Non-convex energy functionals. Application to non-convex elastoplasticity

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/82/010023-07503.00/0

Vol. 9(I),23-29, 1982. Printed in the U S A Copyright (c) 1982 Pergamon Press Ltd

NON-CONVEX ENERGYFUNCTIONALS. APPLICATIONTO NON-CONVEX ELASTOPLASTICITY P.D. Panagiotopoulos

Institute of Applied Mathematics, University of Hamburg, FRG on l e a v e of the School of Technology, Aristotle University, Thessaloniki, Greece (Received 2 July 1981; accepted for print 13 August 1981)

Introduction The energy functionals encountered in engineering sciences are usually convex. To this fact is connected also the convexity of the yield function in elastoplasticity. In this paper we show that, i f the elastic energy is not f u l l y recovered after a loading cycle, i t is possible to define a nonconvex plastic potential (wich generally may be non-differentiable). The developed theory allows for the consideration of non-convex yield functions [1-4]. The variational formulations are not variational inequalities as in convex problems but they have more general forms, which we call "hemi-variational" inequalities. On an extension of Drucker's convexity and normality rules

Drucker has given in [5] a "geometric" proof of the convexity of the yield surface and the normality of the plastic strain increment dEp to i t .

The

same arguments lead to a lack of convexity i f the elastic properties change with the plastic deformation.

In this case, which has been experimentally

verified [6], the yield surface can be concave as llyushin remarked [2]. An example on this problem is given in [1].

The thermodynamic restrictions

imposed by Green and Naghdi ~] allow for star-shaped not-necessarily convex, yield surfaces. Similar star-shaped domains for the admissible stresses were considered in soil mechanics by Salen~on and Trist~n-L6pez [3]. Let us consider a yield surface F(o)=k 2 and let ~ * = { o i j } be a stress on, or inside i t i.e. ~* E K={o/F(o)~k 2} (Fig. la).

23

24

P.D.

PANAGIOTOPOULOS

o-oCTK(o)IN K (o) w

F(o)=k 2

V

(a)

(b)

(c)

FIG. 1 a) The loading path b) The one-dimensional case c) Re-entrant corners Due to a loading, the stress varies from A to D on the path ABCD. The stress P . increment d'o i j produces a p l a s t i c strain increment dEij be "very small".

Let doi j and d ~ j

Then the loading releases doij and is such that the stress

state o* is reached again (path DCEA). Suppose that the e l a s t i c energy is not all recovered.

The non-recovered energy amount is denoted by W and we

assume that i t depends on a*-o and on the actual stress state a at which the p l a s t i c deformation s t a r t s .

Obviously i f o~-o=0, W(o,O)=0.

We assume that

for the work done over the loading cycle

6A= (aij-oij)dE~j +doij d~j +W(o,~*-o)>0 holds.

~/~* £ K

(1)

In [5] WzO, and in [2]

W(~,~*-~)=~Cijhk (oij-oij)(ahk-Ohk) •

(2)

6C denotes the variation of the elasticity tensor C due to the plastic deformation. If we choose o*=a, (1) yields that do..dE p. >0 lJ 13 If

(3)

.

o * # o , daij can be made as small as on Ashes compared to a*-o and thus

(1) yields

(oij-oij)dEPj +W(a,a*-o)_>0

V o* £ K

I f W-0, (4) represents the well-known "Drucker's stability inequality" or "postulate of maximumplastic work", which leads to the convexity of the yield surface and the normality rule.

(4)

NON-CONVEX ENERGY FUNCTIONALS

IN E L A S T O P L A S T I C I T Y

25

Further assumptions on the form o f W

In order to study the v a r i a t i o n a l

form (4) we have to make an assumption con-

cerning the dependance of W on o and 0*-0.

So f o r instance we could assume

that

(5)

W(o,o*-o) = G(o*)-G(o) or

W(o,o*-o) =(gradG(o),o*-o), (e,o) = e i j o i j

i~j=1,2,3 .

(6)

In (5), G is assumed to be a convex, lower-semicontinuous ( l . s . c ) function on the stress space taking values in (-~,+~], G ~ .

In ( 6 ) , i t is assumed that

=lim{[G(o+p(o*-o))-G(o)]/p} as p~0+ exists.

Both cases lead

to well known p l a s t i c i t y laws i f K is convex: (4), (5) give rise to the law dEp 6 aG(~) ,

oE K

(7)

i.e. "a standard generalized material" [7], whereas (4) and (6) imply the law dcPj = (~G~(--~o°)ij ,

oE K

i.e. a plastic potential law.

(8) I t is reminded that ~G(o) denotes the subgradi-

ent of G at o (~G(o)={~/G(o*)-G(o)_>(~,o*-o)}).

However a more general de-

pendance of Won the actual stress state and the.stress variation than the one given by (5) or (6) can be defined.

We write {o'--Go} whenever o ' ~ o and

G(o')--G(o) and we assume that W(o,o*-o) = G+(o,o*-o) = l im sup inf

G(o'+1](o*'-o'))-G(o') !J

(9)

{o 'Go}, ~0+, o* '~o* where G is a l . s . c , function taking values in [-~,+~], G ~ . differentiable, non-convex and non-finite.

