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Louisiana State UniversityLSU Digital Commons
LSU Historical Dissertations and Theses Graduate School
1996
Crystal Plasticity Model With Back StressEvolution.Wei HuangLouisiana State University and Agricultural & Mechanical College
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Recommended CitationHuang, 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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CRYSTAL PLASTICITY MODEL W ITH 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
byWei 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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UMI Number: 9628308
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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
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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
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ....................................................................................... ii
LIST OF TABLES ................................................................................................... vi
LIST OF F IG U R E S................................................................................................... vii
ABSTRACT ................................................................................................................... ix
CHAPTER1 INTRODUCTION............................................................................. 1
2 THEORETICAL BA CKG RO UN D................................................ 52.1 Phenomenological A pproach................................................ 5
2.1.1 Isotropic Strain H ardening...................................... 62.1.2 Kinematic H arden ing ............................................... 62.1.3 Anisotropic H ard en in g ............................................ 7
2.2 Crystal Plasticity Approach ................................................ 82.2.1 Single Crystal Plasticity ......................................... 82.2.2 Concepts of Dislocation and
Backstress................................................................... 102.2.3 Polycrystal P la s tic ity ............................................... 162.2.4 Current Crystal Plasticity Approaches
by Asaro and Needleman and Rashidand N em at-N asser..................................................... 17
2.3 Objectives and Method of A pproach .................................. 21
3 THEORETICAL FORMULATION OFCONSTITUTIVE E Q U A T IO N ..................................................... 223.1 Fundamentals........................................................................... 223.2 The Hardening Rule and the
Evolution of the Backstress X ............................................ 25
4 NUMERICAL IMPLEMENTATION ........................................... 324.1 Application to the Simple S h ea r........................................... 324.2 Numerical R e su lts .................................................................. 35
4.2.1 Comparison Between the Two M odels.................. 364.2.2 Effect of Initial O rien ta tio n ................................... 404.2.3 Evaluation of Coefficients A,
B, C for Model 2 ..................................................... 434.2.4 Evaluation of the Interacting
Factors qj ................................................................ 49
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5 DISCUSSION OF R E S U L T S ......................................................... 52
6 CONTINUUM PLASTICITY THEORY FORPOLYCRYSTALS WITH MICROSTRUCTURAL CHARACTERIZATION................................................................. 566.1 Introduction............................................................................. 566.2 Continuum Mechanics and Dislocation
T h eo ry ..................................................................................... 566.3 Macroscopic Mechanical Behavior
of Polycrystalline Materials ............................................... 59
7 PROPOSED FUTURE WORK FOR THE THREE-DIMENSIONAL CASE ................................................................. 657.1 "Pan-Cake" Type P o ly c ry s ta l.............................................. 667.2 "Hexagonal Column" Type Polycrystal ............................ 68
8 SUMMARY AND CONCLUSION................................................. 72
REFERENCES............................................................................................................. 74
A PPEN D IX .................................................................................................................. 79
VITA ............................................................................................................................. 100
v
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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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LIST OF FIGURES
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 inseveral slip system s............................................................................ 14
Figure 2.4 Dislocation configuration in a tensile tested purecopper sample showing high density of dislocationsin the dislocations cells .................................................................... 15
Figure 2.5 Schematic illustration of configurations and thedecomposition of deformation gradient F ...................................... 19
Figure 4.1 Schematic illustration of the planar deformationmodel. Two slip systems denoted by their directionand 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 c a s e s ............................................................................ 38
Figure 4.2b Comparison of the backstress evolution for thetwo backstress m o d e ls ....................................................................... 39
Figure 4.3a Effect of initial orientation on the change ofstresses during shear deformation .................................................. 41
Figure 4.3b Effect of initial orientation on the change oforientation angle, <j>, during shear deformation ........................... 42
Figure 4.4a Effect of parameter B0 on the stress a duringshear deform ation............................................................................... 44
Figure 4.4b Effect of parameter B0 on the backstress X duringshear deform ation............................................................................... 45
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Figure 4.5a Effect of parameter C0 on the stress a duringshear deform ation............................................................................... 47
Figure 4.5b Effect of parameter C0 on the backstress X duringshear deform ation............................................................................... 48
Figure 4.6a Effect of interactive factors qj on the stressa during shear defo rm ation ............................................................. 50
Figure 4.6b Effect of interactive factors q, on the backstressX during shear deform ation.............................................................. 51
Figure 7.1 Schematic illustration of a "pan-cake" typepolycrystal m o d e l............................................................................... 67
Figure 7.2 Schematic illustration of a "hexagonal column"type polycrystal m o d e l...................................................................... 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
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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
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Finally, a general form of yield surface in polycrystalline materials is proposed that
incorporates the Tresca and von Mises and other models.
