Finite element implementation of gradient plasticity models

Computer methods in applied mechanics and englneerlng

ELSEWIER

Comput.

Methods

Appl. Mech. Engrg.

163 (1998) 33-53

Finite element implementation of gradient plasticity models Part II: Gradient-dependent

evolution equations

S. Ramaswamya, N. Aravas b** “Sundia National Laboratories, Solid and Material Mechanics Department, Materials and Engineering Sciences Center, 7011 East Avenue. Livermore. CA 94S50, USA “University of Thessaly, Department of Mechanical and Industrial Engineering, Pedion Areos, 38334 Voles, Greece Received

3 September

1997

Abstract

The variational formulation and finite element implementation of a class of plasticity models with gradient-dependent yield functions is covered in detail in Part I. In this sequel, attention is focussed upon the finite element formulation and implementation of a different class of plasticity models wherein spatial element solutions are obtained for pathological dependence of the gradient-type model is used. 0

gradients of one or more internal variables enter the evolution equations for the state variables. Finite the problems of localization of plastic flow in plane strain tension and of a mode-1 plane strain crack. The finite element solution on the size of the elements in local plasticity models disappears when the 1998 Elsevier Science S.A. All rights reserved.

1. Introduction A general presented nonlocal

review

of research

on non-local

continuum

models

in the

in the Introduction of Part I of our work (this issue), wherein rate-independent plasticity models involving gradient-dependent

context

of plastic

the formulation yield

functions

deformation

is

of a general class of is described

in detail.

In this sequel, we focus our attention upon a different family of rate-independent gradient-type plasticity models in which spatial gradients of the plastic multiplier and state variables enter the evolution equations for the state variables. As an example to this class of models, we present a gradient-dependent version of Gurson’s [4] plasticity model for porous metals, in which the classical evolution equation for the porosity is now replaced by a gradient-dependent evolution equation. The modified equation involves spatial gradients of porosity and is therefore of a spa&-temporal nature as opposed to an ordinary rate equation in the classical model. Emphasis is primarily placed on the Gale&in finite element formulation and implementation of the gradient-dependent model along with related issues such as the interpretation of ‘plastic loading’ in the discretized problem. Among related works involving the numerical treatment of non-local models in the context of porous ductile materials, we mention the recent papers by Tvergaard and Needleman [6,7], which use an elastic-viscoplastic version of Gurson’s model and describe the evolution of the porosity through an integral equation as presented in [5]. The organization of this paper is as follows. Section 2 outlines the general form of plastic constitutive equations with gradient-dependent evolution equations for the state variables. A review of the standard Gurson model is also presented, which is followed by the presentation of the gradient-dependent Gurson’s model. The

* Corresponding 00457825/98/$19.00

author 0 1998 Elsevier Science S.A. All rights reserved.

PII: SOO45-7825(98)00027-9

34

S. Ramaswamy, N. Aravas I Comput. Methods Appl.

Mech. Engrg. 163 (1998) 33-53

full boundary value problem for an elastic-plastic body with gradient-dependent evolution equations is outlined in Section 3. The description of the mixed finite element formulation used in the solution of the problem is given in Section 4. The specific application to the Gurson’s model together with issues such as the numerical integration of the constitutive equations, the conditions for ‘plastic loading’, and the linearization of the finite element equations are described in detail in Section 5. Finally, Section 6 presents finite element solutions for the problems of localization of plastic flow in plane strain tension and of a mode-1 plane strain crack. It is found that the pathological dependence of the numerical solution on the size of the finite elements used in local plasticity models disappears when the gradient-type model is used. The tensor notation used in this paper is the same as that of Part I.

2. Plastic constitutive equations of the gradient-type Classical rate-independent plasticity models are briefly reviewed in Part I wherein the formulation of a general class of nonlocal plasticity models involving gradient-dependent yield functions is described in detail. In this sequel, attention is focussed upon a different family of gradient-type plasticity models in which the gradient terms enter the evolution equations for the state variables. In particular, we consider plasticity models of the form @(p(a,s,‘.. .,sJCO,

A 20,

/i@=O,

(1)

Dp=/ih'(u,s,,...,s,),

(2)

s,=s,(u,ssl,.. .,s,, Qsa, V2s,, /i, Vh’) , a = 1, . . . , n ,

(3)

where @ is the yield function in stress space, u is the Cauchy stress tensor, s, (a = 1, . . . , n) is a collection of scalar state variables, i is a non-negative plastic multiplier, N and S, are isotropic functions of their arguments. Since we deal with rate-independent materials, the functions ?a are homogeneous of degree one in A’. The presence of gradients in the evolution equations for the state variables requires additional boundary conditions on the boundary of the plastic zone. The general form of these boundary conditions is A=0

onSP,

s, =s, , -

on Sp ,

a=l,...,n,

as u-s 3%

(4)

no13

a=l,...,n,

(5)

on Si,

where S, and S,, are known functions over Sp and SI, respectively, n is the outward unit normal to SE, and Sp U 5’: = Sp, where, as defined in Part I, Sp is the boundary of the plastic zone VP, Sp is the elastic-plastic interface, and SE = Sp n S, where S is the outer surface of the body. 2.1. Example: Gurson’s plasticity

model

This section presents a brief review of the Gurson [4] model for porous metals, and a suggested modification to it of the gradient-type. 2.1.1. Review of the Gurson model The yield function depends on the linear invariant of u and the quadratic invariant of u’. The model involves two state variables (CZ= 2): (i) the equivalent microscopic plastic strain in the matrix material Tp, and (ii) the ‘porosity’ J which is defined as the volume fraction of the voids. The yield condition is of the form @cP? 42

