A •• .-".

EXISTENCE THEOREMS FOR A CLASS OF PROBLEMS IN

NONLINEAR ELASTICITY: EQUILIBRIUM OF

INCOMPRESSIBLE HYPERELASTIC BODIES

J. T. Oden, N. Kikuchi, and M.-G. Sheu

Contents

1. Introduction

2. Mechanical Preliminaries

3. Variational Theory and the Existence of Saddle Points

4. Lagrange Multiplier Theory

5. The Uniform Boundedness of lipII€ Y

6. Existence Theorems for Incompressible Hyperel~stic Bodies

7. An Application to a Model Boundary-value Problem

8. Concluding Comments

References

October, 1978

1. Introduction

In this article, we present some general theorems on the existence and

characterization of solutions to a wide class of nonlinear boundary-value problems

in the theory of finite deformations of incompressible hyperelastic bodies. It

is natural to use mini-max arguments for these problems, since the hydrostatic

pressure, which appears as an unknown in the equations of elastostatics for

such problems, can be identified as a Lagrange multiplier associated with the non-

convex incompressibility constraint, det F = 1.~

Our approach involves some generalizations of the saddle-point theory in

EKELAND and TEMAM [6] to non-convex differentiable functionals defined on weakly

sequentially closed sets in reflexive Banach spaces, and generalizations of the

notion of polyconvex functionals introduced by BALL [3]. In [3], BALL confines

his attention to questions of existence of minimizers of the total potential

energy functional in sets K of kinematically admissible motions which satisfy,

for incompressible materials, the constraint det F = 1, F being the deformation~ '" .

gradient. Such an approach avoids the question of existence of hydrostatic

pressures and, therefore, cannot, in general, be used to characterize minimizers

of the energy as solutions to the weak equilibrium equations.

In the present study, we establish some general conditions sufficient

to guarantee the existence of minimizers in K. These include conditions which

extend and generalize those used by LADYZHENSKAYA [8] in her studies of incom-

pressible flow and those of BABUSKA [2] and BREZZI [4] in their work on the linear

theory of incompressible elastic bodies. We also show that the saddle points of

our Lagrangian can be characterized as solutions of the weak equilibrium equations

for certain incompressible hyperelastic materials.

2

After establishing some mechanical preliminaries in the section following

this Introduction, we review, in Section 3, some concepts from modern variational

theory and we present a theorem which lays down sufficient conditions for the

existence of saddle points of non-convex functionals. In Section 4, we consider

some special results applicable to saddle-point problems encountered in Lagrange-

multiplier formulations of constrained minimization problems. A major difficulty

in our studies of Lagrange multiplier methods for problems with nonlinear con-

straints is to show that subsequences of Lagrange multipliers associated with

a perturbed Lagrangian are uniformly bounded in a real parameter € with respect

to an appropriate norm. We address this problem in Section 5 and develop suffi-

cient conditions for such uniform boundedness. In Section 6, we apply the results

of Section 3 to a class of Neumann boundary-value problems in elastostatics in-

volving two-dimensional, incompressible, hyperelastic bodies subjected to dead

loading. We prove a general existence theorem for this class of problems.

In Section 7, we consider an application of our theory to a model problem.

Specifically, we consider the problem of determining equilibrium configurations

of an incompressible hyperelastic body characterized by a strain-energy function

cr of the form

(1.1)

where Po is the mass density in a fixed reference configuration, El, EZ' and

E3 are material constants, and I and II are the first and second principal

invariants of the Cauchy-Green tensors. A constitutive equation of this form

was suggested by ISIHARA, HASHITSUME, and TATIBANA [7] as being appropriate for

certain incompressible rubbers. The conditions of our theory infer conditions

existence of solutions to our model problem whenever tractions are specified.......

on the elastic constants El

, E2' and E3 which are sufficient to guarantee the

on the bounda ry .

3

An interesting aspect of our approach is that in applying our generalized

Ladyzhenskaya-Babu~ka-Brezzi condition to concrete cases we are led to the

study of an auxiliary problem involving the Monge-Ampere equation in bounded

domains in ~n. In Section 7, we discuss a special construction of a perturbed

Lagrangian L , € = lIn, which makes it possible to apply some of the recent€

regularity results of CHANG and YAU [5] to our model problem for the case in

which the domain n c ~2 is C2 and strictly convex.

Some concluding comments are given in Section 8.

