Associative versus non-associative porous viscoplasticity based on internal state variable concepts

Pergamon International Journal of Plasticity, Vol. 12, No. 5, pp. 629-669, 1996

Copyright © 1996 Elsevier Science Ltd 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. Ma t in* and 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 in final revised form 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, non- associative 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 duct i le meta l s subjected to large d e f o r m a t i o n undergo i rreversible micro- s t ruc tura l changes (damage) which degrade their s t rength and mechanica l proper t ies . U n d e r creep cond i t ions (creep or viscoplast ic damage) , voids or cavit ies nucleate and grow unt i l they l ink or coalesce, leading to the f rac ture o f the specimen or s t ruc tura l c o m p o n e n t (Put t ick [1959]; Rogers [1960]).

In con t r a s t to classical inelast ici ty re la t ions which are based on the incompress ib i l i ty o f inelast ic de fo rma t ion , the overal l s t ructure o f the const i tu t ive equat ions for a po rous ma te r i a l reflects vo lumet r ic inelast ic d e f o r m a t i o n and a m a r k e d pressure dependence . Typical ly , these theories mode l the p o r o u s m e d i u m as a macroscop ica l ly equiva lent iso- t ropic , h o m o g e n e o u s c o n t i n u u m with vo id vo lume f rac t ion as a scalar in terna l var iable . The impl ic i t a s sumpt ion in this c o n t i n u u m t r ea tmen t is tha t the voids in the po rous ma te r i a l are spa t ia l ly r a n d o m l y d i s t r ibu ted and exhibi t r ea sonab ly spher ical shape with r a n d o m size ( i so t ropic damage) .

* 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 ( l _ 0)] sinh (2(2 -- m) ah ] {p 1_ )1/m \ 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

1 1 11 = tr(tr - A), J~ = ~tr(s - a) 2, 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 rate- dependent response of the unit cell model, where the matrix is modeled as a viscoplastic

632 E. B. Marin and D. L. McDowell

Table 1. Coefficients of the elliptic model (porous viscoplasticity)

Viscoplastic potential • (Duva & Hutchinson [1984])

~o~ [3hl~ + h212,] ((1/m)+l)/2 ¢ - 1 -z:2 -2 - ' - L 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 hi = 1 + ~ 0 , h 2 - 2 1 + m l + t g , h3 = (1 - O)U("+I)

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

2 1 (mO_m(1 -- Lg)) 2/(m+l) hi = 1 + ~ 0 , h 2 = ~ \ 1_ ,6 m

Sofronis & McMeeking [1992]

1 (rn~m(1 - ~ ) ) 2/(re+l) hi = (1 + zg) 2 / ( m + l ) , h2 = ~ \ 1 - Om

h3 = (1 - - ~q)l/(l+m)

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

3J~ _ ,, ~gcost./q2II ~ Fd = ~ - -t- zql u~T-~. ) -- 1 -- q3 ~2 = 0 (7) "'d ---0

where F--~ Fd represents the dynamic yield condition and Ra = /~ (~P , ~P) is the dynamic yield strength of the matrix material. The values ql = q 2 : q3 = 1 correspond to the original


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.


II. A CLASS OF POROUS INELASTIC INTERNAL STATE VARIABLE MODELS

In a general f ramework, a class o f viscoplastic constitutive models for a porous 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 = I lDP[lnp = g((F))np

tr = C ( 0 ) : ( D - I l D P l l n p ) - ~ - - ~ tr

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

k : s -- R) P -- a R ( R )

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

The viscosity function g o f the porous material, which is assumed to be a homogeneous funct ion o f the (static) yield condition, F (cf. Fig. 1), defines the viscoplastic flow rule

0 = 0.15 %/v. Jl

" ° 1 ~I/Yo

-a21Y *

O = 0 .05 *z ¢0 d 1

- e 2 V a

O = ,~ * n °2 /Yo ,d u . I u 1

- o Yc

- • | l y ?

~ ) m O *l 'a J1

- o 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).


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=~/3h lJ~+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 pre- scribed 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

O I

~a~,j2°,R,o) /

/

DP n j ~ p

g " no

\ ° : \ . . ~lJm~ ~ 1~0'2 ' y/~ 05 n j

OPlaa Dl, "~. - IDOl l a F l a o l 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.


OF/Oct OF OF I OF ( s - ~x) (10) n~ --ilor/Ocrll, O~ - OI, + - ~ 2

In these general terms, the model can represent either an associative (np= ha) or a non- associative (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 super- scripted "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.

V• i~ 1 (13) nA = ns + 3 X / ~ I


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/~2n s + V/~_~ ~ lira V~hh~2i V ~ ~--*1 (14)

nA = [3J~..i-3-12 h2 lim,~2]1/2 - - 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 1Lhe limit 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):D p, are equal (since voids do not contribute). This assumption leads to

~p = ( 0 " - - A) : D p (0" - A) : np (1 -- 0)Rd - ( ] - 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

l1 3hi 1 ---0 ~ - ~ - 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


0 = v ~ ( 1 - 0)/~I[DPll (19)

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 + Dv p (20)

where DdP and Dv p 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 Substituting expressions (21) into eqn (20), we obtain

where

D p = ,~pllOPll(ns +/3e) = ~p~¢/1 +/32i[OP[in p

= (1-1/3(tr(np))2)l/2.

