystallographic slips on certain specific crystallographic planes and
crystallographic directions. These crystallographic planes and directions are
determined by the crystal lattice structure. Usually, they are the most densely
packed planes and the directions of the closest distance between the neighboring
atoms. In an FCC crystal, the primary slip planes are the most densely packed
{111} planes. In a BCC crystal, the primary slip planes are {110}. InanHCP
crystal, they are the {0001} planes. This is because the number of atomic bonds
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between these plane layers is the least to break during slip. In an FCC crystal, the
primary slip directions are the < 110> directions. In a BCC crystal, the slip
directions are the < 111 > directions. In an HCP crystal, the slip directions are
the < 1120 > directions. The possible combination modes of a slip plane with a
slip direction are called the slip systems. In an FCC crystal, the four {111} slip
planes and six <110> slip directions compose the twelve possible combinations of
them (24 if the negative shear directions are included). These systems are listed in
Table 2.1.
Table 2.1 Designation of slip systems in FCC crystal.
Plane 111 III 111 111
Direction Oil 101 110 o il 101 110 O il 010 no oil loi 110
System al a2 a3 bl b2 b3 cl c2 c3 dl d2 d3
The crystallographic slip occurs when the resolved shear stress of a slip
system due to the applied load reaches a critical value. Schmid (1924) introduced
a Schmid critical shear stress law from his experimental results of tension test to
single crystals. It is observed that for a given crystal at a certain temperature,
when the resolved shear stress on a slip system denoted by its slip direction s and
normal n reaches a critical value t0 , the crystal yields. Let angle 0 and X denote
the angles between the tensile stress axis with the normal to the slip plane n and
the slip direction s, respectively. We can then derive the following expression
Tq = aM = <7 cos
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where M is the Schmid factor and a is the magnitude of the applied tensile stress.
In the most general case of any applied stress tensor a, the Schmid law can be
expressed as follows
tq = Taylor and Elam (1925) later perfected this law to include work hardening. The Taylor-Schmid law is based on the shearing of a single crystal during com pression tests. The critical shear stress can be generally expressed as ry = f (y) where y is the plastic shear strain. When y = 0, Ty equals r0. For a work hardening material, ry increases with the increment of plastic shear strain y. When the difference of hardening in each slip system is considered, then the critical shear stress for each system should be considered. Hill (1965) and Hill and Havner (1982) expressed them in the following way d^y = E, Hw dy* (2.8) where y ! is the amount of shear in slip system £. Hkf is the hardening moduli of the slip systems. 2.2.2 Concepts of Dislocation and Backstress It is now universally accepted that crystallographic slip in crystallographic materials results from the creation and movement of dislocations. A schematic illustration of a dislocation in a simple cubic crystal is shown in Figure 2.1. The linear defect BC, or a dislocation BC in this simple cubic material is caused by a shear deformation on the slip plane in an atomic distance b. Macroscopic slip lines and slip bands formed in crystal during plastic deformation are the products of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. KgUre2A created c 1 i t d Tby vshearing l ilIUStrati0” on plane 0f ABC a diS,0Ca,i0" along the BC direction AC in a simple cubic crystal. with permission of the copyright owner. Further reproduction prohibited without permission. 12 creation and motions of many dislocations in the lattice levels. Figure 2.2 shows an example of slip lines formed on two {111} planes in a plastically deformed single crystal of a Cu3Au. Dislocations exist in many configurations such as dislocation pile-ups, dislocation cells, etc., during the plastic deformation. Work hardening in the material is caused by the interaction of dislocations. The formation of what type of dislocation configuration depends on the deformation history and intrinsic material properties such as the stacking fault energy. To fully discuss it in detail is beyond the scope and the purpose of this study. A few examples of different dis location configurations are given here for the interpretation of backstress concept. These photos were taken from thin slices of several different tested materials by using a JOEL 100CX transmission electron microscope (TEM) in LSU Life Science Building. Figure 2.3 is a transmission electron microscopic photo showing dislocations which occurred in several slip systems in an alpha brass single crystal after a tensile test. Figure 2.4 is a TEM photo showing high density of dis locations in the dislocation cells in a tensile tested pure copper material. In this work, we are interested in the backstress and its evolution during the plastic deformation. As aforementioned, backstress is the residue stress embedded in the material due to plastic deformation. Physically, it is caused by the dislocation interactions in the crystalline materials. The above illustration of dislocation configurations showed the source of the backstress and complexity of their interactions. It is known that each dislocation forms an elastic stress field due Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 50 /i-m Figure 2.2 Microphoto and schematic illustration showing two groups of slip lines along two different {111} planes in a plastically deformed Cu3Au single crystal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Figure 2.3 Dislocation configuration in a tensile tested alpha brass single crystal showing dislocations in several slip systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 m W Figure 2.4 Dislocation configuration in a tensile tested pure copper sample showing high density of dislocations in the dislocations cells. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 to the crystal imperfection in its core. The interaction between dislocations through these elastic stress fields of the dislocations of different configurations will hinder the plastic deformation and work harden the material. 2.2.3 Polycrystal Plasticity The overall response of a polycrystal solid to a macroscopically uniform external load is the combination of the accumulative response behavior of each single crystal. There has to be an averaging method to link the microscopic response of each grain to the macroscopic mechanical behavior. The generally used method to calculate the macroscopic quantity is the volume averaging method introduced by Bishop and Hill (1951a, 1951b). It can be expressed as follows: (2.9) where, < A > and A are the average and the local values of a physical quantity respectively. V and V 1 are the overall volume and the volume of i-th grain respectively. For a moderately fine grained metal such as the grain size of about 0.1 mm, there are approximately one thousand grains in one cubic millimeter space. For these grains, their boundary surfaces are irregularly shaped and randomly oriented. This large number of grains interact with each other elasto- plastically during the deformation to keep the grain boundaries in contact. To physically simulate such a complicated process in real detail is almost impossible. Some simplification has to be made for practical applications. Sach (1928) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 proposed that for a macroscopically uniform load, the stress should be homo geneously distributed in each grain. This would result in the strain in each grain to be different for the same stress value due to the crystal anisotropy. Therefore, the boundary contact is almost impossible to maintain. Taylor (1937) proposed that the plastic strain in each grain is identical under a uniform load by neglecting the small amount of elastic strain. This assures that the contact between grains is maintained. Taylor’s model was further developed and elaborated by Bishop and Hill (1951a, 1951b). More advanced models that incorporate elasto-plastic behavior were later proposed by Lin (1964), Lin and Ito (1966), Kroner and Teodosiu (1972), Hill and Rice (1972), Hutchinson (1976), Asaro (1983), Peirce et al. (1982, 1983), Nemat-Nasser (1983), Rashid and Nemat-Nasser (1992), and Havner (1992). The development of such a theory gives us a better understanding of the real physical process that occurs inside the crystals. With the advent of computer technology, the crystal plasticity theory has made tremendous progress. This is reflected in the works of van Houtte (1976), Asaro and Needleman (1985), Needle- man et al. (1985), Mathur and Dawson (1989), Nemat-Nasser and Obata (1986), and Rashid and Nemat-Nasser (1992). 2.2.4 Current Crystal Plasticity Approaches by Asaro and Needleman, and Rashid and Nemat-Nasser Currently, the most used crystal plasticity approaches are those proposed by Asaro and Needleman (1985), and Rashid and Nemat-Nasser (1992). These Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 approaches have quite accurately predicted the elasto-plastic behavior as well as the texture evolution of crystals in some cases. In their model, the deformation gradient F is decomposed into a crystallographic slip part Fp and a lattice distortion part F*: F = F* • Fp (2.10) Figure 2.5 illustrates the configuration of the material for different deformation gradients, F, F*, and Fp. The plastic part of the rate of stretch is given by LP = f • F 1 - F* • F *'1 = Dp + Wp (2.11) where Dp and Wp are further expressed by the summation of crystallographic slips as follows: Dp = Ef P“ (2.12) Wp = E 7a n a (2.13) The second order tensors P“ and are defined as follows: p“ = (s*“ n*a + n*“ s*“ )/2 (no summation over a) (2.14) 0 “ = (s*“ n*a - n*“ s*“ )/2 (no summation over a) (2.15) where s*“ and n*“ are the slip direction and normal of slip plane in the current configuration. Through a lengthy mathematical manipulation which is omitted here, the following relation is obtained, f = C : D - X) R“ ya (2.16) a where C is the elastic moduli, and f is the Jaumann rate of the Kirchoff stress. R“ is a second order tensor such that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Figure Figure 2.5 Schematic illustration of configurations and the decomposition of deformation gradient F. