Variational formulations for the plane strain elastic-plastic problem for materials governed by the von Mises criterion

Pergamon

International Journal of Plasticity, Vol. 12, No. 4, pp. 54%560, 1996 Copyright © 1996 ElsevierScience Ltd Printed in Great Britain. All rights reserved 0749-6419/96 $15.00 + .00

s0749.6419(96)00019-8

VARIATIONAL FORMULATIONS FOR THE PLANE STRAIN ELASTIC-PLASTIC PROBLEM FOR MATERIALS GOVERNED BY THE VON MISES CRITERION Antonio Capsoni and Leone Corradi Department of Structural Engineering, Politecnico di Milano, Italy

(Received in final revisedform 25 November 1995) A~'act--Path-dependent materials, complying with Drucker's postulate requirements and governed by an internal variable rate plasticity model, are considered. A variational principle for the small strain, rate plasticity problem is established in this context and extended to cover finite loading steps. Results are subsequently specialized to plane strain solids made of elastically isotropic materials with a plastic behavior governed by the von Mises criterion, accounting for combined isotropic and kinematic hardening. By exploiting previous results, the formulation is fully reduced to the plane. Further generalizations of the statements are also provided, which can be regarded as extensions to the elastic-plastic, plane strain problem of the Hu-Washizu principle in elasticity.

I. INTRODUCTION Displacement-based finite-element solutions of (among others) plane strain problems often exhibit an excess of stiffness and when deviatoric strains become dominant, as typically happens in metals when plasticity spreads, locking phenomena may occur (Nagtegaal et al. [1974]). The use of mixed variational principles is suggested as a remedy: in addition to displacements, other fields are independently approximated and models can be chosen so as to eliminate inconveniences. Even if the effectiveness of the resulting scheme is confined by a stability condition on one side (Babuska [1973]; Brezzi [1974]) and by limitation theorems on the other (Fraeijs De Veubeke [1965]; Stolarski & Belytschko [1987]), efficient mixed finite-element schemes can be designed and their superior performances with respect to displacement models have been assessed by a rather extensive numerical experience, both in the elastic and in the inelastic ranges. If dJLsplacements are mantained as an independent field, for elastic problems stresses and/or strains can only be added and mixed variational principles are basically limited to the Hellinger-Reissner or Hu-Washizu types. When dealing with elastic-plastic materials, however, additional fields enter the formulation and the number of possibilities is increased. In fact, several mixed variational statements were produced, which could be used for establishing finite-element schemes. Some of them (e.g. Simo & Honhein [1990]; Weissraan & Jamjian [1993]) were actually employed to this purpose, but it is felt that the subject is still open to study and the present paper aims at contributing to this line of research. 547

548

Antonio Capsoniand LeoneCorradi

With reference to a fairly general law for elastic-plastic materials, a new variational theorem is introduced and proved for the rate plasticity problem and subsequently extended to cover finite loading steps, as required by computations. Attention is then focused on solids, in plane strain conditions, made of materials governed by a particular, but significant, constitutive model, and both theorems are reformulated in this context. When specialized to the elastic case, the statements reduce to the minimum potential energy theorem and, in fact, with respect to a completely kinematic formulation, only part of the plasticity law is relaxed. A similar rate principle was proposed in a more restricted context by Corradi [1983]; its use entailed some benefits, as it was demonstrated with reference to some examples, but the formulation remains basically kinematic and it is expected to share the troubles experienced with displacement models when strains become nearly deviatoric. In the last part of the paper two actually mixed principles, generalizing the Hu-Washizu theorem to the plane strain (rate or finite step) elastic-plastic problem, are derived from the previous ones. Plane strain formulations refer to elastically isotropic materials with plastic behavior governed by the von Mises criterion, accounting for combined kinematic and isotropic hardening. They exploit the results established in Capsoni and Corradi [1995], where the relevant constitutive relationships were formulated in terms of in-plane variables only, thus making the plane strain problem equivalent to a plane stress one except that for the parameters governing the elastic-plastic model (the main steps of the procedure are summarized). The variational principles are therefore specifically oriented to the solution of a particular, even if relevant, class of problems, but this limitation appears more than compensated by the simplifications with respect to the general case that become possible. Even if the motivation of the study is that of providing a suitable basis for finite-element models, in this paper attention is focused on the variational statements themselves. Details on the approximation assumed for different fields, the discussion on the conditions ensuring stability for the resulting model and its capability of actually eliminating locking, are left to a subsequent contribution. II. THE ELASTIC-PLASTIC MODEL Let the state of the material be known at time t. An infinitesimal (or rate) process leading from t to 7= t+dt, is assumed to be governed by the following constitutive model (summation on repeated indices is understood)