(22)

/3 1 np - ~ e 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)

1 D p - ~ IlDPl[ns

D ~ - /3 1C?%~ Im~lle

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

v 1 X/~/3 iiDPllo. (24)1_ 8 o r = C ( 0 ) : D , ~ I tDP l IC_( ' 0 ) : (ns+/3e)

---- V/~ c[(1 - 0)bnA - A]xdiDPll - aA(A)A

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

= v6(1 - o) ~ IIoPII


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 damage- coupled 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 - o t , 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.


OF/Off v ~ OF ~ OF e + - - - - ns ( 2 9 )

no = [ lOF/Ocr l l - 6 0I, 6 0 J ~

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 - f f ,~2 OF/OJ~ (30)

With 13 defined by eqn (30), the explicit expression for the void growth law, eqn (8)6 , is given by

_ 3 ( 1 - 0 ) OF/OIl Dp I

OF/O~ d (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 pre- scribed 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 1 0 ~

hi 2V/~22 0II []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 p z9 = v/6 0 hi ~ IIDdII (34)

and


(ii) Gurson-Tvergaard form:

R q211 = V~2 3- 0 ( 1 - 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) i 1 ) f ' ,36) 0---- V3 s l n h ~ ( 2 ~ m ) ~ (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[ 1 ] 2 ~ m ( 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-associative porous 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 (40) h~3v/~2

(ii) Gurson-Tvergaard form:

1 R • . [ ' q 2 I I ' ~ = ~ s l n n / - - / /3 - ~ O q l q 2 (41)

V/3~ k, 2R ]

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-associat ive 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, Dd p. Void growth is computed using an explicit void growth law. The specific equations of this very simple model framework are

F = 3 v ~ - (1 - 0 ) R

D,~ = I loPl ln~

O v p = / 3 1 [ O P [ I e

D p P p V/1 + / 3 2 = O d + O v : IloPl[np v or = c ( 0 ) : O - v/1 +/32 II~IIC~(0) : n p - - x / ~ / 3 l l D P l l o " (43)z-8 v a = C(bns - ,x)IIDPll - ~ ( , x ) o ~

k = ~,R(RS - R)IIDPll - nR(R)

0 = v~(1 -- 0)/311DPll

where the kinetic equation specifies the evolution for I JDdPl [, i.e.


'IDa'[ = g ( ( ( 1 : O ) V / ) (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 damaged- coupled 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 (45)

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) = C9exp(--~r ) [MPa]

ks(T) = Cl lexp(-~r) [1 /MPa- s]

Kd(T) = C,3exp(-~-~) [1/MPa]

B(T) = C, sexp(--~r ) [MPa]

Ks(T) = C,Texp(--~) [1/MPa - s]

*T is absolute temperature.


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~ -h3(R+ Y(T)) ~-((7~) (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

o 1 2 tx)2 (47) 11 ~ tr(cr - A), J2 -- ~tr(s -

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

OF/Os OF 2 ns = IlOF/Os[l' - ~ : s - ~ o c ( 4 8 )

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 ma 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.

Internal state variable concepts

Fd = ~ /3h l~ + h2~ll - h3(Y + R + Vg -1 (IIDPlI)) = 0

645

(51)

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

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

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 = 6.90 x 10 -2 [MPa] C2 = 0 [K] C3 = 1.60 x 102 [MPa] Ca = 1.62 × 102 [K] Cs = 1 [s -l] C6 : 0 [K] C7 = 1.91 [1/MPa] C8 = 6.94 x 102 [K] C9 = 1.03 × 103 [MPa] Cl0 = 0 [K] Cll = 0 [1/MPa-s] C12 = 0 [K] C13 = 4.42 × 10 -2 [1/MPa] C14 • 8.56 x 102 [K] C15 = 8,34 x 10 [MPa] Cl6 = 0 [K] Cl7 = 0 [1/MPa-s] C18 = 0 [K]


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

1.5

a / a o 1.0

0.5

0.0 0.(

I I I I

J2-C

J2- M-

M-M Bammann's Viscoplastic Model

Uniax. Tension of 3D Single Element. P0=0.99

I i I P

0...3 0.6 0.9 1.2 £

(a)

.5

50 I I

J2

~ 20

" t /Barnmann's Viscoplastic Mod / Uniax. Tens.-3D Single Elem. I (b )

0) ]0 / P°=0'99 4

>0 M - M - / i

0 J2-C :>

0 I J I i 0.0 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.


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

a /a o

1.5

1.0

0.5

0.0 0.0 1.5

J Ja-C

............... p c-c

\ C-CA

Bamrnann's Viscoplastic Model Jniax. Tension of 3D Single Element

P0=0.99

I I I I

0.5 0.6 0.9 1.2 E

(a)

50 I / I I 1

/

5

.o 2 0 - "~ 0 /Bammann's Viscopiastic Mode]

] Uniax. Tens.-3D Single Elem. (b) o~ / P°=0"99

o 10 ~ _ - ~ C-C

o c

0 I L i t 0.0 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.


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

c r /¢ 0

2.0

1.5

1.0

0.5

I I I t

Bammann's Viscoplastic Model Jz

\ i \ , '%, Po=0.999

Po=0.99

0.0 i i

0.0 0.5 0.6

J2-CA

with D p v o without D v

I I

0.9 1.2

(a)

.5

I00 I I I I

3ammann's Viscoplastie Model Uniax . T e n s . - 3 D S ing l e E lem.

