Verification of the prediction of deformation-induced anisotropy forsimple deformation modes: uniaxial state and pure shear state of

stress

C.H. Lee a, D.Y. Yang a,*, Y.-S. Lee b

a Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, ME3214, Science Town, Daejeon 305-701, South Koreab School of Mechanical Engineering, Kookmin University, 861-1, Chongnung-dong, Songbuk-gu, Seoul 136-702, South Korea

Received 20 November 2001; received in revised form 28 February 2002

Abstract

Deformation texture with preferred orientation is developed by external disturbance applied to the grain during the deformation

process such as rolling. The formation of deformation texture is strongly influencing the mechanical property of the product, and

material anisotropy is observed from the deformation texture, macroscopically. Therefore, the proper consideration and analysis of

deformation texture is required. In the present work, the method for prediction of deformation-induced anisotropy employing the

phenomenological yield potential is proposed. The proposed algorithm is applied to the anisotropic evolution for simple

deformation modes, such as uniaxial stress state and pure shear stress state in X �/Y direction. In order to verify the effectiveness of

the method, the result from the proposed algorithm is then compared with that from the crystallographic texture analysis.

# 2003 Elsevier Science B.V. All rights reserved.

Keywords: Deformation-induced anisotropy; Barlat’s phenomenological yield potential; Crystallographic texture analysis

1. Introduction

In the metal forming processes such as extrusion,

drawing, and forging etc. plastic deformation is one of

the major causes for change in mechanical properties.

From the deformation process, orientation of slip

systems is rearranged with preferred orientations. De-

formation-adaptive preferred orientation is also called

‘deformation texture’, and it is expressed by the material

anisotropy in deformed material. Material anisotropy

induced by deformation processes is affected to the

material property and it is major parameter of process

formability.

In order to use the material anisotropy in finite

element (FE) analysis, anisotropy is usually interpreted

as the material constitutive relations with various

methods. Conventional constitutive law which describes

plastic deformations of metals consists of three parts:

yield functions, stress�/strain (or hardening) functions

and the associated normality flow rule. Up to date,

anisotropic characteristics are interpreted by selection of

proper yield potential implemented the material aniso-

tropy.

The method of examination of material anisotropy

has two major branches; one is a macroscopic method

using a specimen test such as uniaxial tension or

compression test, and another is microscopic observa-

tion of material properties, using orientation distribu-

tion function (ODF), pole figure. From the macroscopic

observation, material anisotropy is represented by

phenomenological yield potential using the anisotropic

coefficients, moreover, from the result of microscopic

observation, crystallographic orientation is represented

by polycrystal modeling based on crystal plasticity.

However, because of excessive amount of computational

time, the model based on crystal plasticity theory has a

limitation for the application and possible uncertainty in

prediction of complex deformation processes. More-

over, both of experimental approaches involve complex-

ity in application.

* Corresponding author. Tel.: �/82-42-869-3214; fax: �/82-42-869-

3210

E-mail address: dyyang@mail.kaist.ac.kr (D.Y. Yang).

Materials Science and Engineering A339 (2003) 302�/311

www.elsevier.com/locate/msea

0921-5093/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 1 6 1 - 2

Therefore, the present study is concerned with devel-

opment of anisotropy and change of deformation-

induced mechanical property in deformed material

under the plastic forming process introducing the properalgorithm for prediction of anisotropic coefficients. The

proposed algorithm is based on the prediction of

anisotropic coefficients in Barlat’s phenomenological

yield potential.

2. Background of the study

2.1. Anisotropy in metal forming

Deformation in polycrystalline materials, like most of

metals, is dominated by the slip system. During defor-

mation, each grain rotates so that active slip directions

become more nearly parallel to the axis of tension with

the result that grains become elongated in the direction

of flow. Deformation of polycrystalline metal is char-

acterized by integration of combined deformations ofindividual grains.

Due to deformation, polycrystalline material shows

the non-random distribution of crystal orientations,

which is known as preferred orientation or texture.

Preferred orientation is important because of the effect,

often very marked, which it has on overall macroscopic

properties of materials, i.e. anisotropy. As preferred

orientation develops, the effectiveness of cancellationdiminishes, and the specimen shows directionality in its

macroscopic properties. A strongly developed texture in

a sheet, for example, may result in the considerably

different tensile strength in the transverse direction as

compared with that in the longitudinal direction. There-

fore, the anisotropic characteristic of materials during

plastic forming, has become of increasing interest as the

processing and use of materials are getting moresophisticated, and more quantitative information is

required in all production and design stages.