TP,f)’

_2!--[

%JTP)

1 *

f2fcosh

[

3P ~ 2a*(TP)

1

-(l+f*)=o,

(7)

where q =dm is the von Mises equivalent stress, u’ being the deviatoric part of a, p = -ukkk / 3 is the hydrostatic stress (positive in compression), and rm is the flow stress of the matrix material. The yield function is used as a plastic potential (N = a@/au), so that

S. Ramaswamy,N. Aravas I Compur.MethodsAppl. Mech. Engrg. 163 (1998) 33-53

DP=);$‘y t [F-f

sinh($JZ]

.

(8)

The evolution equation for the first state variable i” is based on the requirement work’ u :Dp equals the microscopic one (1 -f)r,,,iP, so that s,

(ly:,9p

=iP=

=A

urn

35

that the macroscopic

uzN

‘plastic

(9)

(1 -mm.

As the porous metal deforms plastically, its porosity can change due to the growth (or closure) of the existing voids, and due to the nucleation of new voids, i.e. s’, =f=f,,

+f,,,,

(10)

.

Assuming that the matrix material is plastically void-volume change, one can readily show that & = (1 - f)D;f We consider

= A’(1 -f)N,,

plastic strain-controlled

incompressible

and ignoring

the elastic contribution

.

to the

(11) nucleation

such as

fnuc, = s4fP ,

(12)

where, as suggested by Chu and Needleman [3], the parameter .z&!is chosen so that the nucleation a normal distribution with mean value eN and standard deviation sN:

strain follows

(13) f,, being the volume

fraction

of void nucleating

particles.

2.1.2. A gradient-type modification to the evolution equation of porosity In this section, we discuss different forms of the evolution equation for porosity which involve additional terms that account for such effects as void diffusion, interaction, and coalescence. The derivation of these evolution equations is discussed in the following. We consider an arbitrary volume 0 of the porous elastic-plastic continuum. The amount of ‘void space’ in 0 can change due to the growth (or closure) of the existing voids, the nucleation of new voids, and the flux of voids that enter L! by crossing its boundary IX!. Therefore, we can write (14) where n^ is the outward normal to 80, andj is the void flux vector. The negative sign on the right-hand side of the last equation is needed because J,, j . ri dS is the outward void flux. Using Green’s theorem and taking into account that the volume L! is arbitrary, we conclude readily that

The equation that describes j depends on the physical mechanism that causes the flux of voids. Here, we consider two such mechanisms: (i) diffusion, and (ii) void interaction and coalescence in a plastically deforming metal. In the case of diflusion, the simplest form for j would result if we assume that the flux is proportional to the porosity gradient: j=

-cVf,

(16)

where c is a constant diffusion coefficient (c > 0) with dimensions (length)2 /time. The negative sign in the above equation is due to the fact that voids will move in a manner which will decrease the porosity gradients (see Fig. l(a)). In this case, Eq. (15) becomes f=h’(l-f)Nk,+.diP+cV2f.

(17)

36

S. Ramaswamy,

N. Aravas

I Comput. Methods Appl. Mech. Engrg. 163 (1998)

33-53

fA

0000

000

0000

;;z

0000 0000

-

J

r

n

(a)

f

I iW

Fig. 1. Schematic

representation

of flux of voids.

Next, we consider the case of interacting voids in a plastically deforming matrix. Here, one can argue that, as the matrix deforms, larger voids ‘attract’ smaller voids and, eventually, coalesce with them (see Fig. l(b)); i.e. the void flux now is in the opposite direction of that in diffusion. A simple expression forj that has this feature is

j=h$Vf,

(18)

where 8 is a material length scale, and g0 a reference stress. In the above equation, the void flux is taken proportional to the ‘amount of the rate of plastic straining’, as measured by i, which controls the magnitude of Dp. We also note that j is now non-zero only when plastic flow occurs (h’ # 0). In this case, Eq. (15) becomes

(19) We conclude this section by emphasizing the very different effects that the gradient terms in Eqs. (17) and (19) are expected to have. In the case of diffusion, the term -cV’f in (17) will tend to eliminate any existing porosity gradients; whereas the gradient terms in (19) will have a ‘destabilizing’ effect in the solution, in that they tend to intensify any existing porosity gradients during plastic flow. The presence of the gradient terms in the evolution equation of porosity necessitates additional boundary conditions forf on the boundary of the plastic zone. On the elastic-plastic interface Sp, the value of porosity is specified. In particular, if a point A on Sp yields for the first time, the boundary condition at that point is f =f,, f, being the initial porosity in the material; on the other hand, if point A has yielded previously, the corresponding value of the porosity equals the value off when the point was last actively yielding. We also assume that no voids can ‘enter’ the continuum, so thatj en = 0 on the part of the boundary of the plastic zone SE that belongs to the outer surface of the body. This condition is equivalent to

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 33-53

g=O

0nSE.

