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57 2 K S M E I n t e r n a t i o n a l J o u r n a l, V oL 12 , N o . 4, p p . 5 7 2 ~ 5 8 Z 1 9 9 8

T h e I n fl u en c e o f P l a s t i c - S t r a i n - l n d u c e d A n i s o t r o p y

M o d e l e d a s C o m b i n e d l s o t r o p i c - K i n e m a t i c H a r d e n i n g

i n t h e T e n s i o n - T o r s i o n S t r a i n i n g P r o b l e m ( [ )

- - R e p r e s e n t a t i o n o f t h e B a c k S t r e s s a n d A p p l i c a b i l i t y o f

t h e C o m b i n e d l s o t r o p i c - K i n e m a t i c H a r d e n i n g M o d e l -

Y e o n g S u n g S u h *

( R ece i ved A ugus t 2 , 1997)

I n o r d e r t o e x p r e ss th e p l a s t i c - s t r a i n - i n d u c e d a n i s o t r o p y a t f in i te d e f o r m a t i o n o f d u c t il e

m e t a ls , a c o m b i n e d i s o t r o p ic k i n e m a t i c h a r d e n i n g m o d e l w h i c h i s a p a r t i c u l a r f o r m o f

a n i s o t r o p i c h a r d e n i n g , i s a n a p p r o p r i a t e m o d e l b e c a u s e o f i ts s i m p l e a n d c o n v e n i e n t m a t h e m a t i -

c a l f o r m u l a t i o n . T h i s p a p e r e x a m i n e s t he a p p l i c a b i l i t y o f th e m o d e l i n th e c o m p u t a t i o n o fg e n e r a l s t r a i n i n g p r o b l e m s b y p e r f o r m i n g a n u m e r i c a l t e n s i o n t o r s i o n t e s t . T h e a n i s o t r o p y

g e n e r a t e d b y p l a s t i c f l o w i s e x p r e s s e d b y b a c k s t re s s . T h e e v o l u t i o n e q u a t i o n c o n t a i n s f o r m

i n v a r i a n t i s o t r o p i c f u n c t i o n s o f p l a s t i c s tr a i n r a t e a n d b a c k s t re s s a n d a l s o i n v o l v e s t h e s p i n

a s s o c i at e d w i th i n d u c e d a n i s o t r o p y . A n u m e r i c a l f i t ti n g p r o c e d u r e a l l o w e d u s t o s h o w t h a t

c i rc l e s m o d e l e d a s c o m b i n e d i s o t r o p i c - k i n e m a t i c h a r d e n i n g a r o u n d t h e lo a d i n g n o s e a r e i n g o o d

a g r e e m e n t w i t h t h e e x p e r i m e n t a l y i e l d lo c i t a k e n f r o m t h e n o n p r o p o r t i o n a l s t r a in i n g . T h e

m e a s u r e o f c h e c k i n g t h e a p p l i c a b i l i t y o f c o m b i n e d i s o t r o p i c - k i n e m a t i c h a r d e n i n g b y a n a l y z i n g

t h e t o t a l s t r e s s h i s t o r y h a s a l s o b e e n d e m o n s t r a t e d b y s i m u l a t i n g a n e x t r u s i o n p r o c e s s u s i n g t h e

f i n i t e - e le m e n t m e t h o d . F r o m t h e c o m p u t e d r e su l ts , t h e a n g le v a r i a t i o n s b e t w e e n t h e p r i n c i p a l

s t re s s d i r e c t i o n a n d t h e m a t e r i a l d i r e c t i o n , i n i t i a l l y a x i a l , w e r e o b s e r v e d i f t h e y a r e s m a l l e n o u g h

i n th e a c t i v e p l a s t i c d e f o r m a t i o n r e g i o n t o e n s u r e t h a t t h e s t r es s p o i n t w i l l m o v e a l o n g t h e p a r to f t h e y i e l d lo c u s e x h i b i t i n g n e a r l y u n i f o r m c u r v a t u r e . T h i s i n d i c a t e d t h a t s tr e s s a n d d e f o r m a -

t i o n c a n b e p r e d i c t e d w i t h c o m b i n e d i s o t r o p i c - k i n e m a t i c h a r d e n i n g a s l o n g a s t h e l o a d i n g i s n o t

r e v e r s e d .