"-" 75 j2_CA

.O with D p P0=0.99 v O w i t h o u t 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 viscoplastic model: material response prediction in a partially coupled framework with and without Dv p using Cocks and Ashby void growth factor (CA) for homogeneous uniaxial deformation of a 3-D single element.


(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.

It I

a

l d l element with initial 1 imperfection O,

I ImlM I lUlII I I I I l lUlUl l I I I lUlII I IIIIIIIIII I I I I I I lUl I I I I lUlII I I I I I lUl l I lUl lUl l I I I lUlUl I lUlII I I I I lUUUll I lUlII I I I IIIIIIIIII I lU l l l l l l I I I I I I I I I I , I l l l l l l l l l ] I I I I I I I I I 1 | I I I I I I I I I i I I I I I I i i i

(a) (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).


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) Dv p, 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 Dv p 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~: : ii!!i!!!ilili~iiiiii!iii!i ~ ~i iiiili i iiiii~iii ~ i~i,iiii!iiiil; '~i ~ ~ii'i~iii i,liiiii:i~i iii,, ~;~J,~,~ ::~ 2 ; , ~:~'~ ~, , :!~! ~!~ '~,~,i~iii,ili~i ...... i:i !:,2,,, ~,

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

~' ' i '~ i'i ~i'?i~i i~i~ ~

II • ffi 0.108 • =0.111 • ffi 0.140 • =0.180

elastic ~gion ~ inelastic ~gion

(a)

17E02 i + 2 . 0 0 £ - 0 2 _ .

+ 1 . 0 4 E + 0 0 :::: : : : .... - 1 . 5 2 E + 0 4

/ + 2 . 0 6 E + 0 0 - 5 . 8 0 g + 0 3

/ (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).


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

0"22

1.8

1.5

1.2

0.9

0.6

0.3

0.0 0.00

I I I o Onset of Necking

' d ' -

' b '

Axisymmetric S p e c i m e n B a m m a n n ' s Viscoplas t ic Model

Part ia l ly Coupled Framework (J2-CA) Po=0.9999 , p01=0.99985

' I t I

0.05 0.10 0.15

£22

(a)

0.20

0.05

0.04

o

o 0.03

S 0.02 m o I> ,u 0,01

0 . 0 0 0.(

I I /I

o Onset

\ H.D.

' c '

' b '

'a'

/ / / / 7 Axisymmetric S p e c i m e n B a m m a n n ' s Viscoplast ie Model

Part ial ly Coupled Framework (Jz-CA) po=0.9999 , poi=0.99985

I I I

0.1 0.2 0.3 0.4

In(Ao/A)

(b)

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.


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.

tM

O b

1.6

1.2

0.8

0.4

I I

o Maximum Load

I I

H.D.

0.0 0,00 0.25

Axisymmetric Specimen Bammann's Viscoplastic Model

Partially Coupled Framework (J2-CA) P0=0.9999 , p0i=0.99985

] I I I

0.05 0,10 0.15 0.20

(a)

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 o Onset of Necking 0.0 i i i t

0.00 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 partially coupled framework J2~A with an initial imperfection of ~oi = 1.5 × 10 -4.


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(•0 = 101.6 x 10 -2 m) and

1.8 I I I

0"22

1.5

1.2

0.9

0.6

0.3

0.0 0.00 0.20


Partially Coupled Framework (Jz-CA) Po=0.99990

1 poi=0.99988 3 po1=0.99900 2 poi=0.99985 4 poi=0.99000

I I [

0.05 0.10 0.15

622

(a)

2.0 I } I I I /

Axisymmetric Specimen ] ~-~ Bammann ' s Viseoplastic Model J

Partially Coupled Framework (Jz-CAp "-" 1.5 Po=0.99990 /1,2,3

1 po =0.99988 3 poi=0.99900 / 0 2 poi=0.99985 4 poi=O 99000 [ co

o Onset of Necking 4//4 a 1.o (b) // 0J E _= o 0.5

H.D material particl

0.0 -'= i i

0.0 0,1 0.2 0.3 0.4 0.5 ).6

ln(Ao/A) Fig. 10. Axisymmetric specimen: local material response during neck development for different initial imperfections tgoi prediicted by the partially coupled framework J2-CA.


the ratio of initial length to initial radius a is lo/a = 4. An initial imperfection of 0oi = 1.5 x 10 -4 is used in most computations. A displacement-rate boundary condition of 2.54 x 10 -2 m/s (strain rate on the order of 2 x 10 -2 s -1) is applied on surface Su (see Fig. 6(a)). The deformed finite element mesh at an averaged axial strain of e = 0.180, where e = ln(1 +u/lo), is shown Fig. 6(b).

Figure 7(a) shows how elastic unloading proceeds at material points outside the neck region as necking develops. Contour plots of void volume fraction and pressure at a deformation level of e = 0.180 are presented in Fig. 7(b).