(1)

% = e~ + Po ~o-

~,

_

~e °

~u_

~e~.j

_

Duktekt

X~

~,_

av

~.q~

~.q~

+(%, X~) = f(%, XD - Cro-< 0

(2a,b) (3) (4a,b)

k = O if ~bO,$
if ~b=O

(5a) (5b)

Variational formulationsfor plane strain plasticity

549

where t 0 is the (infinitesimal) total strain tensor, sum of an elastic (eo) and a plastic (P0) contribution; tr,j is the stress tensor; "q~ and X~ are sets of conjugate kinematic and static !internal variables, accounting for the irreversible structural rearrangements at the microscale; these sets can only include scalars and/or second-order tensors (Coleman & Gurtin [1967]) and the index ot refers to their individual components; + is the (for simplicity, regular) yield function, expressed as the difference between the convex, positive and homogeneous of degree one function f((r,j, ×~) and the initial yield limit %; the instantaneous elastic domain, eqn (3), is then convex; eqns (4) express the normality law and eqns (5) Prager's consistency rule. The above relations assume that thermal coupling can be disregarded and postulate for Helmholtz free energy the additive form

~(eo, "q,,) = U(eo) + V('q,,)

(6)

For elastically linear materials, the strain energy U(e~) is expressed by the positive definite quadratic form

1 U(e o) = ~Dijk, euek~ > 0 Ve,7 ;~ 0

(7a)

where the elastic tensor Duk r has the usual symmetry properties. V(-q~) is the stored energy, assumed to be convex, i.e. 1

02V

----il~ii~

> 0 Vii.

(7b)

Equations (1)-(5) are a special form of the associated laws of plasticity, describing a broad class of path-dependent materials that obey Drucker's postulate requirements (the condition dr~b~ _>0 Vp0 is implied by eqn (7b)). Due to the irreversible nature of plastic behavior, the constitutive law must be expressed in rate form and eqns (2) are replaced by the following relations:

(~ij -

32 U

OeoOekr

Ok1 = Duk, bkr

0z V

k. = - - 4 1 ~ O"q,,O"q~

= H~,~il~

(8a,b)

Note also that eqns (5) can be alternatively written as A

~b < 0

A.

k >0

qbh = 0

(9a-c)

where a superposed cap denotes the values assumed by the relevant quantities at the end of the rate process. In particular $ ^ ^ = qb( tr0., ×.) : ~b + d)dt (9d) Equation (9d) expresses the final value of the yield function as the sum of a finite and of an infinitesimal term, the first dominating unless vanishing. Since, as eqn (3) establishes, it is ~ :~ 0 at time t, the equivalence between eqns (9) and eqns (5) is easily recognized. The rate of dissipation per unit volume is defined by the relation

D(Do-, il~) = tr0~0 - t~ = tr,jp,j - ×~il~ > 0

(10)

where the sign restriction follows from Clausius-Duhem inequality. For any given pair/)~, il., the value of D is uniquely defined and corresponds to the optimal value of the problem D(/i,j, "ft.) = max ((r,~.b,7 - ×.il.) subject to +(tr U, ×~) < 0 (11) ~r~ X,x

550

Antonio Capsoni and Leone Corradi

stating the principle of maximum dissipation rate. The problem (11) can be replaced by the following saddle point statement (Martin & Reddy [1977])

D([~o-,il~) = min k

max {L(/io., il~, }t; cro, X.)} subject to "X> 0

(12a)

cr0, X~

where

L(po,% k; ~0, Xo) :

~0P0 - Xofi~ - k +(~0, Xo)

(12b)

is the Lagrangian function of the problem (11).