K e y W o r d s : P l a s ti c S t ra i n I n d u c e d A n i s o t r o p y , C o m b i n e d l s o t r o p i c K i n e m a t i c H a r d e n i n g ,

Y i e l d L o c u s , T e n s i o n - F o r s i o n , B a c k S t r e s s , L e a s t S q u a r e F i t , E x t r u s i o n

1 . I n t r o d u c t i o n

E x t e n s i v e e x p e r i m e n t a l e v i d e n c e h a s i n d i c a t e d

t h a t m a t e r i a l s e x h i b i t s i g n i f i c a n t a n i s o t r o p i c h a r -

d e n i n g i n d u c e d b y p l a s t i c d e f o r m a t i o n . S u b s e -

q u e n t y i e l d su r f a c e s, w h i c h e x p a n d a n d t r a n s l a t e

i n s t r e s s s p a c e , a r e a l s o s u b j e c t t o s h a p e c h a n g e s .

T h e s e a r e m o r e e v i d e n t a t l a r g e d e f o r m a t i o n . I n

t h i s c o n t e x t , it is i m p o r t a n t t o d e v e l o p s a t i s f a c t o r y

* D e p a r t m e n t o f M e c h a n i c a l E n g in e e r i n g , H a n n a m

U n i v e r s i ty , I 33 O j u n g - d o n g , T a e d u k - g u , T a e j o n ,

Korea, 306-791

e l a s t i c - p l a s t i c c o n s t i t u t i v e r e l a t i o n s t o e x p r e s s t h e

i n f lu e n c e o f p l a s t i c - s t r a i n - i n d u c e d a n i s o t r o p y a t

f i n i t e d e f o r m a t i o n , s i n c e v i r t u a l l y a l l d u c t i l em e t a l s e x h i b i t t h i s c h a r a c t e r i s t i c .

T h e e l a s t i c - p l a s t i c c o n s t i t u ti v e r e l a t i o n s m u s t

i n c o r p o r a t e t h e i n f lu e n c e o f th e v a r i o u s c o m p o -

n e n t s o f p l as t ic s t r a i n - i n d u c e d a n i s o t r o p y , s u c h

a s t h e s e l l - e q u i l i b r a t i n g i n t e r n a l s t r e s s e s c a u s e d

b y d i s l o c a t i o n p i l e - u p a t i m p e r f e c t i o n s , o r i n c l u -

s i o n s i n c r y s t a l l i t e s o r a t g r a i n b o u n d a r i e s o f

p o l y c r y s t a l l i n e m e t a ls , o r w i t h r e s i d u a l - t y p e s t re s-

s e s g e n e r a t e d i n t h e c r y s t a l l i t e s o f a p o l y c r y s t a l -

l i n e m e t a l d u e t o m i s m a t c h p l a s t i c s tr a i n s i n t h e



The Influence of Plastic-Strain Induced Anisotrop) ... ( I ) ,573

variously oriented grains. These can be modeled

by the back stress, a second order tensor internal

variable which represents the kinematic shift of

the yield surface. Another component of the in-

duced anisotropy is associated with the differen-tial hardening of various slip systems and texture

evolution which are partly responsible for the

shape change of the yield surface. Since the

micromechanisms which generate them are em-

bedded in the material, the evolution laws for the

internal variables which are used to model such

mechanisms must reflect the influence of both

deformation and rotation. Several workers

presented different descriptions of distortional

hardening (Kurtyka and Zyczkowski, 1985,

1996 ; Mazilu and Meyers, 1985, Gupta and

Meyers, 1986, 1992 ; Helling and Miller, 1987 :

Voyiadjis and Foroozesh, 1990), but most of

them focus on the fitting of the distorted yietd loci

rather than providing an evolution model for

numerical prediction of material deformation. In

this paper, rather than taking account of the

distortion of yield loci, which is a complete

description of the geometry, only the expansion

and the translation of the yield surface were

considered based on combined isotropic

-kinematic hardening associated with von Mises

yield surface, which is much more practical and

easier for implementation in a numerical scheme.

Experiments carried out by Helling et al. (1986)

indicate that for smooth changes of the strain

vector, the subsequent yield surface maintains an

effectively circular nose geometry, which can be

analyt ically expressed as a combine d isotropic

kinematic hardening with yon Mises yield crite-

rion in the axial stress - generalized shear stress

space. In the major deformation regions of mostmetal-forming processes, drastic changes in the

ratios of the strain rate vector components are not

likely to occur. This suggests that in many metal

forming processes, the stress point remains in the

neighborhood of the nose of the yield surface.

This simplifies the selection of an accurate con-

stitutive relation such that stress analysis can be

associated with combined isotropic-kinematic

hardening, which can represent expansion and

translation of the varying yield surface.

