Mixed FEM for elastoplasticity

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 74 (1989) 177-206 NORTH-HOLLAND

COMPLEMENTARY MIXED FINITE ELEMENT FORMULATIONS FOR ELASTOPLASTICITY

J.C. SIMO and J.G. KENNEDY Applied Mechanics Division, Stanford University, CA 94305, U.S.A.

R.L. TAYLOR Department of Cit~l Engineering, University of California, Berkeley, CA 94720, U.S.A.

Received 17 December 1987 Revised manuscript recieved 11 October 1988

A global formulation of the principle of maximum plastic dissipation is systematically exploited to construct complementary mixed finite element formulations for elastoplasticity. Completely general return ~apping integration algorithms are obtained as Euler-Lagrange equations of a temporally discret~ed Lagrangian functional. The flow rule is no longer enforced point-wise, but rather at the element level in a weak fashion that couples all Gauss points within an element. The resulting algorithms can be linearized exactly in closed-form for an arbitrary functional form of the yield condition and hardening law. The present approach enables one to extend successful superconvergent quadrilaterals to elastoplastic analysis. Numerical simulations involvint; plane-stress elastoplasticity are presented.

Introduction

Over the past ten years, a general methodology for the formulation of integration algorithms for elastoplastic constitutive equations has been developed, which finds its point of departure in the radial return algorithm of Wilkins [1] for plane strain J2-flow theory. From a computational perspective, considerable work has been devoted to the accuracy analysis of this basic algorithm, see [2, 3], and its extensions to more general plasticity models, see [4-10]. Currently, these algorithms are viewed as a class of product formula algorithms emanating from an elastic-plastic operator split. Within this framework, consistency is enforced by means of a closest-point-projection of an elastic predictor state onto the elastic domain. In [11, 12], this point of view is exploited to develop general purpose algorithms capable of accommodating arbitrary functional forms of the flow rule, hardening law and yield condition. From a mathematical perspective, an independent development in a rather general setting starts with the work of Moreau [13], who coined the expression catching up algorithms. Unfortunately, it is fair to say that, with the exception of the work of Nguyen [14], this and subsequent formal developments, see [15] or [16], have passed largely unnoted in the computational literature.

Implementations of existing return mapping algorithms rely crucially on the view of the discrete elastoplastic problem of evolution as a strain driven problem. As a result, within the

0045-7825/89/$3.50 1989, Elsevier Science Publishers B.V. (North-Holland)

178 J.C. Simo et al., Complementary mixed finite element formulations

framework of displacement-like finite element formulations, plastic loading is tested independently at each quadrature point of the element, and an independent return mapping algorithm is performed at each of the quadrature points for given incremental displacements. Effective- ly, such an algorithmic treatment corresponds to a strain space formulation of elastoplasticity, where total and plastic strains are the independent variables, while the stress is regarded as a dependent variable which is computed from the ~lastic strains by means of the stress-strain relations. In contrast to this view, a sizable body of the mathematical literature on plasticity has been concerned with the formulation of the elastoplastic problem with the stress field as the independent variable, the so-called stress problem, see [15-18, 21]. Mathematically, this problem is considerably simpler than the strain-problem which requires careful definition of the smoothness of the strain field, the so-called space of bounded deformations, and elaborate tools of convex analysis, see [23, 46] for a comprehensive review.

In this paper, we explore an alternative algorithmic treatment in which stress and displacements are independent fields and are interpolated independently through a mixed variational formulation of the Hellinger-Reissner type. The central issue addressed here is not simply the derivation of a mixed formulation of the elastoplastic problem, indeed many such formulations exist, but rather the precise development and implicit implementation of the algorithmic structure associated with the discrete mixed problem. Our development hinges on a variational formulation of elastoplasticity based on the principle of maximum plastic dissipation, often credited to Von Mises (see [24]), and weU-known to be equivalent to normality. This principle plays a central role in the mathematical treatment of the elastoplastic problem by methods of convex analysis, as in [15, 16, 25-27, 45].

Following standard procedures, we obtain a complementary variational formulation of the continuum elastoplastic problem by a Legendre transformation that introduces the stress tensor as a basic independent variable. The novelty in the present approach lies in our subsequent algorithmic treatment of this functional, which is transformed into a time-discrete functional by means of an implicit backward-Euler difference scheme. We then show that the first variation of this discrete functional yields the weak form of the appropriate retui'n mapping algorithm formulated in stress space, along with the discrete algorithmic versions of the hardening law and the consistency condition. By introducing finite element interpolations for the stress and the displacement fields, we obtain a discrete optimization problem which can be solved by convex mathematical programming techniques. In particular, we consider the solution procedure by the classical Newton method in detail. This leads to the extension to the present context of the notion of consistent elastoplastic tangent moduli proposed in [6].

From a computational standpoint, the present formulation results in a finite element architecture that is substantially different from current algorithmic implementations. The fundamental difference is that the plastic return mapping algorithm can no longer be formulated independently at each Gauss point, in contrast with displacement-like methods. Here, the closest-point-projection iteration that restores consistency is performed at the global element level and involves all the Gauss points within the element. The present approach enables one to extend successful mixed finite element interpolations formulated within the framework of the Hellinger-Reissner principle to the elastoplastic regime. As an application, we consider an interpolation recently proposed by Plan and Sumihara [28], which is highly insensitive to mesh distortion, free from locking in plane-strain quasi-incompressible elasticity, and appears to yield superconvergent results in bending dominated problems.

J.C. Simo et al., Complementary mixed finite element formulations 179

1. Variational formulat ion of plasticity

The variational formulation of plasticity considered in this paper relies crucially on the principle of maximum plastic dissipation, often credited to Von Mises (see [24, p. 60]), and subsequently considered by several authors, e.g. [29, 44]. For recent discussions within the context of finite strains see [30-33].

1.1. The principle of maximum plastic dissipation; local form

We summarize the basic results needed in our variational formulation of plasticity. Further details in a somewhat more restricted context are given in [34]. According to ~e principle of maximum plastic dissipation, the actual state is the one among all possible admissible states (i.e., states that satisfy the yield condition) that maximizes plastic dissipation. A precise formulation goes as follows.