III. VARIATIONAL F O R M U L A T I O N S OF THE RATE P R O B L E M

Consider a solid of volume 1] and boundary ~[l=~f~+~oll, where ~9~ll and ~uI~ are the free and constrained portions, the latter supposed to be fixed for simplicity. The solid is acted upon by body forces be in ll and by surface tractions f on ~J} and its response to infinitesimal increments (or rates) of external agencies is sought. For displacements and strains small enough to be treated as infinitesimal quantities, the compatibility conditions read

&~=~(ui J 1• where uu =

+@)

in 1~

/t~=0

on

~uf~

(13a,b)

3u/3xj. Equilibrium may be enforced in terms of virtual work by writing

fa~,j~:odx-fab,~i4,dx-fa~a~Si4idx= 0 V B~U=l(g/~iJ+8/~j.~)

inl-l,

8/~ = 0

on

Ou11

(14)

Equations (13) and (14) must be supplemented by the material constitutive law, expressed by eqns (1) (in rate form), (3,4) and (8,9). To produce a variational formulation for the above problem, note that eqn (10) implies, at any instant 0 = f (~+D-o',j&,7)dx=

•Ja

f (~+D)dx-

ga

f

gn

bii~idx- f firdx =0. jo,~a

(15)

(The second expression follows from the principle of virtual work.) Equation (15) holds for the actual evolution; however, if reference is made to solutions merely fulfilling the compatibility eqns (13) and the constitutive law, the functional O = I'00 dr acquires a meaning analogous to potential energy for an elastic system, to which only adds dissipative terms. This suggests that a functional governing the rate process can be constructed on this basis, to be minimized with respect to variables complying with the above conditions. The nonlinear constitutive law, on which the rate of dissipation D depends, is cumbersome to handle as a constraint, but D can be computed by either optimization problems (11) or (12), thus replacing minimization with a saddle point characterization of the solution. To simplify the domain of definition of the functional,

Variational formulations for plane strain plasticity

551

the nfinimax problem (12) is preferable, in that the only constraint to be added to compatibility is the sign restriction on k. Therefore, for the rate process t---)}= t+dt the following functional is considered

o. = fn ( ~ - * + f) dt)dx- fa ~b,duidx- f ~ d u ,

dx

= fn (~ - * + &odp¢ - x~&q~- $dX)dx- fa b, du, dx - f ~ ? d u , dx

(16)

where d( )= (')dt. Note that, consistent with eqns (9), finite quantities are evaluated at the end of the rate process. Let the free energy t~ be expanded in a Taylor series about ~. One obtains, up to second-order terms:

= ~ + 3U deo + _3V _ d "q~ + 1 3ZU de0dekt + 1 _ _32V dxl~drl ~ 3eij 3Tic ~ 2 3e~bekr 2 3"q~3"% = ~ + trijdeij + x~dxL + ~-D,~ktde,Tde,z+

H~dxLd'%

(17a)

All quantities can be expressed as the sum of their values at time t and increments in the step (e.g. ~re=tre+doru). Moreover, since values at t correspond to the actual solution and eqn (13) hold for displacement and strain increments, the principle of virtual work establishes

fn,,jdeodx= fnb, du,dx +f 3,7~fduidx.

(17b)

If eqns (17) are introduced in eqn (16) and the resulting expression is divided dfl, the functional becomes

(~ "~-- l 06 ~ij + 06 Xeal~ qdx - f bii4idx -~-~r~ fiil i dx

dt

(,--~¢~

~

) _1

"n

(18a)

where ~b in eqn (16) was expressed as 6+d~dt, with qb=(b~/b~,j)~r0+(b6/b×~)X ~. Equation (18a) contains the rather unusual expression (6/dt) h, which must be interpreted as identically vanishing in the subregion lip of li where 6 = 0 at t, while in the remaining portion I~, of li, where 6<0, it must be k=0 for the functional to be defined. (6/dt)k can be regarded as a penalty term enforcing the above condition; its introduction eliminates the necessity of distinguishing between the regions of li which are elastic or potentially plastic in the rate process. The following theorem is now proved: The solution of the elastic-plastic rate problem corresponds to the saddle point of the problem min /G P,j, fi~,, k

max (O0 b~, k~,

(18b)