Modeling strain- or deformation-induced

anisotropy has long been a subject of interest

amo ng researchers (Loret, 1983: Dafalias, 1983,

1985a, b : Zbib and Aifantis,.1987, 1988a, b ;

Paul un a nd Pecherski, 1985, 1987 ; Van DerGiessen et al, 1992, Kuroda, 1995) since Mandel

(1978) set up a framework for a relevant con-

stitutive theory. Most of the works have adopted

the concept of plastic spin appearing in Mandel's

framework to describe strain induced anisotropy.

In more recent work, Ning and Aifantis (1997)

described deformation induced anisotropy in

polycrystal line solids at both the grain an d a

aggregate levels. They have demonstrated defor-

mation induced anisotropy by adopting a scale

-inv aria nt approach and compared their theoreti-

cal predictions with combined tension-torsion

data. These works can be regarded to

phenomenological approaches. Among the con-

tinuum approaches, an elastic-plastic theory

which takes account of the spin associated with

plastic-strain induced anisotropy through a com-

bined isotropic-kinematic hardening model has

been proposed by Agah-Tehrani et al. (1987).

This theory, which is based on the multiplicative

decomposition of the deformation gradient into a

plastic followed by an elastic deformation, uti-

lizes the representation theorems tbr symmetric

and anti- symmet ric second order tensors to pres-

ent a general isotropic or form-i nvari ant function

in terms of certain basic functions. ]-'he resulting

isotropic function of the back stress expresses the

direct effect of the rate of strain and the influence

of the rotation associated with the straining of

lines of material elements which carry the embed-

ded back stress. Alth ough the import ance of all

the tensorial base functions utilized in the evolu-tion law for the back stress is as yet undetermined,

Lee and Agah Tehrani (1988) and Suh et al.

(1991) have demonstrated that incor porat ing the

base functions in a simplified form representative

of the influence of such a r otatio n term is of major

importance in the stress analysis of nonpropor-

tional, homogeneous and nonhomogeneous

straining problems. It was also shown in these

papers that inclusion of this rotation term

eliminated the oscillation of the stress-strain



574 Yeong Sung Suh

curve in shear at large strains, which was errone-

ously predicted during the simple shearing of a

rectangular block (Nagtegaal and de ,long, 1982).

In this paper, the basic theory of the evolution

model for plastic strain- induced anisotropy isbriefly demonstrated, and the representation of

the back stress in the deviatoric stress space is

presented. The applicability of the combined

isotropic-kinematic hardening is then examined

by fitting the yield loci obtained from the tens ion -

torsion experimental data provided from Cheng

and Krempl (1989). Final ly, an extrusion prob-

lem is simulated by the finite -elemen t method in

order to introduce a means of checking that the

loading stress remains in the neighborhood of the

nose of the yield surface during forming.

2 . S t r u c t u r e o f B a c k S t r e s s E v o l u t io n

I n v ol vi n g P l a s t i c - S t r a i n - I n d u c e d

A n i s o t r o p y

As plastic strain increases, the plastic-strain-

induced aniso tropy becomes more significant9

The micro mechanisms which generate the plastic-

strain induced anisotropy are embedded in the

material and are convected in both translation

and rota tion with the chang ing deformat ion (Lee,

1985). The Bauschinger effect is modeled by

intr oduc ing the back stress a, a second order

tensor i nternal variable. Physically the back stress

a is a residual stress field embedded in the poly-

crystalline material at the crystallite or crystal-

lattice level due to deformation of the agglom-

eration of the anisotropic crystallites and to dis-

location pile-ups in the crystallites. This back

stress a affects the magn it ud e of the super-

imposed applied stress needed to produce addi-tional plastic flow and thus produces the Baus-

chinger effect. A complete analysis of the rotation

and variati on of the back stress a calls for a

micromechanical analysis of the generation of

residual stress in the crystallites of the polycrystal-

line material as the differently oriented single-

crystal crystallite s deform heterogeneously 9 This

is further complicated by the localized residual

stresses associated with the blockage of the glide

of dislocations within the crystallites due to im-

purity inclusions, particularly in dispersion and

precipitation hardened alloys (Lee, 1985).

