~UU'UC.VU(nl

A: U+U~dXh9(V~,lP

ON THE APPROXIMATION OF PROBLEMS IN NONLINEAR ELASTOSTATICS

J. T. Oden

The University of Texas at AustinAustin, Texas

ABSTRACT

An abstract approximation theory for nonlinear problems defined on reflex-ive Banach spaces is presented. It is shown that finite element approximationsof a large class of nonlinear elasticity problems fall into the types of approx-imate methods covered by the general theory. General convergence theorems areproved and techniques for obtaining error estimates are discussed.

NOMENCLATURE

a (generally nonlinear) operator mapping U into U'a material volume element, as a material surface elementa mesh parameter for a finite element mesh representing n

- g(V~(~),!)- the value of the constitutive functional defining thePiola-Kirhhoff stress at the displacement gradient Vw(X) at particle X

• ~ (~) - the surface traction applied at 1$ £an2 - - -real reflexive separable Banach spacethe topological dual of Uthe space U is continuously embedded in Vthe space of admissible displacements corresponding to the responsefunctional g(Vw,X)

- (v :g(V~,J9: !ELl(n), ~. ~ on anI}

V real reflexive separable Banach spaceV' the topological dual of Vva,p(n) Sobolev space of order m,p· (u: a~uELP({l); lal <m, m>O, l<p<"'}

w:,p(n) - (uE:W-,p(n) :'a~u(!) - 0 on an, I~I~m-l} - - - --

~ - (xl,xl.x3) • the particles labels for n· 3

a~u • al~lu/a~l X~2 x~3 ; ai

- integers greater than or equal to 0,

I~I • al +a2+a3

a re~lar....2Pen~omain in ~n. n - I, 2, or 3, with a SDlOoth boundaryan. an • aOlU ao2, aOl ()a02 • e

66

n closure of flp f • p (X)f(X) ; the body force vector at X; p being the mass density in0- 0 - - - , 0

the reference configuration

1. INTRODUCTION

In this paper, we consider a number of theoretical questions that arise inthe approximation of boundary value problems in nonlinear elasticity, In par-ticular, we will consider some very general results in existence and approxima-tion theory of equations in abstract spaces which apply to boundary-value prob-lems of the following type: Find a displacement field ~ E U(fl)

( g(V,=,~) : V:,dX - ( po~':,dX + i ~'~ds V ~ E U(fl) (1.1)1n ~ a02

where A: B = tr ABT for any two second-order tensors A and B. Equation(~) is a'statement of the principle of virtual work, and, under appropriatesmoothness assumptions, it is equivalent to the classical elastostatics prob-lem:

DivQ(Vw(X),X) + p (X)f(X) ; 0 , X E fl I____ 0----~(~) - Q , 1.' E aOI

(1. 2)

9(V~(!,),~)'~(~) = ~(!') , ~ E afl2

Here r: is the unit outward normal to afl

A key requisite to the development of an approximation theory for problemsof the type (~) is the availability of a corresponding existence theory.Until very recently, however, no acceptable existence theory for problems ofnonlinear elasticity were known, and the theory of their approximation had notprogressed beyond special cases involving monotone operators (1).

In a recent communication (2). we developed a collection of generalexistence theorems for nonlinear operators defined on reflexive Banach spaces,and we showed that these results cover a large class of problems in nonlinearelasticity. In (3), we give concrete examples of applications of the abstracttheory to model problems in elastostatics. In these theories, we do not assumestrong ellipticity, as is done in the work of ANTMAN (e.g. (3» and BALL (5),and, therefore, we may include cases of the type studied by KNOWLES andSTERNBERG (6) in which the strong ellipticity condition can be violated,

We review some of these results in Section 2 of this paper. In the re-maining sections of this paper, we develop a general theory of approximationfor problems of the type (1.1). We show that conventional finite elementmethods are among those which can be successfully used to approximate suchproblems, and we describe a recent result of KIKUCHI (7) which establishesthat, under the conpition that the approximations are defined on a family {Uh}of closed subspaces of the "solution" space U whose union U ~ is every-

hwhere dense in U, the approximations converge strongly to solutions of thegiven problem. Finally, we discuss, in a qualitative way, how a priori errorestimates can be obtained for certain problems.