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Antonio Capsoni and Leone Corradi

under the constraints eu = -~-1(~/j + uji). in f~

/ti = 0

on Oaf

k _> 0

in f

(18C)

To prove the assertion, variations with respect to kinematic fields are first considered. With reference to the unconstrained ones, the functional must be stationary and the relevant conditions read

3,Tft

VS~o = 1(8/~i. j + 8/~j.i) in f~

8/~i = 0 on buff

8pOi = ffl [- Dukt(¢:k~-- b~t) + iro]8[~ijdx 8"/IO1 • ~n (H~'/I~ - ~(~)841~dx = 0

= 0

VS~bo.

VS"fi~

(19a)

(19b)

(19c)

Equations (19b) and (19c) enforce eqns (8). By substituting, as eqn (19b) legitimates, /r,7 -- D0k~(~k~-,bk,), eqn (19a) reduces to the equilibrium condition eqn (14). The variation of the functional with respect to the sign constrained variable k must comply with the condition (Panagiotopoulos [1985]): 8xO, = -

f•( - ~dp-

+ $ ]8kdx = -

f~dt

8kdx > 0

VJt + 8)t _> 0

(19d)

where k_> 0 is the value from which the variation 8k is performed. 8k must be nonnegative where k=0, but is not constrained in sign if k is positive. Therefore, eqn (19d) implies locally (b<0 qbJt=0 and, together with the a .priori constraint k>0, enforces eqns (9). Note that, when minimized with respect to h_> 0, the penalty term -(qb/dt)k imposes h = 0 in the region f ~ , where ~b<0. The fields of static rates being unconstrained, the functional must be stationary with respect to them. One obtains the conditions

8 o, = fo(b -0+

k)8/r¢dx= 0

VSb~

(20a)

which enforce the normality law eqns (4). The saddle-point nature of the statement is now proved. Its minimum nature with respect to kinematic variables is first assessed. Let index ( )1 denote the values assumed by different fields when the conditions implied by eqns (19) (the optimality conditions with respect to kinematic fields only) are fulfilled, while unmarked symbols refer to any feasible values, merely complying with eqns (18c). The following relations are readily shown to apply:

cro. = Dijk/(ek, -- [)~.~) • 1

"1

}(~ = H~'ill~

(21a)

Variational formulations for plane strain plasticity

fn6"l A~:odx = fnb~Ai~dx- fo.nfAi~dx ~ 1 ~-~ 0

~1 ~-~ 0

(~1 ~1 = 0

553

(21b) (22a--c)

where A( )=( ) - ( )1. Suppose that static fields keep their values ~r0, " 1 2~ fixed and that kinematic fields are varied from this solution. By exploiting eqns (21), the corresponding change in the functional is expressed as follows A O 1 = 0 1 -- ( 0 1 ) 1 =

I~['~Diy, t(A %" -

A/~o)(A~, - Ap,,)

+ D~,z(A~,7 - AP/~)(~l' - P~,) + 1H~A'/I~A#I~ + H~A~ld/l~

+ irl A/~°-x~A'il~- ~b~Ak ]dx- fnb~Ai~dx-

The integrand in the above expression is non-negative because of eqns (7), the condition k> 0 and eqns (22). Hence, AOI___0 when only kinematic variables are changed from their optimal values. Now let index ( )0 mark the actual solution, in which different fields comply with all the governing relations, and suppose that static variables are changed from era, " 0 xo to "1 o.u, x l . In this process, kinematic fields also change, consistently with the conditions implied by eqns (19). Since eqns (9) hold for the solution ( )0 and because of eqns (22) and the principle of virtual work, the corresponding change in the functional reads