37

(20)

3. The elastic-plastic

boundary

value problem

We consider an elastic-plastic continuum, which in its undeformed state occupies a volume V, in space. Let V be the corresponding volume of the continuum in its deformed state. We formulate the general problem for the elastic-plastic material of the V.a+b=0, D =;

(21)

[Vv +

(Vr#] ,

(22)

D=D”+DP,

(23)

&=Ce:De,

(24)

(25) @(u,s

iao,

s,)=sO,

I,...)

i,_l=s,(a,s,,...,

s,,

A@=O,

vs,,v2s,,A’,V/i)

)

/i=O

in the current deformed configuration.

(28)

onS,,

(29)

onSp,

s, =s, , -s,,

(30) a=l,...,n

as an

are evaluated

on S, ,

n.a=t^

-&-

(27)

(Y=l,...,n,

in V, where the notation of Part I is used, and all gradients The corresponding boundary conditions are u=ti

(26)

9

cu=l,...,n

on Sp ,

(31)

on SE,

(32)

where ti denotes the prescribed displacements over S,, i the prescribed unit normal n, and S, U S, = S, S being the boundary of V.

4. Variational

tractions

over S, which has an outward

formulation

A variational formulation of the boundary value problem listed in Section 3 is developed in the following. In order to derive the discrete governing equations, we first integrate the evolution equations for the state scheme: variables over the time interval [t,, I,, + I = t, + At] with a backward-Euler As, = s,(a,+,,

(sr),,+r’. . ., &J,+,,

O’SJ,,+JV~~J,+,,

AA, (AN) >

(33)

where we took into account that the functions S, are homogeneous of degree one in h’. We start by expressing the equilibrium equations (21), the traction boundary conditions (29), the yield condition (26a), the evolution equations for the state variables (33), and the boundary conditions (32), in the following variational statement: Find (1) U(X, t) E H2(V) satisfying ujs, = ri , (2) h(x, t) E H’(V) satisfying A],, = 0, and (3) s,(x, 2) E H2(V), a = 1, . . . , n’ satisfying s,lsp =S,, such that for all u* E L2(V) satisfying u *Is, = 0, for all A* EL2(V), for all qz EL2(V), and for all a: EL2(V) satisfying a*,(,? = 0,

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 33-53

38

A&, A, s,, . . . , s,, u”, A*, q;, . . . , q;, a;,

+

I VP

. . . , a;> =

I

1/(qj

+ bi)uT dV +

2 @((a,$,, . . . , s,)A* dV + or=, JAs,-s,(a,+...,

s,,,

I

+ $,

sp

Csa,jij

-in,)a*,

dS

=

O

Vs,,

V2s,,

I

s (it - qjnj)vT

AA,

dS

V(AA))lq*, dV

(34)

>

I

where CT= a(~, A, s,), and a comma denotes partial differentiation

5. Application:

with respect to position,

i.e. A,i = t3A/kxi.

The Gurson model

We consider the case where the evolution coalescence, i.e.

equation for porosity is modified to account for void interaction

i=n(l-f)N,,+&-$7.(Aq).

and

(35)

In order to specialize the variational statement shown in Eq. (34) to the case of the gradient-dependent Gurson model, we start by first integrating the evolution equations for Ep andf over the time interval [t,, t,,, = t, + At] using a backward-Euler scheme to get (1 -f)a;,

AZ’=

Aha:

N,

(36)

Af=AA(l-f)N,,+sPA~p-~V~(AAVf)

(37)

where all quantities are understood to be evaluated at time tn + , . The variational statement is then written as Find u(x, t) E H’(V) satisfying u Is = ti, A(x,t) E H’(V) satisfying i = 0 on Sp, E’(x, t) E L2(V) satisfying satisfying u*Is = 0, for all gplsP = Cp, f(x, t) E H*(V) satisfymg flSP =f such that for all u* EL*(V) ” A* &L2(V), for all qT, q; EL*(V), and for all a: E L2(V> satisfying u?~l,~ = 0, A(u, A. Ep, f, u*, A*, qTI q~, Us) ~

+ I VP (q

+

+

-

a,,,cu)A* dV +

I VP

IV

(a,,.j

f bi)~T

dV-

where u = a(u, A, Ep, f),

a@

crijNij=p-+qap

1

q;dV

(38)

0, (Y= (1 +f* - 2f cosh[3p/(2~,)l}“2,

a@ a4

(~j~j - ~~)uT ds

AEp - AA oyjjnT,.]qT dV

[(l -f)a,

Af-AA(l-f)N,,-i-PAZ”.+$V*(AAVf)

j-,iju; dS =

I s

and

Nkk=--.

a@ ap

(39)

Using Green’s theorem and setting a T = -( e2 /go,) AA q; results in the following variational statement: Find u(x, t) E H1 (V) satisfying u Is = ~2, A&, t) E H’(V) satisfying A’= 0 on Sp, Cp(x, t) E L*(V) satisfying EPlsp = ZP, f(x, t) E H’(V) satisfy&g flsp =j such that for all u* E H1 (V) satisfying u*(~, = 0, for all A* &L2(V), for all qT EL*(V), for all q; ‘E H’(V) satisfying q;lsp = 0,

S. Ramaswamy,

A =

(qjD;

N. Aravas

- biuT) dV-

I Comput. Methods Appl. Mech. Engrg.

