Associative versus non associative porou

Pergamon

International Journal of Plasticity,Vol. 12, No. 5, pp. 629-669, 1996
Copyright © 1996 ElsevierScienceLtd
Printed in the USA. All rights reserved
0749-6419/96 $15.00+ .00

Plh S0749-6419(96)00023-X

ASSOCIATIVE VERSUS NON-ASSOCIATIVE POROUS
VISCOPLASTICITY BASED ON INTERNAL STATE VARIABLE
CONCEPTS
E. B. M a t i n * a n d D. L. M c D o w e l l
George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia
30332-0405, U.S.A.

(Received infinal revisedform 10 January 1996)
Abstract--A general constitutive framework for porous viscoplasticity is used to study the role of
specific void growth models in both associative and non-associative viscoplastic flow rules. Three
particular model frameworks for porous viscoplasticity are identified, denoted as associative, nonassociative and partially coupled. The structure of a specific model framework is defined by the nature
of the inelastic flow rule (associative versus non-associative) and the specific dependence of the yield

function on the first overstress invariant (pressure). As distinct from the great majority of existing
models fi~r flow of porous viscoplastic media, this work considers the physically based models which
employ i~aternal state variables to represent evolving internal structure. Some applications are examined using Bammann's internal state variable viscoplastic model in the context of the three model
frameworks. Copyright © 1996 Elsevier Science Ltd

I. INTRODUCTION
Polycrystal]kine ductile m e t a l s subjected to large d e f o r m a t i o n u n d e r g o irreversible m i c r o s t r u c t u r a l c h a n g e s ( d a m a g e ) which d e g r a d e their strength a n d m e c h a n i c a l properties.
U n d e r creep c o n d i t i o n s (creep o r viscoplastic d a m a g e ) , voids o r cavities nucleate a n d
g r o w until they link o r coalesce, l e a d i n g to the fracture o f the specimen o r s t r u c t u r a l
c o m p o n e n t ( P u t t i c k [1959]; R o g e r s [1960]).
I n c o n t r a s t to classical inelasticity relations which are b a s e d o n the i n c o m p r e s s i b i l i t y o f
inelastic d e f o r m a t i o n , the overall structure o f the constitutive e q u a t i o n s for a p o r o u s
m a t e r i a l reflects v o l u m e t r i c inelastic d e f o r m a t i o n a n d a m a r k e d pressure dependence.
Typically, these theories m o d e l the p o r o u s m e d i u m as a m a c r o s c o p i c a l l y e q u i v a l e n t isotropic, h o m o g e n e o u s c o n t i n u u m with v o i d v o l u m e fraction as a scalar i n t e r n a l variable.
T h e implicit a s s u m p t i o n in this c o n t i n u u m t r e a t m e n t is t h a t the voids in the p o r o u s
m a t e r i a l are s p a t i a l l y r a n d o m l y d i s t r i b u t e d a n d exhibit r e a s o n a b l y spherical s h a p e with
r a n d o m size (isotropic d a m a g e ) .

*Presently a postdoctoral associate at Sibley School of Mechanical and Aerospace Engineering, Cornell University,
U.S.A.

629

630

E . B . Marin and D. L. McDowell

The formulation of these constitutive equations depends upon an expression for the
void growth rate, which is commonly derived from a micromechanical analysis of a
unit cell of the porous material. This approach considers the porous medium as an
aggregate of voids and rigid-viscoplastic incompressible matrix, with the unit cell
representing a characteristic element of this aggregate. Typically, spherical or cylindrical void geometries are used, and voids are assumed to grow isotropically under
applied loading.
In general, these micromechanical models have been used to either (i) find an explicit
expression for the expansion of the void as a function of the imposed stress and strain
fields (explicit void growth model), or (ii) construct a porosity- and pressure-dependent
loading or yield locus (yield criterion) that approximates the macroscopic response of the
unit cell under the applied loading. In the latter case, the void growth process is implicitly
defined by the yield function (implicit void growth model).
A number of explicit expressions for the evolution of damage in ductile metals have
been proposed in the literature (Cocks & Ashby [1980]; Budiansky et al. [1982]; Eftis &

Nemes [1991]; Lee [1991]). A common feature of these different models is that all of them
predict an evolution of porosity which depends on the current state of damage; the stress
triaxiality (ratio of the hydrostatic stress, crh, to von Mises equivalent stress, ~rm); a strain
rate sensitivity parameter, m; and the inelastic equivalent strain rate of the surrounding
matrix, ~P. Such models typically assume matrix strain-controlled cavity growth, neglecting diffusional growth behavior. Taking the void volume fraction ~9 as the internal variable to characterize isotropic damage, these evolution equations can be represented in a
general context as

~)=~(0,°'h,rn,{ p)

(1)

O"m

In the case of strain hardening behavior, the void growth rate may additionally depend on
the exponent on stress in the plasticity stress-plastic strain relation.
An example of such an expression is the explicit void growth model due to Cocks and
Ashby [1980]

[i

1
1_ )1/m

( l _ 0)] sinh (2(2 -- m) ah ] {p
\ 2-7-t7/ O'm/

(2)

which describes the enlargement of spherical grain boundary cavities in a power law viscous solid subjected to multiaxial stress states. Many of these explicit void growth models
have been included in the constitutive description of ductile metals using the mass conservation equation. This equation, together with the inelastic incompressibility of the
matrix, defines the volumetric component for the inelastic rate of deformation of the
porous material. Introduction of such explicit void growth laws, along with deviatoric
viscoplastic straining based on an incompressible flow potential, leads to a non-associative
structure of the inelastic flow rule. Applications of such constitutive equations include the
description of damage evolution in bulk forming processes (drawing and extrusion)
(Mathur & Dawson [1987]; Lee [1991]) and dynamic fracture (high strain rate applications) in ductile metals (Eftis et al. [1991]; Nemes & Eftis [1992]; Bammann et al. [1993]).