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 (2.17) To solve an application problem, a forward numerical technique is used in calculating the slip rate by using a rate dependent form first used by Teodosiu and Sidoroff (1976) expressed as follows M (2.18) where £0 is a fixed reference value, ga is the reference slip stress on slip system a which depends on the total plastic slip, slip rate history and temperature. r“ is the resolved shear stress in this slip system. The other physical quantities such as Dp and Wp as well as the elastic distortion part F*, can then be calculated. Individual lattice spin and the texture prediction of polycrystal are calculated through the rotational part of F*. One of the disadvantages of this approach is that the rate dependent plas ticity model is used even if the material is rate independent. For a rate indepen dent material, a very large parameter of M is adopted such that when r“ < g“, y a -* 0 is obtained. The advantage of this approach is basically for computational purposes. Whether this approach is sound physically or not is still questionable. Another problem in the approach is that the backstress evolution has been neglected in these models. Therefore, it can not be used to model kinematic type hardening materials. To solve this problem, Cailletaud (1992) decomposed the microstructural hardening into two parts: x“ for the kinematic part and r“ for the isotropic part. The slip rate is then given by, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 M (T“ - X“) - Te sign (xa - x“) a (2.19) r = e0 K Hu and Teodosiu (1992) also proposed the decomposition concept. In their approach, the yield function takes the form F(a, P, R, X) = |a-X| -(R q + |P | + R) = 0 where Rq , | P j and R are the dislocation configuration related internal variables. 2.3 Objectives and Method of Approach The objectives of this research are outlined below. The first one is to propose a robust crystal plasticity model based on the concept of crystallographic slip behavior of materials by modifying the existing models. In this model, rate independent model will be proposed. This way one avoids the use of rate dependent models to solve problems in rate independent materials. The second one is to test this approach to model the crystal plasticity. The third one is to predict the change of crystal orientation. Lastly, the final one is to check the feasibility of the proposed model in constitutive modeling the mechanical behavior of materials as well as the evaluation of the parameters. To achieve these goals, a two dimensional planar slip deformation with two independent slip systems is first implemented. The problem of simple shear is used in order to evaluate the effectiveness of the proposed model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 THEORETICAL FORMULATION OF CONSTITUTIVE EQUATION 3.1 Fundamentals A Cartesian coordinate system is used in this work with C 0 and Ct repre senting the configurations at times t 0 and t, respectively as shown in Figure 2.1. The configuration is based on an embedded crystal lattice frame. The deformation gradient associated with C 0 and Ct is expressed as follows: F = F*5 • Fp (3.1) where superscripts e and p denote, respectively, the elastic and plastic components. The corresponding associated velocity gradient is given by L = F • F 1 = F6 • F5' 1 + F* • Fp • FP"’ • F 6’1 (3.2) The above expression may also be expressed as follows: L = Le + F e • L 0P • F e_1 (3.3a) where (3.3b) where s0“ and n0“ are the unit vectors of the ath slip direction and normal to the slip plane in the configuration C0. y a is the slip rate in slip system a. Making use of the elastic rotation tensor Re(t), such that Fe = Re • IF (3.4) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 where 1/ is the right elastic stretch tensor, we can express the unit vectors s(t) and n(t) as follows: s(t) = Re(t) • s0 (3.5a) n(t) = Re(t) • n 0 (3.5b) The corotational stress rate with respect to the crystal lattice is defined by %* = a - (We • a) +(a • We) (3.6) where a is the Cauchy stress, %* is the corotational rate with respect to the lattice spin, and We is the elastic spin by the substructure elastic rate of distortion. The constitutive expression is given here in terras of the Kirchhoff corotational stress rate, t , and De the symmetric part of the elastic part of the velocity gradient Le such that t * = C : De (3.7) where C is the elasticity moduli tensor. The Kirchhoff stress r may be expressed in terms of the Cauchy stress a and the Jacobian of deformation such that, a* + a Dkk = C : De (3.8) The Jaumann rate of the Cauchy stress, a is expressed as follows: £ = ff-W -ff + or t = (a* + W e • which leads to the following relation °a = a* - (W - We) • a + a • (W - W e) (3.9c) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 Rashid and Nemat-Nasser (1992) proposed a numerical method to deal with the right stretch tensor IP while accounting for the rotation of the unit tensors s and n. Since IP is relatively small, it may be expressed as IP = I + Y (3.10) where I is a second order identity tensor. Y is a second order tensor whose second-order or higher terms in Y may be neglected such that the inverse of IP may be simply expressed as U ^ 1 =I-Y (3.11) Making use of IP and its inverse, Equation (3.3a) may be rewritten as follows: Le = Re • R6’1 + Re • Y • Re'' (3.12) From equation (3.12), one includes that W e = Re • Re 1 (3.13) De = Re • Y • Re"‘ (3.14) We define N LP = £ y a sa n“ (3.15) a=l and correspondingly we also have Lp = Lp + Re • Y • Re 1 • Lp -Lp • Re • Y • Re_1 (3.16) The symmetric and anti-symmetric parts of the velocity gradient may be expressed as follows: D = De + Dp + (B : a) • Wp - IVP • (B : a) (3.17a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 W = We + WP + (B : a) • DP - DP • (B : ff) (3.17b) where B = C 1 is the inverse of 4-th order plastic moduli tensor C. The resulting stress rate to be used in the analysis is given as follows: if = W e • a - a • We + C : (D - Dp) + C : [WP • (B : 3.2 The Hardening Rule and the Evolution of the Backstress X The proposed model in this work incorporates the evolution of the back stress, X, into the formulation. The yield criterion is proposed by the following expression which in fact is based on the work of Hill (1965) and Hill and Havner (1982) with the incorporation of backstress, ( where t0“ is the initial critical shear stress of the slip system a when backstress X = 0. T“ is the hardening of the resolved shear stress in slip system a. The plastic strain rate and plastic spin are given as follows: N DP = £ 7“ P“ (3.20a) N w p = y , ya fl“ (3.20b) where P“ = 2. (s“ n“ + n“ s“) (no summation over a) (3.21a) 2 « “ = 1 (sa n“ - n“ s“) (3.21b) 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 To model the material behavior, the evolution equation for the backstress is required. Numerous expressions for backstress X are available in the literature such as those proposed by Cailletaud (1992) and by Hu and Teodosiu (1992) for rate dependent plasticity. In this work, two models of backstress evolution are proposed. The first one properly extends Prager’s rule for small strain theory equation (25) into the finite strain theory such that, A X* = a(y) DP (3.22) a (y) = a 0sech" (3.23) Ts - To where a(y) is a coefficient that is a function of the total amount of shear y. The N shear y is given by the following relation y = £ 17 “ |. In equation (3.23), rs is a=l the saturation shear stress for t0, and Dp is the plastic strain rate. In the corotational rate of backstress, equation (3.22), instead of the total spin, W, the elastic spin rate, We, is used based on the physical grounds that the corotational rate of backstress, X , is with respect to the crystal lattice, defined as follows: A * . X = X - We • X + X • We (3.24) The evolution expression for T“ in equation (3.19) is linearly dependent on 7“ with the proper material coefficient t a/3 such that it is similar to the Hill and Havner (1982) hardening rule, T “ = £ W y* (3.25) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 where j8 is the active slip system. The material coefficient t aj3 is the interactive hardening rate at time t. It is affected by the amount of slip in the material such that H (7) for a =(3 a/3 (3.26) H (y)q for a (3 where q is an interactive coefficient for hardening. The hardening coefficient H(y) is given as follows: H0T H(y) = H 0 sech' (3.27) Taking the time derivative of equation (3.19), one obtains QoS / - & dk( (3.28) where Qa/S is defined as follows: Qafl = tap + a(7) ^ ■ P“ + C : : P“ + C : [(B : a) • fl*3] : P“ - C : [0* • (B : a)] : P“ (3.29) and 0^ is defined as follows: *kl = Cijkf " *ij Py (3.30) The summation in equation (3.28) is over the active slip systems /?. Let M be the inverse of Q, then one can solve for 7 “ from equation (3.28) as follows: = Dsj = Mag et> : D r (3.31) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 An alternate second expression for the backstress is proposed here by assuming that the backstress in a single crystal is derived from crystallographic slips in each slip system: N X = £ (A“ P“ + B“ N“ + C“ S)“ “=1 (3.32) where A“(y), B“(y), C“(y) are crystal lattice related parameters which depend only on the total amount of shear strain and N“ = n“ n“ (3.33a) S“ = s“ s" (3.33b) The basic physical idea of this backstress expression is that the backstress is directly related to the amount of crystallographic slips in the crystal. Equation (3.32) could be further modified by taking its corotational rate of equation (3.30) such that A N X* = Y ( A “ P“ + B “ N“ + C“ S“) (3.34) a = l One notes that the corotational rates of P, ft, N and S are zero due to the fact that they are defined in the current lattice configuration. The derivative rates of parameters A“, B“ and C“ are defined as follows: N A “ = Y AO/3 V3 (3.35a) P-l N B “ = Y Ba/3 y 13 (3.35b) 0-1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 N c a = £ C “P 7^ (3.35c) JS-1 where the interactive parameters A0^, B a<3 and C“^ are given as follows: for a = P (3.36a) I A'qj 't for a ^ /? ’ B' for a = /3 Ba/3 = (3.36b) ^ B'q2 for a ^ /? C' for a = (3 CaP = (3.36c) C'q3 for a ^ |3 and q; are material parameters representing the interaction between different slip systems such that 1.4 > q > 0. In equations (3.36), the parameters A', B' and C' are expressed as follows: A07 A '(t) = A0 sech" (3.37a) Ts “ To B07 B'(y) = B0 sech 2 (3.37b) C07 C'(y) = C0 sech" (3.37c) Ts - To where A0, B 0 and C 0 are the initial values of A', B' and C', respectively, at y = 0 . rs is the saturation value of r0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Following the same derivation procedures as in the case of the first model of the backstress, one obtains the following relation similar to the relationship expressed in equation (3.28), £ {tB0 + C : P0 : P01 + £ ( F P ^ + P N 5 P « + S5) : P“ - [C : • (C’1 : o)] : P“ + C : [(CT1 : a) • G^] : P} y 13 = 6a : D (3.38) Let us define Qap = W + c : p/3: ^ + £ v 5 + N* 6 + C5/3 S6) : P“ + C : [(CT1 : a) • ft0] : P“ - C : [fl0 • ( C 1 :«■)]: P“ (3.39) and 0a = Po':C-cr:Pa I (3.40) then the slip rate 7“ can be solved from the following relation: y a = 0“ : D (3.41) where, similar to the first model, are the components of the inverse of Q = (Q'l)aP Substituting the obtained slip rate into the equations in this chapter, the Dp, Wp, De, We, a and X can thus be calculated. Based on the above formulations, a Fortran code of crystal plasticity modeling program CPMD was developed for the numerical implementation of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. model. This program along with the flow chart of the program are attached in Appendix. This program is aimed to be used with a finite element package to solve complex metal forming problems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 NUMERICAL IMPLEMENTATION 4.1 Application to the Simple Shear Similar to the approach used by Peirce et al. (1983) and Nemat-Nasser (1983), a two-dimensional pseudo-crystal structure is used for the feasibility study of the proposed models. As shown in Figure 4.1, two slip systems (s1, n1), (s2, n2) exist in this crystal structure. These s, n vectors can be expressed in terms of the relative angles 4> and ^ as follows: s1 = [cos n 1 = [- sin cp, cos 4>, 0]T (4.1) s2 = [cos (0 + ip), sin ($ + \j/), 0]T n 1 = [- sin ( The deformation gradient field F is given by 1 kt 0 F = 0 1 0 (4.2) 0 0 1 and therefore F 1, F and F-F 1 can be expressed as 0 k 0 F = 0 0 0 (4.3a) 0 0 0 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 X 2 X i Figure 4.1 Schematic illustration of the planar deformation model. Two slip systems denoted by their direction and normal to slip planes (s1, ir), (s2, n2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 1 - k t 0 F ' 1 = 0 1 0 (4.3b) 0 0 1 and 0 k 0 FF ' 1 = 0 0 0 (4.3c) 0 0 0 _ The strain rate and spin rate are thereafter given as 0 k/2 0 D = k/2 0 0 (4.4a) 0 0 0 1 0 k/2 0 W = - k /2 0 0 (4.4b) 0 0 0 A cubic structured single crystal usually does not have isotropic elasticity. However, to test the model’s physical feasibility, an isotropic type material with a planar double slip crystal model is used with its elastic moduli given by (4.5) Cjjk? ~ 2 (l + v) ^ ^ ^ ^ where E is the elastic modules and v the Poisson’s ratio. Let us define the components of Cjjkf in terms of Lame’s constants X and n, C u n - C 2222 ~ C3333 — X + 2/t C 1122 - C 1133 “ C2233 “ C 2211 “ C 3311 “ C 3322 “ ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 C 1212 “ C 1221 ~ C 2121 _ C 2112 ~ C 1331 ~ C 1313 ~ C 3113 ~ C 3131 = C 2323 = C 2332 ~ C 3223 ~ C 3232 = M while the rest of Cjjk£ = 0. The inverse of is given by p-l — P 'l — P "1 c 1111 - ^ 2222 ~ c 3333 = 1/E r - i — P 'l 1122 ~ c 1133 ~ 2233 = C l 2211 = C’^ l l = C’!3322 = - vIE p-i — p-l c 1212 _ c 1221 ~ c 2121 = Cl2l\2 = C^mi = C l1313 ~ C’SllS — P-l — P "1 _ 3131 ~ 2323 = C ’12332 = C_13223 = C_13232 = (1/2)|* The non-zero components of the stresses in the case studied are < 7n , a12, a22 with the added constraint that (Tn = ‘ °22 (4-6) The material’s properties used here are given by E = 1.17 x 10 11 (Pa) v = 0.33 k = 1.0 x 10 "3 1/sec (4.7) tq = 2.76 x 10 8 (Pa) (4.8) 4.2 Numerical Results The numerical calculations conducted here are aimed at studying the physical feasibility of the proposed models to describe the material behavior of metals without attempting to relate it to any specific type of metals. The goal of this study is the potential feasibility of the proposed two models in depicting various kind of material plasticity behaviors. This study also attempts to identify Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 the change in the plastic behavior of the materials by adjusting the material parameters. In this study comparisons are made between two different backstress models in order to depict the appropriate hardening law. The effect of different initial crystal orientation is also studied. A parametric study is conducted in order to first evaluate the coefficients in the second model, namely Aq, B0 and C 0 and also to study the interactions between the different slip systems for different values of the interactive factors q;. The list of the different numerical problems conducted here is shown in Table 4.1. 4.2.1 Comparison Between the Two Models Samples with same initial orientations and hardening rates H 0 are com pared, namely Hl-0 and H2-0. In Table 4.1, 1-0 represents the "ideal plastic crystal material", while Hl-0 and H2-0 are for models 1 and 2 respectively. The crystals with neither isotropic hardening nor backstress are termed "ideal plastic crystal materials". This is different from the phenomelogical ideal plasticity because in the former case, the crystal orientation will affect the macro-stress and the material no longer exhibits the so called phenomelogical ideal plasticity. In Figure 4.2a, the ideal plasticity crystal shows no hardening for the case of initial orientation of hardening and saturation in the shear stress. The hardening rates are different I between models 1 and 2. However, they are non-comparable because the para- i meters Aq and a 0 f°r the backstress models 1 and 2 are quite different in nature. ! i i L ... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 Table 4.1 List of different numerical problems conducted. 4> 'I' H0 Ao “ 0 BO CO It Name Model q (°) n (Pa) (Pa) (Pa) (Pa) (Pa) 9i 1 1-0 0 60 0 - - - - 1 -- 2 1-30 -30 60 0 --- - 1 -- 3 Hl-0 0 60 108 - 109 - - 1 - 1.4 4 Hl-30 -30 60 108 - 109 -- 1 - 1.4 5 H2-0 0 60 108 109 - 0 0 2 1.4 1.4 6 H2-30 -30 60 108 109 - 0 0 2 1.4 1.4 7 H2-0 0 60 108 109 - 109 0 2 1.4 1.4 8 H2-0 0 60 108 109 - -109 0 2 1.4 1.4 9 H2-0 0 60 108 109 - 0 109 2 1.4 1.4 10 H2-0 0 60 108 109 - 0 -109 2 1.4 1.4 11 H2-0 0 60 108 109 - 0 -109 2 0 0 12 H2-0 0 60 108 109 - 0 0 2 0 0 13 H2-0 0 60 10s 109 - 109 0 2 0 0 14 H2-0 0 60 108 109 - 109 0 2 1.4 0 15 H2-0 0 60 108 109 - 109 0 2 0.2 1.4 16 H2-0 0 60 108 109 - 0 0 2 0.2 1.4 17 H2-0 0 60 108 109 - 0 109 2 0 0 18 H2-0 0 60 108 109 - -109 0 2 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 1.6E+9 Comparison of Models ( o v s y ) ( m odel 1 ) ( m odel 1 ) ( model 1 ) 1.2E+9 ( m odel 2 ) ( m o d e l2 ) ( m odel 2 ) ( id e a l) o.. ( ideal) a,, ( ideal) to CL 8.0E+8 0 cn £ 4.0E+8 A-A-A-& A A A- A —A-A A a ArA-A-A a a a-a-a-a - CO A—A A—A A-A- A -A-A A—A A—A A—A A—A A—A A-A A - $-0-0-0 0-0- 0-0-0-0- 0 -O'0-0-0 0-0 0—0-0— 0-0 0 0 - 0 - 0 0- 0 0 - 0—0 0 -0 0 - 0 - 0 - 0 - 0 - 0.0E+0 I I I I -I- -I I I + - + -I I I I I -I- + ~ 4 ^ - 1 - + -4.0E+8 0.00 1.00 2.00 3.00 Shear y Figure 4.2a Comparison of stress changes during shear deformation for two different backstress models and the ideal plastic crystal cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 6.0E+8 Comparison of Models (Backstress X vs. Shear y) — X„ (model 1) X,: (model 1) — |— X3? (model 1) —<$>- X,, (model2) 4.0E+8 — - A - *13 (model 2 ) —|— XJ3 (model 2) (O Q. X co C / 3 2.0E+8 — L.0 co a ; A -A -A A A- A-A -A A A A A-A -A A A A A-A -A : o CO ca O.OE+O -2.0E+8 0.00 1.00 2.00 3.00 Shear y Figure 4.2b Comparison of the backstress evolution for the two backstress models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 The normal stress components of stress on and a22 behave quite differently as observed in Figure 4.2a. But, as this will be explained later, the parameters A0, a 0 and q; will significantly affect the stress strain curves. In Figure 4.2b, the backstress curves are shown for the two proposed models. The shear part of back stress X 12 increases and finally saturates with the increase of shear deformation. The non-deviatoric parts of the backstresses Xn and X22 are close to zeros for the backstress model 1, whereas for model 2 they are not. However, as will be dis cussed later, the coefficients B 0 and C 0 significantly affect these stresses. The results in Figures 4.2a and 4.2b show that models 1 and 2 can both be used to simulate various kinds of plasticity behaviors of materials and simultaneously obtain reasonable results. 4.2.2 Effect of Initial Orientation Comparing the two different initial orientations 0° and -30°, it is clear from Figure 4.3a that the initial orientation angle, The primary slip occurs in the system that lies in the 0° degree plane. The second slip system is activated at a rather late stage of strain when the resolved shear stress is big enough to induce it. The amount of slip in the second slip system is negligible. The (A4 < 1°). For a crystal with an initial -30° orientation, one observes as shown in Figures 4.3a and 4.3b that both slip systems are activated almost simultaneously at the onset of plastic deformation. The shear yielding stress is higher in this case Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced CL Stress G -4.0E+8 — -4.0E+8 4.0E+8 — 4.0E+8 — 8.0E+8 O.OE+O 1.2E+9 Figure 4.3a Effect of initial orientation on the change of stresses stresses of change the on orientation initial of Effect 4.3a Figure 0.00 during shear deformation. shear during 1.00 Shear y Shear o - o - o - o ^ 2.00 0 . . 