163 (1998) 33-53

39

iivr dS I S

(40) where u = a@, A, Cp, f) and D; = l(w$} ,

h*(x)> = ~Nx)lbq~,

s; = LW~J(~~~

9

sT = LW)J(w:)

(43) (44)

9

where [N(x)], Lh ,(x)1, Lh,(x)l and Lh,( x )A are arrays containing are the vectors of nodal quantities of the form

tw,j=tu;,u;,u;,A’, Ep’,f Lw$] = LuT’, u$,

u;‘,

A* NNODE , qy-E,

standard element shape functions,

{w,} and {w:}

‘, . . , uyNoDE, uroDE, ~~~~~~~~~~~~~~ EPNNoDE,fNNoDEl ,

(45)

A*‘, qy’,

q;‘,

. . . , vTNNoDE,u

t

9

and the notation

La]= {a}’ is used. Using

the above

(47)

{VA*(x))= b,(x)l{wf),

(48)

Ps;cx)> = b,Wlw?,

(49)

Let {w} and {w*} be the corresponding global vectors of nodal unknowns. Substituting the above equations into the variational equation (40), and taking into account that the resulting equation must be satisfied for arbitrary {w*}, we arrive at a non-linear equation for {w} of the form

{G> = ON I

where

(50)

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 3’3-53

40

[BIT{ U}

W+NzM(j--

e

dV - 1

e

[N=](b) dV - / s [N](i) dS e

(51) where the subscript e denotes the element-number, and NELEM is the total number of elements in the finite element mesh. The last equation is a nonlinear equation that must be solved for the nodal unknowns {w}. 5.2. Numerical

integration

of the constitutive

equations

In a finite element environment, the solution is constructed incrementally with the integration of the constitutive equations being performed at the elemental Gauss points. At a given material point, the solution (F,, a,,, A,, gi,f,) at time t, as well as the deformation gradient F,,, = AF OF,,, and the increments AA, AZ’ and Af corresponding to the time increment [t,, t, + , = t, + At] are known, and the objective is to determine the stress a, + , . It is to be noted that, similar to the situation in Part I, the yield condition and evolution equations for the state variables are satisfied through variational statement and therefore do not figure explicitly along with the other rate equations to be handled at the material level. Likewise as in Part I, the values of AA and AZ’ at each node are required to be non-negative and the distinction between plastic loading (AA > 0) and elastic response (AA = 0) is made as described in Section 5.3 that follows. More specifically, the constitutive equations to be integrated are as follows: D=D’+D’,

(52)

:=Ce:De,

(53)

Ce=(K-5G)H+2G$

(55)

where

with K and G denoting the elastic bulk and shear moduli, respectively, and Z and $ representing the second- and fourth-order identity tensors, respectively. Likewise as in Part I, under the assumption that the Lagrangian triad associated with the incremental deformation gradient A&t) remains fixed in the time interval [t,, t,, ,I, the constitutive equations (52)-(54) can be written as k’=S”+&‘p,

(56)

&Ce:&,

(57)

which are analogous to those of a ‘small-strain’ theory. In the remainder of this section, we discuss the procedure to perform the numerical integration strain equations (56)-(58) along lines similar to ones outlined in [l]. Integration of (57) gives

of the finite

&+1 = a, + C’ : ALP = U” + C” : (LIE - ALP) = &-” - C’ : Al? ) where &-” = Us + C” : AE is the (known) elastic predictor method is used to integrate the plastic flow rule (58):

in co-rotational

(59) stress space. The backward

Euler

s. Ramaswamy, N. Aravas

I Comput. Methods Appl. Mech. Engrg.

163 (1998) 33-53

41

(60)

where

(61) part of &n+ , . In deriving u are the same as those of & Furthermore, introducing the notation

u^ A+ 1 being the deviatoric

Eq. (60) we took into account that the invariants

p and q of

(62) in (60), we have the expression

for the incremental

AEp=fAEP1+AE,R,+, Substituting

plastic strain given by

.

the expressions

(63)

for C’ and At?’ in (59), we find

G”+, =&-e-KAEpI-2GAEyn^,+, Considering

logarithmic

the deviatoric

.