1.8

1.2

°22 0.9

I I I

[ o Onset of Necking 1 I

' d '

0.6 I J2-c J2-CA Jz-CAn

0.3 Axisymmetric Specimen Bammann's Viscoplastic Model

Po=9.99990 , Poi=0.99985 0.0 i i l

0.00 0.05 0.10 0.15

£ge

(a)

0.20

0.0 0.1 0.2 0.5 0.4 0.5 0.6

in(Ao/A)

Fig. 11. Axisymmetric specimen: local material response during neck development predicted by the associative (C-C), non-associative (C-CA) and partially coupled (J2-C, J2-CA, J2-CAn) frameworks, using an initial imperfection of 0oi = 1.5 x 10 -4.

2.0 ' ' ' ' l

Axisymmetric Specimen l ~-. Bammann's Viseoplastic Model /

= 1.5 "°o°9999°'= poi=0.99985 /

Onset of Necking ] O

(~ j2_CA n 1.o- kCA (b)

~ 0.5

? o.o


Figure 8 shows the local stress-strain and porosity evolution responses at material points located at positions a, b, c and d (see Fig. 6(a)) along the axis of the axisymmetric specimen. Note that the local response at these material points starts to deviate from the initial uniaxial homogeneous deformation (H.D.) path at the onset of necking due to the induced triaxial stress state. Note also that when necking starts, elastic unloading initiates at the material points on the boundary Su (e.g. point a, see Fig. 8(a)). The subsequent elastic state at these material points then prevents further void growth (Fig. 8(b)). On the other hand,

0 b

1.6

1.2

0.8

0 . 4

I I I I

o Maximum Load

f

J

H . D . \ @

4 ~ ~ 2 , 1

3 /


Partially Coupled Framework (Jz-CA) P0=0.99990

1 P0i=0.99988 3 pot=0.99900 2 poi=0.99985 4 p01=0.99000

I I L I 0.0 0.00 0.05 0.10 0.15 0.20 0.25

In(l +u/l o )

(a)

Cq ¢5

o b

5~

1.5

1.2

0.8

0.4

I I I I

o Maximum Load

f /

S.D.

Jz-CAn \ jz_CA ~Jz - (

C-C

0 .0 i

0.00 0.05 0.10 0.15 0.20 0.25

Axisymmetric Specimen 9ammann's Viscoplastic Model

P0=0.99990 , pot=0.99985 L I L

In(1+u/1 o)

(b)

Fig. 12. AxisFmmetric specimen: load versus averaged axial strain (global) response during neck develoment for (a) different initial imperfections ~oi using the partially coupled framework J2~CA and (b) several model frameworks using an initial imperfection (void volume fraction) of ~oi = 1.5 × 10 -4.


for material points located in the necked region (e.g. point d), the true (Cauchy) stress and void growth rate increase due to the constraint imposed by the triaxial state of stress accompanying necking. It should be pointed out here that a non-proportional history of triaxial stressing is induced at these material points by the neck curvature.

The global response of the axisymmetric specimen is presented in Fig. 9, where the load and cross-sectional area (logarithmic) strain ln(Ao/A) are plotted as a function of the averaged axial strain, e. Note again that necking deviates the specimen mechanical response from the homogeneous deformation path. In particular, Fig. (9) clearly shows that neck development limits the ductility of the specimen, with ductility being defined as the final reduction in area at the minimum section.

The effect of the magnitude of the initial imperfection on the local response of the specimen is examined in Fig. 10, where the local (a) stress-strain (for material points a and d) and (b) porosity evolution (at material point d) responses are plotted for 0oi = 1.2 x 10 -4, 1.5 × 10 -4, 10 × 10 -4 and 100 x 10 -4. These figures show that a somewhat wide range of initial imperfection magnitudes can be used to predict the onset of necking without significantly affecting the local response.

I

l f element wilh initial 1 imperfection O.

(a) (b)

Fig. 13. P1 ane strain specimen: (a) finite element mesh and element with imperfection 0oi , and (b) deformed mesh at an averaged extensional strain of e = ln(1 +u/lo) = 0.180. The mesh consists of 330 four-node quadrilateral elements, type ABAQUS-CPE4. The element aspect ratio along the symmetry plane at the center of the specimen is approx. 3:1. Displacement-rate boundary conditions are applied on Su.


Figure 1 l[(a) compares the prediction of the local stress-strain response at material points a and d and Fig. 1 l(b) gives the porosity evolution response at material point d, for the different model frameworks mentioned above. An initial imperfection of 0oi = 1.5 x 10 -4 is used here. It is observed that the point of elastic unloading at material point a (onset of necking) is predicted at the same strain level (approx. 0.1066) by all models. Also, the stress-strain response at point d after the onset of necking differs slightly among the models. However, the predicted porosity, which is of the order of 0.01 at the onset of necking for all models, changes significantly among models as necking develops. The tendency in porosity evolution response predicted by these model flame- works is the one observed in the 3-D single element studies.