Assume the existence of a Helmholtz free energy function, ~(e t, e p, at), of the form ,, j a , (1.1)

Here, e t := V'u t is the strain tensor, e p is the plastic strain tensor, and at is a vector of suitable internal plastic variables, all evaluated at time t. Here, VSut denotes the symmetric part of the displacement gradient. Next, via the Legendre transformation1, introduce the internal variable vector qt,

O(q , ) := - ~( ,~, ) - ~ , . q , , (~.2a)

so that

q~ = -O , ,~(at ) , a t - -aq@() . (1.2b)

Observe that qt = -O, ~(e t, e p, a t) may be interpreted as the thermodynamic force (affinity) conjugate to at. By definition, the local plastic dissipation is given by

Dp[e,, e,", ~,; ,~,P, ,~,] := - aci,(e,, EP, a,) . ~_ a~Ce,, eP, ,~,) 0e p ' 0a 4,. (1.3a)

Standard arguments exploiting the Clausius-Duhem inequality (along with the assumption that unloading is elastic) show that the stress tensor is given by o" t = O~/aEt-Vqt(et- eP). It then follows from (1.1-3a) that

(1.3b)

Let the yield function be defined in stress space by 0(o', q) = 0. The admissible set of stress states, then, in the convex set defined as

1 Introduction of this parameterization is essential to obtain a symmetric discrete return mapping algorithm and algorithmic elastoplastic tangent moduli in Section 3.


~,,q := {(d', ~) ER6 x Rq I 4~(&, #)~


REMARK 1.1. It follows from (1.2b) that dt t = -O:qO(qt)#t . Consequently, the local hardening law (1.10b) can be rephrased in the equivalent form

4,=-.i,,[02,o(q,)l-'oj,(o.,, q,), (1.11)

which serves to define the generalize d hardening moduli h(~r, q). If one postulates a quadratic form for O(qt) , i.e., O(q,) = q,.D-lq,, then [~2q0]-1-D is constant.

1.2. Complementary variational formulations of elastoplasticity

The complementary variational formulation of elastoplasticity is obtained through two steps: (i) A global formulation of the principle of maximum plastic dissipation, followed by (ii) a Legendre transformation that eliminates the strain tensor e, in favor of the stress tensor ~r t as an independent variable.

1.2.1. Notation Let us start by introducing the space of kinematically admissible variations (virtual

displacements) defined for simplicity as 2

v := {,~. n->R' l HE[H*(n)] ~ ; ~/la.n =0}, (1.12)

where N ~


relations

V~(E,- e, p) = ~r,, Vx(~rt) = e , - ..~P,. (1.15)

By substitution of (1.2a) and (1.14) ~nto (1.1) and inserting the resu?~ into II(ut, ~.t, ~.P, at) defined by (1.13a) we obtain the complementary functional

where u t

L l l (u ,o ' ,eP q)t "= { -X{ ,~: ) -O(q) -o~'q+o ' : [VSu-e ,P ]} ,dn+I lext (U , ) , -- I] V, (o't, qt) EO(~gq and r~text(~t) is defined by (1.13b).

(1.16)

1.2.3. Variational characterization of plastic response As shown below, a variational characterization of the evolution of plastic flow in stress

space is accomplished by introducing the Lagrangian functional associated with the plastic dissipation over the entire body ~2 C R 3. From the local expression (1.3b) for the plastic dissipation, by appending the constraint through a Lagrange parameter as in Proposition 1.1, we obtain

"= fo , , , + q, . - ( , , , , q , ) ] t in , (1.17)

where ~,, E K p is the Lagrange parameter, in physical terms the plastic consistency parameter, and K p is the proper positive cone defined by (1.8).

2. Discrete variational problem

To derive discrete governing equations we first construct the discrete complementary functional by integrating the dissipation functional L p over the time interval [0, t,,+l] C R+ with a backward-Euler difference scheme. Then, discrete variational equations are obtained as Euler-Lagrange equations of the discretized functional. Explicitly, the discrete complementary functional for elastoplasticity is defined as

ftn + i l=l(u, o', e p, q)[. "= l](u, o', e p, q)l.+t + L p d~', (2.1a)

jr,,

which is a discrete statement of the first law of thermodynamics for isothermal quasi-static problems. That is, (2.1a) may be interpreted in physical terms as follows

Potential energy[, = Potential energyl, + t + Dissipationl~ + t. (2.1b)

One should also note that the second law of thermodynamics (or Kelvin's dissipation inequality) implies Dissipationl~ +t ~0. Making use of a backward-Euler difference scheme, we define


f t'+t f~ "----" " (~n+l jt n LPdT [On+ 1 "P-P)-ATn+lt~(O.n+l, qn+l)+qn+l.(Ol '+l-Oln)]d,(" ~ (2.2)

where ATn+I := '~n+l -- q/n" By substituting (2.2) into (2.1a) the potential energy II" at time t" becomes a function of the state variables at t'+l, which we shall denote by the symbol 0-n+,- Accordingly, substitution of (2.2) and (1.16) into (2.1a) yields the explicit expression

~''+1 "----f~ [--X(''+I)- O(qn+l)- a" 'qn+l - A~'+I~/)'+I] da

+ fo [~r"+l: (VSu'+l- ep)] da + I Iext (Un+l ) (2.3)

Enforcement of the stationarity conditions for (2.3) through application of the directional derivative formula 3 leads to the following discrete Euler-Lagrange equations for the variables (o', q) E K,,q, AT E K p and u with u - ti E V:

t" t" Dff . '+l " 1~ --- JD [On+ 1" VSl~ - pb " ~] da - Ja [" ~ dr = 0

gn

DO_'+, 8~ - f~ 8o': [(V'u -eP) - 8X(~r'+,) - A%+,8~4}'+,1 da = O,

DB.,,+ 1" 8q = fn

DB.'+ I " By = fo

Bq.[-Oq@(q'+ 1) "~" {~qO(q~) -- AT,,+ lOq~tn+l] da = O,

~(o',,+,, q,,+z)Sy dKt O.