2.2. Historical background in application of anisotropy

The study related with material anisotropy may be

categorized into two major branches. One is microscopic

approach by observing crystal plasticity, and the other isrelated with experimental observation at the macro-

scopic level. The observed characteristics are so inter-

preted as to be described in the yield potential which

represents the material anisotropy.

2.2.1. Consideration of anisotropy based on crystal

plasticity

The major cause of mechanical anisotropy in poly-crystalline materials is crystallographic texture, i.e. the

individual crystals tend to rotate towards certain stable

orientations during plastic deformation. Therefore,

when the crystal slip system is observed along the

deformation path, it gives important information in

material anisotropy.

Until now, there have been many research works todescribe their characteristics of crystal slip systems.

Some works describe the single crystal deformation

model [1�/5], and then, the single crystal model is then

extended and applied to polycrystals [6�/10]. The

method of approach is relatively accurate but requires

tremendous computation, and it may also result in

unreliable prediction for the analysis of complex plastic

deformation. They are major disadvantages for applica-tion to industrial problems in the plastic forming

processes.

2.2.2. Consideration of anisotropy based on

phenomenological yield potential

It is well known from the plasticity theory and the

experiments that plastic deformation is described by the

yield surface. The approaches for formulating the yieldsurface are divided into two major groups: One

approach is to set the elastic limit using the limit of

some physical properties. For example, von-Mises yield

criterion is based on the limitation of elastic distortion

energy, and Tresca yield criterion is based on the

limitation of maximum shear stress. There are many

kinds of other yield conditions for isotropic approaches.

Another approach is to formulate a proper yieldfunction, which describes the best approximation of the

data from experimental results or material constants of

material model based on the physical concept. In FE

calculations, the anisotropy of material is usually taken

into consideration through a phenomenological yield

potential. This enables relatively accurate predictions

taking moderate calculation time. Moreover, considera-

tion of deformation-induced characteristics such asanisotropy has recently become more important to

simulate and observe the plastic forming process be-

cause of the latent anisotropy due to reorientation of

crystal slip system along the deformation process.

Therefore, nowadays, there are many approaches to

describe material anisotropy using the new formulation

of phenomenological yield potential.

This yield function describes the stress states corre-sponding to yielding of the six-dimensional stress space.

The concepts of the yield surface and plastic potential in

the stress space are based on basic assumptions in the

classical theory of plasticity [11]. A common approach is

to assume that the yield function and plastic potential

are identical, and that the plastic strain rate is normal to

the yield surface, i.e. associated plasticity. One of the

first attempts to take into account the anisotropy of thematerial through an anisotropic yield function was

carried out by Hill in 1948 [12,13]. Since then, several

anisotropic yield functions have been proposed i.e. by

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311 303

Hill [14,15], Bassani [16], Budianski [17], Karafillis [18]

and Barlat [19�/21].

2.3. Background and purpose of the study

In the phenomenological yield potential, yield surface

is described by the deformation characteristic such as

stress and strain-rate etc. and several coefficients related

to the material anisotropy. However, anisotropic coeffi-

cients are very sensitive to many parameters such as a

type of the deformation process and the degree of

deformation and so on. Due to dependence on diverseparameters, the determination of coefficients is mostly

obtained by various macroscopic experimental methods

such as uniaxial tension or compression test with proper

specimen, and microscopic observation is also carried

out using the ODFs and pole figures. When the yield

potential is determined by macroscopic observation

using material test, then it is validated by comparing

the crystal plasticity model using the result of micro-scopic observation. Therefore, texture measurement is

also an important tool among the traditional material

characterization methods. Texture data are normally

presented in the form of pole figures or ODFs. Both can

be used as inputs in numerical or analytical predictions

concerned with forming and product properties.

However, experimental methods have some disadvan-

tages. For example, when the deep drawing process issubjected to analysis using the finite element method

(FEM), the experiment to characterize the material

anisotropy is carried out for the initial material pro-

cessed by the forming process, typically rolling.

Although material anisotropy is observed and applied

to the next forming stage with a proper experimental

method, it is difficult to acquire the information of

processing stages during the forming process. Thisapproach has a limitation to consider the material

anisotropy, and it is only applied to observe the effect

of potential characteristics related with material aniso-

tropy to be developed in the course of analysis for the

next forming stage.