4

2. Mechanical Preliminaries

Most of the notations we adopt here are standard in continuum mechanics.

Indices take on values 1 through n, n = 1, 2, or 3, the summation convention is

adopted, and vectors and tensors are denoted by boldface characters. The inner

Ttr(AB ), where tr A denotes the trace"'"

andproduct of two vectors

tensors A and B by

of A.

A:B

Y is denoted'"

T= tr(A B) ='" '"

x·y'" '"

and of two second-order

We consider the deformation of an elastic body !IJ relative to a fixed

reference configuration ~so that the position of each material particle

X € !l1 in the reference configuration is ~ = ~ (X). The deformation of the

body carries particle to the current spatial position-1

:: x(X),X ~ = ~(5) (X»'" '"

where ~ denotes the current configuration. The deformation gradient F at

X is given by

F (X) = V'x (X)t"V1"t..I rvrOJ

(2.1)

where V' denotes the material gradient. The left Cauchy-Green tensor is given

by

B

and the three principal invariants of B are given by

I = tr B

(2.2)

II 1 ( 2 2)"2 ( t r~) - t r (~) =

III = det B

adjV'x:adj\lx'" '"

(2.3)

Here adj V'~ denotes the transpose of the tensor of cofactors of ~; e.g., if

we denote by xk K, the Cartesian components of V' x, then'"

5

(adj ~)Kk =

x2 2x3 3-x2 3x3 2,xl 3x3 2-xl 2x3 3:xl 2x2 3-xl 3x2 2" "1" "," "

I

x x -x X I X X -x X I X X -x x2,3 3,1 2,1 3,3. 1,l3,3 1,3 3,1. 1,3 2,1 1,1 2,3

I

(2.4)

For plane problems,

(adj ~Kk

o o

o

o

x x -x x1,1 2,2 1,22,l

(2.5)

II = I + III - 1

We will consider ~ to be composed of an incompressible hyperelastic

material homogeneous in its reference configuration. Then all motions of ~

are subject to the incompressibility constraint

J 1, J == det F~ (2.6)

Moreover, there exists a strain-energy function a

the Cauchy stress in such a material is given by

a(x)'"

~(B(x)) such that"''''

T 2pB ~ - plj'"oB '"

(2.7)

where p is the mass density in the current configuration, ~ is the unit tensor,

and p is an indeterminate hydrostatic pressure. If the material is isotropic

and homogeneous in its reference configuration, the strain-energy function

can be expressed as a function of the invariants I and II: a = a (I,ll).

Since

6

B (2.8)

we have, for isotropic homogeneous incompressible hyperelastic materials,

(00 ocr ) ocr 2

T = 2p dI + I ill ~- 2p ill ~ - pl (2.9)

In (2.2)-(2.9), it is understood that these relations hold at each ~ € ~(~;

throughout this study, we frequently refrain from indicating the explicit de-

pendence of various quantities on X for simplicity, unless confusion is likely.

The first Piola-Kirchhoff stress is defined by

Q = T(adj 9x)T'" '" '"

The body ~ is in static equilibrium at each particle X if

(2.10)

Div Q + PO! = 0 v X E n (2.11)

where Div denotes the material divergence operator, Po is the mass density

in the reference configuration, f is the body force per unit mass, given as~

a function of ~, and n = int(~ (~) .

In addition, various conditions can be imposed on the boundary on = ~(O$) .

Boundary conditions of place involve the specification of the displacement

U E X - X'or x on on ; boundary conditions of traction involve the specification'"

of the traction

on on (2.12)

where being a unit outward normal to on at ~ € on ; mixed boundary con-

ditions involve the specification of boundary conditions of place on o~ c on

7

and traction on o~ c on, where oQ=01\ uo~, O!\ n o~ =0; etc. In the present

study, we consider traction boundary conditions. Consequently, the data f

and ~ must be compatible; i.e., the applied external forces must be such that

the body is globally in equilibrium.

The classical boundary-value problem of elastostatics of an isotropic

incompressible hyperelastic body subjected to body forces f and to surface~

tractions ~ on its boundary consists of seeking the vector-valued function

x = x(X) and the scalar-valued function p = p(X) such that~ N~ ~

J(x) - 1~

o in n

o in n

= t on on"'0

(2.13)

where T = T(~, p) is given by (2.9).