(2.4a)

(2.4b)

(2.4c)

e'P+ 1 -- f " p "1" A~. . '+lOtr~(O'n+l , q '+ l ) , ~lf'+ 1 ---1 Or" "b ATn+10~q0(O' '+l , q '+ l ) ,

in which (eP+ 1, an+,) are eliminated by means of the complementary relations

=vsu. - 0 .x (~.+ ) a.+~ = -0 ,O(q .+, ) E'+I +1 1 ' 3 The directional derivative formula takes the form

d I f.(... z + a8o~,...). Df 'Bm:=~ ~--o

(2.5)

(2.6)

REMARKS 2.1 P and the internal variables a'+l at the current (1) Note that the plastic strain tensor e'+x

time t.+, are no longer present in variational equations (2.4). Equation (2.4b-d) may be viewed as the weak form of the following closest point projection algorithm formulated in stress space:

These four variational equations hold for any ~EV, 8~rE[L2(n)] 6, 8qE[L2(n)] q and By E K p.

(2.4d)


A main implication discussed below is that within the present stress based framework one can no longer obtain perfectly plastic behavior, since in the absence of hardening the elastoplastic problem becomes ill-posed in stress space.

(2) If equations (2.4b-d) are enforced point-wise one recovers the displacement model. Within the stress based variational framework presented in Section 3, the consistency condition and hardening law are enforced point-wise, but the flow rule is enforced in a weak sense. Alternatively, the consistency condition and hardening law may be enforced in a weak sense as discussed in the Appendix. A related approach within the restricted framework of J2-plasticity is recently discussed in [37]. Weak enforcement of the flow rule, on the other hand, leads to a closest-point-projection iteration which is no longer performed independently at each Gauss point as in typical displacement implementations, but rather is performed in a global element iteration that couples all the quadrature points within an element, as is discussed below.

(3) By resorting to a Hu-Washizu type of variational formulation, it is possible to enforce point-wise the flow rule, hardening law and consistency condition while, at the same tim~, obtaining a weak formulation of the stress-strain relations. In fact, this is the proper variational setting of assumed stress methods widely used in current inelastic calculations. We refer to [38] for a detailed account.

3. Mixed finite element formulation

The mixed finite element formulation discussed below will be based on discontinuous stress interpolations over the elements that define the finite element discretization. Our motivation for this approach is quite explicit. Although the developments that follow are general, the actual implementation o~tlined in Section 4 for plane stress is based on a discontinuous stress interpolation proposed by Pian and Sumihara [28]. This interpolation does not exhibit "locking" behavior in the incompressible limit for plane strain and is highly insensitive to mesh distortion.

3.1. Approximation; discrete Kuhn- Tucker conditions

We consider discontinuous stress interpolations defined by the finite dimensional approximating subspace

s I -- = S


Sim~'llarly, point-wise enforcement of (2.4c) yields the discrete hardening law

-o,o(e.+,) + o,O(q~)= a~+,o,#,(~r.+,, q.+,) . (3.3)

3.2. Structure of the general return mapping Within the context of the discontinuous approximation (3.1) for o', a general solution

strategy proceeds as follows. Equations (2.4b), (3.2a) and (3.3) are solved at the element level to obtain o'n+l, A3',+ 1 and qn+l for a given strain VSUn+l which, subsequently, are substituted into the momentum balance equation (2.4a).

To formulate the discrete problem resulting from (2.4b), (3.2a) and (3.3) after substitution of the interpolation (3.1), we introduce the following notation

r Ee(~n+l) :--- JQe st(x)O~rX(S(X)#~n+I) da ,

L(/3n+1, qn+1, 3'n+i) := fa, St(x)a~Orn+1, qn+1)3'n+1(X) da,

Etrial fae n+1 :m st(x)[V sun+ I - e p] d~,

(3.4a)

(3.4b)

(3.4~)

where, hereafter, 3',+1 is used in place of the more appropriate symbol A3'n+ 1. In the case of plastic loading within the element (i.e., 3'n+1 > 0), equations (2.4b), (3.2a) and (3.3) then yield the discrete problem

: --'-- l~trial ---- 0 R(~, q, 3')n+1 -Ee(~n+l)- L(~n+l , qn+l, 3'n+1)+ ~"n+l , o~O( q.+,) - a,O( q.) + v~+,o,(~+,, q~+,) = o,

3'.+xO(S/3n+'J, qn+1) = 0,

(3.5a) (3.5b) (3.5c)

where 3', +1 ~ 0 and 0n +1 ~< 0. The system (3.5) constitutes the appropriate return mapping for the elastoplastic problem.

As a consequence of the discontinuous stress interpolation (3.1), (3.5) may be solved (locally) at the element level for/3n+1, 7n+1 and qn+l. A general algorithm for the solution of (3.5) for an arbitrary form of the yield function O(zr, q) is given below.

3.3. Numerical solution strategy; general yield condition The solution of (3.5) at the element level may be accomplished by a systematic application

of Newton's method as follows. We regard (3.5a, c) as a nonlinear system of equations in/3n+ 1 and Y,,+I in which qn+l is treated as a function of both /3n+ 1 and 7n+1 through (3.5b). A separate Newton iteration for qn+l is then performed on (3.5b) once /3n+ 1 and 3'n+1 are determined. The linearized system corresponding to (3.5a, c) is obtained through a systematic application of the directional derivative formula and (at each time step t,+l) takes the form

o = R *) + ~d I ~--o R(P(*) + a A~(*~, V(*) + a A3'(*)) (3.6a)


or, equivalently

R(,) _ [~(k ) ] - I A~[](k) _ fo e StN(k) AT(*)(X ) d~. 0 -

In addition we have

0= &(*)+ ~aad I,~=o ~('B(k) + a A~ (k), T (*) + a AT(k) )

(3.6b)

(3.6c)

or, equivalently, 0-- 0 (k) + N(k)'s Aft (k) - [0qc~(k)]t [02qqO .4_ T(k)C~2q~(k)]-l[Oq~(k)] AT(k) ' (3.6d)

where the superscript (k) refers to the iteration index in the local Newton iteration and

