finite element implementation of gradient plasticity models part i: gradient-dependent yield...

Computer methods in applied

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a . __ __ F!!!!! ELSEVIER Comput. Methods Appl. Mech. Engrg. 163 (1998) 1 l-32

Finite element implementation of gradient plasticity models Part I: Gradient-dependent yield functions

S. Ramaswamya, N. Aravasb3* “Sandia National Laboratories, Solid and Material Mechanics Department, Materials and Engineering Sciences Center,

7011 East Avenue, Livermore, CA 94550, USA

‘Universiry of Thessaly, Department of Mechanical and Industrial Engineering, Pedion Areas, 38334 Voles, Greece

Received 3 September 1997

Abstract

Theories with intrinsic or material length scales find applications in the modeling of size-dependent phenomena such as, for example, the

localization of plastic flow into shear bands. In gradient-type plasticity theories, length scales are introduced through the coefficients of

spatial gradients of one or more internal variables. The present work undertakes the variational formulation and finite element

implementation of two families of gradient-type plasticity models in which higher-order gradients of the state variables enter the yield

function (in Part I) or the evolution equations for the state variables (in Part II). As an example, the application to a gradient-type version of

the von Mises plasticity model is described in detail in the present paper. Numerical examples of localization under plane strain tension are

considered using both the gradient-type (non-local) model and its corresponding classical (local) counterpart. An important consequence of

using the non-local model is that the numerical solution does not exhibit the pathological mesh-dependence that is evident when the standard

von Mises model is used. 0 1998 Elsevier Science S.A. All rights reserved.

1. Introduction

Classical (local) continuum theories possess no material/intrinsic length scale. The typical dimensions of length that appear are associated with the overall geometry of the domain under consideration. In spite of the fact that classical theories are quite sufficient for most applications, there is ample experimental evidence which

indicates that, in certain specific applications, there is significant dependence on additional length/size parameters. Some of these instances, as selected from the literature, include the dependence of the initial flow stress upon particle size [6], the dependence of hardness on the size of the indenter [33], the effect of wire-thickness on torsional response [15], the development and evolution of damage in concrete [5], the failure

by localization in soils and rocks 136,371, etc. All these investigations highlight the inadequacy of local continuum models in explaining the observed phenomena, thereby motivating the need to introduce non-local continuum models that have length scales present in them.

A variety of methods by which length scales have been introduced to form non-local continuum models are available in the literature. These methods include the approaches of Vardoulakis [36], de Borst [lo] and Fleck and Hutchinson [ 141 in framing micropolar continuum theories where rotational degrees of freedom are added to the conventional translational degrees of freedom. Alternatives to the micropolar continuum models lie through non-local constitutive models which are either of an integral- or a gradient-type. Integral-type constitutive models have been used in [28,17,35], wherein the evolution of certain internal variables is expressed by means

* Corresponding author.

0045.7825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved. PII: SOO45-7825(98)00028-O

12 S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 11-32

of integral equations. Gradient-type non-local constitutive models, in the context of linear elasticity, have been

introduced by Mindlin [19]. In the context of plastic deformation, such models have been used by Aifantis [l-3], Bammann and Aifantis [4] and Coleman and Hodgdon [8]. Several subsequent studies involving

gradient-type models have been published in the literature. Among these, we mention the works of

Triantafyllidis and Aifantis [34], Schreyer and Chen [30], Zbib and Aifantis [38-401, Vardoulakis [36] and

Vardoulakis and Aifantis [37]. An interesting feature of the gradient-type plasticity models is that the presence

of spatial gradients of the state variables necessitates additional boundary conditions for the relevant quantities

on the boundary of the plastic zone [22].

The finite element implementation of small-strain versions of micropolar and gradient-type plasticity models

has been the subject of several publications in the recent years [27,13,22,11,12]. Advances on the finite strain

implementation of gradient plasticity models include the recent work by Mikkelsen [ 181, who studied the

localization and post-necking behavior of thin sheets.

The present work discusses a general finite element formulation and implementation of a finite-strain version

of two classes of gradient plasticity models. In Part I of our work, we consider elastic-plastic models in which

gradients of one state variable enter the yield function. In Part II (this issue), we deal with models in which

gradients of the plastic multiplier and the state variables enter the evolution equations for the state variables. The

developed mixed finite element formulation is based on Galerkin’s method and includes the displacement, the

state variables, and the plastic multiplier as the nodal degrees of freedom. Emphasis is placed on the finite-strain

aspects of the problem and on the appropriate conditions that define ‘plastic loading’ in the discretized version

of the problem. In the present finite element formulation, when plastic flow takes place, the yield condition is

enforced through the variational equation of the problem. It is shown that the discrete ‘plastic loading/

unloading’ Kuhn-Tucker conditions are exactly the same as those of the continuum formulation, provided that

the corresponding ‘nodal values’ of the yield function are defined properly.

A brief outline of Part I of our work is as follows. Section 2 gives a brief summary of the standard local form

of plastic constitutive equations, whereas Section 3 describes a class of plastic equations of the gradient type

together with an application to the von Mises model with a gradient dependent yield stress. Section 4 outlines

the general form of the boundary value problem for an elastic-plastic body with a gradient-dependent yield

function. Section 5 describes the mixed variational formulation of the elastic-plastic boundary value problem.