AOl = (OO' - (Ol)° = fn [1Dok,(A~- Abo)(A~:k, - A[~,,) +

•

•

H~A'q~A~I~ +

"1 "1 ~ijPo -

7/ "0 • "1 "1 cr0Ab0 - ×,~1~ + X°A~I~ dx

(23)

where now A( )=( )1_( )0. Solution ( )1 complies with all the governing relations, except 1Lhose implied by eqns (20); hence, for the process under consideration, the following relations apply Ab 0 = Du,r(Ai~k, - Ap,r)

A2~ = H~A'/I~

J i Ab'~A~°dx = 0

(24)

Moreover, since the normality law holds for the actual solution, and taking account • 1 "1 " 1 - 1 - - ' 1 " -1 • +'1"0 of the definition of~b, one also has o~apu-×dq~-craA/)0-×~A'q~ + k and, hence "1 " "0 " t~p o -o'/jApo

-

"1"1 x,;q,~ + ~(°A'/I,, = Ab'~A.b,~ A~(,,A'/I~+~lk°

= AbaA/~,j- Ab,j(A/~0- APa) - Ak~A~ + ~,'k ° Since kc=0 in the region where it is ~b<0 when the rate process takes place, one can also write ;blk°=(,~,/dt+~,J)k°=(,bl/dt)k °. By substituting into eqn (23), and taking account of eqns (24), one therefore obtains

554

Antonio Capsoni and Leone Corradi

AOl=f~qI-1Dijkl(A~:ij-ApiJ)(A~:kg-mpkl)-2Ha[3a'+l~~a~+ °ldJX'dl The integrand in the above expression is non-positive because of eqns (7), the condition k°>0 and eqn (22a). This shows that the value of the functional decreases when static • 1 "I fields are changed from their optimal values to cr U, X~, which completes the proof.

IV. THE FINITE-STEP INCREMENTAL PROBLEM

Consider now a finite step ~--+}=~-+At, in which loads undergo increments Abi, Af from their values b~,f at 7. The compatibility and equilibrium equations (13) and (14) are linear and hold for the finite step as well. The constitutive relationships, however, are nonlinear and some hypotheses are required. Their equivalents for the finite step will be derived by means of a variational procedure. Consider the functional eqn (16) and let finite increments replace infinitesimal ones in it. By virtue of eqns (7a), (2a) and (1) one can write

0 - 1.7= Dijkte-ktAeo. + ~DijklAeijAekl

= ~0(A% - APo.) +

Dokt(A % - Apij)(Aekt - APk/).

Since quantities at t- correspond to an actual solution and increments within the step are compatible, the principle of virtual work establishes

fa(u-O+&~AP,j)dX-fnbiAu~dx-fa,,?Au~

dx

:f.[~Do,,(aeu-Ap,j)(,Xe,,-~Xpk,)+,X.ru~Xpoldx-£~b, zXuidx-faoafAu, dx Then, the following expression for the functional is arrived at

Ol = f~[1D~:k,(A%- APo)(Aek:- Apk,) + V ( ~ + A'q~) + AcroAp U

- (y
(25a)

where $=#(80, ~ ) is regarded as a function of static variables. In writing eqn (25a), the term V--V(¥1~) was dropped since it is unaffected by variations. Consider the saddle-point problem min

Aui, Ap,j, A-q~, AR

max (O1) A~,j, A×~

(25b)

under the constraints |

Ae o = 2(Auio + Auj.i) in O

Aui = 0

on 3uO

Ah > 0 in ~.

(25c)

It can be verified that its optimality conditions include the following set of relationships

Variational formulationsfor plane strain plasticity

(26a,b)

Acro. = D/jk,(Aek, - Apk/)

A p ~ - ~cro.~d~ 6"o. Ah > 0

Ah

555

Arl, -

~b < 0

3~b

Ah

~bAh = 0

(27a,b)

(28a--c)

which define the constitutive law assumed for the finite step. Equations (27) are normality conditions, with normals evaluated at the final position, and can be interpreted as a backward difference approximation of eqns (4). Equations (28) replace eqns (9); note that the two sign restrictions and the complementarity condition imply Ah= 0 if ~<0, showing that the irreversible nature of plastic behavior is no longer accounted for within each step, even if unloading from a plastic state is possible at the beginning of the subsequent one. In other words, the material is considered as step-wise reversible, or holonomic. In addition, the saddle-point conditions include equilibrium, still enforced via the virtual work principle, and eqns (25) are a variational formulation of the finite-step, incremental elastic-plastic problem (its saddle-point nature can be assessed as was done for rates).