I t may be non-

The r.h.s, of (9) is the defi-

nition of G+(o,o*-o) which is called "upper subderivative" and has been introduced by Rockafellar (see e.g. [8]) in the non-convex optimization. Roughly speaking G+(o,o*-o) is a kind of differential quotient which gives much more information than (5) or (6) on the behaviour of G at the current stress state o.

I f G is Lipschitzian,i.e. i f for every o and o*

G(o*) -G(o) _
c>O ,

(i0)

then G+(o,o*-o) = l im sup

{0' Go}, w-0+

G(o' +IJ(o*-o))-g(o' ) P

(II)

26

P.D.

i.e.

W takes a more simple form. (~-a*,d~ p) + G+(a,~*-o) _>0

Recently F.H. Clarke (cf.

PANAGIOTOPOULOS

From (4) and (9) we obtain (12)

V a* £ K .

[8-.10]) defined the notion of "generalized gradient".

This notion w i l l be used here:

The generalized gradient ~G(a) of G at a is

the set of all X'S such that for every o* G+(a,o*-o) > (x,a*-a)

(13)

and since (12) is a consequence of (4) and (9) we are led to the material law d~p £ ~ ( ~ )

,

a ~ K

(14)

I f G is convex, the generalized gradient §G(o) coincides with the subgradient aG(o)

i.e.

the law (7) results, i f G is concave ~G(o) =-a(-G(a)),and i f

G is d i f f e r e n t i a b l e ~G(a)=gradG (a) and (14) represents (8).

I f G is con-

vex, aT is the superpotential of Moreau [ I i ] . I f W-O, (4) leads to the convexity of K and the normality of dEp to K at the point o [5].

I f W~O and (9) holds, then the vector (d~P,-l) is normal at

the point (o,G(o)) to the set epigraph G(epiG) . epi G= {(a,~) / G(o)_<~}.

I t is reminded that

Indeed [I0] we may write

~G(a)={d~P/(d~ p , - I ) E N e p i G ( o , 8 ( o ) }

,

(15)

where Nepi G(a'G(a)) denotes the normal cone to epi G at (o,G(o)). As in the classical time rate eP. ~PE ~ ( a )

p l a s t i c i t y theory i t is more convenient to work with the

In this case we may consider a law of the form ,

~ E K

(16)

Here R has the same properties as G and physically represents the function of power dissipation.

In order to visualize the properties of (16) the

reader is referred to Fig. ib where an one-dimensional law (16) is depicted. I t is worth noting that we are led to the law (16) not only by the procedure followed by Drucker but also by modifying s l i g h t l y the basic hypothesis of Naghdi and Trapp [12].

So i f we assume that "the external work done on the

body by body forces and surface tractions in any closed loading cycle is not smaller than a quantity W depending on o and a*-o

(in [12] W=O) such

that (9) holds" we o b t a i n a l a w involving generalized gradients. tential

R in (16) w i l l be called "superpotential

The po-

in the sense of Clark~' [13].

NON-CONVEX ENERGY FUNCTIONALS

IN E L A S T O P L A S T I C I T Y

27

I f R satisfies (10), ~R(o) can be "approximated" by a sequence {grad R (o)} and the nondifferentiable multivalued law (15) by classical potential laws: Indeed in this case ~ , ~-I~(~) is the convex envelope of the set {×/there exists o~o such that G is differentiable at a and gradR(o ) - × } . Non-convex y i e l d surfaces in p l a s t i c i t y

I t is pointed out that (4) does not imply the convexity of the yield surface K.

Here we shall consider the law (16) with G(o)=IK(O) ={0 i f o E K, ~ i f

o ~i K}, where K is a closed subset of R6 (the space of pointwise stress tensors) i.e. ~P ( ~ K ( O ) ,

K closed .

(17)

I f K is convex this law corresponds to a r i g i d perfectly p l a s t i c body [7]. From (17) results [8], on the only assumption that K is closed, that

~P £ NK(O)

(18)

i.e. the plastic strain rate ~P is element of the normal cone to K at o. o ~ i n t K , ~P=o, otherwise ~P may have a non-zero value.

If

Thus i f

K= {o/F(~)_
~>_0, F_
(19)

I f K is the intersection of the domains Fi(o)
~iFi=O,

i=l,2,...,m,

then

FiO .

(20)

We obtain thus for K generally non-convex the same results as for the convex case. I t is obvious that (20) describes the r i g i d - p e r f e c t l y plastic behaviour for a non-convex y i e l d surface including also reentrant corners.

I f in the neighbor-

hood of ~ the y i e l d surface is convex~then (18)-(20) coincide with the convex p l a s t i c i t y results.

For a non-convex but d i f f e r e n t i a b l e part of the y i e l d

surface (19) holds.

At a reentrant corner of K, relations (20) are v a l i d ,

and obviously NK(O) and K may have common points, i f (3) remains valid.