=(~,+, , .~(k1~ ~(k) (3.7) A( ' ) (k) :----"' ,n+l - - ( ' ]n+l '

[~(k ) ] - l . fo est[O2x + T02o~ 2 2 2 02qtro](k)s d~'~ , --" -- T Oo.q~[C~ qqe "[" TO2qq~] -1

2 "' (k) s := - +

At each step (k) of the Newton iteration, the linear system (3.6b, d) is solved for Aa(~') and "-'/"n + 1 ,.(,) is then performed. Details of the step-by-step (k) A separate Newton iteration for ~,,: AT,+~.

solution algorithm for this general problem can be found in [39]. Observe that the symmetry of -.+t=' 0 are material constants. Following [7], we define the function f(zr) as

f(o.):-~[o.teo.] 1/2 , ,[2,

-1 2 0 0

(3.9)

Observe that for this choice 0~,f= [o "t e ~r] -~/2 P o', and, by construction, ]]0~,fH = 1. Con- sequently, (3.8b) becomes linear in ~, and takes the form

1. C. Simo et el., Complementary mixed finite element formulations 187

Note that the Legendre transformation (1.2) is unnecessary in the ease of a single scalar internal variable since a symmetric algorithm emanates directly from a formulation in q. We complete our model problem by further assuming that the elastic response is linear, that is

X(lflr) "= Io ' t c - lo " , (3 .10)

where C := [O2~X(o')] -1 is the elasticity tensor, assumed to be constant (and point-wise stable, i.e. g' t> llgll 2, for some constant a >0).

3.4.2. Return mapping algorithm Because of the linearity assumption on the elastic response, Ee(/3.+ 1)= Hell.+1 is linear in

gl,,+ 1, with H e defined as

H" "= fn, s ' (x )c - l s (x ) da . (3.11)

In addition, the discrete yield function 4}(er.+ 1, q. + 1) becomes linear in 3'. + 1 (recall that ,/. + 1 is used in place of the more appropriate symbol A y.+ 1) due to the linearity of the hardening law. As a result, in the ease of plastic loading within the given element (i.e.T.+I >0), (3.5a-c) yield the following discrete return mapping problem,

"ffi -H ~1.+1 - L(~.+I, T;,+I) + --,,+~ , Rn+l 17trial ~ 0

~.+1 := f(er,,+,)- ~ K(q . ) - 3ZH'T,,+; =0,

(3.12a)

(3.12b)

where backward-Euler integration has been used on (3.8b) to yield the following linear discrete hardening law

q .+ l - qn = V r] %+1. (3.13)

.(k) Linearization of (3.12a) about the state (fl, T, q).+l yields

where

[ 'sdal' ' n+t m H -1 .I.+1

]-1 := H e + f ~t~2 [(k) ~. (k) l'-~ J-r n+lJ ~, v t r~ Jn+ l . . ,Tn+ l lat j d~ '~ . ja,

Aft

188 J.C. Simo et a!., Complementary mixed finite element formulations

Box 1 Integration algorithm. Elastic predictor. Model problem

(i) Initialization (k = 0):

H e := fo s,(x)c,sx) d~ ,

fn, ' VS ~tr ia l :__. S (x)[ u,,+l - eP] d ~"n + 1

(ii) Calculate elastic predictor:

#(0) __ jg le - l~t r ia l aa a.~n+ 1

(iii) Check for yielding at the element:

f(S(xs)# '))

IF ~)(Xs) ~< 0 for all x 8 E ~2, THEN Elastic Process; EXIT

ELSE Plastic Process; GO TO Box 2

ENDIF.

REMARKS 3.1 (1) In Boxes 1 and 2, the notation (.)(k), Xt 8 and N 8 refer to the following:

(.)(k) . __/. ~(k) k In+l ,

x s := x E n e at Gauss point g,

N s := number of Gauss points in/~,,

f~:= domain associated with element e.

(2) In addition, the notation (@) stands for the Macauley bracket of @, i.e., (@)= @ if @>0, and (O}ffi0 if ~


Box 2 Integration algorithm. Plastic corrector. Model problem

(iv) Enforce discrete Kuhn/Tucker conditions to compute 3,(k):

1 'Y~)=~H'--'-; ('~'k'(xe))' g=l " " 'Ne '

I,~, := {ge {1,2,. . . ,Ng} [ g(k)(xs)>O) .

(v) Compute constitutive residual and check convergence:

L (k):- 2 [st(x)a~ffk)Ls~(sk)']sWg, gElat

R(k) 2_ _He~(k) -- L(t) + j~trial ,

IF IIR"II > TOL THEN

(vi) Compute consistent tangents and get Ap(k):

[~( , ) ] - t := H + ~. t 2 (k) (k) [s a..f s]..~. :,w,, glact

&.8(.) = (,..--(k))-, + IH--"; ~' [$t(O'f(k))(a"f(k))t$]'J~We R(')" gelat

(vii) Update/3 (k) and compute ~(k+ ~):

#(k+,) = p(h) + Ap("),

~(k)(Xe):-- ~ K(q,,(X.)), g 1,.. N e f(S(xs)p ~) - = ., ,

SET k = k + 1 and GO TO (iv)

ELSE

(viii) SET

q.+,(=.) = q.(~.) + ~ "v. , , (~.),

e,,+,(x.} ?,,+,(xe)a.f,,+,( .)

ENDIF,

EXIT.

matrix emanating from the weak form of momentum balance (2.4a). Assume a C(n) displacement approximation defined by

v" :- { h e [C(n)]N I re. }

= ~, NA(x)SdA, 8d A e. R N C I7. (3.16) A=1

By introducing the [N (N Nen) ] matrix of shape functions N := [Nl(x)l"" fcNen(X)l], . 8dN~,] , we obtain where 1 is the (N N) unit matrix, and setting 8d t " - - [Sdt l " t


~hJo" = N(x)Sd, s h ... V ~ JOe B(x)Bd, (3.17)

where, following standard notation (see e.g. [40]), we have denoted by B "= V'N.