The application to the von Mises 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 6. Finally, Section 7 presents numerical examples on strain localization under plane strain

tension using both the classical von Mises model and the gradient-dependent von Mises model. A key

observation while using the gradient-dependent model is the elimination of the pathological mesh-dependence of

the numerical solution that is observed in the context of the classical model.

Standard notation is used throughout. Boldface symbols denote tensors the orders of which are indicated by

the context. All tensor components are written with respect to a fixed Cartesian coordinate system, and the

summation convention is used for repeated Latin indices, unless otherwise indicated. The prefixes tr and det

indicate the trace and the determinant respectively, a superscript T the transpose of a second-order tensor, a

superposed dot the material time derivative, a prime ’ the deviatoric part of a tensor, and the subscripts s and a

the symmetric and anti-symmetric parts of a second-order tensor. Let a and b be vectors, A and B second-order

tensors, and C a fourth-order tensor; the following definitions are used in the text (ub), = a$,, (A - B)ij = A,,B,,

A :B = AiiBi,, (AB),,, = AjiB,,, (8A/tU3)ijk, = dA,,/8Bkl, and (C:A), = C,,,,AkI.

2. ‘Local’ versus ‘gradient-type’ plastic constitutive equations

The present work is concerned with a large-deformation analysis of isotropic elastic-plastic materials. We confine ourselves to situations in which the deformation rate D is written as the sum of an elastic and a plastic

part:

D=D”+D’. (1)

The flow rule, that defines Dp, and the evolution equations of the state variables s, (a = 1,. . , n) for a local rate-independent model are of the form

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) 11-32 13

Dp = iN(u, sJ, (2)

i, = /ii&q sp> ) ff,p=l,..., n, (3)

where u is the Cauchy stress tensor, i a non-negative plastic multiplier, and (A’, S,) isotropic functions of their

arguments. In view of the assumed isotropy, all state variables s, are scalar. In anisotropic models, some of the state

variables s, are tensorial quantities; in such cases, the corresponding material time derivatives sb are replaced by

appropriate co-rotational derivatives that account for the evolution of the underlying substructure that defines the

anisotropy [9]. The methodology presented in the present paper can be extended easily to the case of anisotropic

elastic-plastic materials; however, for simplicity, we restrict our attention to isotropic models.

The plastic parameter h and the yield function @ satisfy the ‘plastic loading-unloading’ criterion given in a

Kuhn-Tucker form by

@(V,S,)dO, i 20, /i@=O, (4)

where @(~(a, se) is the local form of the yield function in stress space.

As mentioned in the Introduction, we consider two different versions of plasticity theories of the gradient-

type. The first version is the subject of Part I of this work and deals with models in which the yield function (4a)

depends not only on u and s,, but on gradients of the state variables, such as V’s, and V’S, as well. Part II considers elastic-plastic models in which the yield function has a local form, whereas the evolution equations of

the state variables (3) include terms of the form VS~ and V’s,.

3. Plastic constitutive equations of the gradient-type

We consider plastic constitutive equations in which the yield function depends on gradients of one state

variable (say s, ). In particular, we deal with plasticity models of the form

@((a,s,,vs,,v*s,)~0, i 20, /i@=O, (5)

Dp= ti(a,s,), (6)

s, =/is",(a,s& cY,p= l,...,n. (7)

The yield condition is now a partial differential equation as opposed to an algebraic equation, which is the case

in the local model. The presence of the gradient terms in the yield condition requires boundary conditions for the state variables on the boundary of the plastic zone. A detailed discussion of the appropriate boundary conditions

for gradient-type plasticity models has been given by Muhlhaus and Aifantis [22].

For the case of a yield function of the form (5), the general form of the required boundary conditions on si is

s1 = s, on Sp , (8)

where S, and S, are unknown functions over Sp and SE, n is the outward unit normal to SI, and Sp U SE = Sp, where Sp is the boundary of the plastic zone.

In the following, we present the example of the von Mises model with a gradient dependent yield stress, and discuss in some detail the required additional boundary conditions for the state variable s, .

3.1. Example: The von Mises model

In its local form, the yield condition is written in terms of the quadratic invariant of the stress deviator CT’. The model involves only one state variable, the ‘equivalent plastic strain’ Zp, i.e. cx = 1 and s, = 2’. The gradient-type form of the yield condition is


@(q 2P) = a, - uy(~p, VTP, V22P) = 0 ) (10)

where V~ =vm is the von Mises equivalent stress. In the applications that follow, we consider a yield

stress a; of the form

ffY = gOg(%P) + +r&(ZP)~V~P( + &,&(~p)v*Zp ) (11)

where a, is a reference stress, (g,, g,, g3) are dimensionless functions, and ( 8, 6, ) are material length scales. In

the above equation, the term a,,g(T’) defines the usual ‘local yield stress’ (T:~‘. The two gradient terms in (11)

modify the material yield stress, and can ‘harden’ or ‘soften’ the material depending on the sign of the value

their sum has at a given point.

Plastic normality is assumed, i.e. N = a@/du, so that

. a@ 3/i u’ DP=Az=~--g-.

