Endochronic Description of Plastic Anisotropy in Sheet Metal

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Please cite this paper as follows:

Han-Chin Wu and Hong-Ki Hong, Endochronic Description of Plastic Anisotropy in Sheet Metal,

International Journal of Solids and Structures, Vol.36, pp.2735-2756, 1999.

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\

PERGAMON International Journal of Solids and Structures 25 "0888# 1624Ð1645

Endochronic description of plastic anisotropy in sheet metal

Han!Chin Wu

a\

_{\ Hong!Ki Hong}

b

a_{Department of Civil and Environmental En`ineerin`\ The University of Iowa\ Iowa City\ Iowa\ U[S[A[} b

Department of Civil En`ineerin`\ National Taiwan University\ Taipei\ Taiwan\ R[O[C[ Received 13 June 0886 ^ in revised form 0 April 0887

Abstract

It is shown in this paper that an extended form of Hill|s quadratic yield criterion for anisotropic sheet metal can be derived from an endochronic theory of plasticity[ The extended form considers the combined isotropicÐkinematic hardening and the {anomalous behavior| observed in the anisotropic plastic behavior of sheet metals can be accounted for by the concept of kinematic hardening[

This form of anisotropic endochronic theory can accommodate the usual requirement of normality between the plastic strain rate and the yield function[ In addition\ the theory leads naturally to the expressions for back stresses[ This work provides an additional example to show that the form of the intrinsic time is directly related to the form of the yield function[

0[ Introduction

In the case of rolled sheet metal\ when principal axes of anisotropy are the axes of reference\ the yield function f proposed by Hill "0837# has been widely used[ The yield function f is

1f "G¦H#s1

x−1Hsxsy¦"F¦H#s1y¦1Ns1xy0 "0#

where "sx\ sy\ sxy# are the in!plane components of Cauchy stress^ and the out!of!plane components

are considered to be zero[ The coe.cients F\ G\ H and N specify the anisotropy of the metal sheet[ This quadratic form "and its variations such as the one for planar isotropy# has been repeatedly used in applications[ It is generally satisfactory for predicting the sheet metal behavior for R − 0\ where R is the plastic strain ratio of the transverse to the thickness strain and it speci_es anisotropy[ In the case of R ³ 0\ {anomalous behavior| has been observed by Pearce "0857# and Woodthorpe and Pearce "0869# for commercially pure aluminum sheet[ In this case\ the yield stress in the

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equibiaxial tension test is higher than that for the uniaxial tension\ and eqn "0# is known not to predict this behavior[

It was pointed out by Wu et al[ "0887# that the {anomalous behavior| was the consequence of neglecting the role played by kinematic hardening in most anisotropic theory of sheet metals[ Another cause leading to this doubtful result is by expressing the coe.cients of yield function\ i[e[ F\ G\ H and N in eqn "0#\ in terms of the strain!ratio R[ The latter is equivalent to expressing the yield function in terms of the ~ow rule[ This\ of course\ is not in accord with the conventional theory of plasticity[ Wu et al[ "0887# showed that by considering a combined isotropicÐkinematic hardening behavior\ an extended form of eqn "0#\ i[e[\

"G¦H#"sx−rx# 1 −1H"sx−rx#"sy−ry#¦"H¦F#"sy−ry# 1 ¦1N"sxy−rxy# 1_f1 "1#

is useful for predicting the sheet metal behavior for all values of R!ratio[ In "1#\ rx\ ryand rxyspecify

the center of the yield surface and therefore\ represent the kinematic hardening^ f represents the isotropic hardening[ Thus\ the more complex non!quadratic forms of yield function\ such as Hill "0868\ 0889#\ Gotoh "0866#\ Barlat and Richmond "0876# and Barlat and Lian "0878#\ are not needed[

In addition to yield function\ eqn "1#\ ~ow rule and hardening rules are required for plasticity[ These are separately proposed rules\ although some of the parameters may be inter!related[ In this paper\ the endochronic theory is used to derive all equations\ i[e[\ yield function\ ~ow rule and hardening rules\ by use of a uni_ed approach which is based on irreversible thermodynamics of internal state variables[