We will be concerned with the determination of conditions for the exis-

tence of solutions to systems of the type (2.13), or of alternate weaker forms

of it described in Section 4.

3. Variational Theory and the Existence of Saddle Points

In this section, we establish several results connected with the problem

of determining saddle points of functionals defined on real reflexive Banach

spaces. We begin by reviewing a few definitions and properties of differen-

tiable functionals; we then establish some theorems which are the basis for

our existence theory described in Section 5.

We adopt the following conventions and notations:

8

iU is a real reflexive Banach space wi th norm 11·11~ iU

K is a nonempty weakly sequentially closed subset of ~~

'V is a real reflexive Banach space with norm 11'1111

M is a nonempty? closed, convex subset of vlf

L is a real functional defined on K xM

(3.1)

If iU' and OY''"

are the topological duals of iU and~, respectively,.-

(. , ·)iU and (., 'ly shall denote duality pairing on the respective spaces iU' x ru~ ~

and "'I' x oy; 1.e., if *l E ~I and~ *q E "V', we will wri te

and * *)q (q) = (q , q 'V (3.2)

whe re r E ~ and q E 0\/.

We next review some aspects of variational theory.

·Let g be a real functional defined on a real Banach space ~ and

let iU1 denote the dual of ru. If, for a x E ru there exists a linear func-~ ~ ~

* *tional x = x (x) E iU' such that the following limit exists for any y E ~,""J _. '""J

* > 1(x (x),y = lim -e (g(x + ey) - g (x))'" '" ~ E4 0+ '" '" '"

(3.3)

then g is said to be Gateaux differentiable at x. The functional x* is

referred to as the Gateaux derivative of g and is denoted

og(x)~dz *x (x).

'"(3.4)

If g is cateaux differentiable at every x EKe iU, it is said to be Gateaux~ '" '"

differentiable on K.

'--

9

'We recall that a real functional g defined on a subset K of a re-

flexive Banach space ~ is lower semicontinuous on K if~

lim inf g(~) ~ g(~'k-+oo

VuEK'"

(3.5)

for any sequence (~) c ~ converging strongly to u. The functional is

weakly lower semicontinuous on K if (3.5) holds for any sequence~

converging weakly to u.~ Moreover, such a functional g defined on K is'"

(weakly) ~ semicontinuous on K if~

lim sup g(~) S g(;9k -to00

v u'"

EK (3.6)

for any sequence converging (weakly) strongly to u.'"

The following results are proved in VAINBERG [15J.

Lemma 3.1. A real functional g defined on a convex subset M of a

reflexive Banach space OY is weakly upper (lower) semicontinuous on M if it

is concave (convex) and upper (lower) semicontinuous on M. 0

Lemma 3.2. If a functional g: ~ c ~ -toR is Gateaux differentiable on

K and if its GAteaux derivative~ og/oz is monotone on~ K;'"

i.e., if

>0 V~ y"'EK (3. 7)

then g is weakly lower semicontinuous on K. []

'--

10

The following theorem establishes another sufficient condition for weak

lower semicontinuity of differentiable functionals which is of particular im-

portance in existence theories of nonlinear elasticity:

THEOREM 3. 1 . Let CU and ~ be Banach spaces and K be a weakly se-'"

quentially closed subset of ~. Let G; ~ -+ ~ be a weakly sequentially continu-

ous map; i.e., if (~) is a sequence from ~ converging weakly to ~ then

converges weakly to G(x)IV

in ~. Moreover, let Ag be a real convex

Gateaux differentiable functional defined on a convex set M in ~ containing

the range G (~ of G. Then the functional g: K -+ R defined by the composition'"

(3.8)

is weakly lower semicontinuous on K.

Proof: Let (~) be a sequence in K which converges weakly to a point

X E K • Then G(~) converges weakly to G(x) in M. Since ..... is convex and~ ..... '"g

~teaux differentiable, we have, for any Gl and G2 E M,

(3.9)

Setting Gl = G(~) and G2 = G(~, we have

Hence,

(3.10)

lim inf g(~) ? g(~k-+oo

o

11

We remark that the representation (3.8) of g as the composition of a

convex differentiable functional g and a weakly sequentially continuous map

G is a generalization of the notion of polyconvex functions introduced by BALL

[3]. Polyconvexity, as defined by BALL, involves the following local properties:

Let D be an open subset of the space Mnxn of all n x n matrices F and

define the map Q: D ~ Y by

Q(F) ::: r , Y=]R if n = 1~

Q(t) = (r,det D , Y = M2x2 x ~ if n = 2

Q(D = (~adj ~ det t), Y = M3x3 x M3x3 xR if n = 3

where adj F is the transpose of the matrix of cofactors of F. Then, a function- '"g: D ~ 'R representable as g:::g p Q is polyconvex if and only if g: Q (D) ~ JR.

is convex on Q(D). BALL then chooses the underlying space CU such that the

operators 't.. ~ adj vx. and 't.. .... det 'il't.. are weakly sequentially continuous from

CU into suitable spaces, where ~ denotes deformation gradient in continuum

mechanics.

Now we come to the problem of the existence of saddle points of ~teaux

differentiable functionals. Let K and M be any two sets and L be a real

functional de·finedon KxM. Then a point (x,p) E KxM is said to be a saddle'"

point of L if

L(::s,q) ~ L(~p) ~ L(X"p) V t E ~ and V q EM (3.12)

We first record the generalized Weierstrass theorem; for a proof, see,

for example, VAINBERG [15].

12

Theorem 3.2. Let K be a nonempty weakly sequentially closed subset of~

a reflexive Banach space ~ with norm II·I~. Let g be a real functional de-

fined on K which is weakly lower (upper) semicontinuous. Assume that K is

bounded or that g is coercive in the sense that

lim (Iltl~r00 g iJ = (±) 00(3.13)

Then g is bounded below (above) and attains its infimum (supremum) on K.'" o

Next, we come to an important theorem on the existence of saddle points

of G~teaux differentiable functionals. A similar theorem, in which somewhat

stronger hypotheses are assumed, has been proved by EKELAND and TEMAM [6, p. 164].

In their study, condition (i) below is replaced by the hypothesis that, for every

q E M, X. ~ L(;:,q) is convex and lower semicontinuous. They also require that

K be a closed convex subset of ~~ We will show by a procedure slightly dif-

ferent from that given in [6] that these restrictions can be weakened.

Theorem 3.3. Let ~ ~,o/, and M be as defined in (3.1). Let L be

a real functional defined on KxM such that the following conditions hold:

(i) q E M, t ~ L(x.,q) is weakly lower semicontinuous

(ii) t E ~ q ~ L(X, q) is concave and upper semicontinuous

(iii) Either K is bounded or there exists an element ~ E M

such that

lim L(x, Clo) = +00, tEKIlyll~ 00~~

(iv) Either M is bounded or

lim inf L(y, q) ;:::-00, q E M.Ilqll~ 00 tE~

Then L has at least one saddle point in KxM.,..

(3.14)

13

Proof. Assume that condition (iii) of (3.14) holds. Then the weakly

lower semicontinuous real functional L(·,~ is bounded below on ~ for any

fixed q and it attains its infimum on ~ by Theorem 3.2. Denote such a mini-

mum value of L(',q) by g (q) and, for given q, its set of minimizers by E .",q

Then E C K and for any e(q) E E ,~ '" ~

L(e(q),~.'"

(3.15)

We next show that condition (ii) implies that g is weakly upper semicon-

tinuous on M. Pick a sequence (qk) C M which converges weakly to q E M.

Then

Hence, using condition (iih we obtain

lim sup g(qk) ~ lim sup L(~(q),qk) .'S L(e(q)'q) = g(~k....oo k,...oo

which shows that g defined by (3.15) is weakly upper semicontinuous on M.

Thus, again by Theorem 3.l2, g(~ is bounded above and attains its supremum at

some point p E M such that

g(p) = max g(~qEM

= max min L(X;~qEM t~

(3.16)

g(p) 5 L(X;p) 'r/ "i E ~.

From (it), we have, for any "i E!S q E M and 8 E (0,1), L(v (l-8)p+8~ >

Setting y = ~ - e((1-8)p + 8q)'"

gives

g(p) ? g((1-8)p + 8q) = L(~, (1-8)p + 8q)

? (l-8)L(~,p) + eL(~,~ ? (l-8)g(p) + 8L(~~pq)

14

from which we obtain

g(p) ? L(~, q) 'V q E M. (3.17)

The inequality (3.l7) and condition (iii) imply that ~ is bounded for

any 8 E (0, 1) . Also, by hypothesis, K is weakly sequentially closed.~ For each

8, we can identify a representative ~ E ~(1-8)p+9q and thereby construct a

Since ~ is reflexive, we can extract a

converging weakly to a point

bounded sequence (~} as 8 ~ O.

subsequence (~} with 8k ~ 0k

x E K.'"