3.5.1. Residual force vector With the above displacement interpolation the weak form of momentum balance (2.4a)

takes the form

E D[.+, .~h= ~ a h ' where a~ = 8dt[R~-f~]. (3.18)

e l l

The vector f~ gives the element contribution to the external loading (emanating from H,,t(u)). On the other hand, the element internal force vector, R e, is given by

Re "= fl~ Bt(x)o'n+l d~'~ -" Gtpn+l , (3 .19)

where we have set

G "- fn. St(x)B(x) dn. (3.20)

Note that the element internal force vector R, becomes completely determined from (3.19) once/]~.~ is obtained, from a given strain e,+~ ffi V'u, by the return mapping algorithm in Box 1 and Bvx 2.

3.5.2. Consistent tangent stiffness matrix The element tangent stiffness matrix is obtained by (exact) linearization of the weak forms

(2.4) through a systematic application of the directional derivative formula. This, in turn, involvea the linearization of (3.19) and the linearization of the algorithm in Boxes I and 2 as follows. Here, we regard all the fields (t r, q, y) as functions of uhE V h. Then, for any incremental displacement field AuhE V h such that Au h := N(x)Adn+l, the directional derivative of the discrete weak form Ge(d,+~) is computed as

d__ a . (d .+, +. Ad.+,) = 8d' ' Ad.+, d~ ~-o ad.+1

(3.21)

Thus, the element tangent stiffness matrix is given by K~'=Gt[OlJn+l/dn+l]. The matrix [~t3n+l/~d,+l] i,~ obtained by linearization of the discrete system (3.5). To this end, from (3.4c) and (3.20) we first observe that

Etrial fo e n+l ~--" Gd~+l w

~ik7 trial a-*n+ |

S(x)e p dl~ ~ ads+ ' = G . (3.22)

Y.C. Simo et al., Complementary mixed fini:e element formulations 191

For simplicity, hereafter we restrict our attention to the model problem. Differentiation of (3.12a, b) then yields

,-,- ~ f t #7,,+~ dn= 1 (~n+l ~t. Jne S Ou.(~tn+ 1 (~dn+l =,,+x #d.+ t

6;, (3.23a)

a/],+l 2 07,+1 =0, (3.23b) [a,,.f]t$ 8d.+t 3 H' 8d.+~

where ~+1 is defined by (3.14b). Solving (3.23b) for [Oy~+llad.+l] and substituting in (3.23a) yields

a.s,,+, = [=._, + 3 f,,. t t 1-1 S (oo.(k,,+l)(a~.ck,,+,) S d13 G. (3.24)

Finally, by inserting (3.24) into (3.21) one obtains the expression of the element tangent stiffness in the form

S'(a,,k,,+x)(O=,ck,,+x)tS dfJ]-I G. (3.25)

REMARK 3.2. The tangent stiffness for (3.5) is found by differentiation of (3.5a, c), again considering q. +1 as a function of both/3,, +1 and % + 1 via (3.5b), and proceeding along the same lines as above to obtain ([39])

o [ - , fo ~--" ~n+l -- -1

i s ~T'n+ l~rn+ 1 s dD G, ~n+ t t

.+, : - 1/[(0.)'(0', ,e + (3.26)

where ~.+t and N.+t are defined in (3.7). Notethat Ke in (3.26) is symmetric for arbitrary functional forms of the yield criterion (3.2) and hardening law (3.3).

Global expressions for the residual force vector and the tangent stiffness matrix are obtained through the standard assembly procedure.

4. Application and numerical simulations

We consider stress interpolations for a quadrilateral element with bi=linear isoparametric interpolation for the displacement field. The stress interpolation is a modification of that proposed by Pian and Sumihara [28]. For linear isotropic elasticity the resulting element appears to be optimal. Its performance under distortion and in the nearly incompressible limit is excellent. The proposed modification enables one to achieve all the computational advan- tages reported in [41]. The performance of this element, 11owever, appears to be superior to that obtained with the interpolation proposed by these authors.


4.1. Stress interpolation for a quadrilateral element

Let g~e C R2 be a typical quadrilateral element with nodes x A E R 2 (A = 1,2,3, 4), and denote by qt. g] ---> g~e the isoparametric mapping, where g] : = [ - 1, 1] x [ - 1, 1]. We have

t~ ~ b ---> ~,(~) = ~, fc~(~)x~ , A=I

(4.1)

where ~A(~) are the standard bilinear interpolation functions on the unit square/]. Let .~ "= qt(0) be the centroid of /~e and /~ the Jacobian of the isoparametric mapping at I!he centroid, i.e.,

o,t,(~) ] [Pl P~] (4.2)

Define ~rh(~) as the stress tensor tr convected by the constant mapping #, that is ~.h := l~-lO.l~-t. We consider a stress interpolation, discontinuous between elements, and defined in terms of five parameters as

, , h 1 ----'/'o + A(~ 2 - ~2)[0 (4.3) where %h, A, B are constant and ~' are defined by the expression fa, (~ ' - ~ ' )d~ = O. The interpolation for the stress tensor is defined by mere transformation (push-forward) of (4.3) with F. Replacing ~.h by the alternative constant matrix tr~ : - l~.~t , the final result may be expressed in reduced matrix notation as (recall that x = ~(~) )

0.11 0 .22 0.12

h

= [ I1 (6 ~- ~2)a, I(' - ~ ' )a2]g -S(x )O, , (4.4)

where I := Diag[1, 1, 1] is the (3 x 3) unit matrix, tS~ is the (5 x 1) vector of stress parameters, and at, a2 are constant (5 1) vectors defined in terms of the components of 1~ as

a, := [(~',y (~)" (~:,)l-'-' ', (4.5)

REMARKS 4.1 (1) The stress interpolation outlined above is obtained from that proposed by Plan and

Sumihara [28], by replacing fi with (~:i- ~) , i = 1, 2. For linear elasticity where the tangent elasticities C are constant, this modification results in a block diagonal weighted compliance matrix H ~-1, i.e.,

H~- J _ "vol(/'Je)C -1 0[32 ] ]

(4.6)

J.C. Simo et aL, Complementary mixed finite element ~ormulations 193

where O-1 is a (2 x 2) matrix. Thus, He can be inverted in closed form. For the elastoplastic case this computational advantage is lost, even if a~,, f(o') is constant, as in the case of the Von Mises yield condition.