The endochronic theory of plasticity was initially proposed by Valanis "0860\ 0879#[ Further development of the theory in the case of initially isotropic materials was due to Wu and Yip "0879\ 0870#\ Wu and Yang "0872#\ Valanis and Lee "0873#\ Im and Atluri "0876#\ Wu et al[ "0884b# and others[ The case of deformation induced anisotropy was investigated by Wu and Yeh "0876#\ Wu and Lu "0884# and Wu et al[ "0884a#[ In these papers\ the distortion of the yield surface was considered together with the combined isotropicÐkinematic hardening[ It was shown in Wu et al[ "0884a# that the form of the yield function depends closely upon the expressions used to de_ne the intrinsic time\ which is a time!like parameter used to register the history of deformation in the endochronic theory[ This idea is further explored in the present paper[ Also\ in the works of Wu and Yeh "0876#\ Wu and Lu "0884# and Wu et al[ "0884a#\ the plastic strain rate is necessarily pointing along the radial direction\ emanating from the center of the yield surface\ and it is not normal to the yield surface after the yield surface has su}ered a distortion[ In this paper\ it is shown that it is possible to formulate an anisotropic endochronic theory that obeys the normality rule\ if the yield surface is expressed in a quadratic form given by "1#[ The present paper addresses the problem of initial anisotropy together with the deformation induced anisotropy[

1[ The endochronic constitutive framework

In the formulation by use of Helmholtz free energy c\ see Valanis "0864#\ the free energy is a

function of the current strain oijand n number of internal state variables q

r

ij\ where r 0\ 1\ [ [ [ \ n[ The internal state variables are phenomenological variables used to specify the current state of material internal structure and the free energy is

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c 0

1sr Ar

ijkm"oij−qrij#"okm−qrkm# "2#

where Ar

ijkmare constants[

For rolled metal sheet\ let x denote the rolling direction "RD#^ y the transverse direction "TD#^

and z the normal direction "ND#[ The nonzero strain components are ox\ oy\ oxyand oz[ No energy

is stored due to oz[ Thus\ "2# reduces to

c 0

1sr ðAr

"ox−qrx#1¦Br"oy−qry#1¦1Dr"oxy−qrxy#1¦1Cr"ox−qrx#"oy−qry#Ł "3#

where Ar

\ Br \ Cr

and Dr

are constants[ For a stable material\ any deformation will cause the free

energy density c to increase[ A consideration of uniaxial straining in the x!direction leads to Ar

× 9

and a consideration of uniaxial straining in the y!direction leads to Br

× 9[ Finally\ pure shear

leads to Dr

× 9[ The stress components are

sx 1c 1ox s r ðAr "ox−q r x#¦C r "oy−q r y#Ł "4a# sy 1c 1oy s r ðBr_"o y−qry#¦Cr"ox−qrx#Ł "4b# sxy 1c 1oxy s r ðDr_"o xy−qrxy#Ł "4c#

and the evolution equations for the internal variables are

1c 1qr ij ¦br ijkm dqr km dz 9 "r not summed# "5# where br

ijkmis the dissipation tensor representing the viscosity of the material[ The evolution of the

variables is with respect to a time!like parameter z which is often referred to as the intrinsic time or the endochronic time[ The intrinsic time is monotonically increasing and is de_ned in terms of the plastic strain[ The intrinsic time measure for sheet metals will be further discussed in a later

section[ In the rolled sheet metal\ the components of internal variables are "qr

x\ qry\ qrxy# and it is assumed that there are no coupling e}ects among these components\ so that the dissipation tensor has the following form

b ¼ r

&

br x 9 9 9 br y 9 9 9 br xy

'

"6#

Using "3# and "6#\ "5# reduces to dqr x dz ¦P r_qr x¦U r_qr y P r_o x¦U r_o y "7a#

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dqr y dz ¦Q r_qr y¦Vrqrx Qroy¦Vrox "7b# dqr xy dz ¦R r_qr xy R r_o xy "7c# where Pr_Ar_:br x\ Ur Cr:bxr\ Qr Br:bry\ Vr Cr:bry and Rr Dr:brxy "8#

Note that "7a# and "b# are coupled in qr

x and qry[ A standard procedure may be used to decouple

the equations[ The resulting equations are dq¹r x dz ¦l r 0q¹rx C rox¦D roy "09a# dq¹r y dz ¦l r 1q¹ry E rox¦F roy "09b# where lr 0 and l r