(64)

part of the last equation,

we can show readily that [l]

3&-“’ n^,I+ I =--known, 2q”

(65)

where qe = (3/2)&-“’ : be’, and be’ is the deviatoric part of Ge. With tin+, known, Eq. (64) makes it clear that the determination of bn+, reduces to finding AE,, and AE4. Projecting Eq. (64) along Z and n”, respectively, and taking into account that &,,+, : Z = -3p, +, , &a+, : ri, +, = 4 ,,+,, and nl,,,, : ii,+, = 312 we find p=p”+KAE,

and

q=q’--?GAE,,

(66)

where pe = --Ge :Z/3. The unknowns AEP and AE, are determined as follows. Using the definitions a@/ap and &P/aq, and using the expressions (66) for p and q, we find 3fAhh AEP = - sinh “In

3(p” + K AEJ 2%

Eq. (67a) is a nonlinear algebraic equation (67b) can be solved for AE, to give AE, =

2AAq’ r+f,+6AAG

where

.ri,+,

2Ah (qe - 3G AE,) . AEy = 4i

for AEP, which is determined

by using Newton’s

the derivatives

(67) method, whereas

= known.

With AEP, AE, and n^,+, known, increment a,,, , is found from 0 n+l =R,+,

I

and

(62), evaluating

.R;+,

&,,+, is determined

from (64). Finally,

>

R,+ , is the rotation tensor associated with AF, + , .

the true stress at the end of the

(69)

42

S. Ramaswamy,

N. Aravas

I Comput. Methods Appl. Mech. Engrg.

163 (1998)

33-53

5.3. A note on plastic loading/unloading Part I discusses the procedure for developing and enforcing the discrete Kuhn-Tucker conditions which are appropriate for the present finite element formulation. For purposes of completeness, we will briefly outline the Kuhn-Tucker conditions in the context of the modified Gurson model. In a continuum formulation, the Kuhn-Tucker conditions @(x)CO,

Ah(x) 3 0,

AA(x)@(x) = 0 ,

(70)

must be satisfied at every point of the continuum. In the present formulation, since the yield condition and evolution equations enforced globally, the loading/ unloading conditions are also enforced in a ‘global’ We start with the finite element interpolation (41b) for AA and isolate the nodal element that refer to AA, denote the corresponding column-vector by {Ah:}, and element as AA(X) =

for the state variables are sense. degrees of freedom of the rewrite (41b) within each

Lh, (x)~(AA:}

(71)

where L%,(x)1 is the row-vector of shape functions that are used in the interpolation of AA(x). We also use the notation {AAN} to denote the corresponding global vector that contains all nodal degrees of freedom in the structure that refer to AA. Next, we define the global ‘nodal vector of the yield function’ {a} so that AA(X)@(X) dv= LAA"~{@

(72)

,

which implies that NELEM {a}

=

c

where

{zf}

{zC} =

S2=l

v {h,(x)}@(x) dV , I c

(73)

and the notation of Part I is used. The shape functions in (7 1) are chosen so that

G,WI 3 08 >

(74)

where the notation {a} 2 {0} means that all the components of {a} are non-negative. Using a procedure similar to that of Part I, we find that the discrete Kuhn-Tucker conditions every nodal point as follows: AA”aO,

+=O,

AAre=

For the case of the gradient-dependent

(nosumoveri),

Gurson

model discussed

can be written at

i=l,...,NTOTAL.

(75)

in Section 5, we have that

AA(q - Q_X) dV

LAA”J{%} = 1

(76)

VP

where (Y= (1 +f’

- 2fcosh[3p/(2~~)1}“‘.

From the last equation

{%} can easily be concluded

to be (77)

which is exactly the term in Eq. (51) that corresponds to the yield function. In our finite element calculations, the discrete Kuhn-Tucker conditions are enforced as follows. The solution is determined incrementally, and, at every increment, each node is labeled as either ‘elastic’ or ‘plastic’. The condition AA = Asp = Af = 0 is enforced at all ‘elastic’ nodes, and the set (50) of nonlinear equations is solved for the nodal unknowns by using Newton’s method. Once a converged solution is obtained, that solution is accepted if (1) the components of (} are such that zi < 0 at all ‘elastic’ nodes, and (2) the calculated nodal unknowns are such that AA 2 0 and AZ’ > 0 at all ‘plastic’ nodes. If either of the above two conditions is violated at some nodes, then these nodes are relabeled and the solution

S. Ramaswamy,

N. Aravas

I Comput. Methods Appl. Mech. Engrg.

for the increment is repeated. This process is terminated, that satisfies both (1) and (2) above, is obtained.

5.4. Linearization

when an acceptable

163 (1998) 33-53

solution for the increment,

43

i.e. one

of the finite element equations

The set of nonlinear equations (50) is solved for the vector of nodal unknowns {w}, i.e. for the nodal values of u, A, Sp and A by using Newton’s method. In the following, we discuss briefly the determination of the corresponding Jacobian. We start with the variational equation (40), i.e. A@, A, Ep, J u*, A*, qT, q:) = 0, the linearization of which is expressed formally as A(u, A, E’, A u*, A*, qT, q;) + DA&, A, Ep, J u*, A*, q:, 9;).