The rate of the equivalent plastic strain is defined as

(12)

-;P = (13)

Using the flow rule (12), we can write the above equation as

s’ 1

= -;P=j_A’S 1 ’

i.e. S1 = 1 in this case.

(14)

We identify Sp with the part of the boundary of the plastic zone that lies in the interior of the body, i.e. with

the so-called elastic-plastic boundary; the remaining part, SI, is the part of Sp that belongs to the outer surface of

the body (see Fig. 1).

On the elastic-plastic boundary Sp, the value of the equivalent plastic strain is specified. In particular, if a

point A on Sp yields for the first time, the boundary condition at that point is 2’ = 0; on the other hand, if point

s = S” u s,

Fig. 1. Schematic representation of the region occupied by the elastic-plastic continuum.


A has yielded previously, the corresponding value of the equivalent plastic strain equals the value of 2’ when the

point was last actively yielding.

The boundary condition for 2’ on Si is assumed to be

(15)

4. 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 gradient type in the deformed configuration and write

V.a+b=O, (16)

D =; [Vu + (VU)~] , (17)

D=D”+D’, (18)

:=Ce :De, (19)

Dp = iN(u, s,) , (20)

@(u,s,,v~,,v2s*)~o, Iho, /i@=O, (21)

s’, = ii&q so) ) C&p=1 ,...) II, (22)

in V, In the above equations b is the body force per unit deformed volume, u is the velocity field, C” is the

fourth-order isotropic tensor of elastic moduli, and a superposed V denotes the Jaumann co-rotational derivative.

The elastic equations (19) correspond to linear hypoelasticity; when the elastic strains are small, the hypoelastic

equations (19) are consistent, to leading order, with those of a hyperelastic material.

We note that all gradients that enter equations (16)-(22) are understood to be evaluated in the current

deformed configuration V. We put no restriction on the magnitude of the deformation associated

The corresponding boundary conditions in the deformed configuration

u=ti on S, ,

n*a=t” on S,,

s, = s, on Sp ,

ap -- an - in, 3 on Sf: ,

with the motion from V, to V.

are

(23)

(24)

(25)

(26)

where ti denotes the prescribed displacements over S,, t^ the prescribed tractions over S, which has an outward unit normal n, and S, U S, = S, S being the boundary of V.

5. Variational formulation

A variational formulation of the boundary value problem discussed in Section 4 is developed in this section. In what follows, all equations are written in the current deformed configuration V, t stands for time, and x denotes the current position of a material point in V.

We start by expressing the equilibrium equations (16), the traction boundary conditions (24), the yield condition (21a), and the boundary condition for the state variable sr (26) in the following variational statement: Find


(1)

(2)

(3)

u(x, t) E H2(V) satisfying r41sU = ti, where Hk is the space of functions with square-integrable derivatives

through order k,

A&, t) EL”(V), where L2 is the space of square-integrable functions, and

si(x, t) E H2(V) satisfying s,lsp = s”,, ^ ^ such that for all u* EL’(V) satisfying u*l,” = 0, for all A* EL’(V), and for all ST EL’(V) satisfying ST Isp = 0,

A@, A, sl, u*, A*, SF) = I V

(q, + b,)$ dV + (ii - qjiinj)uT d&s

+ f VP

@(a, s,, vs,, V2s,)h* dV + (+QjZj - S,)sT d&S = 0) (27)

where u = a(~, A, s, ), and s, = s,(u, A, si ) (a = 2, . . . , n). In the above equation, VP denotes the plastic zone, and a comma denotes partial differentiation with respect to

position, i.e. A = aA / axi.

Similar variational formulations within the framework of local plasticity theories have been discussed by Nyssen and Beckers [26], Pinsky [29] and Simo et al. [31].

6. Application: The von Mises model

In the case of the von Mises model, where ff = 1 and si = 2’ = A, the variational statement takes the form: Find u(x, t) E H’(V) satisfying uIs = u^, and ;‘(x, t) E H2(V) satisfying Zplsp = zp, such that for all u* E L2(V)

satisfying u*Is U = 0, for all A* E i2(V), and for all Zp* E L2(V) satisfying kp*Isp = 0,

A(u,TP, u”, A*, :‘*) = (r+j + bi)u; dV +

+ I VP

[a, - a,,g(2p) - ao~,g,(:p)~V:p( - ~o~2g2(~P)~~jjlA* dV

+ I SiZ qijTp* ds = 0 ) (28)

where ct = a@, 2’).

If we use Green’s theorem, and set

Tp* = u0 e2g2(Zp)A* , (29)

without loss of generality (as both rP* and A* are arbitrary), we can re-write the above variational statement as:

Find U(X, t) E H ’ (V) satisfying u Is

satisfying u*ls,

= u^, and TP@, t) E H’(V) satisfying Zplsp = ip, such that for all u* E H’(V)

= 0, and for all A; E H’(V) satisfying A*lsp = 0,

(30)

- I vp [(a, -‘aOg(:‘) - a,[, g, (:‘)IVZ’] + a0~2g;(:P)jVEP]2)A*

+ a, ezg,(Zp)Vgp -VA*] dV= 0,

where u = a@, 2’), Dz = (uzTj + v,*,)/2, and a prime denotes differentiation.