Internal state variable concepts

631

Implicit void growth models, on the other hand, couple damage with the mechanical
response of ductile metals by specifying a yield function which depends on pressure and
porosity. This yield function can be represented as
F = F(tr - A, R, tg)

(3)

where tr and A are the Cauchy stress and the backstress (kinematic hardening) tensors,
respectively, and R is an isotropic hardening parameter representing either the static or
the dynamic yield strength of the matrix material (Perzyna [1966]). For an initially
isotropic material, F depends on tr and A through the invariants of the overstress tensor
(o-A), i.e.
F =/7(I,, J~, J~, R, O)

(4)

where I1, J~, J~ are the first, second and third invariants of (tr-A) defined as
11 = tr(tr - A),

1
J~ = ~tr(s - a) 2,

1
J~ = ~tr(s - a) 3

(5)

Here, s = ~r - 1/3tr(tr)I and ot = A - 1/3tr(A)I are the deviatoric components of tr and A,
respectively. The symbol tr( ) denotes the trace of a tensor and I is the second rank identity tensor. The dependence of F on the third stress invariant, J~, is typically omitted in
specific theories.
The function F serves as a viscoplastic potential in an associative flow rule for inelasticity. In this case, F prescribes the magnitude and direction of inelastic straining under the
condition F > 0. This flow rule, along with conservation of mass, provides the expression
for the (implicit) void growth law. In some models (cf. Becker & Needleman [1986]),
F ~ Fa is taken as a "dynamic" yield condition; as such, F ~ Fd = 0 during plastic
flow and R - - , Ra is identified as the dynamic yield strength of the matrix
(R ~ Ra =/~a(~ p, ~P)). Here, ~P is the accumulated equivalent plastic strain in the matrix
and ~P its :rate. In other models (e.g. internal state variable viscoplasticity (Bammann
[1990])), F ,directly defines the Perzyna-type (Perzyna [1966]) viscoplastic flow rule with
F > 0. In this latter case, R is the quasi-static yield strength of the matrix, usually

expressed as R = k(~P), and the stress point typically lies outside the quasistatic yield
surface. In this approach there is a purely elastic domain, albeit small.
Two main functional forms of F have been used. The first one is represented by the
"elliptic" model
F = 3hlJ~ + h2121 - h]R 2

(6)

where hi =/~i(0,m), i = 1,2,3. Expressions for the coefficients hi have been obtained
using a general expression for the strain-rate or viscoplastic flow potential introduced
by Duva and Hutchinson [1984] in their analysis of a voided non-linear viscous material.
Their study used the potential function for an isolated spherical void in an infinite
medium of incompressible power law viscous matrix (Budiansky et al. [1982]). This
viscoplastic potential form has been used by many other researchers to study the ratedependent response of the unit cell model, where the matrix is modeled as a viscoplastic

632

E. B. Marin and D. L. McDowell

Table 1. Coefficientsof the elliptic model (porous viscoplasticity)

Viscoplastic potential • (Duva & Hutchinson [1984])
[3hl~ + h212,]((1/m)+l)/2

~o~
¢

-

1

L

-z:2 -2

-'-

J

where: hi =/~i(v~, m) for i = 1,2,3; ~o, reference strain rate; m, strain rate sensitivity parameter.

Cocks [1989]
1

1

0

h2-21+ml+tg,

hi = 1 + ~ 0 ,

h3 = (1 - O)U("+I)

Michel & Suquet [1992] and Duva & Crow [1992]
2

1 (mO_m(1-- Lg)) 2/(m+l)

hi = 1 + ~ 0 ,

h2=~\

1_,6 m

h3 = (1

--

~q)l/(l+m)

Sofronis & McMeeking [1992]
hi = (1

1 (rn~m(1 - ~ ) ) 2/(re+l)

+ zg) 2 / ( m + l ) ,

h2 = ~ \

1 - Om

h3 = (1 - O) 1~(re+l)

material. Based on analytical (bound estimates) (Cocks 1989]; Michel & Suquet [1992]) and
numerical procedures (Duva [1986]; Guennouni & Francois [1987]; Duva & Crow [1992];
Haghi & Anand [1992]; Sofronis & McMeeking [1992]; Zavaliangos & Anand [1993]),
specific expressions have been derived for the coefficients hi as explicit functions of the
void volume fraction and rate sensitivity exponent of the matrix material (see Table 1 for
examples). Self-consistent approaches have also been used to derive the form of the flow
potential for porous materials (Ponte Castaneda [1991]). Some applications using these
rate-dependent elliptic models with isotropic hardening have been reported in the literature (Eftis et al. [1991]; Nemes & Eftis [1992]; Zavaliangos & Anand [1993]).
Another class of functional forms for F, mostly used with power law inelasticity, is given
by an approximate yield criterion originally derived by Gurson [1977a, 1977b] in the context
of rate-independent plasticity and later modified by Tvergaard [1981]. This yield function
has been extended to solve rate-dependent problems (viscoplasticity), generally of the form
_ zql
,, ~gcost./q2II
Fd = ~3J~
- -tu~T-~.)~ -- 1 -- q3~2 = 0
"'d

---0

(7)

where F--~ Fd represents the dynamic yield condition and Ra = / ~ ( ~ P , ~P) is the dynamic
yield strength of the matrix material. The values ql = q2 : q3 = 1 correspond to the original

Internal state variable concepts

633

model of Gurson [1977a, 1977b], while the values ql = 1.5, q2 --- 1, q3 = ql 2 were introduced by Tvergaard [1981] to improve the prediction of shear localization. Other modifications to this model have also been proposed (cf. Perrin et al. [1990]). Numerous applications
using the Gurson-Tvergaard model have been reported in the literature (cf. Pan et al. [1983];
Becker & Needleman [1986]; Tvergaard & Needleman [1986]; Becker [1987]; Needleman &
Tvergaard []L987]; Becker et al. [1988]).
Recently, a general constitutive framework for compressible (porous) viscoplasticity
using an internal state variable formalism has been formulated which treats implicit void
growth models using an associative flow rule (McDowell et al. [1993]). In this paper,
this framework is extended to include explicit void growth models. Both associative and
non- associative viscoplastic flow rules are considered. Since the emphasis of the present
study is on the treatment of the void growth model in defining the structure (associative
versus non-associative) of the constitutive model, void nucleation effects are not considered. Such effects are known to be vitally important in some practical problems and
may be easily included (McDowell et al. [1993]).
A finite strain formulation of the constitutive equations is employed, with the current
configuration taken as the reference state for the next increment of deformation. Small
elastic strains are assumed. These equations are written in terms of strain rates, assuming
the additive decomposition of the rate of deformation tensor D into elastic, D e, and viscoplastic components, DP, i.e. D = D e + Dp. The elastic response of the porous material
is represented by an hypoelastic law under the assumption of small elastic strains. Also, as
is usually assumed, this framework models the porous medium as a homogeneous, isotropic continuum with its instantaneous response at a material element determined only
by the present values of a small set of macroscopic state variables in addition to the
Cauchy stress tensor, tr; these variables include the backstress tensor, A, the isotropic
hardening variable, R, and the scalar damage variable, ~9 (assuming isotropic damage).
The Jaumrnann stress rate is used as the (objective) co-rotational rate for the tensorial
variables cr and A; e.g. ~- = 6" - W-tr + tr-W, where W is the continuum spin tensor.
Specifically, the general internal state variable framework considered in this study
incorporates the following basic elements:
(i)