(Au, AA, AC’, Af) = 0,

(78)

where DA@, A, Ep, f, u*, A*, q:, q;>. (Au, AA, AZ’,

Af)

1

~A(u+tAu,A+~AA,~p+~AiP,f+~A~~*,A*,q~.q:)

E=O.

(79)

It can be shown easily that DA. (du, dA, dC’,df)

L* : (da - o.dl=

=

+ dl,, a)dV-

d

(q - a,cr)A* dV >

(l-f)a,AEP-AA

-d (1 -d

VP

(i[ VP

p$+q$f

[

(

>I >

Aj-+(l-f)$-&Asp]q;d”)+d(~v~~AAV~.Vq:dV)

q:dV

(80)

where L* =Vv*

and

dl=V(du).

(81)

The quantity da plays a crucial role in the evaluation of the first term in the right-hand side of Eq. (80). A brief outline of the calculations is given in the following. Using the expression CT= R * &. RT, we find da=(dR.R=).u-u+lR.R=)+R.d&R=,

(82)

where d&,=

acT,a&, au,, aEk, au,,

au,

aej a(;, dup+xdA+zdCP+Mdf.

aey (83)

The derivatives a&-/8E, a&-/ ah, a&-/ZP, and d&/df are evaluated easily by using the expressions given Section 5.2, where the algorithm for the integration of the elastoplastic constitutive equations is outlined, and listed in Appendix A. The reader is referred to Part I for the derivation of the expressions for dR . RT, aE/ and (alJ,,,,,/au,) du,. The expression for da is finally given by substituting the expressions for dR. RT and d& into (82) follows:

in are XJ as

(84) where

%t

c,_ = Ri, -

and the expression

for 2 is also provided

in Part I.

af Rjl9

(85)

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 33-53

44

The second, third, fourth and fifth terms (differentials) in the right-hand side of Eq. (80) are evaluated in a way similar to that outlined in Appendix A of Part I; the details of these calculations can be found in [8] and will not be repeated here. Substitution of these expressions and the expression for da into (80) with the subsequent introduction of the finite element interpolations results in DA(du, dh, dgP, df) = tw*h{w} with [J] being the desired Jacobian

6. Numerical

,

(86)

matrix.

examples

In this section, we present results from two example problems solved using the proposed scheme. The first example deals with the localization of plastic flow under plane strain tension, and the second is concerned with the problem of a mode-1 plane strain crack. The material used in our calculations is chosen to be elastic-plastic with the plastic behavior modeled using Gurson’s plasticity model. The yield stress of the matrix material ~~(2’) is described by CP IIN a,=Uo

( ‘+,

)

(87)

’

where co is the initial tensile yield stress of the matrix material, The evolution equation of porosity is of the form

e. = no/E,

and E is Young’s modulus.

The following values are used in the calculations: E = 3OOa,, N = 10, and Y = 0.3, Y being Poisson’s ratio. The plastic strain-controlled nucleation is described by the void volume fraction of void nucleating particles f, = 0.1, the mean strain for nucleation lN = 0.3, and the standard deviation sN = 0.1. The results of our calculations are presented hereunder. 6.1. Localization

under plane strain tension

The geometry used for the localization problem is the same as the one used in Part I. The aspect ratio of the specimen is chosen to be L/H = 2, where L and H are its original length and width, respectively. A schematic representation of one quarter of the specimen is shown in Fig. 2. Each material point in the specimen is identified by its position vector X = (X,, X,) in the undeformed configuration. The specimen is subjected to plane strain tensile loading along the X, direction. The deformation is driven by the prescribed end displacement fi, and the lateral surface on X, = H/2 is kept traction-free. As we are interested only in those solutions that are symmetric with respect to both the X, and X, coordinate directions, we analyze only one quadrant of the specimen as shown in Fig. 2. The applied boundary conditions are: (1) u2=0, cr,,=OonX,=O, (2) u1 =0, glZ=O on X, =0, (3) T, = T, = 0 on X, = H/2, and (4) u1 = 0 = known, u,~ = 0 on X, = L/2, where T, and T, are the components of the nominal traction vector. In order to trigger the initiation of non-homogeneous deformation in the specimen, small imperfections are introduced in the material properties. In particular, the initial porosity distributionf(t = 0) is assumed to vary in the specimen according to the expression

where f. = 0.05. The above equation

indicates

that the initial

porosity

level is highest

in the middle

of the

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 33-53

-1

IFig. 2. Schematic

representation

of one quarter of the specimen.

A typical finite element mesh is also shown.

1.5

1.0

P --

0.5

8xJZaah

--l6x64mah ---UX%UEBh -----32xl28mab tlomgmonll 0.0 La.00

I

0.10

I

0.20

1

0.30

Eln Fig. 3. Load-extension

curves for the ‘local’ material

(P = 0).