From the Ref. [23], an extruded plate of the aluminum

AA7108 alloy material has been investigated using the

texture analysis and by the uniaxial tensile test. On thebasis of the experimental results, crystallographic and

phenomenological yield surfaces have been calculated.

For use in the FE-simulations employing the present

anisotropic material, the Barlat’s phenomenological

yield function is recommended. Therefore, in this study,

Barlat’s stress-based yield potential is employed to

predict anisotropic yield potential from deformation

during the forming process. In order to check theproposed algorithm, simple deformation modes are

considered, such as uniaxial stress state and pure shear

stress state in X �/Y direction. Barlat’s tri-component

yield potential for two-dimensional stress state is thus

selected.

3. Introduction of the proposed algorithm

3.1. Introduction and basic assumptions in the prediction

of anisotropic coefficients

The method of prediction proposed in this study is

based on the prediction of anisotropic coefficients in the

Barlat’s phenomenological yield potential. Such predic-

tion enables an effective simulation of the plastic

forming process considering material anisotropy

throughout the process allowing the implementation inthe finite-element analysis with considerably smaller

computational effort. The present method is a conve-

nient method to predict and implement material aniso-

tropy developed during the plastic forming process.

The basic assumption of the present study is as

follows.

a) Yield locus is changed smoothly not abruptly.

b) When material is deformed during the process, yield

potential changes with various possibilities of varia-tions in anisotropic coefficients.

c) The direction of the change in anisotropic coeffi-

cients is varied dominantly along the direction of

Fig. 1. Schematic flow chart of the proposed algorithm.

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311304

the yield surface gradient with respect to the

anisotropic coefficients.

d) The size of the change in anisotropic coefficients is

determined by the stabilization of the yield surfacebetween the upper bound and the lower bound.

Under these assumptions, the algorithm for predic-tion of anisotropic coefficients is proposed as follows

(Fig. 1).

3.2. Algorithm for prediction of anisotropic coefficients

3.2.1. Basic introduction of Barlat’s tri-component yield

potential

Barlat’s phenomenological yield criterion is used for

many applications in forming of aluminium alloys. The

yield function describes yield stresses in general states of

deformation, which are relative values measured with

respect to a reference yield stress. A typical expression of

the yield function is given as follows:

F�F(s)� sm (1)

where F, s and s are the yield function, effective stress

and Cauchy stress, respectively. The exponent m is a

real positive number. Even if the plastic behavior of

materials is not linear, the associated flow rule given in

Eq. (2) can be generalized to represent the behavior of

other isotropic and anisotropic materials.

oij � l@F@sij

(2)

In order to describe the plastic flow behavior of

orthotropic polycrystals, the yield function must be

expressed in the six-dimensional stress space. It must

have convexity and normality following the plasticity

theory. From the current research works [25], hydro-static pressure also plays a key role to the plastic

deformation process such as extrusion, forging etc.

Therefore, the yield potential considering the effect of

hydrostatic pressure should be considered in the future.

In addition, it must be reduced to Eq. (3) or to an

equivalent formulation in the isotropic case.

F� jS1�S2jm� jS2�S3j

m� jS3�S1jm�2sm (3)

where Si (i�/1, 2, 3) represent principal values of stress

tensor.

In this study, special stress conditions such as a

uniaxial stress state and pure shear state are considered.

Therefore, Barlat’s tri-component yield potential is

selected for application. Barlat and Lian [19] proposed

a yield condition for the case of three-dimensional plane

stress, which is very often assumed in sheet formingproblems:

F�ajK1�K2jm�ajK1�K2j

m�(2�a)j2K2jm

�2sm (4)

K1�s?xx � hs?yy

2(5)

K2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�s?xx � hs?yy

2

�2

�p2s?xy2

s(6)

where a , h , p and m are material constants. This

anisotropic yield function has been obtained by sum-

ming up two convex functions, a jK1�/K2jm�/a jK1�/

K2jm and (2�/a)j2K2jm , and using the linear transforma-tions of stresses. Consequently, it is a convex function

which generalizes the plane-stress yield function. Lege et

al. [20] showed that this yield criterion was particularly

accurate for the description of the constitutive behavior

of 2008-T4 aluminum alloy sheet. In this case, the yield

surface in 2D is represented as follows:

R(u)��

2

ajcosujm � ajhsinujm � (2 � a)jcosu� hsinujm�1

m

s

(7)

where u means a plane angle varying from 08 to 3608.When m is 2 or 4, it represents von-Mises yield locus,

and when m is 1 or � it represents Tresca yield locus.