:2. SONE ABSTRACT:'PSTENCE THEOREMS

In (2), several existence theorems were proved which establish sufficientconditions for the existence of solutions to a class of nonlinear equationsdefined on reflexive Banach spaces. In this section, we record a number ofresults of (2) and show conditions under which they apply to problem (~).Throughout this discussion, we us~ the following notstions and conventions:

U,V • reflexive separable Banach spaces equipped with norms1I'lIu' 1I'lIv' respectively

66

UC.V;i.e, Umoreover,

A : U + U· is adual U·

is continuously and densely embedded in V;the injection i: U + V is compact(nonlinear) map from U into its topological j

(2,1)

In general, we also are concerned with situations in which there exists aHilbert space H, identified with its dual H' for which V ~ H = H' C. V· ,but we do not use this fact in the general theorems. Throughout this discus-sion, we assume that all spaces are real.

We are concerned with the following abstract problem: Find u € U suchthat

A(u) = f

where f is a given linear functional in U· Let us denote the value of thelinear functional f at v ~U by f(v) = (f,v); i,e, the bracketsf,v + (f,v) denote duality pairing on U' xU. Then the abstract problem canbe put in the equivalent form: Find u € U such that

(A(u) ,v) - (f,v) VvE,U (2,2)

We recall that an operator A: U + U' is said to be hemicontinuous if

lim (A (u+6v) ,w) = (A(u) ,w) V u,v,w E: U6-1{)

(2.3)

and that A is bounded if it maps bounded sets in U into bounded sets in U',An operator A: U+ U~ is coercive if

(A(u) .u) + + ... as lIullu + ... (2.4)

We next state a fundamental theorem proved in (2):

Theorem 1. Let conventions (2,1) hold, and let A be a bounded, hemicon-tinuous, coercive operator from U--Ynto U· '+ Mo+eover, let there exist acontinuous, non-negative-valued function H:~ x m ·+lR+ with the ?roperty

1 +lim i "(x,6y) - 0 \lx,y€,me+o

such that for every u,v in the ball

Bµ(O) = {w E U: Ilwllu< µ} µ > 0

we have

(A(u)-A(v), u-v)~- H(µ,lIu-vllv)

(2.5)

(2.6)

(2.7)

Then, for every feU', there exists at least one solution u E U to problem~), 0

This theorem can be generalized to cover cases in which A can be decom-posed into two parts, only one of which satisfies (2.7). This generalizationproves to be useful in applications to elasticity problems involving incompres-sible materials or nonconservative loads, In such cases, we encounter operatorsrepresentable in the form

A(u) = A(u,u)

We then have (cf. (2»:

u,v .. A(u,v) : U x U + U' , (2.8)

Theorem 2. Let conventions (2,1) hold and let A: U + U satisfy (2.8).Further, let A(',') have the following properties: ---

(i) \I v E U, u-+ A(u,v) is hemicontinuous from U into U(ii) V U,v e Bµ(O), where Bµ(O) is defined in (~),

(A(u,u)-A(v,u) ,u-v)~- H(µ,llu-vllv) , (2.9)

67

1 (1. e, H is

element u ~ U,

] (2.10)v V,w E U

Vv€ulim inf <A(v,u ) - A(v,u) ,u - u) > 0rr- n n-

lim inf <A(v,u ) -A(v,u) ,w) = 0n

then

where H is a function with the properties described in Theoremcontinuous, non-negative, and satisfies (2.5»

(iii) If {u } is a sequence in U ~verging weakly to ann

and

Then, for every fEU , there exists at least one solution to problem (2,2).-LJ

Remark, ~ key aspect of these results is that, under the conditions stated,the operator A can be shown to be coercive and pseudomonotone. The fact thatit is surjective then follows from the theorems of BREZIS (8) and LIONS (9). [J

Condition (2.7) (or 2.9» is a key property in our theory, In specificapplications, we usually obtain a slightly stronger result: V u,V ~ U ,

<A(u) -A(v) , u-v)'::' F(llu-vllu) - 1I(lIullu,llvllu,lIu-vllv) (2.11)

where + +F: R ... ~ is a continuous, non-negative, non-decreasing, real-valuedfunction such that

lim F(x) • 0 ,x-+O+

(F(x) > 0 • x j 0) (2.12)

+ 3 +H: (~) +~ is a continuous, non-negative, real-valued function such~at 1 +

lim B H(x,y,8z) s 0 V x,y,z E: ~ (2.13)8+0+

and H(·."z) is non-decreasing for every zE: lR+ in the sense thatxl~x2' Yl~x2::;=' H(xl'Yl'z) ~H(x2'Y2'z)

Obviously, the conditions (~) and (~) are satisfied by (2.11) since

F(x) ~ 0 and H{u,u,z) ~ H(u,z) .