1 are eigenvalues of the matrix

$

Pr _Ur

Vr _Qr

%

[ Note that since U

r

9 and Vr 9\ there

are always two real eigenvalues[ q¹r

xand q¹ r y are related to q r xand q r

y through the eigenvectors of the

matrix by the following relations qr x U r "q¹r x¦q¹ r y# "00a# qr y"−Pr¦lr0#q¹rx¦"−Pr¦lr1#q¹ry "00b# and C r "−"Pr #1_¦lr 1P r_−Pr_Vr_¦lr 0V r #:Ur "lr 1−l r 0# D r_"−Pr_Ur_¦lr 1Ur−PrQr¦l0rQr#:Ur"lr1−lr0# E r"Ur_Pr_¦Ur_Vr #:Ur "lr 1−l r 0# F r ""Ur #1_¦Ur_Qr #:Ur "lr 1−lr0# "01#

Note that "09# are now decoupled in q¹r

x and q¹

r

y[ These equations may be integrated with results

g

M N doxy dz?

%

dz? "05c# where G"z# s r Gr_e−lr_z with G"9# s r Gr₀ _"06a# Gxy"z# s r Gr xye −lr xyz with G xy"9# s r Gr xy0 "06b# K 1₂"F¦G¦H# and M "H¦F#"G¦H#−H1 "06c# Constitutive equations "05# are suitable for use in sheet metals[ These are expressions for the stress

components in terms of the histories of total strain components ox\ oyand oxy[ In the equations\ F\

G\ H\ Gr_{\ G}r

M N dop x dz?

%

dz? "07c#

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The forms of expressions in the square brackets on the right hand side of "05# and "07# are assumed to be the same[ These forms are established based on plastic deformation which is further discussed in Appendix B[ The coe.cients F\ G\ H and N involved are those of yield function "1#[ The kernel

functions r"z# and rxy"z# can be determined from the knowledge of G"z# and Gxy"z# using the

method of Laplace transformation[ Denoting the Laplace transformation of a quantity by an overhead bar for simplicity\ i[e[\ s¹ L"s#\ etc[\ "05a\b\c# may be transformed into

s¹x K−2:1GÞð"H¦F#po¹x¦Hpo¹yŁ "08a#

s¹y K −2:1_G Þð"G¦H#po¹y¦Hpo¹xŁ "08b# s¹xy M K2:1_NGÞxypo¹xy "08c#

with the initial values ox"9# oy"9# oxy"9# 9[ The parameter of Laplace transformation is

denoted by p[ Similarly\ "07a\b\c# are transformed into

s¹x K −2:1 r¹ð"H¦F#po¹p x¦Hpo¹ p yŁ "19a# s¹y K−2:1r¹ð"G¦H#po¹py¦Hpo¹pxŁ "19b# s¹xy M K2:1_Nr¹xypo¹ p xy "19c# with o¹p x"9# o¹ p y"9# o¹ p

xy"9# 9[ The plastic strain components in "19# are now expressed by the

di}erence between the total strain and the elastic strain[ Since the expressions in the brackets of "05# and "07# have been obtained from the consideration of plastic deformation\ the elastic behavior derived from "05# is not exact and it may be obtained from "05# by setting z : 9[ A further approximation is made to retain only terms of elastic strain in the direction of the applied load so that for loading in the x!direction\ the plastic strains are

op x ox−o e x ox− sxK 2:1 H¦F and o p y oy−o e y¼ oy "11a#

and for loading in the y!direction\ the plastic strains are

op

x¼ ox and opy oy−

syK2:1

G¦H "11b#

The shear component is

op

xy oxy−

sxyK2:1N

M "11c#

Substitute "11a# into "19a#\ "11b# into "19b# and "11c# into "19c#[ The resulting equations and "08# are then combined\ respectively\ to yield the following expressions

r¹  GÞ

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r¹xy GÞxy

0−pGÞxy

"12b# Using "12# and after the inverse Laplace transformation\ it may be shown\ following Valanis "0879# and Wu and Yang "0872#\ that the kernel functions are

r"z# d"z#¦r0"z# "13a#

rxy"z# d"z#¦rxy0"z# "13b#

where d"z# is the Dirac delta function^ and r0"z# and rxy0"z# are given by

r0"z# s n−0 r0 Rr_e−arz "14a# rxy0"z# s n−0 r0 Rr xye−axyrz "14b# where Rr_{\ R}r

M N doxy dz?