The shape of yield potential is determined by the

anisotropic coefficients, a , h , p and m and deformation

characteristics represented by stress, sij . When the initial

isotropy is assumed, the yield potential on P-plane mustshow a circular shape. However, from the deformation

along the working path, the deformed material shows

the change in microstructural orientation and mechan-

Fig. 2. Comparison of isotropic yield locus.

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311 305

ical properties, and the shape change of the yield

potential conforming to the deformation is investigated.

The change can be predicted by employing the stabilized

yield potential which represents the current deformationstate (Fig. 2).

3.2.2. Formulation for prediction of anisotropic

coefficients

As mentioned above, deformation texture evolves by

various deformation processes. Even from the initial

isotropic condition, anisotropy is induced during the

forming process developing the specific directionality as

in extrusion and in rolling, etc. In this work, the

evolution of anisotropic characteristics due to deforma-tion is studied. If all constants a , h , p and m were

known then the stress-based yield potential is the

function of stress only:

F�F(sij; ci) (8)

However, the yield potential is treated as a function of

the anisotropic coefficients under certain deformation

i.e.:

F�F(ci;sij)�F(ci;sij �known)�F?(ci) (9)

When the deformation history is known from the

result of FE-analysis in a certain deformation process,

the yield potential varies with respect to directional

constants, a , h , and p can also be known. Therefore, the

prediction of stable yield potential can be implemented

by a proper searching procedure using the gradient

function @F=@ci:/In order to predict the anisotropic characteristics due

to a deformation process, first of all, a proper initial

guess is required. In most of plastic forming processes,

the initial isotropy of the starting billet is usually

assumed. Using the initial guess, the yield potential is

temporarily determined, and then, the calculation of the

upper bound and the lower bound is required at the

same stress state. In this case, von-Mises yield potential

is used as the upper bound, and Tresca yield potential isselected as the lower bound. Searching is implemented

between these boundary values in the direction of the

yield potential gradient with respect to anisotropic

coefficients. Eq. (10), Eq. (11) and Eq. (12) show the

expressions for the yield potential gradients.

@F@a

� jK1�K2jm� jK1�K2j

m� j2K2jm

(10)

@F@h

�amjK1�K2jm�1

�syy

2�

(sxx � hsyy)syy

4K2

�

�amjK1�K2jm�1

�syy

2�

(sxx � hsyy)syy

4K2

�

�2(2�a)mj2K2jm�1 (sxx � hsyy)syy

4K2

(11)

@F@p

�(�amjK1�K2jm�1�amjK1�K2j

m�1

�2(2�a)mj2K2jm�1

)ps2

xy

K2

(12)

The calculated gradient is assumed as a dominant

direction of change in anisotropic coefficients, and it will

be used to find the optimal values for anisotropic

coefficients. Under the assumption of the same stressstate, the anisotropic yield potential is bounded by the

Tresca yield potential and the von-Mises yield potential.

From this line of thought, the range of Barlat’s tri-

component yield potential is assumed as follows:

RTresca5RBarlat��@F@ci

�Dci5Rvon-Mises (13)

�@F@ci

��

�@F@ci=k @F@ci

k� (14)

In the Eq. (14), k@F=@cik means the length of one-

dimensional gradient vector @F=@ci; so, [@F=@ci] repre-

sents a unit vector in the direction of gradient vector,

@F=@ci: From Eq. (13), the variation of anisotropiccoefficients is also calculated as follows:

Dc?i�Dclower�t(Dcupper�Dclower) (15)

where, Dclower and Dcupper are determined by Eq. (13),

and parameter t has the range of 0�/1 (05/t 5/1). Then,the anisotropic yield potential is only varied by the

parameter ‘t ’. In the iteration process, the solution norm

is examined from the calculation of yield potential with

Fig. 3. Change of yield potential with variation of parameter, t .