0 - A - - o - — 0 ” Com parison of Differenet of parison Com - + - —A Initial Orientations, Orientations, Initial o o o - o - o - o I — ‘'ideal plastic’’ ‘'ideal O) s 0 ( i , O a„ , =- ) 0 -3 = 0 ( ,, o ( n o r;(0=O) O = 0 ( cr,; ; ( ;; o ) 0 = 0 ( 0 0 =- ) 0 -3 = 30) 0 -3 = 0 3.00 41 ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced - e • Angle Figure 4.3b Effect of initial orientation on the change of orientation angle, angle, orientation of change the on orientation initial of Effect 4.3b Figure - - -30.00 -40.00 20.00 10.00 10.00 0.00 0.00 ,deformation. shear during 1.00 Shear Shear Effect of Initial Angle <)> Angle Initial of Effect y 2.00 3.00 42 43 compared to the 0° orientation. In Figure 4.3b, it is shown that the plastic deformation makes the -30° slip plane rotate toward the stable orientation of 0°. The material exhibits softening due to orientational changes as shown in Figure 4a. This is the mechanism for texture development in polycrystalline materials. 4.2.3 Evaluation of Coefficients A, B, C for Model 2 N For the second model, the backstress evolution is given by, X = a=\ A“P“ + B“N“ + C®S“, where the coefficients A", B“ and C“ are history depen dent parameters representing the current dislocation configuration. Since the dislocations are formed by the crystallographic slips in the crystals, therefore, A®, B® and C“ are functions of slip amount in each slip system. Different slip systems may interact with each other, which can be described by the interactive factors q l5 q2 and q 3 respectively for A®, B® and C“. Among the coefficients A®, B“ and C“, it should be noted that A® is related to the pure deviatoric shear in crystals since it is the coefficient that relates to the slip tensor P“. However, the B“ and C“ are related to the non-deviatoric parts since they are the coefficients for tensors N® and S®. Assuming that the disloca tion slips create a deviatoric shear stress controlled by A(y) along with some non- deviatoric stresses controlled by B(y) and C(y), therefore a fixed value of Aq with a range of values of B 0 and C 0 and 0° for $ are used in this study. In Figure 4.4a and Figure 4.4b, the stress vs y and the backstress vs.y curves are plotted respectively. Both positive and negative values of B 0 are used. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 1.6E+9 a vs v Effect of Be —O— °n t _bd ) 1.2E+9 — A A- A-A-A -A A A A- A A A- A A A A A—A— 8.0E+8 — (0 CL 0 4.0E+8 — to to a) W 0.0E+0 ^ O A -4.0E+8 — & O - 0 - 0 - 0 O O O - 0 - 0 - 0 - O - i -8.0E+8 0.00 1.00 2.00 3.00 Shear y Figure 4.4a Effect of parameter B 0 on the stress a during shear deformation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 1.5E+9 x vs y Effect of 0^ - 0 - M-B.) - 2&- 1.0E+9 — 1— -O - X„(*Bb) co CL A- A-A-A A A A A-A -A — 5.0E+8 — A - a a— —A A A A—, X C/3 C/3 L.0 ^ 0.0E+0 o CO CD -5.0E+8 — -1.0E+9 0.00 1.00 2.00 3.00 Shear y Figure 4.4b Effect of parameter B 0 on the backstress X during shear deformation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 It is found that the change of value and sign of B 0 has rather minor effect on the shear stress a12 and shear part of backstress X12. However, it significantly changes the magnitude and shape of normal stresses au and u 22 as we^ as t^e normal component parts of backstresses Xn and X22. This agrees with the fact that B 0 is the coefficient of the non-deviatoric backstress. In the backstress curves, it is also noted that B 0 hardly changes the Xn . However, it drastically changes the X 22 from positive to negative values. Figures 4.5a and 4.5b are the stress and backstress curves for positive and negative values of C 0 respectively. Similarity between the effect of C 0 and that of B0 is noticed. The C 0 has rather minor effect on the shear stress a 12 and shear backstress X12, but has a significant effect on the non-shearing components of stress and backstress an , a22 and Xn respectively. However, C 0 drastically changes the Xn part from positive value (for positive value of C0) to negative value (for negative value of C0) whereas it hardly changes the value of X22. Therefore, it may be concluded that the B 0 controls the X 22 part whereas C 0 controls the X n part for the case of orientation 0°. It should be noted that since the Xn and X 22 are the non-deviatoric stresses, they are associated with the climbing of dislocations and the formation of dislocation cells. More detailed investigation in the relationships of B 0 and C 0 to the dislocation configuration will be studied in the future in more detail. I Ii I I i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 1.6E+9 Ov» Y Effect of B0 1.2E+9 -O- -A- a -A A A- A-A-A -A A A A- A A -A A A a - a — 8.0E+8 (0 Cl 0 4.0E+8 + + !/) (/) 0/3 0.0E+0 -4.0E+8 ^ O- 0 -0 -0 -0 O -8.0E+8 0.00 1.00 2.00 3.00 Shear y Figure 4.5a Effect of parameter C 0 on the stress a during shear deformation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 1.5E+9 x v s y Effect of CQ 1.0E +9 A-A -A-A A A A-A-A-A — 5.0E + 8 — A V V a a a a—a_ a — CO Q. X 0.0E + 0 -5.0E + 8 ^<^-0-0-0 <> o o-o-o-o -I -1.0E + 9 0.00 1.00 2.00 3.0 0 y Figure 4.5b Effect of parameter C 0 on the backstress X during shear deformation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 4.2.4 Evaluation of the Interacting Factors qj As mentioned in the previous section, the q; factors are related to the interaction between different crystallographic slip systems. To study this effect, different values of q; are used ranging from qL = 0 for the no interactive case, to q} = 0.2 for the weak interactive case and finally to qj = 1.4 for the strong interactive case. It is observed that for the case of q; = 1.4 in model 2, that due to the strong interaction between different slip systems every time a new slip system is activated or deactivated the backstress exhibits discontinuity. Figure 4.6b shows discontinuity in the backstress curves. It is also found that at q; = 1.4 with particular values of B 0 or C 0 one can cause the backstresses to become negative (Figure 4.6b). This would imply that during the loading, the backstress instead of resisting the plastic deformation, would rather enhance the plastic deformation due to its negative value. This phenomenon can not be accepted. Therefore, to avoid this kind of phenomenon, it is advised to set a lower bound for qj such that q; < 0.5 in order to avoid over-interaction. For the cases of weak interaction between different slip systems, q; = 0.2, and no interaction, qj = 0, Figure 4.6a and Figure 4.6b show that no discontinuity in the backstress curves. Therefore, it seems more appropriate to select q; < 0.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 1.6E+9 Comparison of Interaction Factorq, (B 0,Co = O) _ 1 - £ r ~ o,j(q, = 1.4) 1.2E+9 — |---- ° :j fa.= 1 4) --- —0 ~ o Ttfa. = 0) “A" °i:(q, = 0) ---- 1— °::fa. = 0) a A: A- A-A-A A A A A-A-A - . j t A “ 8.0E+8 — J CO Q_ -A A A A A A—A A A A- A A — 4.0E+8 F-H—t—I I I' + 0 ^ O-O—O -0 O O 0-0-0-0 O O O-0-0-0 0.0E+0 + -F -I I—h-F -F -F-+-+H- -t- + +--t~F-F -0—0—0—0- 0—0- -4.0E+8 0.00 1.00 2.00 3.00 Y Figure 4.6a Effect of interactive factors qj on the stress a during shear deformation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Backstress X (Pa) -5.0E+8 -1.0E+9 -1.5E+9 -2.0E+9 5.0E+8 — 5.0E+8 0.0E+0 1.5E+9 1.0E+9 — 1.0E+9 Figure 4.6b Effect of interactive factors q; on the backstress X X backstress the on q; factors interactive of Effect 4.6b Figure 0.00 Effect of qt on on qt of Effect c tes X stress. ack B during shear deformation. shear during -A 1.00 A A-— A A A - A - A ^ - A A -A A - A - A r A ^ Y 0 - 0 - 0 0 0 0 k "i z Ik-A— zknk—S — k n k —z —A Ilk-- A zk "aik jk 2.00 0 - 0 - 0 o i o o -o “ + + - 3.00 51 CHAPTER 5 DISCUSSION OF RESULTS As mentioned in the previous chapter, the aim of this study is to point out the importance of the proposed kinematic hardening models with their respective evolution equations for the backstresses. The main focus of this study is to see how these models can be incorporated into existing well known models such as those proposed by Asaro (1983), Peirce et al. (1983) and Rashid and Nemat-Nasser (1992). From this point of view, the proposed models are successful in inter preting crystal plasticity more effectively. Also these models along with the numerical results have opened a new channel to further investigate the kinematic hardening of materials with their appropriate backstress evolution. The models proposed here are also aimed at both improving some of the backstress models used presently as well as giving the physical understanding to many phenomena which cannot be explained by other models especially those based on the phenomelogical approach. Zbib and Aifantis (1987, 1988) proposed the following backstress a = a mM + a nN, with its corresponding evolution equation as follows: (5.1) The corresponding plastic spin tensor is given as follows: Wp = - — (a • DP - DP • a) (5.2) a n 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 In the above expression for the backstress a, the model is limited to a single slip system. Consequently, the given backstress evolution is therefore limited to just one slip system. No simple formula expression for two multi-slip systems can be derived from this model. It is also known that the backstress is mainly caused through the dislocation configurations by the plastic deformation. Therefore, the coefficients a m and a n should depend on the deformation history. However, the relationships between a m, a n and the plastic deformation is not given explicitly. Also, in the expression of the plastic spin, one may encounter two possible problems. The first problem is that if a n is very small or zero, the plastic spin would be infinitely large. It is known that the a nN term is a pressure dependent term related to climb of dislocations. This term may become very small or negligible in certain cases, which will result to an unreasonably high plastic spin. The second problem is that when Dp and a are proportional to each other, the plastic spin is canceled out. Whether this is true or not is questionable. In the proposed backstress models, multiple slip systems in the crystal are assumed. The plastic spin is defined on physical grounds and not indirectly derived from mathematical equations with numerous assumptions. The evolution of backstress is also physically defined by the co-rotational spin rate with the crystal lattice. Because of their physical nature, all the quantities and parameters can be traced back to their physical background. Therefore, the models that are proposed here have many intrinsic physical advantages over the phenomelogical models. These proposed models are easier to understand. They are also