%

dz? "15c#

Equations "15# are the endochronic constitution equations for sheet metals expressed in terms of the histories of plastic strain[ These equations are valid in the plastic region only[ In order that the strainÐhardening may be discussed\ the intrinsic time z is further scaled by introducing another intrinsic time z through the relation

dz dz

f"z# "16#

with initial conditions z 9 and dz:dz 0 at z 9[ In "16#\ f"z# is a scaling function and will be identi_ed later with isotropic hardening[ Thus\ f"z# may be referred to as the isotropic hardening function[ When z 9\ no plastic strain has as yet occurred and at this state z 9 and f"9# 0[

M N doxy dz?

%

dz? "17c#

eqns "15a# and "15b# may be further written as

$

"H¦F#do p x dz ¦H dop y dz

%

sx−rx K−2:1_f "18a#

$

"G¦H#do p y dz¦H dop x dz

%

sy−ry K−2:1_f "18b#

which may then be solved for

Mdo p x dz "G¦H#"sx−rx# K−2:1_f − H"sy−ry# K−2:1_f "29a# and Mdo p y dz "H¦F#"sy−ry# K−2:1_f − H"sx−ry# K−2:1_f "29b# Also\ "15c# is rewritten as M N dop xy dz sxy−rxy K−2:1_f "29c#

Therefore\ in an anisotropic sheet\ it takes a multiaxial stress state to produce a single plastic strain component[ Equations "29a\b\c# may be considered as the {~ow rule| using concept of the ~ow theory of plasticity[ Comparing "29# with "B1#\ it is seen that the plastic strain increments for the two cases are along the same direction\ i[e[\ the normality condition is satis_ed[ Finally\ the plastic incompressibility is assumed so that

dop

x¦dopy¦dopz9 "20#

2[ The de_nition of intrinsic time and the yield function

1 ¦1N K2

0

M N dop xy dz

dz1₉ _"22#

Thus\ either dz 9 and the quantity in the bracket " # 9\ or dz 9 and the bracket " # 9[ The case of dz 9 corresponds to the elastic behavior and the case of dz 9 corresponds to the plastic behavior[ In the latter case\ after simpli_cation\ one obtains

"G¦H#"sx−rx# 1 −1H"sx−rx#"sy−ry#¦"H¦F#"sy−ry# 1 ¦1N"sxy−rxy# 1_f1 "23# This is exactly the same as "1# and is an extension of the yield criterion for sheet metals proposed by Hill in 0837[ This equation also identi_es f as the isotropic hardening function\ because it

speci_es the size of the yield surface[ It also shows that rx\ ryand rxyspecify the center of the yield

surface and therefore\ represent the kinematic hardening[ These are also known as the components of the back stress[

The coe.cients F\ G\ H and N are determined from experiments[ Several tests have been used in the literature for this purpose[ They range from tests that determine yield stresses to those that determine the width to thickness plastic strain ratio R[ The latter tests do not have the same degree of accuracy\ however and they involve assumptions and other factors[ Thus\ the values of the coe.cients determined entirely from the yield!stress tests are di}erent than those determined from the plastic strain ratio tests or those determined from the mixture of the two types of tests[ This\ of course\ is not acceptable\ because F\ G\ H and N are material constants and for a given metal sheet\ their values are _xed[

Of the two types of tests\ the yield!stress tests are the simpler tests[ Although the yield has several de_nitions\ i[e[\ the proportional limit\ proof strain\ or backward extrapolation\ as long as the de_nition is chosen\ a rather well!de_ned yield stress can be determined from the experiments and the result is rather repeatable although subjected to some degree of expected experimental scatter[ On the other hand\ there are many factors of uncertainty associated with the plastic strain ratio test[ Experimentally\ the measured value in the thickness strain does not have the same degree of accuracy as in the width and longitudinal strains in a sheet metal due to its thinness[ Therefore\ an experimentally determined ratio between the width and thickness strain is not of high degree of accuracy[ Experiments show that the R!ratio varies with strain[ According to Mellor "0871# for titanium 004\ the R!ratio varies greatly with the increasing plastic strain if it is de_ned as the ratio of the width to thickness strain[ The ratio varies the most at the transition zone from the elastic to plastic strain[ But if the R!ratio is de_ned as the ratio of plastic strain increments\ then it almost