0.60

45

S. Ramaswumy,

N. Aravas

1 Compur. Methods Appl. Mech. Engrg.

163 (1998)

33-53

47

specimen (X, = 0) wheref(0) =f,, and lowest at the loading edge (X, = L/2) wheref(0) = 0.97f,. This makes the specimen initially ‘softer’ at its center (X, = 0) and ‘harder’ at its loaded edge (X, = L/2). Four node isoparametric plane strain elements with 2 X 2 integration points are used in the discretization. The initial (undeformed) finite element mesh is uniform in both directions. Four different meshes are used, namely 8 X 32, 16 X 64,24 X 96 and 32 X 128 where the first and second numbers denote the number of elements along the X, and X2 directions, respectively. Two sets of calculations are carried out. The standard Gurson model (e = 0) is used in the first set, whereas the second set is carried out using a gradient-dependent Gurson model with e = 0.2L. For each of the two sets of calculations, solutions are obtained using all four different finite element meshes. Figs. 3-6 show results corresponding to the ‘local’ Gurson model (t! = 0). Fig. 3 shows the ‘load-extension’ curves as calculated using the four different meshes. The normalized load $ plotted in Fig. 3 is defined as F = F/(bH/2), where F is the calculated total axial force from the analysis of the one quarter of the specimen, and b its undeformed thickness. The solid curve in this figure corresponds to the homogeneous solution. It is found that the numerical solutions obtained using different mesh sizes agree with each other up to a macroscopic At that strain axial strain E’” = 0.13, where the ‘macroscopic axial strain’ is defined as 6’” = ln[l + c/(L/2)]. level a shear band forms, and beyond that point the obtained numerical solutions exhibit a strong meshdependence, as is evident from the curves shown in Fig. 3. The contours shown in Fig. 4 demonstrate the evolution of ,f in the specimen up to the point of shear banding as calculated by using the 24 X 96 mesh. It should be noted that, although only one quarter of the specimen is analyzed, the contour plots presented in the following are shown for the whole specimen. In the early stages, f is almost uniform with a very slight inhomogeneity due to the presence of initial imperfections. At the macroscopic strain level of 6’” = 0.13, the inhomogeneity begins to grow into a shear band as shown in the figure. Fig. 5 shows the variation off along X, = 0 and X, = H/2, at a macroscopic strain level of E’” = 0.165, as calculated using the four different meshes. The width of the shear band tends to zero as the mesh is refined, and the strong mesh dependence of the solution is clear. It should be noted, however, that this mesh dependence appears only beyond the critical strain E‘” = 0.13. At smaller extension levels, the corresponding profiles off are almost identical for all four different meshes. Fig. 6 shows contours off at a macroscopic strain level E’” = 0.165, as calculated using the four different meshes. Once again, the strong mesh dependence of the solution is evident. Figs. 7-9 show results for the case of a gradient-dependent Gurson material with 8 = 0.2L. Fig. 7 shows ‘load-extension’ curves as calculated using the four different meshes. The solid line corresponds again to the homogeneous solution. The calculated load-extension responses obtained using different mesh sizes are found to converge on a single curve as the mesh is refined. Fig. 8 shows the evolution profiles off along X, = 0 and

Fig. 5. Variation off different meshes.

for the ‘local’ material ( f? = 0) along X, = 0 and X, = H/2 at a strain level E’” = 0.165, as calculated

using the foul

S. Ramaswarq,

N. Aruvas

15

I Comput.

Methods

Appl.

I

Mech. Engrg.

163 (1998)

I

I

I

49

33-5.3

-y----l --1.0

P 0.5

Fig.

Load-extension

-8X32mnb - - 16r64d ---24x%mcsb -..-.- 32rl28meab _ homogc-

curves for the gradient-dependent

Gurson material

with 6 = 0.2L.

0.30 0.25 0.20

f

a15

f

a15

a10

a10

a05

a05

0.30

035 0.20

f

a15 a10 a05 0.00

1

a0

0.5

1.0

1.5

X/cm) Fig. 8. Evolution off for the gradient-dependent as calculated using the four different meshes.

Gurson material with 6 = 0.2L along X, = 0 and X, = H/2 up to a strain level

l“’ = 0.185.

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 33-53

51

Fig. 10. Finite element mesh used for the mode I crack under plane strain.

X, = H/2, up to a macroscopic strain level 61n = 0.185, as calculated using of the shear band is now independent of the mesh size when the mesh contours off at a macroscopic strain level 6’” = 0.185, as calculated using contours shown in Fig. 9 make it clear that the solution is independent of enough.

the four different meshes. The width is refined sufficiently. Fig. 9 shows the four different meshes. Again, the the mesh size when the mesh is fine

4.2. Mode I crack under plane strain We consider the problem of ‘small scale yielding’ of a mode-1 crack under plane strain conditions. Fig. 10 shows the finite element mesh used in the calculations, and Fig. 11 shows the details of the mesh in the region near the blunt crack tip. Because of symmetry, only the region above the crack plane is analyzed. A total of 1333 plane strain 4-node isoparametric elements with 2 X 2 Gauss integration points are used in the calculations; the corresponding number of nodes is 1408. The outermost radius of the mesh is R = 15OOr,, where r,, is the initial radius of the circular notch. The symmetry conditions are enforced ahead of the crack plane, and displacements consistent with the leading term of the asymptotic expansion of the near-tip solution for a mode-1 crack in an isotropic linear elastic material are applied on the outermost radius on the outer boundary of the mesh (e.g. see [2]). The applied displacement field is of the form U =s