(2) It can be easily seen that the interpolation outlined above passes the mixed patch test in the form recently discussed by Zienkiewicz et al. [42]. In particular, the one element test with minimum displacement constraints gives 2 4 - 3 = 5 displacement degrees of freedom, which is the number of free stress parameters/3.

"4.2. Numerical simulations, extension of a perforated plate

Two numerical examples are considered to demonstrate the applicability of the new variational formulation of elastoplasticity as well as the reliable performance of the corresponding numerical implementation. All calculations are performed on a Convex C1 computer by inplementing the algorithm shown in Boxes 1 and 2 in an enhanced version of the nonlinear finite element computer program FEAP, developed by R.L. Taylor, and described in [40, Chapter 24].

4.2.1. Exter&~ion of a perforated plate A perforated plate characterized by the plane stress/,_-flow material model in Section 3.4

with isotropic hardening is considered first. Displacement controlled boundary conditions are imposed, as depicted in Fig. 1 and the material properties are: Young's modulus E - 70, Posson's ratio v -0 .2 , uniaxial tensile yield stress K 0 -0.243, hardening modulus H ' -0 .2 .

Figure 2 shows a comparison of the load-displacement curves (load calculated from nodal reactions), predicted from the mixed formulation and the standard displacement formulation, each using an identical mesh of 4-node quadrilaterals. The plane stress version of the classical radial return employed here is due to Simo and Taylor [6]. The mixed formulation provides more accurate load-displacement predictions than the displacement formulation as seen by comparison of the solutions for the 72 element mesh and the 722 element mesh. Aside from the added compliance of the mixed solution, however we observe that the load-displacement curves are in good agreement for the two formulations.

.4 t t t t t .~ a

lllll

18

I_ J r I0 ~1

Fig. 1. Geometry, mesh and loading configuration for a perforated plate.


2.2

2.0 ""

1.8-

1.6-

1.4- o,

1.2 "

1 .0 "

0 ,8 "

S .

(a) Oispl. method, 722 elmts ,, (b) Mixed method, 722 elmts . . . . () Displ. method, 72 elrn;s

{d) Mixed ,method, 72 elmts

0.6 u u t 0 2 4 6 8

Displacement

Fig. 2. Load-displacement curves for perforated plate with: (a) 722 element mesh, displacement formulation; (b) 722 element mesh, mixed formulation; (c) 72 element mesh, displacement formulation; (d) 72 element mesh, mixed formulation.

Figures 3-6 compare the evolution of the plastic zones predicted by the mixed and displacement formulations, again using an identical mesh of 4-node quadrilaterals for each. Specifically, the figures contain contour plots of ~ values, where

= f (o ' ) Ko := [K(q)]q_o. (4.7) X.V~'~ K 0

The legend .shown in Fig. 3 applies to all of Figs. 3-6. In each figure, the plastic zone is characterized by ~k ~ 1.0. Again, the mixed formulation gives somewhat better (more flexible) results than the displacement formulation. With a 72 element mesh, the mixed formulation is better able to predict the semicircular elastic region neighboring (x, y) = (10, 0) as shown for

= 0.15. Aside from this added accuracy, the yield zones predicted by the two formulations are in good agreement.

As expected, the rate of convergence exhibited by the global Newton iteration is excellent, as shown in Tables 1-4. In discussing the rate of global convergence for this problem, it is important to recognize two facts. Firstly, Tables 1-4 suggest that the radius of covergence for this problem is such that quadratic convergence is attained only for energy norms below 10 -6 . Secondly, as the mesh becomes more refined for either formulation, the rate of global convergence experiences some degradation, as shown in Tables 1 and 2 or 3 and 4. Consequently, since the mixed problem is more flexible than the displacement problem for a


l-'-1 ~ < .90 .90 < ~)< 1.0

! i 1.o Ii Fig. 3. Yield zone evolut ion for displacement formulat ion with 722 elements. (a) ~-0.15. (b) t i -1 .65 . (c) ti = 3.15. (d) ti = 4.65. (e) ti = 6.15.

Table 1 Global Newton iteration energy norm convergence rate for displacement formulation, 722 e lement mesh

Load step

1 5 10 17 I terat ion (~ -- 0.03) (~2 - 0.15) (ti --- 2.65) (ri -- 6.15)

1 0.905e + 00 0.929e + 00 0.258e + 03 0.257e + 03 2 0.214e - 04 0.120e - 04 0.150e - 02 0.996e - 03 3 0.124e - 05 0.449e - 06 0.827e - 04 0.238e - 04 4 0.831e - 08 0.A.A~A.e -- 08 0.303e - 05 0.149e - 06 5 O. 133e - 13 0.680e - 12 0.730e - 07 0.916e - 07 6 O .190e- 24 0 .228e- 19 0 .535e- 11 0 .211e- 09 7 ~ 0 .226e- 30 0 .717e- 19 0 .177e- 14 8 m n 0.406e - 28 0.135e - 24

196 J.C. Simo e: al., Complementary mixed finite element formulations

[~ ~ < .90

.90 < ~ < 1.0

BI 1.o II Fig. 4. Yield zone evolution for displacement formulat ion with 72 elements. (a) ii - 0.15. (b) Ii -- 1.65. (c) ii = 3.15. (d) ti =4.65. (e) ti = 6.15.

Table 2 Global Newton iteration energy norm convergence rate for d isplacement formulation, 72 e lement mesh

Load step

1 5 10 17 Iteration (~ = 0.03) (~ = 0.15) (~ ffi 2.65) (~ ffi 6.15)

1 0.270e + 00 0.294e + 00 0.817e + 02 0.717e + 02 2 0.111e - 04 0.104e - 04 0.138e - 02 0.628e - 03 3 0.119e - 05 0.177e - 07 0.705e - 04 0.4~ ~e - 05 4 0.233e - 09 0.305e - 12 0.623e - 06 0.637e - 09 5 0.161e - 16 0.258e - 21 0.124e - 09 0.269e - 16 6 0,127e - 30 0.217e - 31 0.240e - 16 0.273e - 29 7 - - - - 0.662e - 29


$ < .90

.90 < $ < 1.0

/ 1.o

Fig. 5. Y ie ld zone evo lut ion for mixed formulat ion with 722 e lements . (a) I i - -0 .15 . (b) t i - 1.65. (c) t i - 3.15. (d) ti - 4.65. (e) ri = 6.15.