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remains constant over the whole plastic strain range tested\ up to a strain of 02)[ However\ the experimental results are not su.ciently accurate to allow computation of the ratio of plastic strain increments closer to the initial yielding zone[ The curve for R rises slightly with decreasing strain\ but an extrapolation of this curve to zero strain is not recommended[ Experiments were also conducted by Lin and Ding "0884# for as received and cold!rolled aluminum sheets using cruciform plate specimen[ A similar conclusion was also obtained regarding the R!ratio at small plastic strain[ The authors stated that R could not be reasonably determined when the plastic strain is in_nitesimal due to the quite severe scattering of the plastic strain increments[ Finally\ it should be pointed out that the method of determining the coe.cients F\ G\ H and N by use of the R!ratios does not conform to the traditional method of plasticity[ Traditionally\ the yield function is determined from the yield stresses and not from the ~ow rule[ Since R is determined by the ~ow rule\ the determination of the yield function in terms of R would have the same e}ect as in terms of the ~ow rule[ Due to the aforementioned reasons\ it is believed that the coe.cients of the yield function should be determined from tests that determine the yield!stresses and will be subsequently discussed[

These coe.cients may be determined at the condition of initial yielding[ Thus\ rx ry rxy 9

and f 0 and "23# reduces to "G¦H#s1 x−1Hsxsy¦"H¦F#s 1 y¦1Ns 1 xy0 "24#

Note that x denotes the rolling direction and y the transverse direction[ In a tension test along the

x!direction\ the stresses are sx 9\ sy sxy 9 and "24# reduces to

G¦H "sY

x#−1 "25a#

Similarly\ a tension test along the y!direction leads to

H¦F "sY

y#−1 "25b#

Yielding under equibiaxial tension occurs when sx sy sYB[ In this case\ "24# reduces to

G¦F "sY

B#−1 "25c#

Due to the usual assumption that hydrostatic stress does not a}ect yielding\ sY

B is also the

compressive yield stress perpendicular to the sheet[ The through!thickness compression test was carried out by Naruse et al[ "0881#[ Cylindrical specimens were prepared from discs of each sheet material glued together with an epoxy adhesive[ The specimens were tested in compression using Te~on sheet and graphite grease for lubrication between the specimen ends and platen of the test machine[ Factors of uncertainty arose from the epoxy adhesive\ the Te~on sheet and graphite grease[ Therefore\ the equibiaxial tension test is preferred over the aforementioned compression test[ Equations "25a\b\c# can be solved for G\ F and H to yield

1G 0 "sY x# 1− 0 "sY y# 1¦ 0 "sY B# 1 "26a# 1F 0 "sY y# 1− 0 "sY x# 1¦ 0 "sY B# 1 "26b#

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1H 0 "sY x#1 ¦ 0 "sY y#1 − 0 "sY B#1 "26c# Finally\ the coe.cient N may be determined from the tension test of a specimen cut at 34> angle

with the x!direction[ The tensile yield stress for this specimen is denoted by s34 and

sx sy sxy 0

1s34[ Using this condition\ it may be found from eqn "24# that

1N

0

s34 1

1

−1 −"sY B# −1 "26d#

Since s34 is easily determined experimentally\ N is thus determined from "26d#[ The shear yield

stress sY

xyis determined in pure shear with material element parallel to the orthotropic axes[ It may

be shown from "24# that

sY xy

0

z1N "27#

However\ in sheet metals\ pure shear is di.cult to realize experimentally[ Also\ the simple shear test is sometimes used in the literature to determine N[ The stress state of this test is never simple\ however and the test can at best be used as an approximation[ It has thus been shown that the coe.cients of the yield function can be determined by a set of tension tests[ Well!controlled tension tests are simple to perform[ The equibiaxial test can be carried out by use of cruciform specimens as in Makinde et al[ "0881# and Lin and Ding "0884#[

In summary\ using this formulation\ the initial material anisotropy is speci_ed by material constants "F\ G\ H\ N#[ The strain hardening is speci_ed by an isotropic hardening function f

and the kernel functions r"z# and rxy"z# which characterize kinematic hardening[ The kinematic

hardening describes the deformation induced anisotropy[ The constitutive equations are given in "15# with the back stress expressed by "17#[ From "15#\ the yield function and ~ow rule are derived and given in "23# and "29#\ respectively[ Finally\ the intrinsic time is de_ned by "21#[

3[ Material parameters of the endochronic theory

The constitutive equations of the theory are "15a\b\c# with the intrinsic time de_ned by "21#[ In these equations\ F\ G\ H\ N\ K and M are known as described in the previous section[ The

determination of parameters associated with kernel functions r0"z# and rxy0"z# need to be discussed[

In the _rst place\ the yield stresses will be identi_ed by setting z 9¦

in "15#[ In the case of uniaxial stress sxloaded from the initial state\ "sy−ry# is zero so that\ from "18b#\