6.1. Finite element formulation

In the context of the formulation discussed above, the approximations used for the displacement u and the equivalent plastic strain fields rP must be continuous, i.e. in C”(V). The domain V is discretized, and standard element interpolations are introduced. The solution is obtained incrementally, and within each element we write at every instant

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) II-32 17

bw~ = WWl{w,~ 3 TP = Lh<x>l(w,> t (31)

{v*W> = bw)I~W~~ 1 A* = Lh(x>l(w,*), (32)

where [N(x)] and L/z<x>J are arrays containing standard element shape functions, {w,} and {w,*} are the vectors of nodal quantities of the form

Lw,J = tu;, u;, z& 9,. . . , U;NNODE, UyoDE, UrNoDE, zNNoDE_l) (33)

Lw:l = Lu:‘, $1, u$‘, A*‘, . . . ) UyNODE, U2*NNODE, U3*NNODE, A*NNODEJ) (34)

NNODE is the number of nodes per element, and the notation Laj = {a}’ IS used. Using the above equations we

can readily write

{D(x)] = W)l{w,] 9 P*(x)) = [fw1~w,*> t (35)

and

WP(NJ = r~wb,> T -m*w1 = bwl~w%~~ (36)

Let {w} and {w*} be the corresponding global vectors of nodal unknowns. Substituting the above equations into

the variational equation (30), and taking into account that the resulting equation must be satisfied for arbitrary

{w*}, we arrive at a set of nonlinear equations for {w} of the form

W({w>)> = {01> (37)

where

(G) = NY, r=l

(I, e

[BIT{a} dV- j-” [NIT(b) dV- l,< [NIT(i) dS

(38)

where 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}.

6.2. Numerical integration of the constitutive equations

In a finite element environment, the solution is developed incrementally and the constitutive equations are

integrated at the element Gauss integration points. Let F denote the deformation gradient tensor. At a given

Gauss point, the solution (F,,, a,, 2:) at time t, as well as the deformation gradient F,,, , and equivalent plastic

strain Tf: + , at time t, + 1 are known, and the problem is to determine the stresses a,,, , . For the case of the isotropic elastic-plastic von Mises material under consideration, the constitutive equations

that need to be integrated are

D=D”+D’, (39)

:=C”:D”,

Dp zz &l,

where

(40)

(41)

c~=(K-~G)*z+2G9, (42)

K and G are the elastic bulk and shear moduli respectively, Z is the second-order identity tensor, and 4 is the fourth-order symmetric identity tensor with Cartesian components 9ij,, = (&$ + e&,)/2, Sij being the Kronecker delta.

It should be noted that in the standard displacement-based finite element formulation of local plasticity


models, one has to consider the yield condition in addition to (39)-(41) at the Gauss point level, in order to determine the equivalent plastic strain increment A;’ = ?E+, - 25: locally. However, in the present formulation,

the yield condition enters the variational formulation, ZE+ 1 is a nodal degree of freedom, and, as a consequence,

we deal with a simpler set of equations at the Gauss point level. The distinction between ‘plastic loading’

(AT’ > 0) and elastic response (Ah’ = 0) is now made at the nodal points, where AZ’ is required be

non-negative, as discussed in Section 6.3 below. The value of SE+, at a Gauss point is obtained from the nodal

values using the finite element interpolation functions, which are chosen in such a way that, when the nodal

values of Agp are non-negative, the condition ATP(x) 2 0 is met everywhere. Therefore, in the present

formulation, when the material calculations for a time increment start at a Gauss point, it is known already

whether ‘plastic loading’ (AT’ > 0) or elastic response (ATP = 0) takes place at that point over the increment.

The time variation of the deformation gradient F during the time increment [t,, t,, ,] can be written as

F(t) = AF(t) + F, = R(t) . U(t) . F, , t, s t d t, + , , (43)

where R(t) and U(t) are the rotation and right stretch tensors associated with AF(t). The corresponding

deformation rate D(t) and spin W(t) tensors are

D(t) = [g(t) . F - ‘(t)] s = [A@(t) * AF - ‘(t)] s , (4)

and

W(t) = [g(t) *F-‘(t)], = [A@(t). AF -‘(t)], , (45)

where the subscripts s and a denote the symmetric and anti-symmetric parts, respectively, of a tensor.

If we assume that the Lagrangian triad associated with AF(t) (i.e. the eigenvectors of U(t)) remains fixed in the time interval [t,, t,,+,], we can readily show that

D(t) = R(t) * S(t) * RT(t) ) W(t) = k(t) * RT(t) ) and &t)=R(t)-&)d?T(t), (46)

where E(t) = In U(t) is the logarithmic strain associated with the increment (E, = 0), and b(t) = RT(t) - a(t) * R(t) is the co-rotational stress [23,25].

We note that at the start of the increment (t = t,)

AFn=R,=Un=I, &,, = a,, and E, = 0,

whereas at the end of the increment (t = t, + , )

(47)

AF,,+, =F,,+,*F,’ =R,+,*U,,+, =known, and E,+, =lnU,+, =known.