more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 practical and easy to correct any possible errors by simply comparing them to properly designed experimental data. The proposed models are also compared with the bounding surface plasticity model with backstress decomposition proposed by Lamar (1989). In his model the backstress is decomposed into two parts namely the long range backstress and the short range backstress. The long range internal stress is stable and slow to change because it is associated with dislocation cells and subgrain boundaries. The short range backstress fluctuates more rapidly in response to the directional change of loading stresses as it is associated with nonuniform forrest dislocations, dislocations tangles, pile-ups, etc. This approach provides a linking between the phenome- logical model and the physical process inside the material. However, to the authors, it seems more close to the phenomelogical approach which does not directly link it to the slip processes inside the material. Instead, the model focuses more on the bounding surface and its moving directions. Another disadvantage of this is that the model proposed by Lamar (1989) cannot predict the crystal orientation. The model proposed in this work is more physically sound and with the proper adoption of functions and fluctuation for the parameters, the long range and short range backstresses can be included in the proposed models. The numerical results obtained in this work show that both proposed backstress models can depict various kinds of plastic material behavior such as ideal plastic, isotropic hardening, kinematic hardening. They can also depict the anisotropy of materials through the texture development. With minor modification i j I i i i j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 the proposed models may also be applied to rate dependent plasticity. It is a new and simple approach to model the anisotropic hardening of materials and simul taneously incorporate the backstress evolution into the current widely accepted models by Asaro and Needleman (1985), Peirce et al. (1983), and Rashid and Nemat-Nasser (1992). However, comparing the two proposed models, it is found that both of them are physically sound but they offer different choices for some compromise between more pro-macro (or phenomelogical) for the 1st model and more detailed microstructure characterization through crystallographic slips for the second model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 CONTINUUM PLASTICITY THEORY FOR POLYCRYSTALS WITH MICROSTRUCTURAL CHARACTERIZATION 6.1 Introduction A constitutive model for polycrystals is proposed here in order to describe the plastic deformation of metals using microstructural characterization. In this work use is made of the concept of the dislocation tensor as introduced by Mura (1968) in order to describe the microstructural behavior of metals. This proposed approach quantitatively describes the plastic deformation using both the pheno menological and physical approaches. Two types of microstructural materials, namely the single phase homogeneous materials and the precipitate hardened materials as they depict many commercially used engineering materials, are used to study the isotropic hardening of materials. 6.2 Continuum Mechanics and Dislocation Theory Mura (1963, 1965, 1968) and Kroner (1958) used the second-order distor- tional tensor 1? to describe the deformation of material, /?jj = Uj j, where u is the displacement in the original configuration. However, since the plastic deformation is caused by generation and motion of dislocations, the dislocation density tensor a must be introduced because it describes the density and configurations of dis locations. For each crystal, the dislocation tensor a can be defined as a = £ pg t8 ® bg ( 6 .1) g 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 where the summation is over all slip systems g, pg is the scalar dislocation density in the crystal slip system g, tg is the unit vector of dislocation line in the crystal slip system g, and bg is the Burgers vector of dislocation in that system. The slip model used here is based on the edge dislocation model, therefore, t = s x n where s and n are unit vectors along the slip direction and normal to the slip plane, respectively. From the compatibility requirement e ^ u ^ = 0, Mura (1968) showed that the dislocation density tensor has the following relation with the components of the distortional rate 0 “ hi = ehfk P'ki,l = “ €hfk ^"ki.f (6-2) where a superposed dot indicates material time differentiation and 0 is the velocity gradient which can be decomposed into the elastic part 0' and plastic part 0". Since the plastic deformation is caused by dislocation motion, the plastic part of the velocity gradient L can be expressed as L" = (0")T = D" + W" = Y , y g s§ ® n* = E 7g (P + «) (6.3) g where D" = £yg Pg , W" = £yg (6.4) g g and yg = pg b s £g + pg b g v g is the shear strain rate in the slip system g. I and v are the average slip distance and velocity of dislocations, respectively. In equation (6.4), Pg and flg are related to the plastic strain and rotation, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 For the edge dislocation model, the distortional rate tensor 0" has the following relation with dislocations. = £ [/,s (ts x £g) ® bg + p8 (t« x v8) The dislocation stress tensor proposed here for multiple slip system is general in nature and is analogous to the single slip system proposed by Aifantis (1987). The backstress proposed here for multiple slip system is given by the following expression: XD = £ (A“ P“ + B“ N“ + C“ S“> (6.6) a where Na = n“ ® n“ and S“ = s“s“ . The term C“S a is directly related to dis location climbing, which is negligible at low and intermediate temperature ranges. The deviatoric part of XD is the first term of equation ( 6.6), which is closely related to the transition of the yield surface center. The dislocations described above are composed of many individual disloca tions moving along specific crystal directions and on specific crystal planes. To predict which slip system will be activated under the loading conditions, the resolved shear stress must be calculated in order to compare it with the critical shear stress. This yields the Schmid yield criterion for single crystals 7s = s 8 • a • n 8 = ay Py > if (6.7) where r 8 is the resolved shear stress, a is the applied stress t c8 is the critical shear stress of slip system g. The Schmid law is a form of Tresca yield criterion in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 crystalline material since it uses the resolved shear stress instead of the shear components of the stress tensor used by the Tresca criterion. When dislocation stress XD 5* 0, then, the dislocation stress will interact with a. The Schmid criterion for kinematic hardening materials can be expressed as 7s = sg • (a - XD) • n 8 = (a - XD) : Ps ^ rfic (no sum over g) ( 6 .8 ) In single crystals, several slip systems may be activated simultaneously. Neverthe less, Taylor (1934) showed that only five slip system is independent with each other and the combination of them should result in minimum "internal work" of the plastic deformation. 6.3 Macroscopic Mechanical Behavior of Polycrystalline Materials All the scalars and tensors discussed in section 1 such as ps, t s, Pg, etc., are described in single crystal models which assume a homogenous state of deformation within each grain. This is not quite accurate. Zaoui (1986) has shown that within one grain there are some domains with distinct active slip systems to make the grain boundary compatible. But on a macroscopic scale, the average of each grain will give out the macroscopically correct value. Therefore, the averaging formula is given by the following expression 1 N 7 = T7 £ 71 V l v 1=1 where y 1 is any scalar or tensor related to grain I, the average 7 is the macro scopic one, V is the total volume of material and V 1 is the volume of grain I. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 For single crystal materials, the Schmid law fits quite accurately the experimental data. However, for polycrystalline materials, Tresca’s yield condition is not valid since the compatibility requirement in the grain boundaries will significantly affect the strain field within each grain. Von Mises suggested a yield criterion based on the distortional elastic energy and it may be expressed as follows f = T:T-k2 = 0 (6.10) t is the deviatoric Cauchy stress and k is a material property dependent on the yield stress. In the presence of dislocation stress, the von Mises yield criterion may be written as follows: f = (r - X'D) : (r - X'D) - k 2 = 0 (6.11) Comparing the Tresca and von Mises criteria, it is noted that Tresca’s criterion is based on the maximum resolved shear stress TC = M||ff'|| (6.12) where || • || denotes the norm of the tensor a', and M is the Schimd factor. The von Mises’ is based on the distortional yield energy condition, where G is the shear modulus. From equation (6.7) we note that in order to determine the most easily activated slip systems for each crystal, the resolved shear stress should be calculated for all systems and compared. When the critical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 distortional strain energy, Up, caused by this resolved shear stress is reached then yielding occurs for this grain: Up « _ L (rg)2 = _L_ ( a : M«) (cr:Mg) = _ L H® akt (6.14) P 2Gg 2Gg 2Gg J J where H ijkt = p if Pkf = \ (hifkf + hiftk + hjfkf + hjffk) and hftf - “i* «j8 nif Sfg (6.16) Assuming similar types of slip systems have identical shear moduli, Gg, then the proposed macroscopic plasticity yield condition that accounts for polycrystalline structure is given by . N _ i Z [Hift,]' V 1 where V 1 is the volume of the I-th grain. For the case when dislocation stresses exist, a similar yield function is proposed as shown below Hijk, (ffij - XDlJ) (ffkf - XD„) - k 2 (6.18) Hjjkf is defined by crystal orientation. Therefore, by a proper orientational distribution function o}(p,6,(f>) in the Eulerian space, Hjjkf can be calculated through H-* Hjjkf- = X) —“ ’ dv' = f H ijk? sin # dP d Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Hykf is a function of texture structure and will change according to the deformation history. Consequently, the change of Hjjkf will cause the shape change of the yield surface from von Mises yield surface to a distortional surface. Expressions for HjjW based on the phenomenological theories were also proposed by Voyiadjis and Foroozesh (1990). The mechanisms of increase the yield strength of metal materials such as grain boundary hardening, solution hardening, work hardening and precipitation hardening have been quite extensively studied especially in 1950s and 1960s by many material scientists and solid physicists (Weertman and Weertman 1983). A brief discussion of precipitate hardening is given in the following. One starts from the quantitative analysis of precipitate phase. Let N be the volume density of precipitates, X be the mean distance between centers of neighboring precipitates and \ a is the mean free space between neighboring precipitates. Then the following relation exists between the parameters Vf = | irr3N = N asr 3 , X = N ' 1/3 (6.19) where Vf is the volume fraction of the precipitates. a s is the shape constant and r is the radius of the precipitates. The free distance between particles is given by X„ = X - 2 r = X (1 - l l ) = N "1/3 [1-2 (^)1/3] (6.20) X a s For the cases when the secondary phase is not very closely spaced and the particles are not coherent the increment of yield strength due to the precipitates is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 Orowan’s model. The critical yield strength is inversely proportional to the free space between particles. 