We refer to (2.11) as a generalized G2rding inequality because of its similarityto GRrding's inequality of linear elliptic theory.

Let us now return to problem (~). We introduce the notation

<A(w) ,v) ..i<!,:r) .. I

fl

g(V~,~) : v~ dX

po~'~dX + i S'vdXa02

) (2,14)

Then (1.1) reduces to

<A(w),v) - (f ,v) VvE:U (2.15)

The definition of U, of course, depends upon the form of the response func-tional Q(',') and the boundary conditions.

Example 1,

In (3), we show that if

f ka<~(~) ,~) .. .h Q (V~)vk,a dX

ka a [ 2Q (v~) .. aw El (II - 3) + E2(II - 3) + E3(IZ - 3)k,a

(2.16)

68

2where 0 C ~ and II' 12, 13 are principal invariants of Green's deformation

tensor %:S a (Oak+wk,a)(OSk+wk,S) , ll,S,k = 1,2, and El' .."E4 are

material constants, then 4(i) A is bounded and hemicontinuous from wi, (fl) into its dual

w-l,4/3(0) - ~o- _1 4 -1 4/3(11) A is coercive from W-' (0) into W' (0) 1£

-0 -

(2,17)El > IE3' or E2 > 0 and E3 ~ 0

(iii) V U,v E B (0) C wl,4(n), then V E > 0 3 Y > 0 such that~ - -lJ ~ -0 0

4(A(u) -A(v) ,u-v) ~ 2(E2-E)llu-:r1l1 4

- ~ ~ - - - - W-' (fl)

+ 2(El+E3-4E2)lIu-vll\ 2 - Y(E'lJ)lIu-vIl4~3- - w-' (0) - - L (fl)

I 4 - - (2,l8)(iv) 'If U,v,W EO: W ' (0) , 3 C > 0 such that

- -. -0

(A(u) -A(v) ,w) ~ cllwlll 4 lIu-!1I 1 4 g(~':r)- - - - - - w-' (n) - W-' (fl)- -

g(u,v) a (1+lIull\ 4 + 11,,1121 4 )

- - - ~ ' (n; ~' (fl)

are Sobolev spaces; i.e.

) (2.19)

(2,21)

~r,p(n) = (~ - (ul,u2) : uS' a~us ~ LP(fl) , lal ~r, a,S = 1,2) )- (2,20)wi' p (fl)- {u £ WI, P (0) : U - 0 on aa}-0 - - --

We equip Wl,p(O) with the norm_0

lIuliP- Wl,p(Q)

where Iv~IP = [ [la~uSIP. Similarly, ~p(n) = (LP(fl»2. The space1 101-1 S=l,2 1

W ,p(n) equipped with the above norm is a reflexive Banach space and W ,p(O)-0 -0

is densely embedded in LP(fl) with compact injection.In this case, 1£ E; - E > 0 ,

then

El + E3 - 4E2 ~ 0

4F(x) = 2(E2 - dx

4/3H(lJ,lJ,z)- H(lJ,z) - Y(E,lJ)X

(2.22)

(2,24)E > 0o

With A defined by (2,16), the problem of finding w € w1,4(fl) such that- •• 0

(~(~),y)= <!,y) V y EO: ~,4(fl) , (2.23)

where (f ,v) = 1pof.v dX is equivalent to the Dirichlet problem (1.e" the

problem of place) in nonlinear elastostatics. Here afl- aOl ' a02 = ~Notice that requirements of our existence theorem lead to conditions

(2.l7) and (2.22) on the form of the constitutive equations,We can also include singular behavior and the requirement of local invert-

ibility in our analysis; e.g. if J - 113, we add to (2,16) the term

kn aQ - a (E lnJ(w)o wk,a 0 •

69

While the set Mv = (~ ~ ~~,4(0) :J(~) ~ v > 0 a.e, in O} is not convex, wecan identify a closed, £onvex, nonempty subset Kv of Mv which contains theorigin. The operator ~ - ~o +~, where ~ is given by (2.16) and

(~o(~),~) = iQ:(V~)vk,a dX, can be shown to be coercive, pseudomonotone and

hemicontinuous on Kv for 1> v > O. Hence, solutions to the constrained prob-lem exist in Kv' We then argue that if solutions exist. there exists aconstant vf' depending on the,data, such that J(~) ~ vf > 0 a.e. in nBy choosing vf su.:h that 0 < v f < vf ' we arrive at the conclusion that solu-

tions do exist for the case in which the singular term (2.24) is preaent, (]