Then

x E E. Indeed, by (ii) and the definition of en'~ ~p .....0

Since L(,:,g) q) is bounded below by g(q)J taking the limit inferior of thek

last two inequalities as 8k ~ 0 gives

Hence x E E .~p

Now, the limiting case of (3.17) is

g(p) > L(x,q) 'V q EM- ~

which together with (3.16) gives

L(x, q) < g(p)'" - L(x,p) S L(y,p) V Y E K and V q E M

~ '" '"

This completes the proof. [J

15

4. Lagrange Multiplier Methods

In this section, we consider the problem of minimizing functionals sub-

ject to constraints. We will assume throughout this section that conventions

(3.1) hold with M = ~.

Let F : K ~~ be a functional satisfying the following conditions:

(i) F is weakly lower semicontinuous

(ii) limIIll~t00

F (y) = + 00.... VyEKN

(4.l)

Let B K ...cyr be an operator such that....

v q E Y', y -+ (B(y),q) is weakly lower semicontinuous'" "''''Y

(4.2)

Note that K is not the "constraint set." The constraint is introduced by

considering the problem of finding x E N, with N C int K, such that'"

F(x)'"

where the set N is defined by

= inf F(y)lE~ '"

(4.3)

N = {y E K~ B(y) = 0 in o/'} J 0'"

(4.4)

Ac~ording to classical. Lagrange-multiplier concepts, the minimization

problem (4.3) can be converted into the problem of determining saddle points of

a functional L on K x or, where'"

(4.5)

....

The functional L is called the Lagrangian of F with respect to the constraint

16

B(x? = 0 in ~I and the elements q € Vare called Lagrange multipliers.

We note that a saddle point of L, defined by (4.5) also satisfies (4.3);

i.e., a saddle point of the associated Lagrangian is a minimizer of the con-

strained minimization problem. However, the converse may not be true. More

precisely, even though there exists at least one minimizer of the constrained

minimization problem, the existence of a saddle point of the associated Lagrangian

is not necessarily guaranteed.

Under conditions (4.l) and (4.2), the Lagrangian L defiried by (4.5)

satisfies the following conditions:

(i) \I q € or, t -+ L(~, q) is weakly lower semicontinuous

(li)

(iii)

lim L(y,O) = +co V y € KIIy II-+ 00 '" '" '"

"'CU

V Y E K, q -+ L(y, q) is concave and upper semicontin-'" '" '"

uous on OY (indeed, L(y, q) is linear and continuous in q)'"

(4.6)

Thus, all of the conditions sufficient to guarantee the existence of a

saddle point of L are satisfied except one: we cannot conclude that L(l,q)

is coercive with respect to q (i.e., that for some "!.nE~, L("!.n,q)-+ -00 as

IIqll -+ (0). To overcome this difficulty, we introduce the perturbed Lagrangian,q

, a:~l (4.7)

and (ii) of (4.6) are satisfied for L .€

+g: K -+ lR'"

Indeed, iffunc tiona1 .on SY.

Here € is an arbitrary positive number. We easily verify that conditions (i)

We show that -€!Iqlf.is a concave11

= (0,(0) is convex and f: R+ -+ R is

convex and monotone increasing} then we easily verify that the composition fog

is convex. Setting g(q) = IIqll and f(x) = €Xa:,a:2. l, reveals that €lIqlf0/ ~

is convex. Hence _€lIqlla: is concave. Moreover, -€l\qIE is also continu-"Y r

ous on 0/: the norm 11'11 is continuous and -€lIqle is the composition of 11·11~ 0/ ~

and a continuous function. Hence -€II'le is also upper semicontinuous. Finally,0/

we note that for any "!.n€~,

17

- co (4.8)

Hence, there exists a saddle point (x ,p )"'€ ~

of L in K 'l( 'Y :€

L (x , q) < L (x , p ) < L (y, p) V Y E K and V q E 0/ (4.9)€ "'€ - € ~€ € - € '" € '" '"

We next consider a sequence ((x ,p )} of saddle points obtained by"'E €

allowing € to approach zero. We will show that /Ix 11,., is bounded. Since"'€ -u.,.,

(x , p) is a saddle point,"'£ £

:s F(~E? + (B(~€),P€)ay - €/lP€I~:S F(~) + (B(l),P€)"Y - €/lp€n~

from which we have, V 't... E ~ and V q E "\I,

F(x) + (B(x ),q) - €llqJP:s F(y) + (B(y),p >'0/"-'£ rv~ ~ by 'V €

Set q = 0 and pick ~ E ~ so that B(~) = O. Then

(4.l0)

(4.11)

(4.l2)

Since F is coercive, there must exist a constant Co > 0, independent of €,

such tha t .