1/3 V, 1 a G b _ aG b (6 .21) ay = ay° mx: = ~ w 1/3 1 - 2 a c where a = 0.5 is a material constant. From the above equation, the yield strength has a direct relation with particle size and volume fraction. Assuming the effective free distance due to the increase of plastic deformation is given by \ f = \ a (1 - ??e1/n), then the corresponding effective yield stress is as follows: a 'G b e 1/m “ 1/m = <7V + (6.22) ° y = °y 0 + l/n\ ^ct 1 - 1}T l/n where e is the effective plastic strain, c, 77, m and n are material constants. Differentiate equation with respect to effective plastic strain, one obtains dgy = c m n (6.23) de 1 - ije1/n (1 - 17 e1 /n)2 The yield hardening rate in equation (6.23) with respect to the increase of plastic strain is related to the microstructural parameters (free space, volume fraction and density). The equation is very simple for use and the constants can be determined by conducting a series of uniaxial tensile tests of different amount of plastic deformation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 For tensorial consideration note that the second term in equation (6.22) is the magnitude of the backstress. Therefore, for proportional loading, its tensorial form may be expressed as XD- ± ----S------‘ (6.24) X i « Tl/n ' ' a 1 - 7} € where e" is the plastic strain tensor. From the previous assumptions made, equations (6.22) to (6.24) are only valid for small to moderate range of plastic deformations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 PROPOSED FUTURE WORK FOR THREE-DIMENSIONAL CASE The two-dimensional model can be considered as a simplified model in order for one to understand the real physical processes taking place in the crystal materials during plastic deformation. The most commonly used metal materials are of very high order of crystallographic symmetry such as FCC, BCC and HCP. To depict the texture evolution and hence the resulting anisotropic hardening of these crystalline materials, the orientation of each grain will need to be monitored. Traditionally in the texture analysis, three Eulerian angles for each grain 0j, <£, are used for the description of the orientational change. Mathematically, a mapping tensor between the orientation of a grain and the external reference axes are used. This mapping tensor is given as follows: cos cos R = -cos^jsin^-sin^jcos^cos^) - sin (f> j sin 2 + cos -s in ^ 2sin R transforms the tensors from the global system to the local crystal lattice system. The tasks in the three-dimensional modeling are basically the same as the two- dimensional ones except that they have to deal with more orientational parameters. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 The three-dimensional model for a single crystal is an intermediate stage to understand the behavior of poly crystalline materials during plastic deformation. Modeling the behavior of an aggregate of grains and predicting the macroscopic mechanical properties are of paramount importance in metal forming applications. Also, the modeling for polycrystalline material and the texture evolution is the ultimate goal. The polycrystal model proposed here are based on the initial models proposed by Asaro et al. (1985), and Rashid and Nemat-Nasser (1992). Since the contact of grain boundaries has to be maintained, the Taylor type model is adopted which assumes that each grain in the aggregate has to have the same amount of deformation. During the developing process of modeling, modifications to release some of the restriction in the model will be considered. Due to the complexity of the poly crystal materials, it is not possible to model the huge amount of grains and their respective boundary conditions. Simpli fication has to be made for feasible numerical calculation. Two polycrystal models are proposed for use. They are the "pan-cake" type and the "hexagonal column" type given as follows. 7.1 "Pan-Cake" Type Polycrystal A "pan-cake" type crystal model is illustrated in Figure 7.1. In this type of polycrystal, each grain is assumed to be of a sheet shape with significant difference in orientation with the neighboring grains. Slips are restricted only in the plane of each grain. Therefore, during the simple shear deformation, when the Taylor type Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 X Figure 7.1 Schematic illustration of a "pan-cake" type polycrystal model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 model is adopted, the boundary of the deformed layers are still in contact as the shear is not in the thickness direction. The macroscopic mechanical behavior is calculated by averaging all the grains over the entire thickness range. The texture evolution is calculated through an orientational distribution function (ODF) to track the orientation redistribution during the deformation. The number of grains along the thickness direction is to be ten or more. The upper limit of the number of grains will be determined by several factors. These factors include the computer simulation capability, the effectiveness of the numerical approach used, and the accuracy of the calculation vs. the computational effort, etc. 7.2 "Hexagonal Column" Type Polycrystal The "pan-cake" type polycrystal model may be too strict due to the restrictions since the off surface slip between the neighboring grains is prohibited to keep the grain boundary in contact. A "hexagonal column" type polycrystal model which is used by Becker (1991) is proposed for numerical simulation as shown in Figure 7.2. The deformation mode chosen for this kind of polycrystal modeling is a three-dimensional channel die compression with its deformation gradient F given by X 0 0 0 1 0 Xtanxy Xtanxx X-1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 X Figure 7.2 Schematic illustration of a "hexagonal column" type polycrystal model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 where xy and xx are the angles between the deformed cubic edge with respect to their original directions. For simple rolling, xy and xx can be set equal to zero. The shape of grains is assumed to be kept hexagonal during the channel die com pression deformation. This means that the deformation in the thickness direction is uniform and the boundary contact is maintained. The number of grains in this simulation will also be chosen according to the same factors considered like the "pan-cake" type model. The above proposed two models to a certain degree resemble the micro structure of some engineering materials. Figure 7.3 shows that the flattened grains in a cold rolled drawing quality SAE 1008 steel that resembles the "pan-cake" type model and the equi-axial grain structure in a hot rolled SAE 1008 steel that resembles the "hexagonal column" model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.3 Micorphotos showing flattened grains in a cold rolled drawing quality SAE 1008 steel (top) resemble the "pan-cake" polycrystal model and equi-axial grains in a hot rolled SAE 1008 steel (bottom) resemble the hexagonal polycrystal model (400X). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 SUMMARY AND CONCLUSION In this work two backstress models are proposed. Numerical results show that both of them are physically sound. They also offer a choice of compromise between modeling plasticity in macro and crystal level. By incorporating the backstresses into the plasticity model, the proposed models offer tremendous flexibility in accommodating various kinds of plasticity behaviors of materials including kinematic anisotropic hardening and prediction of crystal orientation change. In the proposed models, interaction between slip systems can be incor porated through coefficients qj. From the numerical results, it is advised that one takes a lower value of qj < 0.5 to avoid over-interaction between different slip systems which may result in discontinuity in the backstress curves. For the polycrystal characterization, the relationships between continuum mechanics with the dislocation tensors for multi-slip system polycrystalline materials have been proposed which establish a unique bridge between macroscopic mechanical behavior and microstructural physical process. The differences between the yielding process of single crystal and poly crystalline materials have been discussed in detail, and a general yield criteria f & Hjjkf (ay -X DJ (crkf -X Dkf) - k 2 = 0 is proposed which incorporates Tresca, von Mises and distortional yield surface theories. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 The theory is also applied to the precipitate hardening materials, where the yield stress, work hardening rate, and the backstress are related to the plastic strain and microstructural parameters such as the mean free space between precipitates. The equation is made simple for practical use. A crystal plasticity modeling program CPMD is developed which can be incorporated into a finite element analysis package to solve problems in metal forming operations such as rolling, extrusion, drawing and forging, etc. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES Aifantis, E. C., 1987, The physics of plastic deformation, International Journal of Plasticity, 3, 211-247. Asaro, R. J., 1983, Crystal plasticity, J. of Applied Mechanics, 50, 921-934. Asaro, R. J. and Needleman, A., 1985, Texture development and strain hardening in rate dependent polycrystals, Acta Metall., 33, 923-953. Becker, R., 1991, Analysis of texture evolution in channel die compression - I. Effects of grain interaction, Acta Metall. Mater., 39, 1211-1230. Bishop, J. F. W. and Hill, R., 1951a, A theory of the plastic distortion of a polycrystalline aggregate under combined stresses, Phi. Mag., 42, 414-427. Bishop, J. F. W. and Hill, R., 1951b, A theoretical derivation of the plastic properties of a polycrystalline face-centered metal, Phi. Mag., 42, 1298-1307. Cailletaud, G., 1992, A micromechanical approach to inelastic behavior of metals, Int. J. of Plasticity, 8, 55-73. Dafalias, Y. F., 1983, Corotational rates for kinematic hardening at large plastic deformations, J. Appl. Mech., 50, 561-566. Dafalias, Y. F., 1993, Plannar double-slip micromechanical model for polycrystal plasticity, J. of Engineering Materials, 119, 1260-1284. Dafalias, Y. F. and Aifantis, E. C., 1990, On the microscopic origin of the plastic spin, Acta Mech., 82, 31-48. Havner, K. S., 1992, in: Finite Plastic Deformation of Crystalline Solids, Chpt. 3, Cambridge University Press, 1992. 34-60. Hill, R., 1965, Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 14, 99-102. Hill, R. and Havner, K. S., 1982, Perspectives in the mechanics of elastoplastic crystals, J. Mech. Phys. Solids, 30, 5-14. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Hill, R. and Rice, J. R., 1972, Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20, 401-413. Hu, Z. and Teodosiu, C., 1992, Work hardening behavior of mild steel under stress reversal at large strains, Int. J. of Plasticity, 8 74-92. Hutchinson, J. N., 1976, Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. R. Soc. London, A, 348, 101-127. Kroner, E., 1958, Continuum theory of dislocations and self stresses, In: Ergebnisse der Augervandten Mathematik, 5, Springer-Verlag, Berlin, 1-277. Kroner, E. and Teodosiu, C., 1972, Lattice approach to plasticity and visco plasticity, in: A. Sawezak ed., Problems of Plasticity, International Symposium on Fundamentals of Plasticity, (Noordhoff, Groningen) 45-88. Lamar, A., 1989, Bounding surface plasticity theory with backstress decomposition and material memory, MS Thesis, Georgia Institute of Technology, Georgia, 85 pages. Lin, T. H., 1964, Slip and stress fields of a polycrystalline aggregate different stage of loading, J. Mech. Phys. Solids, 12, 391-408. Lin, T. H. and Ito, M., 1966, Theoretical plastic stress-strain relationship of a polycrystal and the comparisons with the von Mises and the Tresca plasticity theories, Int. J. Eng. Sci., 4, 543-561. Mathur, K. K. and Dawson, P. R., 1989, On modeling the development of crystallographic texture in bulk forming process, Int. J. of Plasticity, 5, 67-94. Mroz, Z., 1967, On the description of anisotropic work-hardening, J. Mech. Physics Solids, 15, 163-175. Mura, T., 1963, On dynamic problems of continuous distribution of dislocations, Int. J. Eng. Sci., 1, 371-381. Mura, T., 1965, Continuous distribution of dislocations and the mathematical theory of plasticity, Phys. Stat. Sol., 10, 447-453. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Mura, T., 1968, Continuum theory of dislocations and plasticity, In: Proceedings IUTAM Symposium on Mechanics of Generalized Continuum, edited by Kroner, Springer-Verlag, Berlin, 269-278. Needleman, A., Asaro, R. J., Lemonds, J. and Peirce, D., 1985, Finite element analysis of crystalline solids, Computer Methods in Applied Mechanics and Engineering, 52, 689-708 Nemat-Nasser, S., 1983, On finite plastic flow of crystalline solids and geo materials, Trans. ofASME, 50, 1114-1126. Nemat-Nasser, S. and Obata, M., 1986, Rate-dependent, finite elasto-plastic deformation of polycrystals, Proc. R. Soc. London, A407. 343-375. Peirce, D., Asaro, R. J. and Needleman, A., 1982, An analysis of nonuniform and localized deformation in ductile single crystals, Act. Metall., 30, 1087-1119. Peirce, R., Asaro, R. J. and Needleman, A., 1983, Material rate dependence and localized deformation in crystalline solids, Acta Metall., 31. 1951-1976. Prager, W., 1956, A new method of analyzing stresses and strains in work- hardening plastic solid, J. Appl. Mech., 23, 495-496. Rashid, M. M. and Nemat-Nasser, S., 1992, A constitutive algorithm for rate-dependent crystal plasticity, Computer Methods in Applied Mechanics and Engineering, 94, 201-228. Sachs, G ., 1928, Zur ablritung einer fliessbedingung, Z. Verein Deut. Ing., 72, 734-754. Schmid, E., 1924, Neuere untersuchungen au metallkristailen, in: Proc. 1st Int. Cong. Appl. Mech., ed. C. B. Biezeno and J. M. Burgers, 343- 353, Delft, Waltman. Taylor, G. I., 1934, The mechanism of plastic deformation of crystal, Proc. Roy. Soc. London, A145. 362-404. Taylor, G. I., 1934, Plastic strain in metals, J. Inst. Metals, 62, 307-324. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 Taylor, G. I. and Elam, C. F., 1925, The plastic texture and fracture of aluminum crystals, Proc. Roy. Soc. London, A102. 634-667. Teodosiu, C. and Sidoroff, F., 1976, A physical theory of the finite elasto- viscoplastic behavior of single crystals, Int. J. Eng. Sci., 14, 165-176. Tseng, N. T. and Lee, G. C., 1983, Simple plasticity model of two-surface type, ASCE J. Engineering Mechanics, 109, 795-810. van Houtte, P. and Aemoudt, E., 1976, Consideration on the crystal and strain symmetry in the calculation of deformation textures with the Taylor theory, Materials Sci. and Eng., 23, 11-23. Voyiadjis, G. Z. and Foroozesh, M., 1990, Anisotropic distortional yield model, ASME, JAM, 57, 537-547. Voyiadjis, G. Z. and Kattan, P. I., 1989, Eulerian constitutive model for finite deformation plasticity anisotropic hardening, Mechanics of Materials, 7(41, 279-293. Voyiadjis, G. Z. and Kattan, P. I., 1990, A generalized Eulerian two-surface cyclic plasticity model for finite strains, Acta Mech., 81, 141-162. Voyiadjis, G. Z. and Kattan, P. I., 1991, Phenomelogical evolution equations for the backstress and spin tensors, Acta Mech., 83, 413- 433. Voyiadjis, G. Z. and Sivakumar, S. M., 1991, A robust kinematic hardening rule for cyclic plasticity with ratchetting effects, Part I: Theoretical formulation, Acta Mech., 90, 105-123. Weertman, J. and Weertman, J. R., 1983, Mechanical properties, mild temperature dependent. In: Physical Metallurgy, edited by R. W. Calm and P. Haasen, Elsevier Science Publisher, 1259-1307. Zaoui, A., 1986, Quasi-physical modelling of the plastic behavior of polycrystals, In: Modelling Small Deformations of Poly crystals, edited by John Gittus and Joseph Zarka, Elsevier Applied Science Publisher, 187-225. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Zbib, H. M. and Aifantis, E. C., 1987, Constitutive equations for large material rotations, in: Constitutive Laws for Engineering Materials: Theory and Applications, C. S. Desai et al., eds., Elsevier Science Publishing Co. Inc. 1411-1418. Zbib, H. M. and Aifantis, E. C., 1988, On the concept of relative and plastic spins and its implications to large deformation theories. Part I. Hypoelasticity and vertex-type plasticity, Acta Mech., 75 15-33. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 PROGRAM SOURCES This appendix contains a flow chart and source code for the various programs used in this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 FLOW CHART OF THE MAIN PROGRAM BEGIN INPUT CCUBIC SLIP 12 MATDATA LOAD T=T+AT ► K=K+1 NAB GAMA-DOT DWEP SIGMAPRJ SCHMID NO YES UPDATE NO FINISHED YES END Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 « H H H « *4 SI H H «« > > H n 03 Z « « H W II r* r» I n ♦ 4 - - »» - «, ,. <— cr n H « « g g H S h <1 «• H H « « . « % « . H —. W H I I g | H* H <* K PH «* < < ' If I H H O H O O 04 * _ _ _ i O —'—'M O —'H H I O > > S Q > S E H N M H N n H f l I • 03 03 O • 03 O O n «h m «h * h t* pr ■* > O —' i • h h h m h m h h E H '——' «■ H — * «. H IRRRRIRRR « X z>nnv>w i « n h CR n H PR l l l l ll l l *>03'-'—'04 03—’O404Z> **«««««• H I I I I I I I I « I W H CR n N n n < I A A AA<-* « * » »( HrIHHMNfRnn : 11 HHWHiHMnHN*'! I W H W H H H H O w —.ww«—— w-wo WHHHHHHMH ! SS H PR OHHHNnNfinH '. < < < <<<<■■ ) + + & CU i y O O o o o o * ; r» n h n W H W W M W H H > i w H HM H M H I i l l l i l l i i » *• © O » 04 0 * 55 : s s s X s s s -RHRRRRRR H t* QiC*0*P*GiCUC*Slt * I o o o H A «"S> «—» «—» M A m» m t a O O O O ( R H n n H N C I H N i b e o k i h h hr ■ • u ^ u « U — u HHHfRNfRnn H D O H n P • «. P P; ft N Hi ft 2 H H Z "Ig U 3 M H H H H H H H S S s S H —■ H W H Si QQ q Q h h > > h SSht-Q hhZw feZ S llliilli RUUU b n n H hi hi *■* X < - ~ * C5 — — H 03 O X to > 03 o - z (M • < • — - u HDS-. u r> «-» ?§ fr» t* *. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 w K —s H * K > H* 01 « > b - *W H S.s to k!>^K O] * 85 * *• * H * M H j I^Im < « H « « C — b b S b* to> b •> b•• O H —’ H H » M ' I H f* 0 i H n n > ^ > > SS .. ** * » CO K 9S 0) [i] H H rlw «• B H (1 H H H B i S SSBi-^B a Hi PP5bb£po Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 CR K s « #•. H E u »*> H W 9 u8 2S 1*5 u ♦ H B « o K H H S H H o l U t> S’** M .££ . o O < o * * y> w w M M * o a aU U 3 H Q w i ' Q Q CR £4 ( H Q H H Q PR w >UI I • H « 4 • H CR I W b? W « ; U H • H U O • O U PR PR o O < ) O <* PR O CR SS J ♦ O O O • ' O * * U O ■ i a i i i r r I © U 4 i b . £. I I I I I I I I I p I I I o W ro ^ H PR n a w t-4 C m !» k tO I n n g u I n H CR r> h pr r> P « •. > H r 4 p g n P < b b P w % a o PR > « » < S m O d o b u «. • Z H < VO O H w H H PR • P* CD CO 00 91 Oi fft © h N H H N H H H O H H M H p H H H * < H H H < i PR H O m Q w ^ w^wwwChOh HH H O u o u U U O U Z * O O u o u U t) u u U U U Z *(JO u u u o u u U L> U U Z P > > > > > >>> > > > > H > > > > > > s> > WWW W> P2 Z£££8d w w WWWw w w w 8S £S8 P w w w w w w w w wE§§^u5SqS ig38“ 8£E8 HW V < o o ffl Q w > to SS1!1! N PR *■ * • H CR O **•' m h % H H X>nOin% o • Z •>• *> •» > > «. b * < pa *— (9 • » I n h o o o ' “ "'Q Pl H r( Q H Q Q H H 328 4 H •O U U ‘HHU*nncR • Cl U ! B w% nnAAAAAAAAAi r m u n « £ Q< uuuuuouoouuuuuooooua Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced u u uouoouuouou o LAST UPDATEi 2/17/94 £ g 01 n o n n rn c« f g HUM E HUH S h O Z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced SUBROUTINE TO CALCULATE SCHIMID STRESS AND SET IYFLAG AS AN IN DICATE THAT A SLIP SYSTEM I S 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced ucjooouuoyu SUBROUTINE TO CALCULATE THE TRANSFORMATION TENSOR R BETWEEN U U)U n H» H z gSsti .