The constitutive equations (39)-(41) can be written now as

E;+j”+jj’P,

&Ce:Ee,

(48)

(49)

(50)

(51)

which are similar to those of a ‘small-strain’ theory. In the following, we discuss briefly a procedure that can be used to integrate numerically the set of equations

(49)-(51). The elastic part (50) of the constitutive equations can be integrated exactly to yield

&I+1 =tT,+C":AE', (52)

where we took into account that &,, = a,,, and introduced the notation AA = A,, I - A,,. The backward Euler method is used to integrate the plastic flow rule (51):

AEp = hP&,,+, .

Eq. (52) then becomes

&+, = ue - 2G bpi+,,+, ,

(53)

(54)

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) II-32 19

where ue = a,, + C’ : AE is the known ‘elastic predictor’ and AE = In U,,, , . Next, we show that $,,+, can be

determined from the elastic predictor we. Using the definition of fin+, = (3 /2)( &’ /a;),+, and taking the

deviatoric part of Eq. (54), we find

n I u n+, = a”‘/[1 + 3GA;P/(a,),+,l, (55)

i.e. &L+, and uer are collinear. Therefore, substituting the above expression for &L+, into the definition of

fin+,, we find

(56)

where crs = (15~~‘fl>‘)“*. With 3n+, known, Eq. (54) defines &n+,, and the stress a,,+, at the end of the

increment is found from

U ?I+1 =&+I. kn+, -R,T+, >

which completes the integration process.

(57)

6.3. The ‘plastic loading/unloading’ conditions

In a continuum formulation the Kuhn-Tucker conditions

A:‘(X) 2 0, @(X)SO, A:p(~)@(~) = 0 , (58)

must be satisfied at every point of the continuum. In the present formulation, when plastic flow takes place, the yield condition is enforced globally, rather than

locally, through the variational equation (30). Therefore, the loading/unloading conditions should be satisfied in

a ‘global’ sense. In the following, we discuss in some detail the discrete counterparts of the continuum

Kuhn-Tucker conditions (58). A brief treatment of that topic is also given in [31].

Starting with the finite element interpolation (31b) for hip, we can isolate the nodal degrees of freedom in

{w,} that refer to A\‘, denote the corresponding column-vector by {AZ:“}, and rewrite (31b) within each

element as

A;‘(X) = Lh(x)_l(A::“} , (59)

where @(x)1 is the row-vector of shape functions that are used in the interpolation of A:‘(X). We also use the

notation {hpN) to denote the corresponding global vector that contains all nodal degrees of freedom in the

structure that refer to AZ’.

Next, we define the global ‘nodal vector of the yield function’ {%} so that

i A:P(~)@(~) dV= LAZ”“_@}, (60)

VP

which, in view of (59), implies that

NELEM

{@} = c {s:) where {@= I, {&+1@(~) dV > (61) P=l e

where {I,} is the local ‘nodal vector of the yield function’ and {zf} is the corresponding global quantity. The shape functions in (59) are chosen so that

{%x)1 F= {0>1 (62)

where the notation {a} 3 (0) means that all the components of {a} are non-negative. Then, the Kuhn-Tucker conditions (58) together with (59)-(62) imply that

{ATpN} G= (0)) {%} G (0) and LA%““J$} = 0. (63)

The conditions (63) can be written in terms of the components of the vectors {A?““} and {$} as follows:

20 S. Ramuwamy, N. Aravas I Compur. Methods Appl. Mech. Engrg. 163 (1998) 11-32

NTOTAL

A7yN > 0 7 qS0 i=l,...,NTOTAL and c A:;” A?& = 0, i=l

(64)

where NTOTAL is the total number of nodes in the finite element mesh. In view of the first two inequalities, Eq. (64~) can be simplified as follows. Eqs. (64a) and (64b) imply that

NTOTAL

c A:;“A+O, i=l

which, together with (64c), yields

A:p”l=O (nosumoveri) i=l,...,NTOTAL.

Summarizing, we write

AE;N>O, T+O, A?pNzi = 0 (no sum over i) i=l,...,NTOTAL.

(65)

(66)

The above expressions are the discrete counterparts of the Kuhn-Tucker conditions (58). It is interesting to note that the discrete Kuhn-Tucker conditions are written now at the nodal pjnts and have the same form as the continuum ones, provided that the ‘nodal vector of the yield function’ {@} is defined properly (Eq. (61)).

For the case of the von Mises model of the gradient type discussed in Section 3.1, we have that

(gC - q,,g - a, l, g, IV;‘( - c0 e2g2V2;p) dV . (67)

If we integrate by parts the last term and take into account that A:’ = 0 on Sy, and ASP/& = 0 on SE, we conclude that

{@> = N;q j-P [(a, - gOg - a,Q,lV>pI + a,~2g,~V?p~2){h} + @2g,[m]T{Gp}] dV, (68) e

which is exactly the term in Eq. (38) 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 AL\s’ = 0 is enforced at all ‘elastic’ nodes, and the set (37) 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 qi G 0 at all ‘elastic’ nodes, and (2) the calculated nodal unknowns are such that A:’ 2 0 at all ‘plastic’ nodes. If either of the above two conditions is violated at some nodes, then these nodes are relabelled and the

solution for the increment is repeated. This process is terminated, when an acceptable solution for the increment, i.e. one that satisfies both (1) and (2) above, is obtained.