x
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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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2
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
III|
}
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3
(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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4
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 o f 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.
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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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6
where rkl is the deviatoric stress. r kl = 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
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:
2.1.2 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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F (akl) = 0 (2 .2)
(2.4)
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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.
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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}. I n a n H C P
crystal, they are the {0001} planes. This is because the number of atomic bonds
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9
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 < 1 1 0 > 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 I I I 111 111
Direction Oil 101 110 o il 101 110 O il 010 n o o i l 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 <j> cos X (2.7a)
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10
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 = <rM = s • a • n (2.7b)
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
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KgUre 2A c 1 it d T v lilIUStrati0” 0f a diS,0Ca,i0" BC created by shearing on plane ABC along thedirection AC in a simple cubic crystal.
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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
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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.
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14
Figure 2.3 Dislocation configuration in a tensile tested alpha brass single crystal showing dislocations in several slip systems.
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15
m
W
Figure 2.4 Dislocation configuration in a tensile tested pure copper sample showing high density of dislocations in the dislocations cells.
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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:
where, < A > and A are the average and the local values of a physical quantity
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)
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(2.9)
respectively. V and V1 are the overall volume and the volume of i-th grain
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 Asaroand 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
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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 = E f 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
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19
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Figu
re
2.5
Sche
mat
ic
illus
tratio
n of
conf
igur
atio
ns
and
the
deco
mpo
sitio
n of
defo
rmat
ion
grad
ient
F.
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
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
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,
M(2.18)
and W p as well as the elastic distortion part F*, can then be calculated. Individual
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21
r = e0 (T “ - X “ ) - T eM
asign (xa - x “) (2.19)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.
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CHAPTER 3
THEORETICAL FORMULATION OF CONSTITUTIVE EQUATION
3.1 Fundamentals
A Cartesian coordinate system is used in this work with C0 and Ct repre
senting the configurations at times t0 and t, respectively as shown in Figure 2.1.
The configuration is based on an embedded crystal lattice frame. The deformation
gradient associated with C0 and Ct is expressed as follows:
where superscripts e and p denote, respectively, the elastic and plastic components.