A general formulation of the evolution equa-

tions for back stress can be expressed in terms of

form-i nvari ant tensor functions, the constantsinvolved being determined by an experimental

program of measuring the stress response to pre-

scribed strain histories or vice versa. Such tests

would involve a generalizati on of both the Prager-

Ziegler strain rate term and the rotation influ-

ence. The generalized Prager-Ziegler law

proposed by Fardshisheh and Onat (1974), ex-

pressed as

d- -h (a , D p) § (1)

is equivalent to using the Jaumann derivative

~, h(cg, D r') (2)

where h can be represented in terms of certain

basic functions (Agah-Tehrani, 1987) such that

" p 2 l 2h ( c e , D p ) 7 ] I D P - - r ] 2 ~ P 0 t @ r ] 3 g ( a ~ t r ( a ) 1 )

p - - p 2+Z]a(czD +13 ~ 3- tr (a DP) l) +z/a (CD p

-p 2 2 , 2+D a 3t r( ~ DP) I)+ ~$/ oI- ~W (3)

where W is an antisymmetric tensor function of

/ 2 D p n p'P V : the back stress a, the rate of

plastic de form atio n D p, and the rate of effective

plastic strain. The relation between the rate of

strain and the rate of deformation is defined by

9 d c _ F Tthe form g = dt 9D 9 F, where g is the rate

of strain, and F the deformation gradient. The

back stress is a type of residual stress embedded in

the deforming material, and the spin term in Eq.

(1) expresses the contrib utio n of the consequent

rotation of the back stress to the material deriva-

tive with respect to fixed axes, in which W is the

spin, the antisymmetric part of the velocity gradi-

ent. The form of the isotropic tensor function, h,

is chosen in such a way that Eq. (3) wil l be rate

independe nt, and therefore h must be linear in D P.

with linea r terms in D ~' has the form

Vr ~bl(~,DP-Dr'a) @ ~ 2 ( g g 2 D , ~ ' - - D P a 2)

+ ~b:~ czDP6~ ae D P a) (4)

The coefficients ~b's and ~'s are functions of invar-



The ln j7uence of P las t ic -S zrain-l nduced Anisotropy . . . ( [ ) 575

iant of ce.

According to Agah-Tehrani el a / . (1987),

is expressed in the simple form

Vr (trN) (c~D~ DPa) [N (a D p- DP6~)

§ (o:D p - Dr'a) N] (5)

1 zwhere N ~bll- -(~ce +~ba[ 2tr(a )l-f f ' ]) . Equa-

tion (5) reflects the fact that Vr can be obta ined

from the basic antisymmetric te ns or( aD ~' -D Pa )

and an isotropic tensor function of a,. The first

five terms in Eq. (3) which do not con ta in Vr are

selected to incorporate the direct effect of the rate

of strain and the second the influence of the

rotation due to the straining of lines of material

elements which carry the embedded back stress.Such rotations can have a significant spin effect

indep enden tly of the spin W. Selection of the first

term in Eq. (3) to express the direct effect of the

rate of strain reduces (2) to

c2=c DP § ~/a -o age (6)

where c is the classical Prager Ziegler hardening

modulus. All the first five terms are directly

related with the loading strain, but it was assumed

that the first term is dom ina nt. Onat (1982) also

proposed a similar form for the same purpose andit plays a similar role as the Prager-Ziegler

kinematic hardening model. This is approximate-

ly verified with the experimental rneasurements as

shown in part II of this paper. Of course, the

second, third and remainder terms may also be

taken into account but this needs more careful

experimental verification. Moreover, from a

compu tati onal point of view, more terms will lead

to more complicated manipulations. Therefore a

simpler form will be preferred if the accuracy of

the constitutive equatio n is not si gnificantly de-

creased. Selecting the first term of Eq. (4) and

non- dimen sion aliz ing by the magnitude of the

back stress, 'W will have the form (Lee and Agah

-Tehrani, 1988)

~ [a D p DPa] (7)

& is a non dimen siona l material d epende nt

parameter which controls the contrib ution of the

spin due to induced anisotropy. (, will be nonzero

if a and D p have different principali direc tions.

Only nonnegati ve values of ,& must be: consi dered

so that the spin of the eigen triad of the embed-

ded anisotropy is not larger than the average spin

of the material particle. Introduction of tile aboveexpression for ~/ has removed the anomaly aris-

ing from the choice of terms added to preserve

objectivity in finite deformation analysis, such as

the oscillation of the shear stress with increasing

shear strain during simple shearing of a rectangu-

lar block (Suh et a l . , 1991).

3. Approach of Verifying the

Applicability of Combined Isotropic-

Kinematic Hardening

3 .1 C i r c l e - f i t o f t h e y i e l d s u r f a c e d a t a m e a -

s u r e d f r o m n o n - p r o p o r t i o n a l s t r a i n i n g

Helling et a l . (1986) observed that the devel-

oping yield surface inflates in the loading direc-

tion while a flattening of the yield surface is

taking place in the opposite direction. Figure I

Fig. 1 Measured yield loci of 70:30 brass after

shear prestress. (BI) : No prestress. (B2) :

v,'3r=167MPa~ 7~=0.024. (B3) : v;3r=

252MPa, ~=0~74. (B4) : ,/3r=3 78MPa,

~?; =0.32. v5• lO ~ strain off'set yield defini-

tlon. From Helling e/ aL (Helling et al.,

1986).