6.4. Linearization of the jinite element equations

The set of nonlinear equations (37) is solved for the vector of nodal unknowns {w}, i.e. for the nodal values of II and Ep, by using Newton’s method. In the following, we discuss briefly the determination of the corresponding Jacobian.

We start with the variational equation (30), i.e. A@, Tp, u *, A*) = 0, the linearization of which is expressed formally as

A@, -Sp, u*, A*) + DA@, Tp, v*, A*). (AU, A:‘) = 0, (69)

where

DA@, :‘, v*, A*). (Au, AT’) = $ A(u + E Au, 2’ -t E A:‘, v*, A*) 1 l =O . (70)

S. Ramaswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) II-32

It can be shown readily that (e.g. see [20])

DA.(du,d;P)= L”:(da-adLT+dL,,u)dV i V

-d [((T, - crOg(ZP) - floe, g*(:P)Iv:P( + g0 e2~;(~P)~v:P]2)A*

\

21

+ o0 ~2g2(:p)V:p .VA*] dVj ,

where

L* =Q)* and dL = V(du) .

The evaluation of the second integral in the above expression is given in Appendix

In the calculation of the Jacobian, the quantity da plays a central role. Therefore, for

(71)

(72)

A.

the rest of this section, we focus on the evaluation of da by using some recent results of Chen and Wheeler [7]; in particular, we show

that da can be written in the form

du=x:dL+sdTP, (73)

where the components _Z,j,, and s,~ are given in Eq. (87) below.

The deviation of the last equation is as follows. Using the expression u = R * &. RT, we find

du=(dR+RT)v-u.(dR.RT)+R.d&RT. (74)

The stress & is determined by using the integration scheme outlined in Section 6.2. The variation d& can be

written as

(75)

The derivatives t~&/aE and ai?/a;P depend on the algorithm used for the integration of the elastoplastic

equations (49)-(5 1) and are identical to those that appear in the linearization of a standard ‘small strain’

formulation. On the other hand, the terms that involve dR . RT, (Xl,, /&Q,) du,, and aE/XJ relate to the finite

kinematics of the increment. We conclude this section with the evaluation of the aforementioned quantities, da,

and the desired Jacobian.

6.4.1. Evaluation of a&-/aE and &%/&P

Using the expressions given in Section 6.2, where the integration of the elastoplastic equations is outlined, we

conclude readily that

a& 3G A;’

aE=C’-P a: >

a& and a~= -2Gfi, (76)

where

9,;,, = ,a,,, - f sijs,, .

6.4.2. Evaluation of aE/dU

For relatively small deformation increments the logarithmic strain E can be approximated accurately by the following expression

E=lnZJ=(Li-I)-~(U-1)2+~(LJ-1)3=-~L+3~-~~2+~~ 3,

from which it follows that

(78)

22 S. Ramaswamy, N. Aravas / Comput. Methods Appl. Mech. Engrg. 163 (1998) 11-32

C9E. v 213 (S~,S,, + Si,S,,) - f (S;,Uj, + Ui,S,, + S, ,“ , , + ui,S, ,>

auk , 2

(79)

In most finite element calculations, the increments used are relatively small and the absolute value of the

elements of ZJ - Z are much smaller than unity; therefore, the last equation provides a very accurate estimate for

dE/ XJ. For completeness, however, a method for the calculation of the exact value of dE/ cXJ is outlined in

Appendix B .

6.4.3. Evaluation of CM * RT and dU,,,, = (XJ,,, / au,) du,

Chen and Wheeler [7] have derived recently the following expressions

1 dR*R'=detZ ~Z+x~V-VdLT)*Z,

and

dU=R’.dL.AF- ~Y.(RT.~L.AF-AFT.~~~T.R).Y.u, (81)

where V is the left stretch tensor of the increment (i.e. hF = V. R = R. U), Y = (trU)Z - U, and Z =

(trV)Z - V. Therefore, we can write

dR:RT=r:dL and dU=A:dL, (82)

where

with

1 1 ‘{I = - det Y,!iRj!i 7 Qij = AFik Yk,U/j > Mij = &,t y - y, ‘Fjk > T, = R,Y,,U,, .

64.4. Evaluation of da

Substituting the expression for dU into (75), we find

n

d&=K:dL+~d_Ep, &?. JE

where Kilk, = v mn A aE,, au,, pqk’ .

(83)

(84)

(85)

Finally, using the expressions for dR. RT and d& into (74), we conclude that

da=x:dL+sd2P, (86)

where

6.4.5. Evaluation of the Jacobian matrix [J]

When the expression for da and the results of Appendix A are substituted into (71), the integrands in that equation become linear in dL and d2 ‘. Finally, when the finite element interpolations are introduced, Eq. (71) can be written formally as

DA. (du, d:‘) = tw*][J]{dw} , (88)

where [J] is the desired Jacobian matrix.