V. P L A N E STRAIN F O R M OF THE VARIATIONAL S T A T E M E N T S FOR V O N MISES MATERIALS

Attention is now focused on the von Mises yield criterion. The relevant yield function reads +(cro, ot o, [3) = f l t r o. - ao.) - 13

f2 = ~3( S o . - eto.)(S o - ao.)

(29a,b)

where S O is the stress deviator and o~o.a tensor of internal variables (back-stresses), accounting for kinematic hardening and also of deviatoric nature. In addition, the set of static internal variables includes the current uniaxial yield limit 13, possibly varying during the elastic-plastic evolution because of isotropic hardening. Since ~d~l~eto.=-~,bl~So.=-~d~l~cro., it follows from eqns (4) that plastic strains po. are themselves kinematic internal variables, to which a further parameter K is to be addedL. The stored energy and eqn (2b) assume the form V('q~) = VK(po.)+ VI(K)

au-

~v~ ~po.

13-

dVl dK"

(30a)

(30b,c)

Let the body be in a state of plane strain normal to the z--x3 axis. The total strain components ~.:=E.33, e_-x=2~3t and %.--2e32 are in this case equal to zero and, for elastically isotrepic materials governed by the von Mises yield criterion, shearing components with index z vanish for all fields. The in-plane components are collected in the vectors

556

Antonio Capsoni and Leone Corradi

t=

%

t

p=

py

t

o=

g,,

a=

a,,

(31a-d)

For shearing strains, the engineering definition, twice the corresponding tensorial component, is assumed (e~y= 2~|2, pxy= 2p12). In addition, p__or_ and a_ are also different from zero. The following equivalent in-plane plastic strain measures are defined

P=

+up

g=

= p + v~p=

(32a,b)

PXy where v is the elastic Poisson ratio. Then, the isotropic elastic law reads

o = D(~-

E v D = (1 + v ) ( l - 2v) l l - ]v

P)

1 -0v v 2 0 1 ( 1 0- 2v)

(33a,b)

(34a)

~r_ : v(crx + or,,) - E p : : v ~ t ' o - E p :

Since plastic flow occurs at constant volume, one can also write P: = - (Px + P.,.) = - Itttp

(34b)

By combining eqns (32) and (34) one obtains 1 p

=

AP

1 12------~la'P

P-- -

I1 -v

--.~ A - 1 - 2v

-

v 0

v

1-v 0

or_ = Vltt'o +

(35a,b)

1

_E2v ltt,P

(35c,d)

On this basis, the elastic-plastic law can be expressed in terms of in-plane variables only. The detailed derivation was presented in Capsoni and Corradi [1995] and only the results are summarized here. If eqns (35) are introduced in the plane strain form of the von Mises criterion eqns (29), the following equivalent in-plane expression is obtained q~(o, g, 13) = F(o - g) - 13 [ M=

F 2 =1(o-

z ) ' M ( o - Z)

2-2v+2v2 -(l+2v-2v2) - ( 1 + 2 v - 2 v 2) 2-2v+2v 2 0 0

(36a,b)

i] (36c)

In eqns (36), o is the vector eqn (31c) of in-plane stress components and g is an equivalent in-plane back-stress vector, defined by the relation F_ _1 I = Xo+Ii = E r P +

aVl o3p

r -

i 1llo/ (1 - 2v) 2 ko 0 o]

(37a,b)

Variational formulationsfor plane strain plasticity

557

The instantaneous elastic range is now defined by the inequality F(a, Z)-13<0, with the current yield limit always given as 13=dV~/dK. Equation (8b), referring to the rate proce,;s, splits as follows .