The corresponding associated velocity gradient is given by
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 R e(t), such that
F = F*5 • Fp (3.1)
L = F • F 1 = F6 • F5' 1 + F* • Fp • FP"’ • F6’1 (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)
Fe = Re • IF (3.4)
22
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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) • n0 (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 W e 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:
£ = f f - W - f f + <r*W (3.9a)
or
t = (a* + W e • <r - a • W e) - W • a + a • W (3.9b)
which leads to the following relation
°a = a* - (W - W e) • a + a • (W - W e) (3.9c)
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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
NLP = £ 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) • W p - IVP • (B : a) (3.17a)
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25
W = W e + 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 • W e + C : (D - Dp)
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 t 0“ 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:
+ C : [WP • (B : <r) - (B : a) • WP] - a (3.18)
(<r - X) : F* > r0“ + T“ (3.19)
NDP = £ 7“ P“ (3.20a)
N
w p = y , y a fl“ (3.20b)
where
P“ = 2. (s“ n“ + n“ s“) (no summation over a) 2
(3.21a)
« “ = 1 (sa n“ - n“ s“) 2
(3.21b)
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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,AX* = a(y) DP (3.22)
a (y ) = a 0sech"Ts - To
(3.23)
where a(y) is a coefficient that is a function of the total amount of shear y. The
Nshear 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, W e, 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 - W e • X + X • W e (3.24)
The evolution expression for T“ in equation (3.19) is linearly dependent on
7“ with the proper material coefficient ta/3 such that it is similar to the Hill and
Havner (1982) hardening rule,
T “ = £ W y* (3.25)
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27
where j8 is the active slip system. The material coefficient taj3 is the interactive
hardening rate at time t. It is affected by the amount of slip in the material such
that
a/3 (3.26)H (7 ) for a =(3
H (y)q for a (3
where q is an interactive coefficient for hardening. The hardening coefficient H(y)
is given as follows:
H(y) = H0 sech' H0T
(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:
r = Dsj = Mag et> : D(3.31)
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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:
NX = £ (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 NX* = 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
NB “ = Y Ba/3 y 13 (3.35b)
0-1
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29
Nc a = £ C “P 7^
JS-1
(3.35c)
where the interactive parameters A0 , Ba<3 and C“^ are given as follows:
for a = P
'q j for a ^ /?I A 't(3.36a)
Ba/3 =’ B' for a = /3
B 'q2 for a ^ /?(3.36b)
CaP =C' for a = (3
C 'q3 for a ^ |3(3.36c)
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:
A '(t) = A0 sech"
B'(y) = B0 sech2
C'(y) = C0 sech"
A07
Ts “ To
B07
C07
Ts - To
(3.37a)
(3.37b)
(3.37c)
where A0, B0 and C0 are the initial values of A ', B' and C ', respectively, at y =
0 . r s is the saturation value of r0.
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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 - c r : P a 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 o f crystal plasticity
modeling program CPMD was developed for the numerical implementation of the
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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.
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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, n 1), (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 <j>, sin 4>, 0]T
n 1 = [- sin cp, cos 4>, 0]T
s2 = [cos (0 + ip), sin ($ + \j/), 0]T
n 1 = [- sin (<t> + rj/), cos ([<j> + ^), 0]T
The deformation gradient field F is given by
F =
1 k t 0
0 1 0 0 0 1
and therefore F 1, F and F -F 1 can be expressed as
F =
0 k 00 0 00 0 0
(4.1)
(4.2)
(4.3a)
32
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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, i r ) , (s2, n2).
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34
F ' 1 =
1 - k t 0
0 1 00 0 1
(4.3b)
and
0 k 0
F F ' 1 = 0 0 00 0 0 _
The strain rate and spin rate are thereafter given as
0 k/2 0D = k/2 0 0
0 0 0
1
(4.3c)
(4.4a)
W =
0 k/2 0- k /2 0 0
0 0 0(4.4b)
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
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,
(4.5)
C u n - C2222 ~ C3333 — X + 2 /t
C 1122 - C 1133 “ C 2233 “ C 2211 “ C 3311 “ C 3322 “ ^
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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 c 1111 — P 'l- ^ 2222 — P "1 ~ c 3333 = 1/E
r - i 1122 ~ c 1133 — P 'l ~ 2233 = C l 2211 = C’^ l l = C’!3322 = - vIE
p-i c 1212 _ c 1221 — p-l ~ c 2121 = C l 2l\2 = C ^ m i = C l 1313 ~ C’SllS
— P-l_ 3131 — P "1 ~ 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 1011 (Pa)
v = 0.33
k = 1.0 x 10"3 1/sec (4.7)
tq = 2.76 x 108 (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
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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 C0 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 H0 are com
pared, namely H l-0 and H2-0. In Table 4.1, 1-0 represents the "ideal plastic
crystal material", while H l-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 <f> = 0 °, whereas both backstress models 1 and 2 show
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-ii
meters Aq and a 0 f°r the backstress models 1 and 2 are quite different in nature.
!
ii
L ...
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37
Table 4.1 List of different numerical problems conducted.