7. Numerical examples

In this section, we study the problem of localization of plastic flow in plane strain tension. The aspect ratio of

the specimen considered is L/H = 2, where L is the length of the specimen and H its width. A schematic

representation of one quarter of the specimen is shown in Fig. 2, where a typical finite element mesh is also shown. We introduce the Cartesian system shown in Fig. 2, and identify each material particle in the specimen

by its position vector X = (X,, X,) in the undeformed configuration. Plane strain tension in the X, direction is

considered. The deformation is driven by the prescribed end displacement fi, and the lateral surface on

X, = H/2 is kept traction-free. We are interested in solutions that are symmetric with respect to both the X, and

X, coordinate directions. In view of the symmetries of the problem, we analyze only one quadrant of the

specimen as shown in Fig. 2. The following boundary conditions are applied:

(1) u2 =O, a;,=OonX,=O,

(2) u1 =O, (T,~ =0 on X, =O,

(3) T, = T2 = 0 on X, = H/2, and

(4) u2 = L? = known, u,~ = 0 on X, = L/2, where T, and T2 are the components of the nominal traction vector.

An elastic-plastic material of the von Mises type discussed in Section 3.1 is used in the calculations. The

‘local’ flow stress TV:?‘“’ = gOg(Zp) is of the form

H 12 4 t

Fig. 2. Schematic representation of one quarter of the specimen. A typical finite element mesh is also shown


(89)

where

uy’(O) = a, ) ufymax = cy’(E,) = urn , uy+) = a, . (90)

The values a,,, = lSu,, E,,, = 0.035 and urn = 0.04~5 are used in the calculations. The corresponding curve

ural(EP) is shown in Fig. 3. The dependence of the flow stress uy on the gradient of 2’ is of the form

a; = Cy’(P) - e*q,v*zp ) (91)

20 I I I I

0.0 I I I I

0.0 0.1 0.2 03 OA

Fig. 3. Local yield stress aya’ as a function of 2’.

Fig. 4. Load-extension curves for the ‘local’ material (e = 0).

S. Ramaswamy. N. Aravas / Comput. Methods Appl. Mech. Engrg. 16-T (1998) 11-32 25

= 0.0253

hn = 0.0257

+ = 0.0255

7.3@E-02

&2eE-02

5.leE-02

4.OOE-02

2.Q@E-02

1.89E-02

&n 1 I 3.0258

Fig. 5. Contours of 2” for the ‘local’ material (e = 0) at strain levels E ‘” = 0.0253, 0.0255, 0.0257 and 0.0258, calculated using the 30 X 60

mesh.

26 S. Ramaswamy, N. Aravas I Comput. Methods Appi. Mech. Engrg. 163 (1998) II-32

which is a special case of the general form of Eq. (11) with 8, = 0 and g2(Tp) = - 1.

We note that the gradient term in the expression for the flow stress ‘hardens’ the material when V2;’ < 0 and ‘softens’ it when V2Ep > 0. At the center of a shear band, where intense shearing occurs, the ‘local’ flow stress

decreases as 2’ increases beyond 6,; on the other hand, 0’:” is negative in that region, thus having a hardening contribution at the center of the shear band.

Linear isotropic hypoelasticity is assumed, with E = 300~“,, and v = 0.3, where (E, v) are the Young’s

modulus and Poisson’s ratio, respectively.

In order to trigger the initiation of non-homogeneous deformation in the specimen, small imperfections are

introduced in the material properties. In particular, the maximum ‘local’ flow stress qm is assumed to vary in the

specimen according to the expression

(92)

i.e. a,, varies quadratically in the X2 direction and is uniform in the X, direction. The ‘weakest’ (smallest a,,,)

region is in the middle of the specimen (X, = 0) where a,,, = 1.5~,,, and the ‘strongest’ is at the loading edge (X, = L/2) where u,,, is 3% higher.

Four-node isoparametric plane strain elements with 2 X 2 integration points are used in the discretization. The

B-bar method is used in order to avoid artificial constraints on incompressible modes [24,16]. The initial

(undeformed) finite element mesh is uniform in both directions. Four different meshes are used, namely 10 X 20,

20 X 40, 30 X 60 and 40 X 80 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. One in which the standard ‘local’ Mises model (e = 0) is used, and

another in which the ‘gradient’ von Mises model with e = 0.2L is used. For each of the two sets of calculations, solutions are obtained using all four different finite element meshes.

Figs. 4-7 show results for the case of a ‘local’ material (& = 0). Fig. 4 shows the ‘load-extension’ curves as

calculated using the four different meshes. The normalized load F? plotted in Fig. 4 is defined as fi = 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 line on that 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 axial strain

E ‘” = 0.025, where the ‘macroscopic axial strain’ is defined as E’” = ln[ 1 + o/@/2)]. At that strain level a shear

band forms, and beyond that point the obtained numerical solutions exhibit a strong mesh-dependence, as is

evident from the curves shown in Fig. 4. The contours shown in Fig. 5 demonstrate the evolution of 2’ in the

0.08

0.06

P 0.04

0.02

i

I I I I

4- 4OxttOmesh

1. 2

Fig. 6. Variation of 2” for the ‘local’ material ( 8 = 0) along X, = 0 at a strain level 6’” = 0.0254, as calculated using the four different

meshes.