A viscoplastic flow rule potential depending on the first and second overstress
invariants (II, ,]2°) and damage (~).
(ii) CorrLbined nonlinear isotropic-kinematic hardening with evolution laws obeying a
hardening minus recovery format, including both dynamic and static thermal
recovery.
(iii) EvoJution of isotropic damage specified by either an explicit or an implicit void
growth law.
(iv) Elasto-viscoplastic damage coupling based on equivalence of both the elastic
strain energy density and inelastic work rate of the matrix and porous materials.
The focus is on isothermal applications. Three model frameworks for porous materials will
be identified. Single element computations are performed using these model frameworks to
compare the predictive capability of several explicit and implicit void growth models. Then,
the constitutive frameworks are used to numerically study the necking process of an axisymmetric and a plane strain specimen under tension. Bammann's state variable viscoplastic
model (Bammann [1990]; Bammann et al. [1993]), considered as representative of rate- and
history-dependent internal state variable theories, is used in these applications.

634

E.B. Marin and D. L. McDowell
II. A CLASS OF POROUS INELASTIC INTERNAL STATE VARIABLE MODELS

In a general framework, a class o f viscoplastic constitutive models for a p o r o u s medium
using internal state variables can be characterized by the following constitutive equations
(McDowell e t al. [1993]):
F = P(tr - A , R , 0 )
D p = IlDP[lnp = g((F))np
tr = C ( 0 ) : ( D - I l D P l l n p ) -

~ - - ~ tr

X = 1/3 C[(1 - 0)bna - A]~ p - f/A(A)A

VL

k :

s -- R) P -- a R ( R )

~) : (1 - 0)tr(np)l[DP]]

The viscosity function g o f the p o r o u s material, which is assumed to be a h o m o g e n e o u s
function o f the (static) yield condition, F (cf. Fig. 1), defines the viscoplastic flow rule

0 = 0.15

%/v.

O =,~u . I*un

Jl

,d
1

~I/Yo - o

"° 1

-a21Y *

O = 0.05

°2/Yo

*z ¢0

- e 2 Va

Yc

- • |ly?

d1

~)mO

*l

-o

'a

J1

YO

Fig. 1. Shape of the yield surface in principal stress space for compressible (0=0.15, 0.10, 0.05) and incompressible (0=0) materials (Doraivelu et al. [1984]). In the present notation, J] = Ii, Yo = Ro (Ro, initial yield
strength of matrix material).

Internal state variable concepts

635

under the condition F > 0. Recall that viscoplasticity (a rate-dependent theory) admits
stress states within, on and outside the (static) yield surface. Plasticity (a rate-independent
theory), on the other hand, admits states only within and on the (static) yield surface.
Neglecting tlhe third overstress invariant J3 °, for an initially isotropic porous material, the
general dependence of F on overstress tr - A can be equivalently expressed in terms of the
overstress invariants /land J2 °. For purposes of this development, the inclusion or
exclusion of It (i.e. a pressure-dependent or pressure-independent flow potential) will define
a damage-coupled F = F(I1, J2 °, R, zg)) or partially damage-coupled ( F = F(J2 °, (1-zg)R))
inelastic model, respectively. In this context, a typical pressure-dependent yield function is
given by the elliptic form

F=~/3hlJ~+h2I~ - h3R

(9)

where R is the quasistatic yield strength of the matrix material, and hi =/~i(0, m), i = 1,2,3.
The hi may be based on either empirical relations or micromechanics with suitable
homogenization concepts (cf. Table 1). Note that this form assumes a region of purely
elastic response defined by F < 0. The particular case R = 0 corresponds to a theory
without a yield surface.
During inelastic deformation, the direction of evolution of inelastic straining is prescribed by the unit vector, np. In general, np is not parallel to the outward unit vector
normal to the flow potential surface, n~, defined by

OI

~a~,j2°,R,o)

/

/
j

g

\

DP

"

°:\..

~lJm~ ~

n

~

no

1~0''y/2~05

nj

OPlaa
"~.

p

-

IDOl

laFlaol

Dl,
nt'

IDPI

Fig. 2. Schematic showing the direction o f evolution of DP in stress space. Associative and non+associative flow
rules are defined by lip = iI~ and np :~ a,,, respectively.

636

E.B. Marin and D. L. McDowell
OF/Oct

n~ --ilor/Ocrll,

OF
O~

OF ( s - ~x)

OF I

(10)

- OI, + - ~ 2

In these general terms, the model can represent either an associative (np= ha) or a nonassociative (np ¢ no.) flow rule, depending on how np is specified (see Fig. 2). Note that we
do not adopt a priori the postulate of generalized normality in terms of evolution of
internal variables in addition to D p. It will be shown below that np is defined by the
specific void growth model used in eqns (8). In state variable viscoplasticity, I[DPI[ is
commonly specified by the kinetic equation

IID II--g

(ll)

where F > 0 for viscoplastic deformation, and V is the drag (friction) strength of the
matrix material which may be specified as temperature dependent or proportional to the
static matrix yield strength R; i.e. V = AoR, where Ao is a constant (Nouailhas [1987];
McDowell [1992]).
Assuming small elastic stretch (adequate for metals), the elastic response of the porous
material is modelled by an hypoelastic relationship, eqn (8)3. In particular, for isotropic
elasticity, the damaged-coupled elastic stiffness tensor C(0) is
C(O) = 2#(0)~ + A(O)I @ I

(12)

where #(0), A(0) are the damaged-coupled Lam6 elastic constants of the porous material,
is the fourth-rank identity tensor and the symbol ® denotes the tensor product. These
Lam6 constants are obtained in terms of those of the matrix by equating the elastic strain
energy density of the matrix and porous materials (Lee [1988]) using the concept of effective stress (Kachanov [1986]). This yields #(tg) = (1-0)/z m, A(0) = (1-0)A m, where superscripted "m" quantities are associated with the matrix. The rate of damage term in eqn
(8)3 accounts for the dependence of the elastic constants on damage in the hyperelastic
form of this equation (Ortiz & Simo [1986]). Alternatively, a hyperelastic stress-strain
relation may be introduced with respect to intermediate configuration (Lubarda & Shih
[1994]) to account for the effect of plastic rotation on the elastic rate of deformation.
The hardening rules for the backstress of the aggregate A, eqn (8)4, and the quasi-static
yield strength of the matrix R, eqn (8)5, follow a hardening minus recovery format, which
has proven very successful to model complex loading histories (Bammann [1990]; McDowell
[1985]; Chaboche [1989]; Moosbrugger & McDowell [1989]). The static thermal recovery
functions f~A and f~R introduce rate dependence in the evolution of A and R. In these
equations, C, b, #g and R s are material constants; C and #R are rate constants, and b and R s
represent saturation levels of the matrix backstress and matrix yield strength, respectively.
The directional index nA gives the direction of the kinematic translation of the yield surface
in stress space. Two expressions have been suggested for nA (McDowell et al. [1993]; Matin
[1993]). The first one is obtained by assuming an equivalence of material response (evolution
of porosity) for both pure isotropic and combined isotropic-kinematic hardening under
proportional loading (Becker & Needleman [1986]; McDowell et al. [1993]), i.e.
nA = ns +