&i(v,B), V--

(90)

where r and 8 are polar coordinates with the origin located at the center of curvature of the notch, K, is the mode-1 stress intensity factor, and U are dimensionless universal functions. The solution is developed incrementally as K, increases from zero corresponding to mode-1 loading. The initial void volume fraction is taken to be uniform at 0.05. Two sets of calculations are carried out. One in which the ‘local’ form of the Gurson model is used (t= 0), and another in which e= rO. Fig. 12 displays the variation of porosity ahead of the crack tip as the applied load increases from zero to K,/(o;,&) = 25.51, for the case of the ‘local’ Gurson model (e = 0). In that figure, x, denotes the current position of a material point ahead of the crack tip, where the origin of the coordinate system is located at the

Fig.

I I. Near-tip mesh used in the crack problem.

52

S. Uamaswamy,

N. Aravas

/ Comput. Methods Appl. Mech. Engrg, 163 (1998) 33-53

Fig. 12. Evolution off for the classical Gurson material (P = 0) up to a load level corresponding to K,/(o,,r,) = 25.51

Fig. 13. Evolution off for the gradient-dependent Gurson material (C = r,,) up to a load level corresponding to K,/(q,,r,,) = 25.5 I ~

center of the undeformed with C = r0 are shown concentration of porosity Detailed calculations of elsewhere.

notch. The corresponding evolution profiles for the gradient-dependent Gurson model in Fig. 13. It is noted that the gradient type Gurson model predicts a smaller at the root of the notch and reduces the spatial gradient off ahead of the crack. this problem, for different values of f are now underway and will be reported

Acknowledgments This work was carried out while both authors were supported by the NSF MRL program at the University of Pennsylvania under Grant No. DMR-9120668.

Appendix A. Expressions involved in d& The partial derivatives of ci with respect to E, A. 2’. andf are evaluated by using Eq. (64) and are as follows:

(A.11

S. Ramaswamy,

N. Aravas

dAE -K+Z-2G-$ri,

8AE

aAE -K+Z-2G-$ti,

aAE

$= a& -_=

-K

dAE

dAE

a&

-1 dA n

df

I Comput.

Methods

Appl.

Mech.

Engrg.

16.3 (1998)

33-53

53

(A.21 C4.3)

2Z_2G2^ v

df n7

(A.4)

where all quantities are understood to be evaluated at the end of the increment, i.e. at t = tn + , . The partial derivatives of AE, and AE4 with respect to E, A, Ep andf, are determined from Eqs. (67a) and (68): aAE

9f AA cash w

-..--L_

dE

2~:

2 AA

dE

a;+6GAA

aAE P= aA aAE Y= aA

(A.3

+ 9fK Ah cash o 81%’

aAE

--L=

ape

aqe

(‘4.6)

dE’

Sfu,, sinh w

_ 2~;

(A.7)

-I- 9fK Ah cash w ’

2q” uif6GAA

6G AA l-

cr;+6GAA

(‘4.8)

) ’

d AE,, _ f AA(9p cash w + 6u,,, sinh o) m du dSP ’ lXp 20-i + SfKu,,, Ahcosh w aAE Y= dCP

4 AA q%,,,

_

(4.9)

da,,,

(A.lO)

(a;, + 6G AA)* dCP ’

aAE P=_ ;?f

6 AA u,,, sinh w 2~;

(A.1 1)

+ 9fK AA cash 6.1

aAE (A. 12)

0 Y

’

with w = 3p/(2u,,,).

The expressions

for dp”/dE,

dq”/aE

and dn^/aE are given by (A.13)

References plasticity models, hit. J. Numer. Methods Engrg. 24 (1987) [II N. Aravas, On the numerical integration of a class of pressure-dependent 1395-1416. 121N. Aravas and R.M. McMeeking, Finite element analysis of void growth near a blunting crack tip, J. Mech. Phys. Solids 33 (1985) 25-49. [31 CC. Chu and A. Needleman, Void nucleation effects in biaxially stretched sheets, Int. J. Engrg. Matl. Tech. 102 (1980) 249-256. Yield criteria and flow rules for porous ductile [41 A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth-I. media, Int. J. Engrg. Mad. Tech. 99 (1977) 2-15. [I J.B. Leblond, G. Perrin and J. Devaux, Bifurcation effects in ductile materials with damage localization, J. Appl. Mech. 61 (1994) 236-242. [61 V. Tvergaard and A. Needleman, Effects of nonlocal damage in porous plastic solids, Int. J. Solids Struct. 32 (1995) 1063-1077. and [71 V Tvergaard and A. Needleman, Nonlocal effects on localization in a void sheet, Danish Center for Applied Mathematics Mechanics, Report No. 523, April 1996. Finite element implementation of gradient plasticity models, Doctoral Dissertation, University of Pennsylvania, [81 S. Ramaswamy, Department of Mechanical Engineering and Applied Mechanics, Philadelphia, PA, USA, 1997.