V € > 0 (4.13)

From (4.l3),it follows that there exists a subsequence (x ,} of (x}"'€ "'€

and an element x E K such that x , ~ x weakly as €I ~ O. Sincerv ~ ~€ ~

18

v q,V ... L (y,q) is weakly lower semicontinuous ,A. €'"

lim€ ' ... 0

inf L , (x .,q) > L (x,q)€ "'€ - ~

v q E OY (4.14)

If we were able to show that lip€ I "0/ we re uniformly bounded in € I, then

we could conclude that there exists a subsequence (p ,,} and apE.:y such that€

P€" --. P weakly in '0/. From the upper semicontinuity and concavity of L(', q),

it would then follow that

v Y E K'" ~ (4.15)

From (4.9), (4.14), and (4.15), we would then have

L(x,q) <L(x,p) <L(y,p)"-I - rv - "V

'r/ l E!< and V q E ey (4.l6)

i.e. (~,p) is a saddle point of L. Thus, we ~ guaranteed the existence of

~ saddle point of L under conditions ~ and ~ if ~ ~ show that

lip€ I lIoy is uniformly bounded in €'.

Theorem 4.1. Suppose that conditions (4.1) and (4.~ hold. Then, there

exists a subsequence of saddle points r(x ,p )} of the perturbed Lagrangian~€ €

L on K x cy which converges weakly to a saddle point (x,p) of L on K 'l( ay ,€ _ IV'-

whenever lipII is uniformly bounded in €. 0€ '¥

To proceed further, we consider the characterization of saddle points

of differentiable real functionals:

Lemma 4.1. Let conventions (3.1) hold and let,.,

q ...L(y,q) be Gateaux"v

differentiable on M for any fixed 1. E~. If (~,p) E K)(M is a saddle point

of L, then

<OL(X,P) ~~ q-p <0aq , (4.17)

19

Moreover, if P E int M, then the inequality (4.17) reduces to an equation, i.e.

if p E int M, then

o VqE")I (4.l8)

Proof. This lemma is proved in, for example, EKELAND and TEMAM [5, p. l58J.

Since the proof is instructive and straightforward, we will include it here.

Since (x,p) is a saddle point and M is convex,'"

Hence

L(x,p+e(q-p)) ~ L(x,p)'" '"

v q E M and VeE [0,1]

1= lim r:dL(x,p+e(q-p)) - L(x,P)]e....o + <:7 '"

<0

If p lies on the interior of M, then for every r E OV, an € > 0 can

be found such that q = p + €r E M". Then

< dL(~,p) ~ <dL(~,P) ~a ' q-p = € a' rq cy q ~y

By replacing r by -r, we conclude that (4.18) holds. []

<0

Lemma 4.2. Let conventions (3.1) hold and let y _ L(y,q) be Gateaux~ ~

differentiable on 1S for any fixed q EM. If (x,p) is a saddle point of L

and x E int K, then'"

(4.l9)

20

Proof. Since x E int K, an € > 0 can be found such that- '"

x + 9y E K V Y E CU and V 0 ~ 9 < €. Because (x, p)'" '" I'\J I'V "" I'V

L on K){M, we obtain'"

is a saddle point of

L(x, p) ~ L(x1{}y) V Y E 'U and V 0 < e < €rv "-'t"V #'\.i I'V

Hence,

1lim n [L(x1{}y) - L(x)] > 0S-. 0 + Q '" '" '"

By replacing l by -X, we conclude that (4.19) holds. []

Let us now introduce the following assumptions:

(i) Condi tions (4.1) and (4.2) hold

(ii) F : K ....R is Gateaux differentiable on dF~, and dl

is of the form

4.20)

where A: K .... CUr is bounded and f is given data in 'U'