"G¦H#do p y dz¦H dop x dz9 "28#

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dz1G¦H K2

$

"H¦F# dop y dz ¦H dop x dz

%

1 dz1 _"39# or

b

"H¦F#do p y dz ¦H dop x dz

b

K2:1 "G¦H#0:1 "30#

Therefore\ when z 9¦_{\ the isotropic hardening function f}

z9 0\ and by the use of "30# and

"25a#\ "15a# reduces to

sx sYx "31a#

Similarly\ in the case of uniaxial stress sy\ when z 9

¦

\ "15b# reduces to

sy s Y

y "31b#

and in the case of pure shear\ when z 9¦_{\ "15c# reduces to}

sxy sYxy "31c#

The parameters of the model can be determined by considering the plane strain extension

condition\ where dop

x 9 but dopy9 and dopxy9[ In this case\ "15a# reduces to

sx K −2:1

$

"H¦F#do p x dz

%

¦K −2:1

g

z 9 r0"z−z?#

$

"H¦F# dop x dz?

%

dz? "31#

The general form of kernel function r0"z# is given in "14#[ It has been found in previous applications

that only one exponential term will capture the essential feature of metal behavior[ The following form is\ therefore\ used in the subsequent discussion]

r0"z# R0e−az "32#

Using "21#\ the intrinsic time for the plane strain extension is

dz 2""H¦F#MK−2 #0:1 dop x "33# so that dop x dz 2

0

K2 "H¦F#M

1

0:1 "34# where the {¦| sign corresponds to loading and {−| corresponds to unloading[ If the isotropic strain hardening function is given by the linear form as

dz

dz f 0¦bz "35#

where b is a parameter that describes isotropic hardening\ then\ in the case of loading\ "31# reduces to the following expression by use of "32#\ "34# and "35#]

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sx

0

H¦F M

1

0:1

6

"0¦bz#¦ R0 "n¦0#b

$

"0¦bz#− 0 "0¦bz#n

%7

"36#

This equation describes the stressÐplastic strain curve[ In the equation\

n a:b "37#

The yield point sy

pl=o=x of this curve may be found by setting z : 9[ Thus\

6

"0¦mo p x#¦

0

s9 x sy pl=o=x −0

1$

"0¦mop x#− 0 "0¦mop x# n

%7

"43# where isotropic hardening sy pl=o=x"0¦mopx# kinematic hardening rx"s 9 x−s y pl=o=x#

$

"0¦mo p x#− 0 "0¦mop x# n

%

"44#

If an experimental stressÐplastic strain curve is available for the plane strain extension case\ then

sy

pl=o=x\ s9x and ET can be experimentally determined and m is known from "42#[ An optimization procedure may be used to determine parameter n by requiring the calculated curve to pass through

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certain experimental points on the graph[ The optimization procedure for the endochronic theory was discussed by Jao et al[ "0880#[

Plane strain extension is not easily accomplished experimentally[ Some published works are Wagoner and Wang "0868#\ Wagoner "0879# and Taha et al[ "0884#[ In Table 0 of Wagoner "0879#\ the saturation model gives the experimental e}ective stress vs e}ective strain relation[ This relation is transformed into an axial stress vs axial strain relation by use of the equations in Appendix C of that paper[ The experimental curve for extension along the rolling direction for 1925!T3 aluminum alloy is plotted in Fig[ 0[ In the data conversion\ the plastic anisotropy parameter r is taken as 9[6 according to Fig[ 2 of that paper[ Figure 0 shows the theoretical plane strain stressÐ strain curve by use of "43#[ The evolution of back stress is also shown and it has been calculated

by use of "44#[ The following parameters were used in the calculation] sy

pl=o=x159 MPa\

s9

x259 MPa\ E

T

537 MPa\ m 0[7 and n 01[

Most equibiaxial tests found in the literature are stress!controlled[ In that case\ sx sy s and

op x o

p

y[ On the other hand\ in a strain!controlled test\ it is possible to perform an experiment so

that op

x opy op\ but sx sy[ The latter test is suitable to the present theory[ However\ no

experimental result has been found in the literature[ In the strain!controlled equibiaxial extension test\ the intrinsic time is from "21# given by