3. APPROXIMA nON THEORY

We now consider properties of approximations of the abstract problem: Findu € U such that

(A(u) ,v) - (f,v) v v ~ U (3,1)

(3,3)

(3.2)

where U(i)(11)

is a reflexive separable Banach space, We assumeThe conventions (2.1) hold }A is a bounded, hemicontinuous, coercive operator from U

into U'We will also assume that either

(iii) A satisfies (I:.l) or }(iii)' A satisfies (2,11).

Then, by Theorem l, solutions (not necessarily unique) exist to (3,1) for everyf € U' -

Let {w.} = {wl,w2" ..} be a countable set everywhere dense in U suchthat for anyJ m ~ 1 , [ WI ,w2' ... ,wm] is the basis of an m-dimensional subspaceU (m) of U, Let us inEroduce as an index set (h E: :m. : 0 < h ::.,l}so that

Uh = U ) and h + 0 as m + ... In this way we obtain a family {Uh

}lm O<h<l

of closed linear subspaces of U such that

T: Uh is dense in UO<h5..l

As an approximation of problem (~), we consider the problem of findinguh E: Uh such that

(3,4)

(3.5)

Theorem 3.1. Let conditions (3.2) and either (iii) or (iii)' of (3,3)hold. Then there exists at least o;e-solution uh of (~) for every ~€ U'and for every h, 0 < h < 1, Moreover, if (3,4) also holds, then there existsa sequence of solutions -{~} which convergences weakly to a solution u of(~), Finally, if ~ ~u in U, then a subsequence, also denoted ~,can be found such that

(3.6)lim (A(~),v) - (A(u).v) V v E: U~O

Proof: If the stated conditions hold for problem (3.1), then it is easilyverified that they also hold in prOblem (~). Hence. solutions do exist to(3.5) for every h> O. The remainder of the proof follows easily from the factthat under the stated conditions A is pseudomonotone. [J

Example 2,

Condition (~) is the key property required of any systematic method ofapproximation. We will now demonstrate that for spaces of the type encountered

70

in most elasticity problems (e,g, the example in the previous section), thisproperty can be met using finite element methods,

We review here some fundamental results of CIARLET and RAVIART (10) (seealso CIARLET (11) or, for a summary account, ODES and REDDY (12».

The construction of a finite element mesh on 0 is envisioned as thefollowing proc~ss: n

1. Let 0 be a fixed domain in ~ and let

it = dia (~) P - sup{diameters of all balls contained in ill .

Let fl be another set in ~n with

h - dia (0) P - sup{diameters of all balls contained in 0 l .

The sets 0 and ft are equivalent if there exists a bijective maponto 0; they are affine equivalent if T is affine; i.e. if Tand of the form

T from (lis bijective

where A is an n x n invertible matrix and b is an n-vector.maps, it is easy to show that if IIAll is the euclidean norm of

For affineA •

and

2. Let P be a finite-dimensional space of functions (dim P = N) and letr denote a set of N linear forms on p, The set r is said to be P-unisol-vent if, given any set of N real numbers Qi' there exists a unique p € P

such that ~i(P) = ai ' l~ai~N ,where r = (</li}~=l 0

3. A finite element is a set (O,p,r) , 0 C~n , 0 i ~, afl Lipschitzian,where r is a P-unisolvent set of linear forms on the N-dimensional set offunctions P.

4. A finite element mesh is the union of the i~ages of a sequence of Ebijective maps {T}E 1 of a fixed master element fl onto non-intersecting

e e=domains (O}E 1 ,0 = T (0) .

e e= e eNow let us assume that the elements p of P are polynomials of degree

k+l p •k > O. and, for simplicity, that the maps T are affine. Let W '(0) and

A ewu,q(O) be Sobolev spaces of functions defined on (l , m~O , l~p,q~'" and

• k+1p mpAlet n be a continuous linear map from W '(fl) into W' (II) which pre-serves polynomials of degree k:

For any set 0 which is affine equivalent to n , let

..........nOv = nv

by which we mean nflv(x) = ft~(~) Moreover, let lui denote the semi-norm WU' q (fl)

Then, under these stated conditions, CIARLET and RAVIART (11) have shown thatAA k+1pA

thereexists a constant C = C (fl,n) > 0 such that for every u E: W '(II),1 1 k+l

lu-nnul ~ C(meso )q-p ~ lui k+l.. wm,q(Oe) e Pe W ,p(n.)