576 Yeong Sung Suh

from Helling et al. (1986) shows the evol ving

yield loci in the axial generalized shear stress

plane during monotoni c shearing of a thin tube in

torsion. The initial yield locus is well represented

by the von Mises circle. As the shear strainincreases, the protuberant nose, which can be

fitted with the circle, progresses forward ill the

direction of increasing shear strain with a flatten-

ing of the opposite side of the yield surface. A

similar result has been also observed by Cheng

and Krempl (1991) in their tensile strain ing of an

AI/ Mg alloy at roo m temperature. They used thin

walled tubular specimens made of an A1/Mg

alloy and conducted tests with a servohydraulic,

computer controlled, MTS axial torsion testing

machine. The stress response was obtained fornon proportional strain paths under simultane-

ous axial and torsional straining. Figure 2

includes the proposed strain path in which a

regu lar- poly gona l strain path is followed by a

square path. At certain predetermined points as

shown in Fig. 2, a series of strain increment

probes was introduced to determine the yield

locus and consequently ascertain the path depen-

dent anisotropy due to non-proportional loading.

The path between the probing points corresponds

to a sum of increments of strain several times the

yield point strain. Since one specimen is used to

complete the whole strain path, in order to obtain

a consistent evolution of stress response, a small

offset strain of 10-4 was used to determine the

yield locus so thai appreciable hardening shouldbe avoided during the yield stress probing.

Details of the testing equipment and procedures

have been described in Cheng and Krempl

(1991).

In this work, the evoluting yield surface was to

be fitted with a circle especially on the nose

geometry in order to examine whether the yon

Mises base combined isotropic-kinematic harden-

ing can be applied to predict non proportional

strai ning problems. The exper imental yield loci to

be fitted were provided by Cheng and Krempl(1991).

3.2 Me asu re of the applicabi l i ty through

FEM predict ion

In most metal-forming processes such as roll-

ing or drawing, commonly a dimension in a

narrow range of orientations, longitudinal or

lateral, is being stretched or compressed through-

out the plastic deformation and thus drastic

changes of strain vector are not likely to occur. Of

course, marked changes can be generated by the

onset of instabiliti es and localizat ion of the strain.

The validity of the application of combined

isotropic kinematic hardening requires that the

Fig. 2 Strain path for tension-tor sion test of a thin

-walled tube. After 1% of axial prestrain, a

combinat ion of axial(z) and generalized

shear(--~7~ ) strains are applied for polygonal

followed by a square path. From Cheng and

Krempl (Cheng and Krempl, 1991).

Fig. 3 Schematic of angle variation between the

principal st ress direction and the material

direction, initially axial, inside the die during

steady state axisymmetric extrusion. Upper

half was shown because of symmetry.



The Inf luence of Plast ic-S train-In duced Anisotrop) . . .. (1 ) 577

stress point traverse a restricted area of the yield

locus. This is achieved if the orientation of the

principal stress does not deviate widely from the

directions of the material elements in the body

which have carried those stress components. Thisprocedure is demonstrated by the finite element

analysis of axisymmetric extrusion of a rod

through a frictionless die. Lines directed axially

in the billet again become axial in the extrudate,

and their direction inside the die are given by the

orientation of the corresponding edges of the

sequentially updated Lagrangian finite elements,

see Fig. 3.

4 . R e s u l t s a n d D i s c u s s i o n

4.1 Circl e-fit to experime ntal yield data

The yield loci probed along the strain path

depicted in Fig. 2 were numer ica lly fitted by

circles. The measurement of yield surfaces in axial

-generalized shear stress space generate yield loci

which exhibit a rounded nose in the direction of

straining and are flattened in the rear as the

literature indicates. The compatibility of the cir-

cle, which corresponds to combined isotropic

kinematic hardening, was assessed by a circle

-fitting program which was developed to compute

the circle through the nose of the yield surface

using a least-squares fit. The program computes

the center and radius of a circle from the scattered

data. The sum of the errors corr espon ding to each

data set is

in m

EEl=El(x , a)2+(y~--b)2-RD21 (8)i- O i- 0

where (a, b) and RD are the center and radius of

the fitted circle, respectively. The integer m indi-

cates the selected number of points used. Now it

is necessary to minimize

rl/ m

H ~Ef 2=Z [(x ~-a )Z- +(y ~-b ) 2 RD2] 2i= 0 i= 0

(9)

The multivariable Newton-Rhap son method

(See, Press et al . 1986 : Yakowi tz and Szidarovs-

zky, 1986, for example) was used to optimize H.