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dz 2""3H¦F¦G#MK−2 #0:1

dop

x "45#

Using the kernel function "32#\ "15a# reduces to

sx syx=bi

6

"0¦m?op#¦

0

s9 x=bi sy x=bi −0

1$

"0¦m?op #− 0 "0¦m?op #n

%7

"46#

where the yield stress is

sy x=bi

1H¦F

M0:1_"3H¦F¦G#0:1 "47#

the stress intercept s9

x=biand the slope of the asymptote ETbiare related by

ET bi""3H¦F¦G#MK−2#0:1bs9x=bi "48# and furthermore\ bz m?op _{with m?}

0

ET bi s9 x=bi

1

"59# eqn "46# is the stressÐstrain relation in the x!direction for equibiaxial extension[ It is then easy to show from "15b# that

sy

0

1H¦G

1H¦F

1

sx "50#

and the yield stress in the y!direction is

sy y=bi

1H¦G

M0:1_"3H¦F¦G#0:1 "51#

Note that the expressions of "47# and "51# satisfy the yield criterion "24#[

4[ The plastic strain ratio

The plastic strain ratio R is determined by use of the ~ow rule[ The ratio Ra for a tension

specimen cut at an angle a with respect to the x!direction is de_ned by

Ra dop y? dop z? "52# where x? is along the longitudinal direction of the specimen^ y? is the transverse direction^ and z? z[ Note that x? is making an angle a with the x!direction[ By use of coordinate transformation\ "52# is transformed into

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Ra −

dop

xsin1a¦dopycos1a−1 dopxysin a cos a dop

x¦dopy

"53# The ~ow rule given by "29a\b\c# is now substituted into "53# to obtain

Ra −"ð"G¦H#sx−HsyŁ sin1a¦ð"H¦F#sy−HsxŁ cos1a

−1Nsxysin a cos a#:"ð"G¦H#sx−HsyŁ¦ð"H¦F#sy−HsxŁ# "54#

Note that the initial yielding is being considered so that rx ry rxy 9 and f 0[ For a tensile

specimen at an angle a to the rolling direction\

sx scos1a\ sy ssin1a and sxy ssin a cos a "55#

where s is the tensile stress applied to the specimen[ Then\ "54# reduces to

Ra −"ð"G¦H#s cos 1 a−Hs sin1 aŁ sin1 a¦ð"H¦F#s sin1 a−Hs cos1 aŁ cos1_a

−1Ns sin1_{a cos}1_{a#:"Gs cos}1_{a¦Fs sin}1_a# _"56#

Substituting a 9\ 89 and 34>\ respectively\ into "56#\ the following expressions are obtained

R9 H G "57a# R89 H F "57b# R34 0 1

0

1N G¦F−0

1

"57c#

For the as!received condition\ rx ry rxy 9 and the tensile yield stress saat any orientation

a may be derived based on the yield function given by "24#[ If the tension specimen is cut at an

angle a\ the stress components are given by

sx sacos

1

a\ sy sasin

1_a

and sxy sasin a cos a "58#

By the substitution of "58# and "26#\ the yield function "24# reduces then to

s1 a

$

0 "sY x#1 cos3_a¦

0

0 "sY xy#1 − 0 "sY x#1 − 0 "sY y#1 ¦ 0 "sY B#1

1

cos1 a sin1_a¦ 0 "sY y#1 sin3_a

%

0 "69#

To each stress state on the yield surface\ denoted by "sx\ sy\ sxy#\ there corresponds a uniaxial stress

state\ denoted by "sa\ a#\ also on the yield surface[ By _xing a\ the uniaxial yield stress is determined

from "69#[ Some special cases are

s9 sYx\ s89 sYy and s34 1

X

0 "sY xy# 1¦ 0 "sY B# 1 "60#

(19)

with respect to a[ Note that "60# are given in terms of the yield stresses and they can be shown to be the same as those of Hill "0889#[

5[ Conclusion

The quadratic anisotropic plane!stress yield criterion and its associated normality rule of the plastic strain rate have been derived based on an endochronic theory of plasticity[ The range of validity of the sheet metal plasticity with quadratic anisotropic yield function is greatly extended by incorporation of kinematic hardening into the model and it can account for the {anomalous behavior|[

It has been proposed that the coe.cients of the anisotropic quadratic yield function be deter! mined by the yield stresses using a set of tension tests[ In addition\ it has been shown that the expression of intrinsic time in the endochronic theory is closely related to the form of the yield function[

Appendix A

"G¦H#doy dz?¦H dox dz?