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311306

various values of ti from 0 to 1. From the iteration, the

value of t is obtained by the minimized value of solution

norm. In the iteration process, the solution norm is

defined as follows:

enorm�jRi�1 � Rij

Ri�1

(16)

Fig. 3 shows the change of the yield potential for

various values of t from 0 to 1 in X �/Y plane. In the

figure von-Mises yield criterion, Tresca yield criterion,

and Barlat’s isotropic yield potential are compared with

the anisotropic yield potential. The dotted line refers to

the paper of Barlat [20], and the coefficient values are

1.25 for a , 1.15 for h , and 1.02 for p , respectively. From

the comparison of the yield potential when the value of t

is 1 and the yield potential for the case of 2008-T4 alloy

sheet with the experimental observation, it shows the

similar trends for the shape change in yield locus.

Fig. 4 shows the variation of the solution norm with

respect to the value of ti from 0 to 1 under the special

stress state like in Eq. (17), and, from the figure, the

value of t is selected as 0.97. Then, the value of t is set as

0.97.

Fig. 4. Determination of the optimal value of parameter, t .

Fig. 5. Change of yield locus under the uniaxial stress state. Fig. 6. Change of yield locus under the pure shear stress state.

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311 307

s?xx s?xy

s?xy s?yy

� ��

s0 0

0 �s0

� �(17)

From the above-mentioned consideration, a new

algorithm is proposed to predict the anisotropic yield

potential considering the deformation characteristics.

The prediction of deformation-induced anisotropy is

described by the value of anisotropic coefficients in the

phenomenological yield potential. The proposed algo-

rithm is now applied to simple deformation modes, such

as uniaxial state and pure shear state of stress, and theeffectiveness and reliability are considered. The sug-

gested algorithm is checked by comparing the computed

results with the polycrystal model based on crystal

plasticity.

4. Application and consideration for simple deformation

modes

4.1. Consideration of uniaxial stress state

To begin with, the proposed algorithm is checked for

the uniaxial stress state. The uniaxial stress state is given

as shown in Eq. (18) with only tensile stress in the Y -direction. In this case, the direction of change in

anisotropic coefficients is calculated as Eq. (19), and

the change of anisotropic coefficients is predicted in the

case of coefficient h , only.

s?xx s?xy

s?xy s?yy

� ��

0 0

0 s0

� �(18)

@F@h

�2mhm�1sm0 (19)

Therefore, [@F=@ci] is expressed as follows in Eq. (20):��@F@a

��@F@h

��@F@p

��� [0 1 0] (20)

The calculated directional vector is applied to find the

range of variation in anisotropic coefficients. Using the

isotropic initial condition, the range of coefficient, h is

shown as in Eq. (21).

h�0:9616�0:0426t (21)

Coefficient, h is calculated as 1.0029 by applying the

selected value of parameter t with the stabilized yield

potential (t�/0.97). Fig. 5 shows the predicted yield

potential for the given deformation characteristics.

From the change in anisotropic coefficients, the pre-

dicted yield surface is changed dominantly along the

instability point. Therefore, the change of the yield

potential will cause the change in its normal vector thatis related with the strain-rate value in the FE-simulation,

and it should be interpreted by the result of deforma-

tion-induced anisotropic characteristics of the material.

When the first quarter of X �/Y plane in Fig. 5 is

considered (sxx , syy ]/0) in computation, the yield locus

is dominantly changed in the range of in-plane angle

from 458 to 908.

4.2. Consideration of pure shear stress state

The following consideration is related with the pure

shear stress state as one of the dominant stress states inthe in-plane stress case. The stress state with pure shear

is expressed as follows in Eq. (22), and the related

coefficient gradient is calculated as Eq. (23), and Eq.

(24). In this case, the change of anisotropic coefficients a

and p is presupposed:

s?xx s?xy

s?xy s?yy

� ��

0 t0

t0 0

� �(22)

@F@a

�2(1�27)t80 (23)

@F@p

�16(1�27)t80 (24)

Therefore, the direction vector for searching, [@F=@ci]

is expressed as follows in Eq. (25).��@F@a

��@F@h

��@F@p

��� [�0:1221 0 0:9925] (25)

The calculated directional vector is applied to find the

range of variation in anisotropic coefficients. Using the

isotropic initial condition, the range of each coefficient

Fig. 7. Change of pole figure in the deformation process [22] (a) Initial

random orientation; (b) 75% uniaxial tension; (c) 75% pure shear in

1,2-plane.

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311308

is shown as in Eq. (26) and Eq. (27) in terms of the

parameter t .

a�0:9733�0:2962t (26)

p�0:9668�0:0364t (27)

Anisotropic coefficients a and p are then calculatedsimilarly as 1.2606 and 1.0021, respectively, by applying

the selected value of parameter t with stabilized yield

potential (t�/0.97). Fig. 6 shows the predicted yield

potential incorporating the deformation characteristics.