It Name 4>(°)
'I'n
H0(Pa)
Ao(Pa)
“ 0(Pa)
BO(Pa)
CO(Pa)
Model 9i q
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
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38
1.6E+9
1.2E+9
toCL 8.0E+8
0cn£ 4.0E+8
CO
0.0E+0
C om parison of Models ( o v s y )
( m odel 1 )
( m odel 1 )
( m odel 1 )
( m odel 2 )
( m o d e l2 )
( m odel 2 )
( id e a l )
o .. ( id e a l )
a , , ( id e a l )
A-A-A-& A A A- A —A -A A a ArA -A -A a a a - a - a - a -
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-
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 Shear y
3.00
Figure 4.2a Comparison of stress changes during shear deformation for two different backstress models and the ideal plastic crystal cases.
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39
6.0E+8Comparison of Models
(Backstress X vs. Shear y)
— X„ (model 1) X,: (model 1)
— |— X3? (model 1)—<$>- X,, (model2)
- A - *13 (model 2 )—|— XJ3 (model 2)
4.0E+8 —
(OQ.
XcoC / 30L .
2.0E+8 —
coa ;oCO
ca
A -A -A A A- A - A -A A A A A - A -A A A A A - A -A :
O.OE+O
-2.0E+8
3.000.00 2.001.00Shear y
F ig u r e 4 .2 b C o m p a r i s o n o f th e b a c k s t r e s s e v o lu t io n f o r th e tw o b a c k s t r e s s m o d e ls .
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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 B0 and C0 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 o f 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, <j> = 0° is quite stable for the crystal.
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 <f> deviates very little during the entire tested strain range of 300%
(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
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Stre
ss
41
1.2E+9
8.0E+8 —
CL4.0E+8 —
G
O.OE+O
-4.0E+8 —^ o - o - o - o 0 . o - o - o - o o o !
3.002.001.000.00Shear y
‘'ideal plastic’’Com parison of Differenet
Initial Orientations, 0
” 0 a „ ( 0 = 0 )
—A cr,; ( 0 = O )
— I— O , i ( 0 s O)
- o - o n ( 0 = -3 0 )
- A - o , , ( 0 = -3 0 )
- + - o ;; ( 0 = -3 0 )
Figure 4.3a Effect of initial orientation on the change of stresses during shear deformation.
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Ang
le
42
10.00
0.00
- 10.00• e -
- 20.00Effect of Initial Angle <)>
-30.00
-40.00
3.002.000.00 1.00Shear y
Figure 4.3b Effect of initial orientation on the change of orientation angle, </>, during shear deformation.
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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 2N
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 q3 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 B0 and C0 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 B0 are used.
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44
1.6E+9a 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 —totoa)
W 0.0E+0
^ O A& O - 0 - 0 - 0 O O O - 0 - 0 - 0 - O - i-4.0E+8 —
-8.0E+8
3.002.001.000.00 Shear y
Figure 4.4a Effect of parameter B0 on the stress a during shear deformation.
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45
1.5E+9x vs y
Effect of 0^
- 0 - M-B.) - 2&- 1—- O - X „ (* B b)
1.0E+9 —
coCL
A - A - A - A A A A A - A -A — A - a a — — A A A A — ,5.0E+8 —
XC/3C/30L .
^ 0.0E+0 oC O
CD
-5.0E+8 —
-1.0E+9
3.002.001.000.00Shear y
Figure 4.4b Effect of parameter B0 on the backstress X during shear deformation.
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46
It is found that the change of value and sign of B0 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 u22 as we^ as t^e
normal component parts of backstresses Xn and X22. This agrees with the fact
that B0 is the coefficient of the non-deviatoric backstress. In the backstress curves,
it is also noted that B0 hardly changes the Xn . However, it drastically changes
the X22 from positive to negative values.
Figures 4.5a and 4.5b are the stress and backstress curves for positive and
negative values of C0 respectively. Similarity between the effect of C0 and that of
B0 is noticed. The C0 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 X n respectively. However, C0 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 B0 controls the X22 part whereas C0
controls the X n part for the case of orientation 0°.