V•
i~

3X/~

1I

(13)

Internal state variable concepts

637

where ns = (s-a)/I Is-~l I, Note that nA is not a unit vector; its definition essentially prescribes the e,volution of the volumetric component of A. On the other hand, the second
expression for nA prescribes a zero volumetric component of A for incompressible plasticity (0 = 0) and is obtained by a simple modification of eqn (13) (Matin [1993]), i.e.
3V/~2ns + V/~_~ ~

lira V~hh~2i

V ~ ~--*1

nA =

[3J~..i-3-12h2lim,~2]1/2

(14)

- - 2 1 3ht ~--,l

This definition assumes that lim(hl/h2) exits as 0 --. 1. Note here that i1A is a unit vector
and hE~hi -~ 0 as 0 --+0 (see Table 1), resulting in liA = lis- In this case, the equivalence of
material response for isotropic and isotropic-kinematic hardening theories will only be
realized in 1Lhelimit as ~ ~ 1. In eqn (14), the deviatotic component of the rate of A
decreases as porosity develops.
The matrix equivalent inelastic strain rate, ~P, is obtained by assuming that the inelastic work rate per unit volume of the matrix, (1-O)R~ p, and aggregate, (o--A):Dp, are
equal (since voids do not contribute). This assumption leads to
~p

=

(0"

--

A) : D p

(1 -- 0)Rd

(0" - A) : np

- ( ] - O))J~-d IIDPll -- X'IIDPll

(15)

where Rd denotes the dynamic uniaxial yield strength of the matrix and is obtained by
inverting the kinetic equation, eqn (11). In particular, for an associative flow rule, i.e. lip
= no, and an elliptic viscoplastic potential, eqn (9), ~P can be expressed as
3hi 1 ---0

~

l1

- ~ - l J (tr(Dp))2

(16)

The evolution equation for damage (voids), eqn (8)6, implicitly specifies a void growth
law. This expression is obtained from conservation of mass, assuming a plastically
incompressible matrix and the same elastic compressibility of both matrix and porous
aggregate; in terms of final results for the void growth rule, this is equivalent to neglecting
the elastic compressibility of both materials (Gurson [1977b]; Haghi & Anand [1992]). If
this void growth law is written explicitly, it typically depends on the current void volume
fraction, ~9, the triaxial state of stress (defined minimally by the overstress invariants/1
and J2°), the strain rate sensitivity exponent, m, and the magnitude of the deviatoric
plastic rate of deformation of the porous material, IIDd p II, i.e.
---0(0, II, V/~, m, IIDPlI)

(17)

As mentioned before, the unit director np is defined by the specific void growth model
considered :in the constitutive framework, eqns (8). To show this relation, we may define
the void growth factor/3 as
/3= 1 tr(D p)
(18)
and express the void growth model equivalently as

638

E.B. Marin and D. L. McDowell

(19)

0 = v ~ ( 1 - 0)/~I[DPll

In present models, typically/3 = 8(0, Ii, (J2°) 1/2, m). If/3 is specified, the void growth law
will then be explicitly known (explicit void growth law). The plastic rate of deformation
O p is decomposed according to
D p = D p + Dvp

(20)

where DdP and Dvp are the deviatoric and volumetric components of D p, respectively. It
can easily be shown that these components are expressed as
D p = ,~p]lDPllns,

D p =/3,~p[lDplle

(21)

where ns -- DdP/I]DaPII = (np-1/3tr(np)I)/{p, e = I/(3) 1/2 and {p = (1-1/3(tr(np))2)l/2.
Substituting expressions (21) into eqn (20), we obtain
Dp =

,~pllOPll(ns+/3e)

= ~p~¢/1 +/32i[OP[in p

(22)

where
/3

np - ~ e

1

q ~ n

s

(23)

Equation (23) shows that the direction of evolution of O p is defined by/3, i.e. by the specific void growth law. Also note that ~p(1 +/32)1/2 _- 1. Equation (23) will be used in the
next section to discuss associative versus non-associative inelastic flow rules in the context
of porous inelasticity.
Using the previous relations, the porous inelastic model given by eqns (8) can be written
in the following alternative form, which reveals the effect of plastic compressibility
(growth of voids) through the void growth factor,/3, i.e.
r = F(/1, J~, R, 0)
Dp - ~
D~-

1

IlDPl[ns

/3
1C?%~
Im~lle

D~ = Da~ + D~ = lID~llnp

ov r = C ( 0 ) : D

1
,~ItDPlIC_('0):

---- V/~ c[(1 - 0)bnA -

A]xdiDPll - aA(A)A

k = ( ~ #R(RS -- R)xdiDP[[ - aR(R)

= v6(1 - o) ~

(ns+/3e)

IIoPII

X/~/3 iiDPllo.

(24)1_ 8

Internal state variableconcepts

639

where
X,[(s - a):ns +/3/vr3tr(cr - A)]

V/1 + f12(l - O)Rd

(25)

Note that for/3 = 0(t9 = 0, F = F(J2 °, R)), this constitutive model reduces to deviatoric
(incompressible) viscoplasticity.
III. MODEL FRAMEWORKS FOR POROUS INELASTICITY

The particular structure of the inelastic model represented by eqns (8) or (24) will
depend on both the direction of evolution of D p, specified by the unit vector np, and the
specific dependence of the function F on the overstress invariants. Accordingly, we can
identify three frameworks for the inelastic analysis of finite deformation problems of
porous materials, which are characterized by
(i)

an associative flow rule with pressure-dependence (fully damage-coupled analysis), i.e.
np = no,

F = F(o" - A, R, 19) = F(Ii, ~ , R, 0)

(26)

(ii) a non-associative flow rule with pressure-dependent deviatoric flow (fully damagecoupled analysis), i.e.
np # no,

r = F(o" - A, R, 0) =

P(Ii~~, R, O)

(27)

(iii) a non-associative flow rule with pressure-independent deviatoric flow (partially
damage-coupled analysis), i.e.
np # ns,

F=P(s-ot,

R,O)=P(~,(1-O)R)