This method requires the evaluation of the first

and second partial derivatives of ]7. The initial

Fig. 4 Experimental initial yield locus with a

numerically fitted circle. This corresponds

to the probing station 1 in Fig. 2.

Fig. 5 Experimental initial yield locus with a

numerically-fit ted circle at sta tion 2, 1%

tensile prestrain.

yield locus was fitted and is shown in Fig. 4,

which indicates that the locus can be accurately

described by the von Mises yield criterion. Figure5 shows the fitted circle for the yield locus for

prestraining of I% in the direction of pure tension

(station 2 in the prescribed strain path, Fig. 2).

The fitting was based on a ten component data set

over the nose geometry of the yieh:t locus involv-

ing a five data set on each side of the axis of the

prestr aining direction. Yield loci with circle-fit at

Stations 6, 10, 14, 18 are shown in Figs. 6--9. In

all cases except the rear part of yield locus (rear

in the sense of opposite the direction of strain-



57 8 Yeong Sung Suh

Fig. 6 Experimental init ial yield locus with a

numerica l ly-f i t ted c i rc le a t s ta t ion 6 .

Fig. 8 Exp erimen tal init ial yield locus with a

num erica lly-fi t ted circle at station 14.


num erically fi t ted circle at station I0.

i n g ) , f i t t e d c i r c l e s w h i c h c o r r e s p o n d t o c o m b i n e d

i s o t r o p i c k i n e m a t i c h a r d e n i n g s h o w q u i t e g o o d

a g r e e m e n t w i t h t h e g i v e n e x p e r i m e n t a l y i e l d l o c i .

T h i s i n d i c a te s t h a t i f t h e d i r e c t i o n o f t h e d e f o r m a -

t i on i s no t t o t a l l y r eve r sed and the l oad ing s t r e ssr e m a i n s o n t h e c i r c u l a r n o s e g e o m e t r y , t h e p r e d i c -

t i o n o f s t re s s r e s p o n s e f r o m a r b i t r a r y d e f o r m a -

t i ons wi l l be ve ry accura t e . I t i s no t ed t ha t t he

v a l i d i t y o f c o m b i n e d i s o t r o p i c k i n e m a t i c h a r d e n -

i n g is w e l l d e m o n s t r a t e d t h r o u g h t h i s w o r k .

Now tha t we have ve r i f i ed t ha t c i r c l e i s a

rea so nab le f i t t o t he von M ise s y i e ld su r face , i t

w o u l d b e v e r y i n f o r m a t i v e i f t h e c e n te r a n d r a d i u s

o f c i r c l e a re r ep re sen t ed a s a func t io n o f t he back

s t re ss so t ha t i t c an be de t e rm ined wi th expe r imen -


num erically fi t ted circle at station 18.

t a l measuremen t s . De f in ing t he back s t r e ss in

s t r e ss space can d i f f e r a ccord ing t o t he measure

adop te d . 1t i s impo r t an t t o r ecogn ize t ha t t he

cen t e r o f y i e ld locus i n t he ax i a l s t r e ss -gen e ra l -

i z ed shea r s t r e ss space canno t be i den t i f i ed d i r ec t -l y a s t he com pon en t s o f t he back s t r ess because o f

i ts d e v i a t o r i c n a t u r e . T h e c o r r e l a t i o n f o r m u l a e

expre ss ing t he von Mise s y i e ld c i r c l e i n t he ax i a l

s t r e ss gene ra l i z ed shea r space have been de r ived

in te rms o f t he s tr e ss and the back s t r ess a s t he

f o l l o w i n g . I n t h e t e n s i o n - t o r s i o n t es t o f a th i n

w a l l e d t u b e , t h e C a u c h y s t r e s s c o m p o n e n t s a r e



T h e I n f lu e n c e o f P l a s t i c - S t r a i n - I n d u c e d A n i s o t ro p y . . .( [ ) 5 7 9

n = 0 (10)

0

and the back stress components are

8 / = ~ O ~ 0 /~ r0 - ( a ~ z + a ~ )

since a is a deviatori c stress. Com bin ed isotro -

pic-kinematic har dening gives the yield condition

2 - 2( ~ ' - a ) : (o r' a ) ~ a = ( c ~ ' ~ j - a , j ) :

(o"lj -- a'Ej) (12)

which is a hypersphere. Here, 6 is the effective

stress. In ten sion -to rsio n testing, the yield condi-

tion is reduced to

(2 . -0 - - -0 -z~)2+( - l o -c~ , , )2+( 1 o-

a~+ a~) 2+2 (r- - a~) 2= 2- ~ 2 (13)