(iii) V q EO(, ';i .... (B(y), q) is Gateaux differentiable on K and'" cy

d (d(B(y),q) >lim ~ (B(y+ez), q)~ = ~ 0/, z

S-. 0 + Q1 '" '" , '" 'U

We observe that V x,z E K, <~(B(x) , q) , z) defines a continuousIV I'V '" oy f'V 0/ "V

'U

linear functional on OV. We will denote its value at q by

C*(x) is, V x E K, a linear operator from K into .'Y'. Likewise, 't/ q E'Y,"J '" IV

(~ (B(!.9, q), ~>'U is a continuous linear functional on 'U with values denoted

2l

*C(x) : cy .... CU'. Then C (x)'" '"

is the transpose of CC:$) in the

sense that V x, z E K and V q E OV,'" '" ~

Note that since (B(X)' q)'V is linear and continuous in q, it is Gateaux dif-

ferentiable with respect to q and

lim d= (Bc.~), q)oy (4.22)~ (B<,~)'P+8q)oy

9-.0+

On the other hand, since ~ is a reflexive Banach space, its norm !I'llyis cateaux differentiable on "Y - (o) (see, e.g., SUNDARESAN[ill) and the func-

tional q .... Ilql~ <X ~ 1, is ~teaux differentiable on 0/. We denote the Ga-teaux

derivative of II·IE by E : oy .... '¥, i.e.,cy

lim ~ lip + eqlE = (E(p), q),~+ cy ~

Then, a saddle point of the perturbed Lagrangian L on K)( <y€ ~

can be characterized by the equations

(A(~€,) ''l}cu + (C(~€,)Pe:"~)qL = (f,'l}CU V y E CU }'" '"(4.23)

(B(~€,),q}CU - e:1 (E(P€,),q}c.y = 0 \:I q E '"Y

provided'~ belongs to the interior of K. Moreover, the weak limit"'€ '"

(x,p) E K xCV of a convergent subsequence of (x ,P )}, if it exists, can be""" "" ""€ €

characterized by

(B(~),q) =0 Y qEay 1 (4.24)

22

We next establish an important property of the sequence ((x , p )} of"-'€ €

saddle points of the perturbed Lagrangian L .€

We recall that x is uniformly"'€

bounded in '!! for every € > O. From the inequali ties (4.10),

Suppose that a = 1. Then, taking q = p + q, q E~, and using the inequality€

we obtain

IlIrll~ - IIsl~1 S IIr-sll~

(4.25)

Then, for arbitrary q E '0/, we have

lim inf (B(x ),q) < 0"'€ -

G+O

By (4.2),

for the subsequence

(B(x) , q) < 0'" - .

(x'} of (x) which converges weakly to x in ~.~ "-'€ '"

(4.26)

Since

q is arbitrary, we arrive at the significant conclusion that

B(x)'"

o in or' (4.27)

That is, the weak limit of the subseguence

N, defined.£1 (4.4). Then, from (4.12),

F(x) < F(y)'" - "-'

(~) belongs to the constrained set

(4.28)

23

for e~ery yEN. Therefore, (4.27) and (4.28) establish that the weak limit~

x of the subsequence (x ,) is a minimizer of the functional F on N, i.e.)'" "'€

X is a solution of the problem (4.3).~

_Theorem 4.2. Let ex = 1 in the perturbed Lagrangian L€

of (4.7).

Then, if (x, p )) E K xOf is a sequence of saddle points of L', there exists~€ e' ~ €

a subsequence Lx I) of (x) which converges weakly to x E K. Moreover, the~€ "'E

weak limit x is a solution of the constrained minimization problem (4.3). []

Remark 4.1. We note that we have not used the uniform boundedness of

lip" to arrive at the above result. []Eoy

It may be (and generally is) possible to say more about the convergence

of the sequence

A func tion

(B (x ) ]"'€

~( .)

under some additional assumptions.

is said to be a ~ function if is defined on

o < r < + 00 and if ~(.) is a real-valued continuous and strictly increasing

function such that ~(O) = 0 and lim ~(r) = 00. Then, by the Hahn-Banachr-+-too

theorem) there exists a duality map j of a Banach space ,y into its dual 'V'

such that

(j (u) , u)""

I/j (u) 1/*

24

IIj (u)" "* Ilu Ily

= ~(I/u"~) J (4.29)

where 11'11* is the norm on the dual space cr' and ~(.) is a gauge function.