Figure 12 presents the predicted load versus averaged axial strain response for (a) different values for the initial imperfection Ooi and (b) several model frameworks with 0oi = 1.5 x 10 -4. It is observed that none of these factors has an appreciable effect on the

ii!iii!i!iii~iii~i!i!i!ii!ii!iiiiiii iiiiiiii!!!iiiii!i!ii!~!ii!~i~ii!~!~ !i?iiiiliiiiiiiiiiii~iiii~i!ii'i,

' ii!i i!iii!ili!i!iiiiiii!iiii ! iiiilili~iiiii!:iiii!iiiii:iiii,'iiii,

!~,iziiiiii!iii!iiiii!ii!iiii~i!iiii

!~i!'i!!i!iliiiiiiiiiiiiiiiiiii!i~i~ii iii,~iiii!iii!i!'iiiii~!iii~iiiiii~

E =0.108 E =0.III E =0.140 c =0.180

elastic region I inelastic region

(a)

iiiiiiiiiiiiiiiiiiiiii

i ......... liiiiii!i +2.001~-02

+6.s5,,-o2 ilili!ili~iiiiiiii!i!i!!!iiii~!!!iii~

+~.11E-0~ !iiiiiiiiiiiiiiii~!ii~ili!i~

. . . . . . . . . :i~ ~ i:~i~i ~'!!i!i ~ . i!iiiiiii!iiii!iiiiiiiiiiiiii!i~!iii! !i~iiiiiiiiiiii!ii;iiii~iii;i

(b)

iiii~i!i~iiiiii~

Fig. 14. Plane strain specimen: (a) elastic and inelastic regions at different levels of extensional strain e during neck development, and (b) contour plots of void volume fraction O(svd29 in percent) and pressure (press in psi) at e = 0.180. Results obtained using Bammann's associative framework with Cocks flow potential ( C ~ ) .


global stress-strain response o f the specimen; however, the ductility depends significantly on the model.

V.2.2. Plane strain tension

The necking of the plane strain specimen is studied using the associative f ramework C-C . The finite element mesh, shown in Fig. 13(a), is similar to the one used for the axisym-

1.8 I I I

1.5

1.2

a22 0.9

0.6

0.,3

0.0 0.00

o O n s e t of N e c k i n g 'd'

'b'

P l a n e S t r a i n S p e c i m e n B a m m a n n ' s V i s c o p l a s t i c Mode l Associative Framework ( C - C )

P0=0.9999 , p0i=0.99985 I I I

0.05 0.10 0.15

E22

(a)

0.20

1.6

% 0.8 b

1.2 / ~

0 . 4 P l a n e S t r a i n S p e c i m e n Bammann's Viscoplastic Model Associative Framework (C-C)

P0=0.9999 , pot=0.99985 0.0 ~ I I I

0.00 0.05 0.10 0.15 0.20

l n ( l + u / 1 o)

I I I I

o M a x i m u m L o a d H.D.

o

0 .25

(b)

Fig. 15. Plane strain specimen: effect of necking on (a) the local stress-strain response at some material particles along the symmetry plane 2-2 (see Fig. 13), and (b) global load-averaged extensional strain response. Results obtained using the associative framework C-C with an initial imperfection of ~oi = 1.5 x 10 -4.


metric specimen. It contains 330 four-node quadrilateral elements type ABAQUS-CPE4. The original length and width of the specimen are 2/0 (lo = 101.6 x 10 -2 m) and 2a, respectively, with an aspect ratio lo/a = 4. A unit thickness is assumed. The displacement- rate boundary condition is 2.54 x 10 -2 m/s (strain rate in the order of 2 x 10 -2 s-l). The deformed mesh at an averaged extensional strain of c = ln(l+u/lo) = 0.180 is presented in Fig. 13(b). Note that in this case the neck curvature is more pronounced compared to the one developed in the axisymmetric specimen (see Fig. 6).

0"22

1.8

1.5

1.2

0.9

0.6

0.5 1

0.0 iz 0.00

I I

4

Plane Strain Speclmen Bammann's Viscoplastic Model Associative Framework (C-C)

#0=0.99990 p0i=0.99985 3 #0i= 0,99900 poi=0.99950 4 0~t=0.99000

I I

0.05 0.10 0.15

I

'd' 4,3,2,1

H.D.

0.20

e2£

(a)

o t~

1.6 i i

o Maximum Load

1.2

0.8

0.4

2 0.0

0.00

I I

o H . D .

(b)

Plane Strain Specimen Bammann's Viscoplastic Model Associative Framework (C-C)

po=0.99990 i pm=0.99985 3 pa=0.99900

pot=0.99950 4 poi=0.99000 J i [ I

0.05 0.10 0.15 0.20 0.25

In(l+u/1 o) Fig. 16. Plane strain specimen: local and global material responses during neck development for different initial imperfections #oi predicted by the associative framework C-C.

660 E.B. Matin and D. L. McDowell

The evolution of the elastic-inelastic region during necking development is shown in Fig. 14(a). Contour plots of void volume fraction and pressure at an averaged extensional strain of e -- 0.180 are shown in Fig. 14(b). Note that at this deformation level, a purely tensile hydrostatic state of stress predominates at all material points in the specimen.

The local stress-strain responses at specific material points along the symmetry plane 2-2 (see Fig. 13(a)) are shown in Fig. 150). The global load-averaged extensional strain curve of the specimen is presented in Fig. 15(b). Again, these figures show that at the onset of

(722

1 . 8 I I I All F r a m e w o r k s 'd '

I. 2 J2- CAn _

0.9

C-C

0.6 2 - c

0.3

0 .0 0 . 0 0 0 . 2 0

Plane St ra in S p e c i m e n Bammann' s Viscoplast ic Model

P0=0.99990 , p0i=0.99950 .