It should be noted that since the Xn and X22 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 B0 and C0 to
the dislocation configuration will be studied in the future in more detail.
I
iII
I
i
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47
1.6E+9O v » 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 + +!/)(/)<D
0/3 0.0E+0
^ O - 0 - 0 - 0 - 0 O <S> o - 0 - 0 - 0 - 0 - j-4.0E+8
-8.0E+8
3.002.000.00 1.00 Shear y
Figure 4.5a Effect of parameter C0 on the stress a during shear deformation.
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48
1.5E+9x v s y
Effect of CQ
1.0E +9
5.0E + 8 — A - A - A - A A A A - A - A - A — A V V a a a a—a_ a—C O
Q.
X0.0E + 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
yFigure 4.5b Effect of parameter C0 on the backstress X during shear deformation.
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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 B0 or C0 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.
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50
1.6E+9
1.2E+9
Comparison of InteractionFactorq, (B 0,Co = O) _
1 -£ r ~ o,j(q , = 1.4)
— |---- ° : 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 —C O
Q_
4.0E+8
0.0E+0
-4.0E+8
J
-A A A A A A —A A A A - A A —
F-H—t—I I I' +
0 O-O—O -0 O O 0-0-0-0 O O O-0-0-0
+ -F -I I— h-F -F -F-+-+H- -t- + +--t~F-F
-0—0—0—0- 0—0-
0.00 1.00 2.00 3.00
YFigure 4.6a Effect of interactive factors qj on the stress a
during shear deformation.
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Back
stre
ss
X (P
a)
51
1.5E+9Effect of qt on B ack stre ss . X
1.0E+9 —
A r A - A - A -A A A - ^ A - A - A A A A-—5.0E+8 —-A
0.0E+0 - + + “
jk "aik zk Ilk-- A —A —z k n k — S-5.0E+8
^ 0 -0 - 0 0 0 0 0 - 0 - 0 -o o o i-1.0E+9
-1.5E+9
-2.0E+9
3.002.001.000.00
Y
Figure 4.6b Effect of interactive factors q; on the backstress X during shear deformation.
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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)a n
(5.2)
52
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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
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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
ij
Iiiij
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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.
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CHAPTER 6
CONTINUUM PLASTICITY THEORY FOR POLYCRYSTALS W ITH 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
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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 ^ " k i . 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" = £ y g P g , W " = £ y g (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.
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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) <g> b8] (6.5)g
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 > i f (6.7)
where r8 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
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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) • n8 = (a - XD) : Ps ^ r fic (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 M aterials
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 N7 = T7 £ 71 V l v 1=1
where y1 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 V1 is the volume of grain I.
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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 - k 2 = 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| | f f ' | | (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
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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® a k t (6.14) P 2 G g 2 G g 2 G g J J
where
H ijkt = p if Pkf = \ (hifkf + hiftk + hjfkf + hjffk) <no sum over §) <6 -15)
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 <ru <r„ = k 2 or Hijk, , tj ak , = k 2 (6.17)V 1 = 1
where V1 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
vi
H-*Hjjkf- = X) —“ ’ dv' = f H ijk? sin # dP d<f>
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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
V f = | ir r3N = N a sr 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
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63
Orowan’s model. The critical yield strength is inversely proportional to the free
space between particles.
1/3a G b _ a G b
ay = ay° m x : = ~ w
V, 1
1 - 2a c
1/3(6 .21)
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:
° y = °y 0 +a 'G b e 1/m = <7V +
l/n \
“ 1/m
ct 1 - 1}T l/n(6 .22)
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 de
m n (6.23)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.
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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 « T l/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.
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CHAPTER 7
PROPOSED FUTURE W ORK 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 0 j, <£, <f>2
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:
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.
cos cos <j>2 - sincj)1 sin4>2cos4>
R = -cos^jsin^-sin^jcos^cos^)
sin ^ js in ^
cos </>2 sin 4> i + sin <j>2 cos </> j cos <£
- sin (f> j sin </>2 + cos <f> j cos <£2 cos cj>
-cos<£jsin0
-s in ^ 2sin<j!>
- cos <2 sin (j>
cos $
(7.1)
65
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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
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67
X
Figure 7.1 Schematic illustration of a "pan-cake" type polycrystal model.