Tab le 3 G loba l Newton i terat ion energy norm convergence rate for mixed fo rmulat ion , 722 e lement mesh

Load step

1 5 10 17

I terat ion (ti -- 0.03) (ti -- 0.15) (ti - 2.65) (~ = 6.15)

1 0.897e + 00 0.921e + 00 0.256e + 03 0.245e + 03 2 0 .220e - 04 0.217e - 04 0.831e - 03 0.185e - 03 3 0.175e - 05 0.196e - 05 0.683e - 02 O. 156e - 04 4 0.449e - 07 0.883e - 07 0.753e - 04 0.189e - 05 5 0.320e - 13 0.719e - 10 0.871e - 05 0.951e - 07 6 0.559e - 23 0.125e - 15 0,275e - 06 0.375e - 11 7 m 0.483e - -27 0.176e - 07 0.210e - 18 8 m ~ 0.192e - 11 0.164e - 25

9 ~ ~ 0 .578e- 19 10 - - m 0.255e - 26 - -


r - ] ~ < .9o .9o < < 1.o

B > 1.0

Fig. 6. Yield zone evolution for mixed formulat ion with 72 elemeats. (a) ii = 0.15. (b) 6 = 1.65. (c) 6 = 3.15. (d) 6 - 4.65. (e) 6 - 6.15.

Table 4 Global Newton iteration energy norm convergence rate for mixed formulat ion, 72 e lement mesh

Load step

1 5 10 17 Iteration (6 = 0.03) (6 - f). 15) (6 - 2.65) (6 - 6.15)

1 0.264e + 00 0.288e + 00 0.800e + 02 0.779e + 02 2 0.222e - 04 0.164e - 04 0.970e - 03 0.196e - 03 3 0.2b0e - ~ 0.165e - 06 0.797e - 04 0.139e - 05 4 0.332e - 08 0r 645e - 1 ] 0,755e - 06 0.278e - 09 5 0 .228e - 14 0.956e - 20 0.222e - 09 0.210e - 16 6 0.172e - 26 0.116e - 29 0.320e - 16 0.564e - 27


given mesh, the mixed problem converges slower than the displacement problem when the same mesh is used. One should note, however, that within a global Newton iteration, the mixed formulation typically requires less computational effort than the displacement formulation, since the mixed method performs the plastic return mapping only once within a plastic element, whereas in the displacement method the return mapping is performed four times (once at each Gauss point).

4.2.2. Bending of a tapered beam In this example we consider a tapered beam with elastoplastic response characterized by

plane stress J2-flow with isotropic hardening. The elastic version of this problem is often referred to in the literature as Cook's Membrane Problem". We impose load controlled" boundary conditions, as depicted in Fig. 7, and assume the following material properties:

1

Fig. 7. Geometry, mesh and loading configuration for a tapered beam.

2~ , , ' ' ' '

2O

s

, I , I ,

0 o 15 20

Elements per Side - n

$ fSSS .S . . . . . " . . . . " " " " "

/ /"

. /

/ / i

] Assumed Stress Formulation I l l ] Displacement Formulation . . . . . i i i i

I ;0 .,5 ~'0 i

35

Fig. 8. Displacement of the top, right corner versus the number of elements per side in the finite element mesh for the (a) assumed stress formulation, and the (b) displacement formulation. Equal numbers of elements in the horizontal and vertical directions are ,~oa tt,.,.,,,,.h....~ . . . . . . . , s , a ~ww~n,ev~L.


Young's modulus E = 70, Poisson's ratio v =0.3333, uniaxial tensile yield stress K 0 =0.243, hardening modulus H' = 0.2. The loading is increased in increments of AF = 0.1 until a final value of F = 1.8 is reached, which corresponds to a state in which nearly the entire specimen is plastic (except for two small elastic zones in the lower left and upper fight comers).

Figure 8 shows the vertical displacement of the top fight node plotted versus number of elements per side, at a load level of F = 1.8, computed with the proposed mixed for~nulation and the standard 4-node bilinear isoparametric displacement formulation with the usual "strain driven" return mapping algorithm. As demonstrated in Fig. 8, for this problem, the displacement formulation exhibits a significant degradation in accuracy over the mixed formulation. In particular, the mixed formulation solution with 64 elements is more accurate

. than the displacement formulation solution with 1024 elements.

5. Concluding remarks

We have presented a discrete mixed variational formulation of plasticity based on the principle of maximum plastic dissipation. The relevant variational principle is simply the discrete statement of the first law of thermodynamics over an arbitrary time-step [t n, tn+~]. Explicitly, we express the potential energy at time t n as the sum of the potential energy at time tn+~ plus the plastic dissipation during [t~, t~+~]. Integration of the dissipation over [t~, t,,+~] with a backward-Euler algorithm yields a discrete Lagrangian whose Euler-Lagrange equations are precisely the momentum balance, the flow rule, the hardening law and the consistency condition, where the latter three characterize a general closest-point-projection algorithm. This results in a discrete optimization problem, formulated in stress space, the solution of which has been considered in detail. Concerning this solution procedure, the following remarks are noteworthy.

(i) Within the context of mixed finite element formulations with independent stress interpolations, the plastic return mapping algorithm can no longer be formulated independently at each Gauss point. The closest-point-projection that restores plastic consistency must be performed at the global element level, and involves all the Gauss points within the element. This is in sharp contrast with traditional displacement-like formulations for which the return mapping algorithm is performed independently at each quadrature t~oint.