Rearranging, this can be expressed as a circle in

the axial stress-generalized shear stress (o'--,/~r)

space :

3 ,2+( ~ - - T a ~ ) ( ~ r - , / 3 a ~ ) 2

2 l ,)= 6 -- 3( ~a ~+ a~ ) (14)

Therefore, the center of the yield locus in terms of

the back stress components in the axial stress

-generalized shear stress (c s- fJ r) space is re-

3presented with (2-a~.z, ff3ch~). If the yie ld locus is

measured and fitted as a circle, the center is

deter mined and thus Cezz and az~ are obta ined .

Once these have been established, the basic infor-

mation is available to assess the terms needed in

the evolution equation for the back stress if, the

most general form of which is given by Eq. (3).

To do so, extensive and elaborated experiments

are required. In this work, a parametric evalua-

tion of Eq. (6), which is a simpler form obtai ned

by taking the first and the rotational terms from

Eq. (3), was carried out and presented separately

in part ll.

4 . 2 A p p l i c a b i l it y o f C o m b i n e d l s o t r o p i c -

K i n e m a t i c H a r d e n i n g i n a n E x t r u s io n

P r o c e s s

The finite element code used for the present

analysis is IFDEPSA (Incremental Finite Defor-matio n Elastic-Pl astic Stress Analyzer) (Sub e t

a l . , 1991). IFDEPSA uses the updated kagran

gian method for solving large deformatiion elastic-

plastic problems. Because of the axial symmetry,

only half of the workpiece was considered. This

symmetry condition requires that there will be

zero shear stress and zero normal velocity along

the axis. The driving force was applied by a rigid

frictionless piston with a prescribed velocity. The

billet was assumed to be snugly fitted into the die

such that there were no ini tial stresses. The calcu-

lations were carried out for a 25% reduction in the

area. The length of the die was 1.2ro, with ro the

initial radius of the billet. The die contour was

assumed to be a smooth curve represented by a

5th-order polynomial with zero slope and curva-

ture at bolh ends. Material for this computation

was 24S-T alu minu m, whose initial and satura-

tion yield stress are respectively 269 MPa and 517

MPa. Young's modulus was 68,950 MPa and

Poisson's ratio was 0.33. The mesh consisted of 58

elements along the axial directi on and I 1 elements

along the radial direction, with progressively

narrow er elements close to the outer surface of the

billet. The finite element mesh is shown in Fig.

10. For the analysis, a total of 580 incremental

steps were utilized such that 40 increments were

needed for each column of elements to advance 3

elernent sizes in the axial direction. A detailed

finite-element formulation for runn ing the pres-

ent analysis is described in Suh e t a l . (1991).

The stresses were obtained with combinedisotropic-ki nematic hardening (f l=0.5 ) with an

appropriale contribution of spin associated with

induced anisot ropy (~b --4.0) which was shown

to generate qu alitative ly close agreement with the

experimental stress pattern, in Suh e t a l . (1991).

/~ is a mate rial -dep ende nt hard enin g parameter

such that isotropic hardening corresponds to /~=

1, and kinematic hardening to ~=0. The compu-

tation was carried out on the centroid of each

rectang ular element. Fi gure I I shows the statisti-



580 Yeong Sung Suh

Fig. 10 Undeformed and deformed mesh configurations. The deformed mesh configuration was obtained

after the piston moved forward by 580 incremental steps with combined isotropic kinematic harden-

ing ( ~ 0 . 5 , ~= 4.0). Due to symmetry, only one half of the billet is illustrated.

5 . Conc l us i ons

Fig. 11 Statistical distr ibution of angle variation

computed under the scheme shown in Fig. 3.

cal distribution of the variation of the angle

between the principal stress and material direc-

tions inside the die, where most of the plastic

deformation is taking place. Out of 132 elements

inside the die, almost 60% of the angles are less

than 15 ~ and even the largest is less than 40 ~

Experimental yield surfaces show that, for such a

mild change of the stress tensor, the stress point

will move along the yield locus exhibiting nearlyuniform curvature. It is conjectured from computa-

tional results that even with larger still practical

reduction ratios, the loading stress point will

reside on the nose circle geometry of the yield

surface. This allows us to use the combined

isotropic-kinematic hardening model for conve-

nient computations in the stress analysis of extru-

sions of appropriate reduction ratios, or possibly

of various metal forming processes.