From the change of anisotropic coefficients, the pre-

dicted yield surface is changed dominantly along the

instability point. Therefore, the change of yield potential

will cause the change in its normal vector that is related

with the strain-rate value in the FE-simulation, and this

is interpreted as the deformation-induced anisotropic

characteristics of the material. When the first quarter of

X �/Y plane in Fig. 6 is considered (sxx , syy ]/0), the

Fig. 8. Comparison of change in yield potential under the uniaxial stress state (a) Flow potential surface evolution based on crystal plasticity [22]; (b)

Change of yield potential on P-plane.

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311 309

yield locus is dominantly changed in the range of in-

plane angle from 08 to 458.

4.3. Comparison with the result from the crystallographic

texture analysis

The result of the proposed algorithm is compared

with the result of prediction using material modelingbased on crystal plasticity. In crystal plasticity, plastic

deformation at macro-level is caused by crystallographic

slip at micro-level. Therefore, in crystal plasticity for the

crystallographic texture analysis, a proper physical

model is required that describes crystallographic kine-

matics by slip mechanism. The accurate constitutive

equation can be obtained by observing the slip phenom-

ena closely, and a crystal model is also required to

explain the hardening mechanism and the change of its

crystal orientation. Fig. 7 shows the change of pole

figure due to the change of crystal orientation by

induced deformation [22].

The change in crystal orientation results in the change

in yield potential. Therefore, the comparison of yield

Fig. 9. Comparison of change in yield potential under the pure shear stress state (a) Flow potential surface evolution based on crystal plasticity [22];

(b) Change of yield potential on P-plane.

C.H. Lee et al. / Materials Science and Engineering A339 (2003) 302�/311310

locus between the crystal plasticity and the proposed

algorithm is useful to validate the proposed algorithm

for prediction of deformation-induced anisotropy. The

yield locus is compared between the different states ofstress. In this case, the yield locus is drawn in the P-

plane.

Fig. 8 shows the comparison of change in yield locus

under the uniaxial stress state. Fig. 8(a) shows the result

obtained by crystal plasticity, and Fig. 8(b) shows the

result obtained by the proposed prediction algorithm. In

the prediction based on crystal plasticity, the instability

point of yield locus along the s3-axis also movescontinuously in the clockwise direction, and it is also

observed in Fig. 8(b). Moreover, the yield locus between

s3-axis and s2-axis shows more hardening characteris-

tics than other range, and it is also similar to the result

from the crystallographic texture analysis based on

crystal plasticity.

Finally, Fig. 9 shows the comparison of change in

yield locus under the pure shear stress state. In this case,a good agreement of the proposed algorithm is also

observed when compared with the prediction based on

crystal plasticity model.

The validity of the proposed algorithm for prediction of

the deformation-induced anisotropy is thus observed by

comparing it with the result based on crystal plasticity.

5. Conclusion

The proposed method in this study enables the

consideration of deformation-induced anisotropy from

the prediction of change of phenomenological yield

potential that conforms to deformation characteristics.

Phenomenological yield potentials have been applied to

predict the presence of anisotropy in the extrudedprofiles in some research works [23,24]. In the present

study, Barlat’s stress-based tri-component yield poten-

tial is chosen to consider the deformation-induced

anisotropy for some deformation modes, such as uni-

axial stress state and pure shear stress state. The change

of yield potential due to deformation is described by the

variation of anisotropic coefficients.

An algorithm for prediction of deformation-inducedanisotropy has been proposed and applied to simple

deformation modes. From the comparison with the

previous study based on crystal plasticity, the validity

of the proposed algorithm is checked. The proposed

algorithm can thus be applied to FE-analysis of plastic

forming processes.

Appendix A: Nomenclature

/s/ effective stress

F Barlat’s stress-based yield potential

/oij/ strain-rate component

/l/ constant in flow-rule

s, sij Cauchy stress tensor, componentm exponent of yield potential FS1, S2, S3 principal value of stress tensor

ci, c anisotropic coefficients, coeffi-

cients vector

a , h , p components of anisotropic coeffi-

cients

s ?xx , s ?yy , s ?xy deviatoric stress component

RTresca, RBarlat,Rvon-Mises

radius of yield locus on P-plane

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