S. Ramaswamy, N. Aravas I Cornput. Methods Appl. Mech. Engrg. 163 (1998) 11-32 27

WZE-02

5.36E-02

4.4QE-02

3.&E-02

2.7BE-02

1 ME-02

Fig. 7. Contours of ip for the ‘local’ material (P = 0) at a strain level E’” = 0.0254, as calculated using the four different meshes.

28 S. Ramaswamy, N. Aravas 1 Comput. Methods Appl. Mech. Engrg. 163 (1998) 11-32

Fig. 8. Load-extension curves for the ‘gradient’ von Mises material with f? = 0.2L.

P

a.!! 1 P _I

Fig. 9. Evolution of gp for the ‘gradient’ von Mises material with C = 0.2L along X, = 0 up to a strain level 6’” = 0.0323, as calculated

using the four different meshes.

S. Ramaswamy, N. Aravas I Comput. Methods Appi. Mech. Engrg. 163 (195%) 11-Z 29

10X :20

Fig. 10. Contours of ip for the ‘gradient’

different meshes.

von Mises material w with t ? = 0.215 at a strain level E’” = 0.0321

20x40

12.47E-02

10.3BE-02

&25E-02

&14E-02

4.03s02

1.91 E-02

as calculated using the four


specimen up to the point of shear banding as calculated by using the 30 X 60 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, 2’ is almost uniform with a very slight inhomogeneity due to the

presence of initial imperfections. At the macroscopic strain level of e’” = 0.025, the inhomogeneity begins to grow into a shear band as shown in the figure. Fig. 6 shows the variation of 2’ along X, = 0, at a macroscopic

strain level of l ‘” = 0.0254, 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’” = 0.025. At smaller extension

levels, the corresponding profiles of 2’ are almost identical for all four different meshes. Fig. 7 shows contours of 2’ at a macroscopic strain level Eln = 0.0254, as calculated using the four different meshes. The strong mesh dependence of the solution is again evident.

Figs. 8-10 show results for the case of a ‘gradient’ von Mises material with 8 = 0.2L. Fig. 8 shows

‘load-deflection’ 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. 9 shows the evolution profiles of 2’ along X, = 0, up to a

macroscopic strain level e’” = 0.0323, as calculated using the four different meshes. The width of the shear band is now independent of the mesh size when the mesh is refined sufficiently. As mentioned earlier in this section,

the hardening provided by the gradient term in the expression for ay balances the softening in aFa’ at the center

of the shear band. A consequence of this is that the shear band spreads to a finite width, which scales with the

‘material length’ & Fig. 10 shows contours of 2’ at a macroscopic strain level e’” = 0.0323, as calculated sing

the four different meshes. Again, the contours shown in Fig. 10 make it clear that the solution is independent of the mesh size when the mesh is fine enough.

Acknowledgments

This work was carried out while the authors were supported by the NSF MRL program at the University of

Pennsylvania under Grant No. DMR-9120668.

Appendix A. Linearization of the yield function equation

The second integral in Eq. (71) can be written as

d (I VP

[(a, - g&‘) - a,8,g,(:p)]V~p] + CJ,,O~~;(~~)(V;~(*)A* + a,&,(Ep)V:p *VA*] dV >

=.I VP [(due - cog’ d:P - a,4?,(g;jV:pl d:‘+ (g,/\VEPj)V?P.d(VZP))

+ u0 e2(g;jV;p)2 + 2g2VZp . d(V:‘))A*

+ g0 e*( g;V:p *VA* d:’ + g, d(V?‘) *VA* + g,V;‘. d(VA*))l dV

+ [(ue - uog - ~@,g,]v:~] + @?2g;(V?p(2)A* + ~~~~~~~~~ .VA*l dLkk dV

where

d(VrP) = V(dT”) - I’?’ . dL ,

64.1)

C4.2)

(A.3)

(A.4) d(VA*) = -VA”. dL ,

and the components s,,,, and s,~ are given in Eq. (87).

S. Ramuswamy, N. Aravas I Comput. Methods Appl. Mech. Engrg. 163 (1998) II-32

Appendix B. Calculation of iYE/aU

Using the definition of the logarithmic strain tensor we can write

E = In U = 5 In hiNiN, , ,=I

where A, are the eigenvalues, and N, the eigenvectors of U, respectively. Therefore, we can write

a A,u(NPN,)

I

Since

aA $ = N,N, (no sum over i) ,

Eq. (B.2) can be written as

N,NiNIN, + In hi & (N,N,) 1

31

(B.1)

V3.2)

(B.3)

(B.4)

The derivative a(N,N,)/X/ that enters the last equation can be evaluated as follows. The dyadic product N,N, is

given by the following expression [21,32]

N{N,=; C2-(I, -A;)C++ ![ I 1 (no sum over i) , (B.-V

where C=FT.F=U2, and

I, = Af + A; + A: ) I, = h;h;A; ) d,=2hf -I,hf ++. I

Using Eq. (B.5) and taking into account that

ac L =; @&j, + U&S,, + &U;, + Qs,,) > au,,

we can calculate readily the desired derivative tl(N,N,)/dU. Finally, substitution of a(N,N,)/dU into (B.4) completes the evaluation of dE/XJ.

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[II L21 131 141

[51

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finite element implementation of gradient plasticity models part i: gradient-dependent yield...

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