(28)

As was mentioned before, the unit vector np is determined by the particular void growth
law through the void growth factor, /3. Therefore, the non-associative structure of the
inelastic flow rule in cases (ii) and (iii) is due to void growth (compressibility) effects.
Model frameworks of these kinds have already appeared in the literature (Tvergaard
[1981]; Perzyna [1986]; Bammann et al. [1993]). However, with the exception of case (i)
(Becker & Needleman [1986]), the implications of applying cases (ii) and (iii) to specific
problems such as localization have not been yet treated. This issue is undertaken in the
application examples presented in this paper. In the following we will discuss how the void
growth factor/3 is specified for the three different cases.
III. 1 Associative porous inelasticity with F = l~(Ii, J2°, R, 0)

This model framework is defined by the constitutive eqns (8) with the conditions (26).
In this model, the void growth law and, hence, the void growth factor, /3, is implicitly
defined by the pressure-dependent function, F. To show this, the explicit expression for
this implicff void growth law, eqn (8)6, will be obtained in terms of F using the associative
flow rule, np = no, where np is defined by eqn (23) and n~ by eqn (10), i.e.

640

E.B. Marinand D. L. McDowell

OF/Off

v ~ OF

no = [ l O F / O c r l l -

0I,

6

e +

~

OF

- -

--

60J~

ns

(29)

where 6 = ]]OF/Ocrl]. Equating the volumetric and deviatoric components of np in eqn (23)
with n~ in eqn (29) yields the void growth factor

V~ OF/Oil
fl - ff,~2 OF/OJ~

(30)

With 13 defined by eqn (30), the explicit expression for the void growth law, eqn
given by
_

3(1 -

0)

OF/OIl Dp
OF/O~
d

I

(8)6 ,

is

(31)

Equation (31) clearly shows that the void evolution is determined by the pressure-dependent
function, F. Therefore, an analysis using this model framework will not require the explicit
form (31) for the void evolution law since it is directly embedded in the formulation of the
constitutive equations through associativity of the flow rule.
Based on eqn (31), one can derive the explicit form of the void growth equation prescribed by typical functional forms for F. To facilitate this, eqn (31) is further reduced
using the following general form for F

F(II,,F2, R , O) = ~(I1,J~2, R, "0) - q~(R, ~)
09 = v/3hlJ~ + f~(Ii, R, 0),

qd -- h3R

(32)

where f~ = (~(I1, ~) = h2I~ for the elliptic model (see Table 1 for expressions of the
coefficients hi = hi(O,m), i = 1,2,3 for specific models), and f~ = ~(Ii,tg, R ) = 20qlR 2
cosh(q211/2R), hi = 1, h3 = (1 + (q10)2) if2 for the Gurson-Tvergaard model (Gurson
[1977a, 1977b]; Tvergaard [1981]). Using this form for F, one can write the void growth
law, eqn (31), as
0=

1 -0

hi

1

0~

2V/~220II []D~]]

(33)

Using the expressions of f~ for these two typical functional forms of F, eqn (33) leads to
the following results.
(i) Elliptic form:
-- O)h2 I1
z9 = v/6 0 hi
~

and

p

IIDdII

(34)

Internal state variable concepts

641

(ii) Gurson-Tvergaard form:
= V~2
3- 0 ( 1 -

R
q211
O)qlq2~sinh(-~llDPdl[
x/3
/

(35)

For this moclel, 3 = ~(0, I1, (J2°) 1/2, R).
On the other hand, eqn (31) can also be used, in principle, to obtain the function f~
which defines F for an associative flow rule once an explicit void growth law is known. As
an illustration of this procedure, consider the Cocks and Ashby explicit void growth
model (Cocks & Ashby [1980]), given by

/2,2m) i1)f

'

0---- V3 s l n h ~ ( 2 ~ m ) ~

,36)

(1--0) l/m

This model describes the enlargement of a spherical grain boundary void in a strain rate
hardening raatrix. To simplify the problem, assume small I1/(3J2°)1/2 (sinh(x) ,~ x for
small x). Then, equating eqns (33) and (36),
~)=(~(Ij ' 0 ) - 8

2+m

1 _ ~)l/m+l

1 hlll 2

(37)

This particular function fits the form of the elliptic potential with

hz=~z(O,m)=~r-~2-m[
2~m

1

(l-~l/m+l

]

1 h,

(38)

Additional model considerations may be necessary to specify expressions for the
coefficients hi and h3.
111.2. Non-associativeporous inelasticity with F = 1~(I1,J2°, R, 0)
The constitutive equations for this model framework are again given by eqns (8), but
with conditions (27) instead. The condition of non-associativity, i.e. np ¢ n,, is imposed in
the structure of the model by specifying a void growth law (or void growth factor) that is
not fixed by the pressure-dependent function F. This means that this model framework
may combine a pressure-dependent flow potential of the elliptic or Gurson type; for
example, with an explicit void growth law, eqn (17).
A number of explicit void growth models have been prosed in the literature (Cocks &
Ashby [1980]; Budiansky et al. [1982]; Eftis & Nemes [1991], Lee [1991]). For the Cocks
and Ashby [1980] model, for example, the void growth factor is given by
3 = 3"-slnn ~3~(2+ m) ~

1 _ O),/m+l

1

(39)

642

E . B . Marin and D. L. McDowell

Other explicit void growth equations can be derived from a pressure-dependent form of
F using the procedure presented in the previous section.
(i) Elliptic form:

/3=v/~ h2 11
h~3v/~2

(40)

(ii) Gurson-Tvergaard form:
1

R

• . ['q2II'~

slnn/--/
/3 = - ~ O q l q 2 ~
V/3~
k, 2R ]

(41)

When using these expressions for/3, a non-associative flow rule is obtained by selecting a
pressure-dependent function F which is different from the one used to derive/3.
It is important to note that the term (3J2°) 1/2, i.e. the von Mises (deviatoric) equivalent
stress, always appears in the denominator of the expression for/3. This fact rules out the
application of this model framework to the case of pure hydrostatic loading, where ,12° = O.
A model that avoids this singular behavior has been developed by Cocks [1989], where/3
is given by
1
1
~
11
/3-- v,~ m + 1 l+vqg9

(42)

with • defined by eqn (32)2. In general, the specification of the explicit void growth relation need not conform to either of the forms for F cited here.
111.3. Non-associative porous inelasticity with F -- F(J2 °, R, 0)
The constitutive equations of this model correspond to those of incompressible inelasticity
(eqns (24) with/3 = 0 and deviatoric backstress tx) with modifications to account for the effect
of damage (void growth) on the yield strength. In this specific model, damage is introduced in
such a way that degrades the elasticity of the material, C(0), and reduces its yield strength, R,
thereby enhancing the deviatoric plastic flow, Ddp. Void growth is computed using an explicit
void growth law. The specific equations of this very simple model framework are
F = 3v~-

D,~=

(1 - 0 ) R

IloPlln~

O vp = / 3 1 [ O P [ I e
D p = O dP + O vp :
v

or =

V/1 + / 3 2

IloPl[np

c ( 0 ) : O - v/1 +/32 II~IIC~(0) : n p

-- x/~/3llDPllo "

v

a = C(bns - ,x)IIDPll - ~ ( , x ) o ~
R)IIDPll - nR(R)

k = ~,R(RS -

0 = v~(1 -- 0)/311DPll
where the kinetic equation specifies the evolution for IJDdPl[, i.e.