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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
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69
X
Figure 7.2 Schematic illustration of a "hexagonal column" type polycrystal model.
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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.
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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).
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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
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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.
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REFERENCES
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Asaro, R. J., 1983, Crystal plasticity, J. o f Applied Mechanics, 50, 921-934.
Asaro, R. J. and Needleman, A., 1985, Texture development and strainhardening 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 o f the plastic distortion of a polycrystalline aggregate under combined stresses, Phi. Mag., 42, 414-427.
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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. o f Plasticity, 8 74-92.
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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.
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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.
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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 o f single crystals, Int. J. Eng. Sci., 14, 165-176.
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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 evolutionequations 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.
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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 andplastic spins and its implications to large deformation theories. Part I. Hypoelasticity and vertex-type plasticity, Acta Mech., 75 15-33.
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APPENDIX
79
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80
PROGRAM SOURCES
This appendix contains a flow chart and source code for the various
programs used in this work.
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FLOW CHART OF THE MAIN PROGRAM
81
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C ............................................................................................................................................................................................................C T O P R E V E N T FRO M E N D L E S S DO L O O P , KM AX- 1 0 I S S E T F O R T H EC MAXIMUM NU M BERS O F L O O P S T O C A L C U L A T E T H E GAMADOT I N O N EC T I K E IN C R E M E N TC ...............................................................................................................................................................................................................
K L O O P -O 3 9 9 K L O O P -K L O O P + 1
I F (N U M Y FL A G . E Q . 0 ) G OTO 4 1 9C A L L N _A B (N U M S L IP , C I J K L , C I N V , S IG M A ,n u m y f l a g , IY P L A G , M O D E L , H A B ,
1 P ,0 M I G A ,N N ,S 9 ,A ,B ,C ,A G A M A ,N A B ,M A B )C A L L G A M A _ D O T (N U M S L I P ,C IJ K L ,D ,M A B ,.P ,G A M A D O T ,S IG M A ,IY F L A G )
4 1 9 C A L L DW EP ( N U M S L I P ,P ,0 M I G A ,G A M A D 0 T ,D P ,D E ,H P ,W E ,D ,W ,1 C IN V ,S IG M A )
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 )C A L L S C H M ID (N U M S L IP ,S IG M A T M P , X B C , P C , I Y F T E M P , T A U ,T A U O )
C .........................................................................................................................................................................................................C C H EC K I F A N Y NEW Y I E L D I N G , I F NO GO U P D A T E 4 9 9C I P Y E S , S E T IY F T E M P " IY F L A G AND GO B A C K T O R E C A L C U L A T EC GAMADOT 3 9 9C ...............................................................................................................................................................* ......................................
I Y D I F F - 0DO 3 1 0 1 - 1 , N U M S L IPI F ( l Y F T E M P ( I ) . N B . I Y F L A G ( I ) ) I Y D I F F - I Y D I F F + 1
3 1 0 C O N T IN U EI F ( ( K L O O P . E Q . I O ) . O R . ( I Y D I F F . E Q . O ) ) G O T O 4 9 9 D O 3 2 0 I » l , N U M S L I P I Y F L A G ( I ) - I Y F T E M P ( I )
3 2 0 C O N T IN U EGOTO 3 9 9
C ...........................................- ......................... - ..........................................................................................................................C U P D A T E T H E O R IE N T A T IO N , S T R E S S , M A T A E R IA L S PA R A M E T E R S E T C .C ......................................................................................................................................................................................................4 9 9 C A L L U P D A T E ( N U M S L I P ,D T ,H A B ,A ,B ,C ,S V ,N V ,P ,O M I G A ,
1 N N , S 3 , S V C , N V C ,G A M A ,G A M A D O T , A G A M A ,X , Y , Z , X B ,1 S IG M A , S IG M A T M P , D P , W E , T A U O , IY F L A G , P H I 1 , P H I , P H I 2 , R , R I N V )
C C H E C K I F F I N I S H E D ...........................I F ( C K * T . G T . 3 . D O ) G OTO 9 0 9 9
C .....................O U T PU T D ATA EV E R Y S E V E R A L I N C R E M E N T S ......................................C ..................... - .................................................................................................. * ...................................................