=

~3 = h~

LOP 2 J h

(38a)

(38b)

-- ~ 2 V l

~)K2

The normality law still applies, provided that plastic strains are replaced by the equivalent m-plane measures eqns (32). Instead of eqns (4), one has P= l ~ * l x = t~b-J

l ~)*l~.

b * ~ . = }~ ~ = - ~[3

- t-~zJ

(39a,b)

Finally, Prager's consistency rule, when written in the form eqns (9), reads _<0

}~_ 0

(~}~ = 0

(40a-c)

Equations (33) and (38)-(40) express the plane strain constitutive law for a von Mises material in terms of in-plane variables only. Note that the in-plane back-stress vector X, eqns (37), is composed of two contributions: g1=bVK/bPaccounts for actual material kinematic hardening, while X0 is an additional term, present even if the material as such is perfectly plastic, accounting for the effects of yielding in the transverse direction. As the structure of matrix r, eqn (37b), evidentiates, the effects of transverse yielding appear as a fictitious, directional kinematic hardening in the plane picture. The: above constitutive relationships must be supplemented by the compatibility and equilibrium conditions eqns (13, 14), where indices now run on x and y only. The solution of the rate problem still corresponds to the saddle point of the problem eqns (18), with variables replaced by the equivalent in-plane measures introduced. By writing, as eqns (36) legitimate

cD = ff(~r- 3) - [3

n - ~/)a

(41a,b)

the expression eqn (18a) of the functional becomes Ol = Sn [~(/~ -P)'D(/~ - P ) + --1P'I-IP h } ~ +2 ~1 2

+ ( # - ~ ) ' ( i ' - nk)-I~(¢- k ) -

~ k l d x - W(~i)

dt

/

(42)

where W is the second-order work of external agencies. Equation (39b) indicates that can be substituted with }~, even if the latter is constrained in sign while the former in principle is not, and the term involving [3 cancels out. Moreover, static variables appear only through their difference = # - ~

and the functional eqn (42) simplifies to the expression

(43)

558

Antonio Capsoni and Leone Corradi

O' : f n l 1 ( ~ -i')'D(k - ~') + l li ' ' hl - I k~ + 2 2

+~'(t'-nk)-~- t kldx-W(fi)

(44a)

Hence, the solution of the plane strain, rate plasticity problem for von Mises materials corresponds to the saddle point of the problem min u, P, k

max {O j } ~

(44b)

subject to /~=Vfi

inl)

ti=0

on31) U

k>0

inf~

(44c)

The first of eqns (44c) stands for the strain~tisplacement relation eqn (13a). Equations (44) are merely a special form of eqns (18) and a formal proof of the statement is not required. It is only mentioned that the saddle-point conditions read fn(~ -P)'D~/~dx

I

~W = 0 VSe = V~i in fl, ~i = 0 o n 3flu

= D(/~- P ) - I-~

P = nk

~ <0

~k = 0

(45a) (45b--e)

with ~=~p+(n's-hk)dt. It is easily recognized that they enforce equilibrium and eqns (38a), (39a) and (40a,c); that is, the governing relations which do not appear as constraints or were not directly introduced in the functional. Analogously, the functional eqn (25a) for the finite-step problem becomes

Ot = ~aII(A,-AP)'D(A,-AP)+

VK(P + A P ) + V i ( h + Ah )

+ A s ' A P - ~'AP + - F ( g + a s ) a k / d x 2

w(au)

(46)

where eqn (36a) was used for ~. The relevant saddle-point statement is straightforwardly established.

VI. GENERALIZATIONS AND COMMENTS In spite of their saddle-point nature, the above statements generalize to plasticity the total potential energy theorem for elastic structures, to which they reduce when P and h are deleted. In fact, besides equilibrium, only some of the constitutive relationships appear in eqns (46) and the mixed character of the statement is merely due to the fact that these relations change their original role of essential constraints to that of saddlepoint conditions. The rate statement eqns (45) can be regarded as an extension to internal variable models of that proposed by Corradi [1983], obtained from the kinematic theorem of Capurso and Maier [1970] by adding only plastic strains as an independent