(43)z-8

Internal state variable concepts

'IDa'[ = g ( ( ( 1 : O ) V / )

643

(44)

Note that F is pressure-independent and is represented by a von Mises flow potential (a
cylindrical ,;urface in stress space); the static yield strength (radius of the cylinder)
decreases with increasing porosity by a factor (1-~9). Note also that the unit vector ns will
be normal to F in stress space. In this model, the void growth factor/3 is specified by
explicit expressions such as those given in the previous section.
IV. INTERNAL STATE VARIABLE VISCOPLASTIC MODEL

Applications of the three constitutive frameworks will be based on Bammann's viscoplastic
model (Bammann [1990]; Bammann et al. [1993]). This is a rate- and temperature-dependent
model that conforms to the structure of the general framework, eqns (8), and has been initially
developed in the context of deviatoric (incompressible) inelasticity for high-temperature
applications (Bammann [1990]). Recently, this model has been applied along with explicit void
growth relations (Cocks & Ashby [1980]) to admit compressibility effects (partially damagedcoupled framework) (Bammann et al. [1993]). In this section, this model is extended to deal
with porous (compressible) inelasticity for both associative and non-associative frameworks.
The inelastic flow rule is given by
DP = g ((h--~V)) ns

Table 2. Bammann's J2 state variable viscoplastic model temperature-dependent material functions*

V(T) = C, e x p ( - ~ )

[MPa]

Y(T) = C3exp(- ~)

[MPa]

f(T)

= C5exp ( - ---~)

[s-1 ]

kd(T) = C7exp(--~)

[1/MPa]

b(T)

[MPa]

= C9exp(--~r )

ks(T) = Cllexp(-~r)

[ 1 / M P a - s]

Kd(T) = C,3exp(-~-~)

[1/MPa]

B(T) = C, sexp(--~r ) [MPa]
Ks(T) = C,Texp(--~)
*T is absolute temperature.

[1/MPa - s]

(45)

644

E.B. Marin and D. L. McDowell

where the constitutive function g is of sinh(.) form and, for our purposes, the function
F = F(tr-A,R, 0) =/~(I1, J°2, R, 0) is assumed to be of elliptic form, i.e.

g= ~/~f(T)sinh(V/3hlJ~+h2I~ ~-(7-h3(R+
~)
Y(T))

(46)

where hi = hi(O,m) (see Table 1) with m = V/Y; f, Y and V are temperature(T)-dependent
matrix material parameters (see Table 2).
In this model, the overstress invariants, lx and J~, are defined somewhat differently than
in eqn (5) (Bammann et al. [1993]) as
11

~

tr(cr -

A),

1
J2o -- ~tr(s
- 2 tx)2

(47)

to provide a straightforward normalization to the uniaxial case. Also, F _> 0 for viscoplastic deformation and R + Y is interpreted as a "quasi-static" yield strength of the
matrix. In eqn (45), the void growth factor fl for the elliptic functional form of F is given
by eqn (40), and the unit vector ns is defined by

ns =

OF/Os
IlOF/Os[l'

OF
-~

2
:

s - ~oc

(48)

Note that D p can be expressed as
(49)
where np is given by eqn (23). Here, when np = na, the flow rule is associative. Otherwise, it
is non-associative• The evolution equations for the Cauchy stress o-, the internal variables A
and R, and the damage variable 0, are given by (Bammann et al. [1993]; Matin [1993])
= c O): (D - W ) A =

(1 - 0)b(r)nA

5i--

llAflA

g

i-

llAflA

(50)

= (1 - O)tr(D p)
where b, kd, ks, B, Kd, Ks are temperature-dependent matrix material parameters (see Table 2)
which can be determined from compression tests, for example, at different strain rates and
temperatures. The damaged elastic stiffness tensor, C(~9), the vector nA and the equivalent
inelastic strain rate of the matrix, ~P, are given by eqns (12), (14)., and (15), respectively. The
dynamic yield strength of the m a t r i x / ~ in the expression for ~P, eqn (15), is obtained by
inverting the scalar viscoplastic flow rule (kinetic equation), eqn (45)1, for F > 0, i.e.

645

Internal state variable concepts

Fd = ~ / 3 h l ~ + h2~ll - h3(Y + R + Vg -1

(IIDPlI)) =

0

(51)

where g-l(.) denotes the inverse of the function g-l(.). From eqn (51), Rd can be identified
as

Rd = r + R + Vg -I (IIDdO[I) = h3

(52)

where ~I, = (3hi J2 ° + h2112) 1/2. Of course, viscoplastic flow occurs when the equivalent
overstress exceeds the static yield strength R + Y.
V. APPLICATIONS

Bamman:a's viscoplastic model in the context of the three model frameworks has been
implemented in the user material subroutine (UMAT) of the displacement-based finite
element code ABAQUS [1993] using a semi-implicit integration scheme of Moran et al.
[1990] based on the rate-dependent yield condition approach (Marin [1993]). Details of
this constitative integration procedure as applied to Bammann's model are presented in
the Appendix. This code is used here to compare the predictive capability of several
implicit and explicit void growth factors (models). Special emphasis is placed on using
Cocks's elliptic model in these different frameworks.
Since different combinations of potentials F and void growth factors 3 are possible, a
specific model computation will be denoted by a set of two letters in the form X-Y. The first
letter, X, indicates the model for the pressure-dependent flow potential wile the second one,
Y, denotes the model for the void growth factor. I f X = Y (e.g. C-C, C = Cocks ([1989]),
Table 3. Consants for Bammann's J2 state variable viscoplastic model 6061-T6 AI

C1
C2
C3
Ca
Cs

=
=
=
=
=

6.90 x 10-2
0
1.60 x 102
1.62 × 102
1

C6

:

0

C7
C8
C9
Cl0
Cll

=
=
=
=
=

1.91
6.94 x 102
1.03 × 103
0
0

C12 = 0

C13 =
C14 •
C15 =
Cl6 =
Cl7 =

4.42 × 10-2
8.56 x 102
8,34 x 10
0
0

C18 = 0

[MPa]
[K]
[MPa]
[K]
[s-l]
[K]
[1/MPa]
[K]
[MPa]
[K]
[1/MPa-s]
[K]
[1/MPa]
[K]
[MPa]
[K]
[1/MPa-s]
[K]

646

E.B. Marin and D. L. McDowell

then model framework (i) is used (associative and fully damage-coupled). This is the case of
an implicit void growth factor. On the other hand, if X # Y (e.g. C-M, M = Sofronis &
McMeeking [1992]), then model frameworks (ii) or (iii) are applied (non-associative and
either fully damage-coupled or partially damage-coupled). In this case, X = J2 (e.g. J2-C, J2
= yon Mises flow potential with the factor (1-0) affecting R, eqn (43)1) indicates a partially
damage-coupled analysis. The use of the single letter J2 denotes deviatoric (~ = 0) J2 flow
theory. To simplify the terminology, model frameworks (i), (ii) and (iii) will be referred to
as associative, non-associative and partially coupled frameworks, respectively.
2.0

I

I

I

I

J2-C

1.5

a/a o

M-M

1.0

0.5

J2- M-

(a)

Bammann's Viscoplastic Model
Uniax. Tension of 3D Single Element.
P0=0.99

0.0
0.(

I

i

I

P

0...3

0.6

0.9

1.2

I

I

50

£

.5

J2

~ 20

"

t

/Barnmann's Viscoplastic Mod
/ Uniax. Tens.-3D Single Elem. I ( b )

0) ]0

/

P°=0'99

/

>0
i

4

M-M

-

0
:>

J2-C
0
0.0

I

J

I

i

0.3

0.5

0.9

1.2

1.5

ln(1 + u / 1 o)
Fig. 3. Bammann's viscoplastic model: material response prediction using the partially coupled (Jz-C, J2-M, J2~A)
and the associative (C--C, M-M) frameworks for homogeneous uniaxial deformation of a 3-D single element.

Internal state variable concepts

647

Two problems are numerically solved using these model frameworks: (1) homogeneous
uniaxial deformation of a 3-D single element and (2) neck development in an axisymmetric specimen and a plane strain specimen under tension.
V.1. Uniaxial homogeneous tension of 3-D element
A cube of side 2a (a = 2.54 x 10-2 m) is considered. Due to symmetry, only 1/4 of the cube
is used which is represented by an 8-node brick element, type ABAQUS-C3D8R. The
2.0

I

I

I

I

J

Ja-C

1.5

a/a o

...............p

c-c

(a)

1.0
\ C-CA

0.5
Bamrnann's Viscoplastic Model
Jniax. Tension of 3D Single Element
P0=0.99

0.0
0.0

I

I

I

I

0.5

0.6

0.9

1.2

I

1

1.5

E
50

I

/

I

/

5
.o 2 0 -

/Bammann'sViscopiastic

"~
0

]

o~

Mode]
Uniax. Tens.-3D Single Elem.

/

o 10

~

_

P°=0"99

-

(b)

~
C-C

o
c

0
0.0

I

L

i

t

0.5

0.6

0.9

1.2

~.5

ln(t + u / 1 o )
Fig. 4. Bammann's viscoplastic model: material response prediction using Cocks pressure-dependent flow
potential in associative (C-C) and non-associative (C-M, C-CA) frameworks for homogeneous uniaxial deformation of a 3-D single element.

648

E.B. Marin and D. L. McDowell

computations are carried out using the elliptic (F or/3) models of Cocks (C) and Sofronis and
McMeeking (M) (see Table 1), and the Cocks and Ashby (CA) void growth factor, eqn (39).
Although the constitutive equations in Bammann's model are temperature-dependent, the
calculations are performed at a constant temperature of T = 21°C. The value of the material
constants CFCIS (see Table 2) used in this example correspond to those of AL 6061-T6 aluminum (Bammann et al. [1993]), as listed in Table 3. The elastic constants for this material are
E -- 69 x 103 MPa, v -- 0.33. An initial void volume fraction of 0o = 0.01 (initial relative
density Po = 0.99) is used and a displacement-rate boundary condition of 0.635 x 10 -2 m/s
2.0

I

I

I

t

B a m m a n n ' s Viscoplastic Model

Jz

1.5

cr/¢ 0

\

1.0

'%,

i\,

Po=0.999

Po=0.99

0.5

(a)

J2-CA
with D p
v
o without Dv

0.0
0.0

I00

i

i

I

I

0.5

0.6

0.9

1.2

I

I

I

I

.5

3ammann's Viscoplastie Model
Uniax. Tens.-3D Single Elem.
j2_CA

"-" 7 5

.O
O

with Dvp

P0=0.99

without D

////

50

~?

/

°

'~

Po=°999

(b)

Po=0.9999

2
0

......

0.0

0.3

0.6

0.9

1.2

.5

l n ( l + u / 1 o)
Fig. 5. Bammann's viscoplasticmodel: material response prediction in a partially coupled framework with and
without Dvp using Cocks and Ashby void growth factor (CA) for homogeneous uniaxial deformation of a 3-D
single element.

Internal state variable concepts

649

(initial strain rate approx. 2.5 x 10-l s-l) is assumed. The relatively high value of 0o is
selected to accentuate differences among various frameworks and models in the case of
homogeneous deformation. The computed results are reported using plots of (a) true stress
versus true strain e and (b) porosity 0 versus averaged extensional strain In(1 + u/L).
Figure 3 shows the numerical results obtained with the partially coupled (J2-C, J2-M,
J2~CA) and associative (C-C, M-M) frameworks. This figure shows that the partially
coupled frameworks J2-C and J2-M give a stiffer stress-strain response when compared to
the corresponding associative frameworks, C--C and M-M. The same tendency is
observed in the porosity evolution response. On the other hand, the partially coupled
framework J2-CA predicts very compliant stress-strain and porosity evolution responses.
Figure 4 presents the prediction using Cocks yield function in associative (C-C) and
non-associative (C-M, C-CA) frameworks. Results with the partially coupled framework
J2-C are also included for reference. It is seen that the material response (strength and
porosity evolution) using the yield function of Cocks [1989] in an associative framework
(C--C) is more stiff than the C-M and C-CA cases. Again, the partially coupled framework using the void growth factor derived from Cocks yield function, J2--C, gives an
"upper limit" stress-strain response.