G O T O 1 9 99 1 0 P R I N T • , 'E R R O R I N O P E N IN G T H E I N P U T F I L E '1 0 1 0 P R I N T * , 'E R R O R I N O P E N IN G T H E S T R E S S E S O U T PU T F I L E ' 2 0 1 0 P R I N T * , * E R R O E I N O P E N IN G T H E S L I P S Y ST E M F I L E 13 0 1 0 P R I N T * , 'E R R O E I N O P E N IN G T H E BACK S T R E S S F I L E '4 0 1 0 P R I N T * , 'E R R O R I N O P E N IN G T H E P AN D O M IG A F I L E '5 0 1 0 P R I N T * , 'E R R O E I N O P E N IN G T H E S T R A IN O U T PU T F I L E '1 0 0 0 F O R M A T ( I X , / ,
• * T H E IN P U T D A T A fc CAUCHY S T R E S S E S O U T PU T * * ****** ./,' T H E IN P U T D A T A A R E S U P P L I E D A S F O L L O W S * ', / ,' E U L E R IA N A N G LE P H I 1 : P H I 1 - , E 1 2 . 6 , / ,' E U L E R IA N A N G LE P H : P H I * ' , E 1 2 . 6 , / ,
E U L E R IA N A N G LE P H I 2 : P H I 2 - » , E 1 2 . f i / ,• E L A S T IC C O M P L IA N C E : S l l - ' , E 1 2 . 6 , / ,' E L A S T IC C O M P L IA N C E ! S 1 2 • ' , E 1 2 . 6 , / ,• E L A S T IC C O M P L IA N C E : S 4 4 « • , E 1 2 . f i , / ,' S L I P R A T E : C K - ' f E 1 2 . 6 , / ,
T IM E IN C R E M E N T : D T - ' , F 6 . 2 , ' S E C * , / ,' I N I T I A L Y I E L D P O I N T : TO - ' , B 1 2 . 6 , / ,' H A R D E N IN G R A T E : HO - ' , E 1 2 . f i , / ,' IN T E R A C T IV E C O E F : Q - , F 6 . 3 , / ,
I N I T I A L C O E F F A : AO - ' , E 1 2 . 6 J .F A C T O R FO R A ( I , J ) : Q1 - ' , E 1 2 . 6 , / ,
' I N I T I A L C O E F F B : BO - ' , E 1 2 . 6 , / ,• C O E F . F O R B ( I , J ) t Q2 ■ ' , E 1 2 . 6 , / ,• I N I T I A L C O E F F C : CO - • , E 1 2 . 6 , / ,' C O E F . F O R C ( I , J ) : Q3 • ' , E 1 2 . 6 , / ,' AGAM A" ' , E 1 2 . 6 , / ,
B A C K S T R E S S M O D EL : MODEL ■ ' , 1 3 )C 1 0 0 1 F O R M A T ( I X , E 1 1 . 3 , 3 E 1 3 . 4 )C 2 0 0 1 F O R M A T ( I X , E l l . 3 , 4 E 1 3 . 4 )C 3 0 0 1 F O R M A T ( 1 X , E 1 1 . 3 , 3 E 1 3 . 4 )C 4 0 0 1 F O R M A T ( I X , E l l . 3 , 4 E 1 3 . 4 )C 5 0 0 1 F O R M A T ( I X , E l l . 3 , 5 E 1 3 . 4 )C 6 0 0 1 F O R M A T ( I X ,3 E 1 3 . 4 )C 7 0 0 1 F O R M A T ( I X ,3 E 1 3 . 4 )9 0 9 9 S T O P
END
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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 EXAM INATION AND D IS S E R T A T IO N 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
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.