** *> H h ( n H / > ) * H P S K O i l b M N OONB t J ^ H Z H m 8 8 eii 8 8 M H m8 « m 2 2H O b H o i i O to h h r> e o n H * Q a i r : " h > ft *. H f-t > * «-* • h .. DuU 04 4 0 u to Cb u u CU W H a a u u o o CO CO CU fb OJ H ra n 04 i O 04ai m n M O M X n n 2 « m < n 2 QUd U o n U Cu >< H H Q • w H Di H « X « « « H H I ■ I H H KKHBiPil M H Q l t fl Q I i ol — O > * » 2 o Q to } Q :8 *-■ a CU X H 2 X cu a* _ X X H H H f* Q { ^CU CU CU ^ {J H X O S H X CO H Qj j X w : H 4 H «. O « _ Ou «— M W CO C4H *Q % 2 ' CU ■*”* —' Cb tm. O CO Q CO Q H to MW 2 2 H X CU X H X /s 4H C H E QUQU O O O CO 2 CO *— h « O * sssas! i « a■ i • b tr ra E s S o E a & | « q H « a 2 4 0 O O X CO OO 00 I R - I R + 1 IF (IYFLAG(J).EQ.l) THEN J R - J R + 1 E L S E E N D IF NABR(IR,JR)*NAB(I , J) J R - J R + 1 MAB ( I , J ) -M A B R ( I R , J R ) I R - 0 IF (lYFLAG(I),EQ.l) THEN E L S E DO 130IF J»1,NUMSLIP (IYFLAG(J) .EQ*1)E L S E E N D IF THEN I R - 0 IF I (IYFLAG(I).EQ.l) R - I R + 1 THEN E N D IF END JR-0 DO 120 I«1,NUMSLIP J R - 0 DO 220 J-1,NUMSLIP E L S E CALL MINV(NABR, MABR, DOM, NUMYFLAG) E N D IF R ETU R N DO 210 I-1,NUMSLIP 130 CONTINUE C MAPPING NAB TO NABR 120 CONTINUE 220 CONTINUE210 CONTINUE C C C C INVERSE THE NABR TOMABR C C MAPPING THE MABR BACK TO MAB Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 M n c i n 3a|a ~S*S S Cu Z W * u D Iz h B • ♦ ** W c3 SU O^ U« o w w n S OS z? o« o J H Cu K S - H H z K ~ o w * os . M I ♦ * m *0 X Q H oOS H- . n n n E 2 • W H i «, •. cu x o H « < H «-t £ cu * R W Ctf ;§§§£ 0} H I l I U« U O W HOW in H K OJ H p bnnp ■ b Z •> * Z «CO O I o o n * H In H O H N H H — ► in Pi wincuEHH'—H'-'H h a ||s H goSitoSSq'So U O qdh - uBzu au ft 3 w 01 Osi r^nuuuouuoooouououuuouuouooooauuuuu J cu cu b cu n O n b A «. H n z CU >9 u o o u H H X 88 H cu % > ci H t a ^ Z H u OWO»HW 0 * cu * H O 05 H H n § h a H 0 i h p ♦ cu * z n I 1 ; •-» w n m m r> c< II>0 (9 0 « » k k (u H o 3 H CU n H s b J H Ian ■ ■ s *-* w * O^W ■* ** 35^VV77Swo” jn n h b CO US O CO CO ^0£gH* o O O O ■ H H J H7liOrIH o o HHNOOOOnH n o o j « M N ft " n n £ tn 0 BO cuwSooooSowl Ihh SSBh *®? oiB o SS HHHQQQQHUI S8 uououuuuu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 ou B! h H Z Z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced LAST UPDATE: 2/18/94 ENDIF 20 CONTINUE SUBROUTINE MATDATA(NUHSLIP,TO,CK,T, ENDIF H O .Q .A O , 0 1 , 8 0 , 0 2 , CO, Q 3 .H A B , A , B ,C , MODEL, AGAMA) R ETU R N 01 o H 8 u £ h O Map O H h 04 U u a r* n rfl 04 & HHZ 04 04 l '-i< ( U W U (D < i - ' M H Cl O O P P toi < CO CJ Y ) I ' r ( « Y Y QS-(r'I}6 QS-(r'I}6 91 92 a *3 hd fed En Q * « 3 3 os a ... ft O f* finristifif) nn « v. "v. ■**«. UJ s « o o o » . kK^kSkSUUHHQQQrlrl ■ « u io u u u O • HI I • • H b 2 ------■IIJOOOKJ S £ £ 5 O O H *-> Bk »3 >3bHf«cir«HctHt>ooHnnoo H H *■-' k U M . *• C-* . » ■* * % ■ • u n N P t b p t n n cu cu H H H g HHZHHHNHdij E Fl fi o o u o 88 & ?M !I I>ScT>Q03cau8S[-'C-'HS§ h a x t> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced LAST UPDATE! 7 / 2 8 / 9 3 (I)-TEHP 94 »4 w >s U o> Q cn Du O U 1 8 B- oom<»ooonH* cu U nr>cu04^^^a*'«-'tu cu § J *- Q O S U U O Q uouuuuouwuoo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission...... ' S E C * , / , , E 1 2, E . 6 1 2 , / . , 6 , /, , E 1 2 ., 6 E 1 , 2 / . , 6 , / , , E 1 2 . f i , / , , B 1 2 ., 6 E 1 , 2 / . , f i , / , J. , E 1 2 . 6 , / , ,F6.3,/, ,/, ,E12.6,/, , E12.6 , /, , E 1 2 . 6 , / , ...... * ' ,F6.2, ' ,E12.6 ' , E 1 2 . 6 P H I * ' S 1 2 • ' Q - Q2 ■ ' Q3 • ' ■ ' , 1 3 ) TO - ' BO - ' CO - • AGAMA" ' , E12.6 , /, B (I,J)t C(I,J): A(I,J): Q1 - • * ' THE THE INPUT INPUT DATA DATA ARE SUPPLIED fc CAUCHY AS FOLLOW STRESSES S*',/, OUTPUT * ' EULERIAN ANGLE' PH: ELASTIC COMPLIANCE! ' EULERIAN ANGLE PHI1: PHI1 ' - SLIP' RATE:' INITIAL HARDENING YIELD RATE: POINT: CK HO -- 'f ' E12.6,/, • ELASTIC• COMPLIANCE: ELASTIC COMPLIANCE: S ll - S44 ' « • • C O E F . F O R ' INTERACTIVE COEF: • ' INITIAL COEF. COEFF FOR C: ' INITIAL COEFF ' B: * * ****** ./, TIME INCREMENT: DT - EULERIAN ANGLE PHI2: PHI2- »,E12.fi/,INITIAL COEFF A: AO - FACTOR FOR BACKSTRESS MODEL: MODEL ...... IF OUTPUT (CK*T.GT. DATA- 3 EVERY .DO) SEVERAL GOTO INCREM 9099 ENTS CHECK CHECK IF F I N I S H E D G O T O 1 9 9 END ...... 910 1010 PRINT PRINT 3010 • ,'ERROR * ,'ERROR IN PRINT INOPENING * OPENING ,'ERROE THE THE IN INPUT STRESSES OPENING FILE' THE OUTPUT BACK STRESS FILE' FILE' C 2010 4010 PRINT *,* ERROE1000 PRINT IN * OPENING FORMAT(IX,/, ,'ERROR THE IN SLIP OPENING SYSTEM THE FILE1 P AND OMIGA FILE' C C 5010 PRINT * ,'ERROE IN OPENING THE STRAIN OUTPUT FILE' 9 0 9 9 S T O P C1001 FORMC3001 AT(IX,E11.3,3E13.4) C4001 FORM FORMAT(IX,Ell.3 AT(1X,E11.3,3E13.4) ,4E13.4) C2001 C5001 C6001 FORMAT(IX,Ell.3 FORMAT(IX,Ell.3,4E13.4) C7001 FORMAT(IX,3E13.4) FORMAT(IX,3E13.4) ,5E13.4) ...... l a g , IY P L A G , M O D EL , H A B , n u m y f ...... - ...... - Z (I)*0.DOTAUO(I)-CRSS IYFLAG (I) -0 GAMADOT(I)*(0.00)X ( I )Y * ( 0 I ) .D " O O . D O NUMYFLAG-0 DO 10 J-l,NUMSIGM SLIP A(I»J)*0.DO T N E W -T + D T IF C A (NUMYFLAG.EQ.0 L L N _A B (N U M) S L IP , C GOTO I J K 419 L , C IN V , S IG M A , CALL SCHMID (NUMSLIP,SIGMATMP, XBC, PC, IYFTEMP, TAU,TAUO) IF (lYFTEMIF P(I).NB.IYFLAG(I)) ((KLOOP.EQ.IO).OR.(IYDIFF.EQ.O)) IYDIFF-IYDIFF+1 GOTO 499 X B (I,J)-0.D 0 T N E W -O .D OD T»D TO CALL GAMA_DOT(NUMSLIP,CIJKL,D,MAB,.P,GAMADOT,SIGMA,IYFLAG)C A L L S IG M A P R J (S IG M A , D E ,H E , C I J K L , S IG M A T M P , D T ) I Y D I F F - 0 K L O O P -O IYFLAG(I)-IYFTEMP(I) DO 310 1 -1 ,NUMSLIP GOTO 3 9 9 DO 320 I»l,NUMSLIP 1 CINV,SIGMA) 1 P,0MIGA,NN,S9,A,B,C,AGAMA,NAB,MAB) 1 1 NN, S3, SVC, SIGMA, NVC,GAMA,GAMADOT, SIGMATMP, DP,AGAMA,X, WE,TAUO, Y,Z,XB, IYFLAG, PH I1, PH I, PHI2,R,RINV) ...... 399 KLOOP-KLOOP+1 310 CONTINUE 10 C CONTINUE TIME INCREMENT 1 9 9 T -T N E W 320 CONTINUE C C C TO PREVENTC MAXIMUM NUMBERS FROM TIKE ENDLESS OFINCREMENT DO LOOPS TO LOOP, CALCULATE KMAX- 10 THE GAMADOT IS 419 IN SETFOR ONE CALL THE DWEP (NUMSLIP,P,0MIGA,GAMAD0T,DP,DE,HP,WE,D,W,C C ...... * IP YES, SET IYFTEMP"IYFLAG AND BACKGO TO RECALCULATE C GAMADOT 399 C C C C CHECK IF ANY NEW YIELDING, IFNO UPDATE GO 499 C C 499 UPDATE THE CALL ORIENTATION, UPDATE(NUMSLIP,DT,HAB,A,B,C,SV,NV,P,OMIGA, STRESS, MATAERIALS PARAMETERS ETC. C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 r> n H 0< w w w o x « K U f O U » O » 0 4 * 04 H H H J h o a Jtn m w 10 H O x m M C/1 - o o o o o o o o o cu u o u u u o u ...... WDM WDM M K l U 2 0 I a wa a iI uw ex u w < J H O I o - 8 ! sis; DUD U X W O * H H e w ■ X M X K >« N CALCULATION AND THE INTERACTIVE FACTORS C i < o H I; D a w M H H IMPLICIT R£AL*8 (A-H,0-Z) GAMA(I)-0.D0 I O H < K n K CK:■*> PARAMETER FOR STRAIN CONTROLLED LOADING C I &4 H O < O K X N DTO:■■> INITIAL VLAUE FOR TIME INCREMENT C INITIALIZING I SET ZEROS < a o - DO 10 1-1,NUMSLIP ouuououuoouaooouooaoouuououauuoooucjuuuoououuuuouou Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced RTEMP»R ( I . KM) * R (J,M M ) 98 w ci w n (*i i H H H n n ( i n n n rl H H H H rt rt d H n n n CJ 05 05 05 tn cn H ■ a a ■ a ■ i a a a a ■ a a a i a a a beebee BBBBBB H N Cl NCICIH^M IHCIrtHNCIHrirtCI a ■ ■ a a ■ i n m o H « n NCICIHHN IMHCINrtnclHCIN . % * p N n C IH H N I H H • H f t n H HnnCICI IflOCIHHCIHrtflfi a a a a o n ci ci h h n w b U >3 a H « n H H n n n c i «t n n n m rl H 11 (1 1*1 Cl H H (I N o o ssi »■< ft fa ri n £Bx % § § § § 5ggggggggggg H f-4 H in M M H M W M I H H W W I u o o o o o H U to U W w SB H H r> *n n n 0 1 H O) *■» e o Q Q O S w 05 © ©• o • Q • Q a j o V UUUUUUUUUUUUUCJOUUU u o u o o u o o o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 X >4 to fa H i n ■ 'S..-. H fa D « fa < p D o . n n z z r> cu n < X >< N OH “ SBKg < a I n h n n h n n I H H H HH H H *>>>>> i CO CO CO CO t o « « • * « I H H * i r t r t n n n i * * I H H I I H H H H H * * > > \ i fa .j M M I. .. - . H «aaaaaaiHi J «— fa X u -S J 1 H H o CO SC r» co B HNn^NnrtNfl > — ►q ♦ co H h n.g O' P H H H N n n n n n . n n n nh M 5 o o n • « q o SOHHHHHHHHH! H H m *3 H • • sss j .2 X a a a a a a X *J w a o o H f f ;H h 3 b ( « H b Q O H l MPa W Q h a 0 a a a : a a o < w • • n i n r> ». X H a n t3oo O H fa ! UoopUOSofafa fa o H fa W H H H l Q Q wra n I H H ffaOHUM Q H H M n n H ^ b . H S S S H f- H ! P O WP h O O !|g H «— H fa H H H — H fa - - X HSgg»£SS 88 SSSBS2S2 E86S SoSfjft o g p fa 8 8 • u s! 8 u J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Wei Huang was bom on May 27, 1961, in Shanghai, China. He graduated from Shanghai Jiao Tong University, China, in July 1982, with a Bachelor of Science degree in Materials Science and Engineering. He then enrolled in a graduate program at Shanghai Jiao Tong University in the same year and obtained a Master of Science degree in Materials Science and Engineering in May 1985. After graduation, he became a research engineer/lecturer at Shanghai Jiao Tong University until May 1989. In June 1989, he joined the graduate program at Louisiana State University where he obtained a Master of Science degree in Engineering Science in August 1991. He then enrolled in the Ph.D. program in Engineering Science in the same year. Since August, 1994, he is working at AlliedSignal, Inc., as a senior metallurgical engineer located at South Bend, Indiana. He is currently a candidate for the degree of Doctor of Philosophy in Engineering Science at Louisiana State University, Baton Rouge, Louisiana. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DOCTORAL EXAMINATION AND DISSERTATION REPORT Candidate: Wei Huang Major Field: Engineering Science Title of Dissertation: Crystal Plasticity Model with Back Stress Evolution Approved: .jor Pro: Dean of the Graduate Scbool EXAMINING COMMITTEE: / K Date of Examination: November 6, 1995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o- 0 -0 -0 -0-j