IImlM
IlUlIIIII
IllUlUll
IIIlUlIII
IIIIIIIIII
IIIIIIlUl
IIIIlUlII
IIIIIlUll
IlUllUll
IIIlUlUl
IlUlIIIII
IlUUUll
IlUlIIIII
IIIIIIIIII

It
I

a

IlUllllll
IIIIIIIIII,
II Il Il IlIlIlIlIlIl1l ]|
II II II II II II Ii iIiI i

l d

with initial
imperfectionO,

element

1

(a)

l
(b)

Fig. 6. Axisymmetric speciment: (a) finite element mesh and element with imperfection t%i used for analysis of
necking process. Mesh consists of 330 four-node quadrilateral elements, type ABAQUS-CAX4. Element aspect
ratio along symmetry plane at center o f specimen is approx. 3:1. Displacement-rate boundary conditions are
applied on S,~. (b) Deformed mesh at an averaged axial strain of e ~ 0.180, where e = ln(1 + u/lo).

650

E.B. Marin and D. L. McDowell

Applications of the original partially coupled Bammann's model (J2-CA) (Bammann et
al.. [1993]) have been made neglecting the volumetric component (Dr p) of the inelastic rate
of deformation D p. Figure 5 compares the stress-strain and porosity evolution responses
using the partially coupled framework with (J2-CA) and without (J2-CAn) Dvp, for
different initial void volume fractions. The difference in prediction between these two
versions of Bammann's partially coupled framework is mainly observed after the point of
maximum load. In this region, when Dvp is excluded, the void growth rate results in a
more compliant stress-strain response.
V.2. Diffuse necking of a bar
Here, the model frameworks presented before are used to study the necking process of
both axisymmetric and plane strain tensile specimens, based on Bammann's viscoplastic
model. In particular, the partially coupled J2-CA and the associative C-C frameworks are

~z~i~!!~ii~:

~iiiiiliiiiiii~iii~i~i,ii!iil;
~;~J,~,~ ::~ 2;, ~:~'~,~
, :!~!~!~ '~,~,i~iii,ili~i

...... i:i !:,2,,, ~,

~...................
.......

ffi

0.108

• =0.111

• ffi0.140

elastic ~gion

~

i~ili!i~iiiiiiiiili~i~i;;
I
i~~!i!ii!iiiiiiiiil
iiii:iii~'~
~' 'i '~ i'i~i'?i~i i~i~~

II
•

: ii!!i!!!ilili~iiiiii!iii!i
~

'~~i~ii'~i,iliii:i~iii,,
i

• =0.180

inelastic ~gion

(a)

17E02

i

+2.00£-02

+ 1.04E+

+2.06E+00

00

::::

/

::: ....

/

_

.

- 1.52E+04
-5.80g+03

(b)
Fig. 7. Axisymmetric specimen: (a) elastic and inelastic regions at different levels o f e = ln(l + u/lo) during neck
development. Note how elastic unloading proceeds as necking develops. (b) Contour plots of void volume fraction ~9 (sdv29 in percent) and pressure (in psi) at e = 0.180. Results obtained using Bammann's partially coupled
framework with Cocks and Ashby's void growth factor and with D~p (Jz-CA).

651

Internal state variable concepts

used to examine the axisymmetric and plane strain problems, respectively. This study
is mainly focused on analyzing the effect of neck development on the local and
global material response. Here, local response refers to the stress-strain behavior and
state of damage (porosity) at specific material points or particles in the specimens.
Global response, on the other hand, alludes to the load-displacement (strain) curve
and associated ductility properties. In addition, a study of the influence of the type
of model framework predicting the onset of necking is carried out using the
1.8

I

I

I

o Onset of Necking
1.5

'd'

-

1.2
0"22

(a)

0.9
'b'

0.6
Axisymmetric S p e c i m e n
B a m m a n n ' s Viscoplastic Model
Partially Coupled F r a m e w o r k (J2-CA)

0.3

0.0
0.00

Po=0.9999 , p01=0.99985
t
I
'I
0.05
0.10
0.15

0.20

£22

0.05

I

I

o Onset

/I

\

0.04

H.D.

o
o 0.03
(b)

'c'

S 0.02

m
o
I>

'b'

'a'

,u 0,01

////7

Axisymmetric S p e c i m e n
B a m m a n n ' s Viscoplastie Model
Partially Coupled F r a m e w o r k (Jz-CA)
po=0.9999 , poi=0.99985

0.00
0.(

I

I

I

0.1

0.2

0.3

0.4

In(Ao/A)
Fig. 8. Axisymmetric specimen: effect of necking on the local response at some material particles along the axis of
the specimen (see Fig. 6). Results obtained using the partially coupled framework J2--CA with an initial imperfection ofOo~ = 1.5 x 10-4.

652

E.B. Marin and D. L. McDowell

associative C-C, non-associative C-CA and partially coupled J2-C, J2-CA, J2-CAn
frameworks.
The material constants for Bammann's model used in these calculations correspond to
those of 6061-T6 aluminum (Bammann et al. [1993]). A constant temperature of 21°C
is assumed for the computations. A uniform initial void volume fraction of Oo = 1 po = l0 -4 (typical value for ductile metals) is assumed. Necking is initiated using an initial
perturbed value of 0, Ooi (poi), at a specific location in each specimen.
1.6
o

I

I

Maximum

Load

I

I

H.D.

1.2
tM
O
b

(a)

0.8

0.4

Axisymmetric Specimen
Bammann's Viscoplastic Model
Partially Coupled Framework (J2-CA)
P0=0.9999 , p0i=0.99985

0.0
0,00

]

I

I

I

0.05

0,10

0.15

0.20

0.25

l n ( l + u / 1 o)

0.5

I

I

I

I

Axisymmetric Specimen
Bammann's Viscoplastic Model
0 . 4 Partially Coupled Framework (Js-CA)

0.,3

P°=0"9999' P°1=0"999/~

/

o

(b)

0.1
j J

0.0
0.00

o Onset of Necking
i

i

i

t

0.05

0.10

0.15

0.20

0.25

In( i+u/1 o)
Fig. 9. Axisyrnmetric specimen: effect of necking on the global material response of the specimen. Results
obtained using the partiallycoupled framework J 2 ~ A with an initialimperfection of ~oi = 1.5 × 10-4.

Internal state variable concepts

653

V.2.1. Asymmetric tension

As mentioned above, the partially coupled framework J2-C is used in this study.
The symmetry of the problem permitted consideration of only 1/4 of the tensile specimen.
The finite element mesh and the element where the initial imperfection 0oi is introduced are
shown in Fig. 6(a). The mesh consists of 330 four-node quadrilateral elements type
ABAQUS-CAX4. The initial length of the specimen is 2•0(â