Let

us further investigate what happens with H i l l ' s principle of maximum work. We use a new notion, the " s u b s t a t i o n a r i t y " :

We say [8] that a direction

o*-o is a direction of approximately uniform descent (d.a.u.d.) of G at o i f for p > 0 and for every neighborhood Y of a*-~, there exists a neighborhood X

29

P.D.

PANAGIOTOPOULOS

of a, ~>0 and ~>0, such that for 0 < ~ < I and every ~' E X with G(a')-G(a)_<~ we have i n f G ( o ' + p ( ~ * ' - a ' ) ) < _ G ( o ' ) - ~ p , over Y.

the infimum being taken

I f there not exists any d.a.u.d, at o, G is called to be "substation-

ary at a".

By means of a proposition proved in [9], (16) is equivalent to the

assertion, that ~P renders the functional g(a*,X) = G(a*)-(a*,X)

substationary at ~ E K

(21)

or to .p 0 E ~-g(o,E ),

oE K.

(22)

I f G is convex, a s u b s t a t i o n a r i t y point is a global minimum point.

Relation

(21) generalizes H i l l ' s maximum principle for a material law of the form (16). I f G is convex, (22) is identical with the generalization of H i l l ' s principle for the "generalized standard materials"

[7].

In the case of (17), g(o',X) =

= -(a',X) and i f K is convex the classical postulate of maximum plastic work is obtained.

I f (18) holds, l~(a,o*-a).. ={0 i f a*-o E TK(O), ~ i f ~*-~ ~1

TK(O)} [8], ~ere TK(a ) denotes the tangent cone to o ( f i g .

Ic).

Accordingly,

i f o*-o E TK(a), W(o,a*-o)=0 and thus the material is stable in Drucker's sense. The B.V.P. and hemi-variational inequalities Suppose that ~ i s a n open, bounded, connected subset of R3, l e t F be i t s boundary (smooth) and IO,T] a time interval.

We consider the following

B.V.P., on the assumption of small displacements ui: in ~x[O,T]

(23)

• .E ~p ~ " ~:EP+c , E ~-R(a), ~ j=CijhkOhk, o E K in ~x[0,T]

(24)

v=O on F and at t=O

(25)

~ i j , j + f i =P ~i '

I +uj ) ~ij=2(ui,j ,i

a(o)=O and v(o)=O

Here vi=u i , a n d u i , i =. ~~u i /~~ x i For R convex such problems have been studied in [14]. Here (24) implies the inequality (13) and thus the variational i n equality approach of [14] cannot be applied. I f f'l E L2(~), Zad ={T/Ti-EJ L2(~), Tij,jEL2(~)} and Uad = [~HI(~)] 3 (the classical Sobolev space) then we

consider the following variational formulation of the problem: Find a E ~ad /~ K, v E Uad such that to satisfy the i n i t i a l conditions and the relations

NON-CONVEX

ENERGY

FUNCTIONALS

IN E L A S T O P L A S T I C I T Y

~CijhkOij(Ohk-Ohk)d~+fR+(o,o*-o)d~+/vi(oij,j,oij,j)d~20 fpvtv~d~+fo.jv*,jd~1 l ~ 1 i The variational

=~ff'v~d~1 1

29

V o'*~Zad~ K (26)

V v E Uad.

form (26) will be called "hemi-variational

(27) inequality".

For

R convex, (26) holds again but with fR+(o,o*-o)d~ replaced by f[R(~*)-R(o)]d~. Then a variational

inequality results.

As far as the author knows, hemi-vari-

ational inequalities have not yet been studied in mathematics. Acknowledgements The author would like to thank Professor V.F. Dafalias, University of California Davis, for helpful discussions. References [ [ [ [ [

1] 2] 3] 4] 5]

A.C. Palmer, G. Maier and D.C. Drucker, J.App.Mech. 24, 464 (1967) A.A. llyushin, Prikl.Math.Mech. 24, 663 (1960) J. Salen~on and A. Trist~n-Lbpez, C.R.Acad.Sc. Paris 290B, 493 (1980) A.E. Green and P.M. Naghdi, Arch.Rat.Mech.Anal. 18, 251 (1965) D.C. Drucker, Proc. F i r s t U.S.National Congress on App.Mech. (Chicago 1951), 487 (1952) [ 6] G. Zhukov, Izv.Akad.Nauk. U.S.S.R. 12, 72 OTN (1956) [ 71 B. Halphen and G.S. Nguyen, J. de M~canique 14, 39 (1975) [ 8] R.T. Rockafellar, La th~orie des Sous-gradients et ses applications a l'optimization, Presses Univ. Montreal (1979)

[ 9] [I0] [ii] [12] [13]

R.T. F.H. J.J. P.M. P.D.

Rockafellar, Can.J.Math. XXXII, 257 (1980) Clarke, Tran.Am.Math.Soc. 205, 247 (1975) Moreau, C.R.Acad.Sc. Paris 267A, 954 (1968) Naghdi and J.A. Trapp, J.App.Mech. 32, 61 (1975) Panagiotopoulos, Superpotentials in the Sense of Clarke and in the Sense of Warga and Applications. To appear in ZAMM 1982 (GAMM-Sonderheft) [14] P.M. Suquet, Quart.Appl.Math. XXXIX, 391 (1981)