71

Now the interpolation ITOu of u on a given element of the mesh must sat-isfy an Inequality of this form, If we use conforming elements (i,e. if we

demand that the projection of u on the partition Ph of 0 be in ~,q(fl»

then we may generate a series of finite-dimensional subspaces U'h of wk+l,p(O)by refining the mesh, These refinements are referred to as regular if thereexists a constant 0> 0 such that h/P e ~ 0 as he + 0, he· dia (£I e)' Ifs·O,l,." is the number of derivatives appearing in the set 1:e of degrees offreedom and 1£

then, V u

wk+l,p(n)c. CS(n)

(~l'P(Oe) ,

(3.7)

<Iu - ITul\/"q(O e) -

Finally, under the assumption that (3,7) holds and that we use regular refine-ments of the mesh, we obtain a family of subspaces

{U } . U C wk+l,p(O)h Q<h~l' h

IcH p -such that VuE: W '(0) there exists a ~ E: Uh for which

II u - - II < Chk+l-ml ul~ \/"p(O) - wk+l,p(O)

Thus, 1£

k+l-m>O

(3.8)

(3.9)

(3.10)

then ~O Uh

is everywhere dense in wk+l,P(fl). Hence, finite element approx-imations constructed in this way can be used effectively to construct approxima-tions of highly nonlinear equations satisfying the conditions descrlbed inSection 2 whenever the underlying spaces U and V are reflexive Sobolevspaces. []

Let us now return to the general problems (3.1) and (~), Following (7),we assume now ~~_L conditions (i), (ii) and (iii)' of (3,2) and (3,3) hold.The operator A: U + U' is then pseudomonotone, which means that-rf {u}converges weakly to u. and n

then

lim sup (A(u ),u - v) > 0n n -n-+<Dv v E U (3,11)

lim sup (A(Un) ,un - v) ~ (A(u) ,u - v) V v E: Un+<»

(3.l2)

Moreover, A is then surjective so that for every f £ U~ ,least one u E: U to (3.1) and a corresponding approximation(3.5). Again, we assume-that (3.4) holds,- Let IJ be a positive numb~such that ~,u,v e: BIJ(0) ,

the ball of radius IJ in U, Then, from (2.11),

there exists at~ satisfying

where B (0) isIJ

so that

Next, observe that

72

(3,13)

(A(~) , uh - u) '" <A(~) 'v -(A(~) ,u)

- (f,uh) - (A{~) ,u)

Hence

lim <A(~), ~-u) - (f,u> - (A(u),u) = 011+0

Thus, from (3,11) and (3.13),

lim (A(uh) , uh - u) = 011+0

o ~ lim (A(u) , ~ - u) + lim sup F(IIu- ~lIu) -lim sup H(IJ,IJ,II u - uhllv)11+0 11+0 11+0

Since ~~·u weakly in U, the first term on the right side of this in-equality is zero, Similarly, the fact that V is compact in U allows us toconclude that a subsequence ~ converses strongly to u in V. Hence,lUi H(IJ,IJ,lIu-~lIv) .. 0, by virtue of (2.13), Therefore, we have11+0

lim sup Fellu - 'i111 U) - 0 .11+0

However, by property (2.12), this implies that

(3.14)

(4.2)

In other words, the ~equence of approximate solutions converges strongly to thesolution u of (b!) ,

4. ERROR ESTIMATES

We shall now outline a method for determining error estimates for approxi-mations of solutions of abstract nonlinear problems for which conditions (i),(ii) and (iii)' hold (see (2.12) and (2,13» under the assumption that for everyu ~ U there exists an interpolant ~ lElUh such that

lIu-~\Iu~hIJg{u), IJ>O (4.1)

where g(u) is a known positive function of u. Clearly, finite element ap-proximations of Sobolev spaces satisfy an estimate of this type.