Variational formulations for plane strain plasticity

559

field. This permitted the elimination of spurious stresses induced by the difference between the total strain approximation (following from the displacement model) and the plastic strain distribution which is implicitly assumed when enforcing the plasticity law. Benefits were demonstrated with reference to frames (Corradi & Poggi [1984]) and to particular plane strain problems (Corradi & Gioda [1980]) and the present, more general formulation is also envisaged to provide them. New,~rtheless, displacement finite-element models often give rise to locking when the incompressibility limit is approached and it can be expected that the same troubles will also be experienced in the present context when plasticity spreads. To overcome the problem, some of the constraints must be relaxed. The sign restriction on ~., however, is inherent and the only meaningful improvement can be obtained by imposing the strain-displacement relation through Lagrangean multiplieXrs. With reference to the rate formulation eqns (44), this can be achieved by considering strain rates as independent from velocities; the first of eqns (44c) is enforced by adding to the functional the term

f n ¥ (Vf~ - ~.) dx

(47)

where the Lagrangean multiplier vector was denoted by iJ since it can be recognized to have the meaning of stress rate. This is an additional static field and no advantage is now associ~Lted to the introduction of g, eqn (43). The modified saddle-point problem reads

02 = ~n[ l(~-f~)'D(~-P) + l p ' H p

+ l 2 hk2 + "'(Vd - e

- g ' # + - ¢ k - (il - ~ ) ' n k ] d x - W0i) dt J min ~i, ~,P, k

max {O2}, subject to ti = 0 on ~u~, 0, X

k > 0 in 1~

(48a) (48b,c)

The statement, which can be proved as before and straightforwardly extended to cover tinite steps, reduces to the Hu-Washizu principle in the elastic case. The solution of the plane strain, rate plasticity problem is brought to the search of the saddle point of an essentially free functional, the only meaningful restriction on it being the sign constraint on h (the condition on ~uf~ could also be enforced variationally and it is kept among the essential constraints only because it is armless). Note that in eqn (48a), ~ can be replaced by the elastic in-plane strain rate e=/~-iJ; then, eqns (48) can be recognized as the form assumed by the statement proposed by Comi and Perego [1995] when specialized to the problem considered, an operation, however, which requires the reduction to the plane of the elastic-plastic model and must not be regarded as trivial.

VII. CONCLUSIONS In 1:his paper, attention was mainly focused on solids made up of elastically isotropic materials with a plastic behavior governed by the von Mises criterion. Two variational statements were produced for the rate plasticity problem and subsequently

Antonio Capsoni and Leone Corradi

t e n d e d to cover finite l o a d i n g steps. T h e first o f t h e m is essentially k i n e m a t i c in Lture, o w i n g its m i x e d c h a r a c t e r only to the r e l a x a t i o n o f p a r t o f the constitutive lationships, a n d can be expected to share some o f the troubles o f completely kinematic rmulations. T h e second one is o f the H u - W a s h i z u t y p e a n d reduces the p r o b l e m to e search o f the s a d d l e p o i n t o f a n essentially free functional, thus leaving a g r e a t ,~edom in the choice o f the a p p r o x i m a t i o n s for different fields, which can be selected as to a v o i d inconveniences. It was m e n t i o n e d t h a t b o t h s t a t e m e n t s can be r e g a r d e d as extensions o r p a r t i c u l a r ttions o f p r e v i o u s results a n d c o n n e c t i o n s with o t h e r existing v a r i a t i o n a l principles n also be established. Nevertheless, the results o b t a i n e d have the distinctive feature at the f o r m u l a t i o n o f the p l a n e strain plasticity p r o b l e m is fully r e d u c e d to the p l a n e d cast in the same f o r m as the e q u i v a l e n t p l a n e stress p r o b l e m , a result achieved by ploiting the p r o p e r t i e s o f the p a r t i c u l a r constitutive law considered. Even if limited such a context, it is felt that the p r o p o s e d variational statements can provide a suitable sis for the n u m e r i c a l analysis o f a class o f p r o b l e m s o f engineering relevance.

lcnowledgements--This research was supported by the Italian Ministry of University and Scientific and zhnological Research (MURST).

REFERENCES $5 i7 70 13 14 ~4 17 10 13 14 15 t7 ~0 ~3 ~5 )5

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