The applicability of combined isotropic

kinematic hardening theory was verified by

showing its conformity with experimental yield

loci, and an FEM calculation of an extrusion

process was carried out as a measure of checking

the applicability of the combined isotropic

-kinemati c hardening model to the general strain-

ing problem like extrusion process. The results

from this work are as follows :

(1) A computer program that fits a circle

through the nose of the yield locus in generalized

stress space has been developed to evaluate the

applicability of combined ha rdening to the experi-

mental yield surfaces obtained from the non-

proportionally prescribed strain path. The center

and radius of the circle, which correspond to the

kinematic and isotropic components of harden-

ing, were obtained by the least-squares fitting

method. It was found that noses of experimental

yield loci that have gone through a nonpropor-

tional straining path are well represented by afitted circle. This indicates that restricted traverses

of the stress point over the nose of the yield

surface, exhibiting nearly uniform curvature, sim-

plifiy the selection of the reliable constitutive

relation.

(2) The general isotropic-kinemat ic hardening

yield condition which expresses a hypersphere in

stress space was represented in two dimension

with center and radius in tension torsion space in

terms of the back stress. This will contr ibu te to



T h e I n fl u e n c e o f P l a s t i c - S t r a i n - I n d u c e d A n is o tr op 3 .... ( I ) 581

e s t a b l i sh i n g a n ew h a r d e n i n g m o d e l b a s e d o n

e x p e r i m e n t a l m e a s u r e m e n t s , s i n c e t h e e v o l u t i o n

e q u a t i o n o f b a c k s t r e s s c a n b e a s s e s s e d i n t h r e e

d i m e n s i o n s f r o m t h e e x p e r i m e n t a l y i e l d l o c i. T h e

m e a s u r e m e n t s e t u p h a s t o b e c a r e f u l l y d e s i g n e ds o th a t u n w a n t e d f a c t o r s s u c h a s m a t e r i a l i n s t a b i l -

i t y o r m e a s u r i n g e r r o r a r e a v o i d e d . T h i s w i l l b e

l e f t a s f u t u r e r e s e a r c h .

( 3) A m e a n s o f c h e c k i n g t h e a p p l i c a b i l i t y o f

c o m b i n e d i s o t r o p i c - k i n e m a t i c h a rd e n i n g in a n a l -

y z i n g t h e t o t a l s t re s s h i s to r y w a s d e m o n s t r a t e d b y

s i m u l a t i n g a n e x t r u s i o n p r o c e s s u s i n g l h e f i n i t e

- e l e m e n t m e t h o d . N u m e r i c a l e v a l u a t i o n s o f t h e

s t e a d y - s t a t e s tr e ss d i s t r i b u t i o n g e n e r a t e d d u r i n g

a x i s y m m e t r i c e x t r u s i o n r e v e a l e d th a t t h e v a r i a t i o n

o f t he a n g l e b e t w e e n t h e c o r r e s p o n d i n g p r i n c i p a l

s t r e s s a n d m a t e r i a l d i r e c t i o n s i n t h e p l a s t i c d e l k ~ r -

m a t i o n r e g i o n i s s m a l l e n o u g h t o e n s u r e th a t t h e

s tr e ss p o i n t w i l l r e m a i n i n th e n e i g h b o r h o o d o f

t h e n o s e o f th e y i e l d s u r f a c e , a n d t h a t t h e s t re s s

a n a l y s i s c a n t h u s b e b a s e d o n c o m b i n e d i s o t r o p i c

- k i n e m a t i c h a r d e n i n g .

T h e c o m b i n e d i s o t r o p i c - k i n e m a t i c h a r d e n i n g

m o d e l , h o w e v e r , h a s s o m e l i m i t a t i o n s i n s t r e s s

a n a l y s e s i n v o l v i n g i n s t a b i l i t y , l o c a l i z a t i o n o r

c y c l i c s t r a i n i n g w h e r e t h e c h a n g e o f t h e s t r es s

v e c t o r c o u l d b e d r a s t i c . T o a n a l y z e t h o s e p r o b -

l e m s , a h a r d e n i n g m o d e l a s s o c i a t e d w i t h s h a p e

c h a n g e s o f th e y i e l d s u r f a c e n e e d s t o b e d e v e l -

o p e d .

Ackn ow l e d g eme n t s

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D r . S. C h e n g o f C o n c u r r e n t T e c h n o l o g i e s C o r p o -

r a t i o n f o r p r o v i d i n g v a l u a b l e e x p e r i m e n t a l d a t a .T h e g e n e r o u s a d v i c e o f D r . E . H . L e e , E m e r i t u s

P r o f e s s o r a t b o t h R e n s s e l a e r P o l y t e c h n i c I n s t i t u t e

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