%

dz? "A3#

Conditions will no be established which will reduce "A0# and "A1# to "A2# and "A3#[ By considering special cases and equating the two sets of equations\ the following relations are obtained C r_Mr_¦Er_Nr_K−2:1_Gr "H¦F# "A4# D r_Mr_¦Fr_Nr_K−2:1_Gr_H "A5# C r_Kr_¦E rLr_K−2:1_Gr_H "A6#

(20)

D r_Kr_¦Fr_Lr_K−2:1_Gr

"G¦H# "A7#

A condition that makes "A5# equal to "A6# is D rMr_¦F

rNr_Cr_Kr_¦E rLr

"A8#

The coe.cient Gr_{of the kernel function G"z# are then de_ned from "A5# by}

Gr D r_Mr_¦Fr_Nr s r "D rMr_¦F rNr # with sr Gr 0 "A09#

Then\ from "A4#Ð"A6#\

F K2:1_s r ð"C r_−Dr #Mr ¦"E r_−Fr #Nr Ł G K2:1_s r ð"D r_−Cr_#Kr_¦"Fr_−Er_#Lr_Ł H K2:1_s r ðD r_Mr_¦Fr_Nr_Ł _"A00#

Similarly\ by comparing "03c# with "05c#\ it may be found that

Gxy"z# ND K2:1 s r Dr_e−Rr_z D 0 Dsr Dr e−Rr_z "A01# where N K 2:1 D with D sr Dr "A02# Appendix B

In the classical theory of plasticity\ the yield function is from "1#

1f "G¦H#s1

x−1Hsxsy¦"F¦H#s1y−1Ns1xy "B0#

Using the normality condition\ the ~ow rule is dop xdl 1f 1sx dlð"G¦H#sx−HsyŁ dop ydl 1f 1sy dlð−Hsx¦"H¦F#syŁ

(21)

dop xydl

1f 1sxy

dlNsxy "B1#

where dl is a parameter[ Equations "B1# may be solved for stress components to yield

sx "H¦F# dop x¦Hdo p y dlM sy Hop x¦"G¦H# dopy dlM sxy 0 dlNdo p xy "B2#

By the substitution of "B2# into "B0#\ one obtains "G¦H# "dlM#1 ð"H¦F# do p x¦HdopyŁ1− 1H "dlM#1ð"H¦F# do p x¦HdopyŁð"G¦H# dopy¦HdopxŁ ¦"H¦F# "dlM#1ð"G¦H# do p y¦Hdo p xŁ 1_¦ 1N "dlM#1

0

M Ndo p xy

1

1 1f "B3#

It appears natural then to de_ne an equivalent plastic strain increment by

do¹ 

6

"G¦H# K2 ð"H¦F# do p x¦HdopyŁ1− 1H K2 ð"H¦F# do p x¦HdopyŁð"G¦H# dopy¦HdopxŁ ¦"H¦F# K2 ð"G¦H# do p y¦HdopxŁ1¦ 1N K2

0

M Ndo p xy

1

7

0:1 "B4#

and to de_ne the equivalent stress seby

se

0

1f K

1

0:1

0

2 1

1

0:1

6

"G¦H#s1 x−1Hsxsy¦"F¦H#s1y¦1Ns1xy F¦G¦H

7

0:1 "B5# so that do¹ dl

0

M K

1

se "B6#

This de_nition of equivalent stress is the same as that of Hill "0838#[ For isotropic materials\ "B5# and "B4# reduce\ respectively\ upon setting N 2F 2G 2H\ to

se s 1 x−sxsy¦s 1 y¦2s 1 xy "B7# and do¹ 0 1"ð1 do p x¦do p yŁ 1 −ð1 dop x¦do p yŁð1 do p y¦do p xŁ¦ð1 do p y¦do p xŁ 1 ¦2"dop xy# 1_#0:1

(22)

z2 1 ""do

p

x#1¦"dopy#1¦dopxdopy¦"dopxy#1#0:1 "B8#

Acknowledgement

The research is supported by the U[S[ National Science Foundation "DMI 86!99092# and R[O[C[ National Science Council "NSC 74!1700!E991!911#[

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Endochronic Description of Plastic Anisotropy in Sheet Metal

Please cite this paper as follows:

Han-Chin Wu and Hong-Ki Hong, Endochronic Description of Plastic Anisotropy in Sheet Metal,

International Journal of Solids and Structures, Vol.36, pp.2735-2756, 1999.

\

Endochronic description of plastic anisotropy in sheet metal

Han!Chin Wu

\ Hong!Ki Hong

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