(ii) The notion of consistent elastoplastic tangent moduli, [6], also carries over to the present stress-space algorithmic framework. As in displacement-like formulations, these moduli are obtained by consistent linearization of the return mapping algorithm. We have shown that a closed-form expression can be obtained for arbitrary functional forms of the yield condition and hardening law.

(iii) A main motivation for the present study is the extension to the elastoplastic regime of recently proposed mixed formulations. The present developments are applicable to any mixed finite element formulation of the Hellinger-Reissner type. In particular, as an application, we have considered an interpolation recently proposed by Pian and Sumihara [28], which is highly insensitive to mesh distorsion and appears to yield super'convergent results for bending dominated problems.

(iv) The present framework carries over to the elastoplastic analysis of plates and shells. Recent work, [43], indicates that successful mixed interpolations for the membrane and

J.C. Simo et al., Complementary mixed finite element formulatio,,.s 201

bending fields can be developed within the context of a Hellinger-Reissner variational setting. As demonstrated in the numerical simulations, use of the traditional displacement formula-

tion of elastoplasticity over the present mixed formulation may lead to significant degradation in the accuracy of the calculations for certain classes of problems, particularly those entailing mesh distortion.

Appendix A. Weak enforcement of the consistency condition

As noted previously, within the variational framework provided by (2.3) and (2.4), the consistency parameter A~, E K p can be spatially interpolated within the element. To this end, we introduce the approximating set

K p' {A~ hEL2(n) ! h = -- A~/ [n e Ft(x)Ye, eye ~Rs and Ye ~0}, (A.I)

where we have used the standard notation % ~> 0 if and only if all the components are positive, i.e. ~,~ I>0 for A-~ 1 ,2 , . . . ,s. In addition, to ensure that KPhc K p we require that the prescribed set of functions F A : n,--, R+, (A - 1, . . . , s), is positive, that is,

FA(X)>~O Vxen, and A f l , . .~ . , s . (A.2)

Since by assumption YeA ~>0 for A = 1 ,2 , . . . , s, (A.2) implies that AT h ffi FA(X)7 ~ ~>0, so that A7 E K p are required. The stresses again are represented by the discontinuous interpolations defined by (3.1).

A. 1. Discrete Kuhn- Tucker conditions

We recall that the consistency parameter A3,.+ t E K ph, and the yield 0. + t : - 0(or. + 1, q. + 1) (with zr. + 1 E S h) must satisfy the Kuhn-Tucker conditions

function

(A.3)

The discrete counterpart of these conditions, which result from the spatial approximation (2.3) and (2.4) is obtained as follows. First, introduce the notation

f 6.+,(x) := r(x)c,(s(x)IJ.+,, q.+,) d. (A.4)

Clearly, since ~b.~ ~~O, for A = 1,2,... ,m, it follows that ~,;i ~


REMARK A.1. It appears that requirement (A.2) on the approximation may be relaxed to the weaker condition

fn FA(x)dfJ>~O for A = 1 ,2 , . . . ,s . (A.6a) This condition also leads to the same discrete Kuhn-Tucker conditions (A.5), as the following estimate shows:

OA "= fa. F+~(x), df~ ~


Equations (2.4b, d) then yield the discrete problem e i~?trial (p..,)-L(p.+,)v.., +=.., =o,

t 7.+1~.+1 =0, lp.+1 ~>0. (A.IO)

A general algorithm can be developed for the solution of problem (A.10) for an arbitrary form of the yield function ~(o', q) and generalized hardening moduli h(o', q). For concreteness, however, we shall again restrict ourselves to the particular form discussed in the model problem.

A.3. Numerical solution strategy To illustrate the numerical solution of system (A.10), we again consider the model problem

discussed in Section 3.4, where, for linear elastic response, Ee(gl~.l) = Hell,,+1 is linear in /3,. t, with H e defined in (3.11) and

Y := fQe r(x)rt(x) d/] . (A.11)

The discrete yield vector 0~+ 1 defined by (A.4) also becomes linear in Y~+l due to the linearity of the hardening law, i.e.,

~,,+1 :ffi fo, l'(x)[f(S(x)[3"+t) - X/]~ K(eP)] d,O - ~H'Y1~,+~. (A.12)

The discrete system (A.10) now reduces to a nonlinear equation in/3,.t E R m, which can be solved systematically using Newton's method. The step-by-step return mapping algorithm closely resembles that in Box 1 and Box 2 and may be found in [39]. The tangent stiffness for this case then becomes

-1 Ke = Gt r ~._ 1 1 _t t G (A.13) L'"+t = ~H; L"+IY L"+I

where G is defined in (3.20).

A.4. Interpolation of consistency parameter for a quadrilateral To complete the foregoing finite element interpolation, we consider the following choice for

r(x),

eh , (A.14)

where ~A E S], A = 1,2,3, 4, are the Gauss points in natural coordinates, 8(.) denotes the delta function and ~ = qr,l(x). Clearly, (A.14) is the simplest choice that satisfies the condition rA(X)30, for x ffi qt(~), as required. In fact, Y becomes a diagonal matrix since


YAs=f~ ~(~- _Ea)8(g- ~s)S(~)d~ 1 d~2 =/~A8 J(ga), (A.15)

where J(~A) is the Jacobian of the isopara_metric mapping evaluated at ~:A and a 2 2 quadrature is used (W 8 = 1). Also note that ~ defined by (A.12) becomes

~A ~- ~A 2 t - (A.16)

Therefore, denoting by fA := f(S(~:,~)/3), the consistency parameter is determined simply as ~/A = (fA - V~ K(e.P)) /}H'.

One should note that, in the case that a 2 x 2 quadrature is used over 1~e(W s = 1), the delta function interpolation (defined by (A.14)) as well as the bilinear interpolation of A~,.+ I between Gauss points results in a formulation identical to that in Box 1 and 2.

Acknowledgment

We are indebted to M.S. Rifai for his help and comments on the subject of this paper, and his suggestion for the simulation in Fig. 8. S!lpport for this research was provided by a grant from the National Science Foundation. J.G. Kennedy was supported by a Fellowship from the Shell Development Company. This support is gratefully acknowledged.

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Mixed FEM for elastoplasticity