I I I

0.05 0.10 0.15

622

(a)

1 . 6 I I I I o Maximum Load

H.D.

........... X 1.2 c-c-- / i . x

.¢ J2-CA'

"~ /..~\~" J.-C (b) b o 0 . 8 c - c

0.4 Plane Strain Spec imen B a m m a n n ' s Viseoplastic Model

Po=0.99990 p~=0 .99950

0 .0 i i I I 0.00 0.05 0.10 0.15 0.20 0.25

In(l +u/l o )

Fig. 17. Plane strain specimen: local and global material responses during neck development predicted by the associative ( C ~ ) , non-associative (C~2A) and partially coupled (J2~2, Jr-C, J2-CAn) frameworks, using an initial imperfection of ~oi = 1.5 × l0 -4.


necking, the material response deviates from the homogeneous deformation path. These computations were carried out using Vgoi = 1.5 x 10 -4. The effect of the magnitude of the initial imperfection on these responses is shown in Fig. 16, where the computed results with Ooi = 1.5 x 10 -4, 5 x 10 -4 , 10 × 10 -4 and 100 × 10 -4 are presented. As in the axisymmetric case, Fig. 16 shows that a wide range of Ooi can be used to predict the onset of necking without an appreciable change in the local and global material responses.

0.4

v 0.5

0

0

0.2

0

> 0.1

0 >

0.0 0.0

l l l 1 I

Plane Strain Specimen Bammann's Viscoplastic Model Associative Framework (C-C)

- p0=0.99990 1 po1=0.99985 3 p01=0.99900 2 po1=099950 4 po1=0.99000

o Onset of Necking / i , 2 , 3 / / 4

H.D. I I I I I

0.1 0.2 0.5 0.4 0.5 0.6

In(Ao/A)

(a)

1.0 I , ~ ] , L

0.8 J~-CAn J~-CA

.0 o 0 . 6

c-cA/ ! ] ] Plane Strain Speeimer (b)

/ Bammann's Model -

i 0.4- / / p0=0"99990 'pOt=0"99950 moto.al / / . Onsoto, Necking/

0.2 - p a r t i c l e / / C / -

0 . 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6

m(Ao/A) Fig. 18. Plane strain specimen: porosity evolution response at center of specimen during neck development for (a) different initial imperfections Ool using the associative framework C ~ and (b) several model frameworks using an initial imperfection of ~9oi = 1.5 x 10 -4.


The effect of the model framework used to predict the onset and growth of necking on the material behavior is studied in Fig. 17, for an initial imperfection of ~9oi = 5 × 10 -4. Note that some differences start to be seen among the models in the predicted material response (mainly the global response) after the onset of necking (post-necking behavior).

The porosity evolution response obtained at a material point at the center of the specimen using (a) different magnitudes of initial imperfection and (b) several model frame-

Z" E 1.2

E .~ 0.8 <

cq

o b 0 . 4 I:::

(~,) I .6 r J , , I .6

o Maximum Load

o

~ o

0 . 0 0 .0 (

Bammann's Viscoplastie Model >arfiially Coupled Framework (J2-CA)

Po=0.9999 , p0i=0.99950 I Homoff. Deform. I

0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0

1.2 2

CO

o.8 v

0 . 4 o b

0 . 0 0 . 2 5

in(l+u/l o)

( b ) 0 .5 J i J

Bammann's Viscoplastic Model t Partially CoupLed Framework (Jz-CA)

0 . 4 P0 =0"9999 ' Poi=0"99957 Homog. D e f o r m . / A X

P

, , , , . j . ..... • .. ......... PS

o . 1

• PS o ~JX 0.0 i I I i

0.00 0.05 0.10 0.15 0.20 0.25

ln ( l+u /1 o)

~ " 0 . 5 <

o < 0 . 2

Fig. 19. Global material responses during neck development of both axisymmetric and plane strain specimens using the partially coupled framework J 2 ~ A with an initial imperfection of ~oi = 1.5 x l0 -4.


works with ~9oi = 5 × 10 -4 is presented in Fig. 18. As was expected, there is a significant difference in porosity prediction among different models after the onset of necking.

Finally, a comparison of the global behavior of both the axisymmetric and the plane strain specimens during neck development is presented in Fig. 19, where the load and the logarithmic strain are plotted as a function of the averged extensional strain. The results were obtained using the partially coupled framework J2-CA with an initial imperfection of tgoi -- 5 × 1 I) -4 . Note that the onset of necking starts approximately at the same strain level. Necking initiates in each case slightly after the maximum load is reached, within 1% additional nominal strain. As was observed before, the plane strain specimen develops a higher neck curvature than the axisymmetric specimen. This results in a higher triaxial stress state in the neck, and hence in higher load-carrying capacity of the specimen (Fig. 19(a)).