Let uh denote the finite element approximation of a solution u of (~),Then, ,if eh - u - ~ is the error, we have

\I~II = lIu-~+iih-uhllu ~ hIJg(u) + II~-iihllu •

Moreover, by setting V = Vh

in (~l) and subtracting (3,5) we have theorthogonality condition

(4.3)

Also, guided by Example 1, we will assume that A satisfies

(A(u) -A(v) ,w) ~ i1wllullu- vllukcllullu,llvllu)

for every u,v,w E U where k(',') is a non-negative valued continuousfunction of the indicated arRuments and is non-decreasinK in each argument.Since, by hypothesis, A also satisfies a generalized G~rding inequality,

(4,4)

(A(u) -A(v) , u-v) ~ F(\Iu-vllu) - H(IJ,llu-vllv)

for every u,ve: BIJ(O)C u , Hence,

73

(4.5)

F(II~-uhllu) - H(µ,lluh-uhI1u)

~ (A(u) - A(uh) , uh - uh) by (2.11) and (~)

< Iluh-uhllullu-iihllk(lIullu,lIuhll) by (4.4)

< II uh - iihIIUhµ g(u)k( II ullU.II u - uhll U+ II ullu)

Let m(u) = g(u)k(1Iullu'II u11u) . Then. as h ....0 ,

Iluh-uhlli/~(II~-iihllu) - H(µ,lIuh-iihllv~ ~m(u)hµ + 0(h2

µ) (4.6)

To reduce this estimate further, let us now assume that F and H are ofthe form su~gested in Example 1; i,e,

a+lF (x) - cx ,a > 0 13+1H(µ,x);Y(µ)x ,6>0. (4,7)

Then (~) reduces to

Clluh-uhll~ - Cly(µ)lIuh-iihll~ ~ m(u)hµ (4.8)

wherein C'Cl > 0 and we have used the fact that U is continuously embeddedin V.

If the left-hand side of inequality (~8) is negative, we can (apparently)only show weak convergence of uh to u. Numerical experiments, however, showthat in most applications strong (and rapid) convergence can be obtained in suchcases. A detailed analysis of strong convergence in such cases must make use ofspecific properties of A and the data and is, at this writiQR, unknown. Ifthe left-hand side of (i.J!) is positive, we determine constants C2 and 0> 0

such that C~II~-iih"~ is less than (CllI~-Uhll~ - Cly(µ)II~-iihll~)Then,

c2liu

h- uh"u 2 (m(u»l/ohµ/o

and the final estimate is

(4,9)

A detailed error analysis for approximations of problems of the type des-cribed here continues to represent a cha.llenging and worthwhile objective forfuture work.

ACKNOWLEDGEMENT

The support of this work by the National Science Foundation under GrantNSF-ENG-7S-07846 is gratefully acknowledged. I also wish to acknowledge fruit-ful discussions of this material with C. T, Reddy, M. G, Sheu and N. Kikuchi.The results on strong convergence in Section 3 of the paper are essentiallydue to Dr, Kikuchi.

REFERENCES

1. ODEN, J,T., "Approximations and Numerical Analysis of Finite Deformationsof Elastic Solids," Nonlinear Elasticity, Edited by D. W. Dickey,Academic Press, N.Y., 1973, pp. 175-228.

2, ODEN, J,T., "Existence Theorems for a Class of Problems in Nonlinear Elasti-city." TICOM Report 77-2, Austin, April, 1977.

3. ODEN, J.T. and REDDY, C.T., "Existence Theorems for a Class of Problems inNonlinear Elasticity II. Analysis of a Model Problem of Finite PlaneStrain," TICOM Report 77-2, Austin, August, 1977.

74

4. ANTMAN, 5.S., "Ordinary Differential Equations of Nonlinear Elasticity 1.Foundations of the Theories of Non-Linearly Elastic Rods and Shells,"Arch, Rational Mech, Anal., 1976, pp. 308-351.

5. BALL, J ,/01., "Convexity Conditions and Existence Theorems in Nonlinear Elas-ticity," Arch, Rational Mech. Anal., Vol. 63, No.4, 1977, pp, 337-403,

6, KNOWLES, J ,K, and STERNBERG, E., "On the Failure of Ellipticity of theEquations for Finite Elastostatic Plane Strain," Arch, Rational Mech,Anal., Vol. 63, No, 4, 1977, pp, 321-336,

7, KIKUCHI, N., "Variational Inequalities in Mechanics," TICOM Report 77-8,Austin, September, 1977.

8. BREZIS, II" "Equat ions et Inequations Non-Lineaires dans les EspacesVectorieles en Dualite," Ann. Inst, Fourier, 18, Grenoble, 1968,pp. 115-175.

9, LIONS, J,L., Quelques Methods de Resolution des Problemes aux LimitesNon-Lineaires, Dunad, Paris, 1968.

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