VI. CONCLUSIONS

In this paper, a general constitutive framework for porous viscoplasticity with isotropic damage was presented. This framework considered either explicit or implicit void growth models along with combined nonlinear isotropic-kinematic hardening rules in a hardening minus recovery format. Both associative and non-associative inelastic flow rules were considered. The non-associativity of the flow rule was due to compressibility (void growth) eftects. Void nucleation effects were neglected. Moreover, the contribution of plastic spin was neglected in taking objective rates of stress quantities.

Using the general framework, three particular model frameworks for porous viscoplasticity were identified. These were denoted as associative, non-associative and partially coupled frameworks. The structure of a specific model framework was 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). The void growth factor, /3, was introduced in the course of the development to characterize a specific void growth model. This variable was used to describe the three model frameworks for porous inelasticity. It was shown that/3 prescribes the direction of evolution of the inelastic rate of deformation associated with void growth; hence, it was used to define, together with a pressure-dependent yield/flow potential, the associative versus non- associative structure of the inelastic flow rule.

A procedure based on the condition of associativity of the flow rule was developed to (i) obtain the explicit form of the void growth model defined by a pressure-dependent flow potential (implicit void growth model) and (ii) derive the flow potential corresponding to an explicit void growth model in order to define an associative flow rule. An explicit void growth model generally defines a non-associative flow rule, unless procedure (ii) is used. Also, a number of explicit void growth models corresponding to the elliptic and Gurson- Tvergaard flow potentials were derived using procedure (i).

A partia![ly coupled porous viscoplastic model was formally introduced in this work. It was defined as a deviatoric viscoplastic model (with a yon Mises flow potential) modified due to the presence of damage (porosity). Specifically, damage was introduced to reduce the strength of the matrix material, enhance plastic flow and degrade the elastic constants. By nature, the partially coupled model is non-associative.

Bammann's partially coupled rate- and temperature-dependent state variable viscoplastic model was extended in this paper to deal with associative and non-associative frameworks. This model originally used avon Mises flow potential along with the Cocks and Ashby void


growth model to define the evolution of porosi ty. The extension o f this model assumes an elliptic flow potent ia l and void growth models o f general nature (explicit and implicit).

Single-element c o m p u t a t i o n s were carr ied out with B a m m a n n ' s viscoplast ic mode l to c ompa re the pe r fo rmance o f explici t and implici t void g rowth mode l s using the associative, non-assoc ia t ive and par t i a l ly coupled f rameworks . These c o m p u t a t i o n s showed tha t for a pa r t i cu la r void g rowth model , the par t i a l ly coup led f r amework gives the stiffest s t rength and po ros i ty evolu t ion response when c o m p a r e d to e i ther an associat ive or a non-assoc ia t ive f ramework .

A part icular appl icat ion o f the model f rameworks with Bammann ' s model was the numerical s tudy of the effect o f necking on the local and global mater ial response of both axisymmetric and plane strain tensile specimens. The numerical results showed no effect o f the part icular f ramework on the onset o f necking, but strong effects on local post-necking behavior and resulting ductility. Clearly, the relative accuracy o f different f rameworks and void growth models is best assessed by careful experiments which examine the post- localizat ion behavior.

Acknowledgements--The authors wish to acknowledge the support of this research by the U.S. Army Research Office, as well as the support of the U.S. National Science Foundation. The authors are also grateful to Dr. D.J. Bammann of Sandia National Laboratories for fruitful discussions and for supplying viscoplastic model constants.

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APPENDIX

This appendix presents the procedure to integrate the constitutive model frameworks presented in this paper. This scheme has the feature of being explicit in the plastic flow d:rection and hardening functions but implicit in the incremental plastic strain (semi-implicit scheme (Moran et al. [1990])). This feature reduces the equation-solving effort during the state update to the solution of a non-linear algebraic equation in the incremental plastic strain, which can be solved by some iterative technique such as Newton's method. For rate-dependent problems, two approaches to this integration scheme can be developed (Marin [1993]). These are denoted the kinetic equation and the rate-dependent (dynamic) yield condition approaches, respectively. In general, both approaches give similar numerical results (Yoshimura et al. [1987]; Matin [1993]). To illustrate the main steps of this integration procedure employing both approaches, we will use the associative version (np = no) of Bammann's porous viscoplasticity framework, eqns (45)1, (46), (51) and (50). For this purpose, Bammann's constitutive model is written as follows:

Kinetic equation:

/0 d = ~--- g(o" - A, R, 0) (A.1)

Rate-dependent yield condition:

Fd(tr -- A, R, 0,/~d) = F(tr - A, R, 0) - h(O,/~d) = 0

Evolution of the Cauchy stress (tr) and state variables (A,R,0):

(A.2)

o" ---- C(zg) : D - V/1 + ~2/~dC(0) : na

A = V q + ~2 XAPd -- XX k = V/1 + 32 Z ~ d -- Xh 0 = V/I + ~2X~&

where #d = [IDdPI[, h = h3VSinh -1 [(2/3)l/2pjj], and the x-factors are given by

2 kd [[A[IA]x, ' x~ __ ~//~ 1 k_s ItAII A XA = V/'~ [ (1 - 0)bnA 31--

XO = (1 --~9)tr(na), + ~ 2 1 -- (tr(no)) 2 X~= 1

(A.3)

(A.4)

In te rna l state var iable concepts 667

Note that the differential evolution equations for tr and A, which involved the Jaummann rate, have been written here in terms of the material time derivative. This assumes that a numerically objective transformation of the constitutive equations have been carried out, which accounts for local rigid body rotation (Lush et al. [1989]). Small elastic stretch and rotation is assumed. For the integration of these constitutive equations, we will consider the configuration of the body at times tn and tn+l with t~÷l = tn + At . Accordingly, subscripts n and n + 1 on variables indicate that the variables are evaluated at times t, and tn+l, respectively. Then, the input to the integration scheme consists of the state (o'(~).A(n),R(,,),zg(.)) plus the strain increment Ae = [%+1 Ddt, and the goal is to determine din the updated state (o-(. + 1),A(n + 1),R(n + 1),Lq(n + 1))"

Kinetic equation approach. Integrating the kinetic equation, eqn (A. 1), from tn to tn + 1 gives

Apd = Atgo(tr, A, R, tg) (A.5)

where A p d := Pd(n + 1)-Pd(n) and the notation go means that the arguments of the function g are evaluated at time to = t~ + OAt (generalized mid-point scheme), i.e.

o" - o'(o ) = (1 - O)cr(n) + 0o"(n+l ) = O"(n ) -{-- 0Ao"

A -- A(0) = (1 - 0)A(,,) + 0A(n+l) = A(,,) + 0AA

R = R(o) = (1 - O)R(n) + OR(n+l) = R(n) + OAR

0 - 0(o) = (1 - 0)0(~) + O0(.+l) = 0(~) + OAO

(A.6)

where A(.) = ( ' ) ( n + l ) - (')(n) and 0 < 0 _< 1. The increments Air, AA, A R a n d A 0 can be expressed in terms of Ap by integrating their corresponding evolution equations, eqns

_ _ S S S (A.3), with respect to At, assuming XA -- XA(n),XA = Xa(~), XR = XR(n), X~ = XR(n), X~ = Xo(~), X, = X, ( , , ) , C ( 0 ) = C ( 0 ( . ) ) = C ( , , ) , / 3 = 3 ( ~ ) a n d n o = n , , ( . )

AO" = C~(n): AC - V/1 + ~/~n)ApdC~(n) : n~(n)

AA = ~/1 + :3~n ) XA(~)Apd -- X~(~)/Xt

AR = 3/1 + ~n)XR(n) Apd -- XSR(n) A t

A~9 ---- 3/1 +/3~n ) XO(n)Apd

(A.7)

Substituting eqns (A.6) and (A.7) into eqn (A.5) will result in a nonlinear algebraic equation for Apd which is solved iteratively using Newton's method, i.e.

Ap(/+l) = Ap(d i) - Ap(i) -- Atg°i) (A.8)

2h~')r ~ :C~(n) : art(n)

where go' (') = dgo/d( . ) and


/_/(oi) 1 [20Ffo i) n~) O1S°i) (O~i) 1 -d-'~')XO(n) ] (A.9)

In eqns (A.8) and (A.9), the superscript i denotes values obtained on the i-th iteration and i + 1 indicates values to be found on the (i+ 1)-th iteration. After solving eqn (A.8) for Apd, the updated state (a(, + 1),A(,+ 1),R(n+ 1),~(n + 1)) is obtained from eqns (A.6) and (A.7) with 0= 1.

Dynamic yield condition approach. In this case, the dynamic yield condition, eqn (A.2), is evaluated at time to, using eqns (A.6) and (A.7) and the approximation abd = ,bd(o) = Apd/At, i.e.

Fd(Apd ) = Fd(0) (0" -- A, R, O,pd) = Fo(cr - A, R, O) - ho(~q,pd) : 0 (A. I0)

Equation (A. 10) is again a nonlinear algebraic equation for Apd. This equation is solved by Newton's method using

F(o i) _ h~ i) ip(di+l):ip(di) 0 [ ~ Ob'(°i) (n~) J[-~-i~8 i,) "4- 10h~i)] ( i . l l )

- L . ( . ) : c ( . ) : n~(~) OAt OPdJ

where 11o (i) is also given by eqn (A.9). The updated state is then found using eqns (A.6) and (A.7) with 0 = 1.

Constitutive (stress) Jacobian matrix. When implementing the foregoing constitutive model and the corresponding integration schemes in implicit finite element codes (e.g. ABAQUS), we require the so-called constitutive (stress) Jacobian J for the global New- ton's scheme (Simo & Taylor [1985]). In this case, J is computed from

_ OAo- - v /1 .o ( , ) ) ® J~- O~n+l 0Ae - C-(n) + OAp (A.12)

where eqn (A.7) has been used. The expression for Op/aAe is obtained from either eqn (A.5) (kinetic equation approach) or eqn (A.10) (dynamic yield condition approach), along with eqns (A.6) and (A.7). The resulting expressions for J are listed next.

Kinetic equation approach:

~ h~oV OFo C . OAt ~ _(n) : n,,(,) ® C~(n) " n,.(n) J .= C(.) - (A.13)

~h3--~z °F° (n~(o) 1 + OAt ~ : C(n) : no(n) + Ha)

Dynamic yield condition approach:


where Ho is given by eqn (A.9) (without the superscript (i)). Note that all quantities in eqns (A.13) and (A.14) with subscript 8 are evaluated at the last Newton iteration.

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