A standard example is the case :y = LP (fl), P > 1, in which

{ }

l/PIIvll= S Iv IPdx = IIvllo<Jt fl ,p

(4.30)

~(r) = rP-l ,

If ~ is a strictly convex reflexive Banach space whose dual is also

strictly convex, then the dual map j of 'V into 'VI with respect to the Gauge

function ~(.) is one-to-one and onto. Moreover, the inverse of the duality map

*j, denoted by j, exists and the duality map of OV' into ~y can be associated

*with a gauge function ~ (.), which is the inverse of ~(.). In view of these

properties, we can obtain some additional results under the following assumption:

.....

yields

The space ~ is a strictly convex reflexive

Banach space, whose dual ~' is also strictly

convex.

Then, returning to (4.25) and choosing

(4.3l)

25

Since

and

= «>*(liB (~) II.)

we have

Then, taking the limit € ~ 0, we conclude that

lim IIB(x ) "* = 0€roO ",g

From (4. 3 1) ,

(4.32)

lim6-+0

IIB(x ) "* = liB (x) "* = 0ov€ '"(4.33)

Summarizing, we have:

Theorem 4.3. Let the conditions of Theorem 4.2 and hypothesis (4.3l)

hold. Then, the sequence (iIB(x) "*) converges to the number IlB(x) "*, where"'€ '"

x is the.weak limit of a weakly convergent subsequence of (x} in K. 0"" "'€ '"

26

5. Uniform Boundedness of IIp€lIey

We have proved that, under mild conditions, there exists a subsequence of

saddle points ((x ),p ) of L on K X. Oo.f which converges weakly to a saddle"-'€ € €

point (x,p) of L on K x CY, whenever lipII is uniformly bounded in € > O.'" '" € .;y

In this section, we shall obtain sufficient conditions for the uniform bounded-

ness of IIp€"~. Throughout this section, we assume that property (4.31)holds;

1.e., "'Y and cy. are strictly convex reflexive Banach spaces.

Let j be the dual map of ay into 6)/1 with respect to the gauge function

~(.). Let M be a closed subspace of 9v. Moreover, let us assume that

For every q E M'J there exists a z E K

such that

-B(z) = j(q) in C'y'

Now, from the inequalities (4.l0), we have

(5.l)

Then, taking q = tp for some positive number€

t in (5.1), gives

i. e.,

(j( tp ),p )0, < - (B (x ),p) + F (z ) - F (x )€ € -r - "'€ € "i' "'€ "-'€

(5.2)

where Z"'€

is the element in K which satisfies (5.1) for q = tp ....... €

Let us suppose that the functionals appearing in (5.~ are coercive in

the following sense:

27

Let z(~ be a solution of (5.1) for given'"

q E M. Then, for some real t > 0,

{F(Z(~)}

lim ~( t IIq II ) - ---fqr- = +00I/qll ~oo 0/ q ~cy

(5.3)

It is clear that if (5.3) holds) the uniform boundedness of lip/I in€ oy

€ is guaranteed; for we have shown that iIB(x)"* ~ IIB(x) "* and that F (x )"'€ "'€" ~€

is uniformly bounded in €. Thus, the functional in braces in (5.3), evaluated

at q = tp , is bounded above by a number independent of €. We record this fact€

in the next theorem.

Theorem 5.1. Suppose that conditions (4.1), (4.2), and (4.31) hold. Further,

suppose that (5.l) and (5.3) hold. Then, lipII is uniformly bounded in €,€ ~

where p is a component of the saddle point (x,p) of the perturbed La-€ ~€ €

grangian L defined by (4.7).€

o

Remark 5.1. Our motivation for seeking conditions such as (5.3) comes

from studies of the existence of solutions to the S to k e s equations de-

scribing uniform, steady flow of an incompressible flu i d. For t his

problem, suppose F, B, and L are defined by

(B(x))q) = - r (div ~) q dxcy 'n

(5.5)

where µ is a positive constant related to the viscosity of the fluid. In this

problem,

and

28

1(we may take ~ = ~ = !£ (n)) (5.6)

(5.7)

1an element ~ E !£) (n) such that

If we suppose that, for every2

q E L (n) such that J qdx = 0, there existsn

div z = q'"

(5.8)

Then the conditions (5.l) and (5.4) are satisfied. Indeed, (5.8~ coincides with

(5.l). It suffices to show that (5.4) holds.

2If! E!: (rt), then, from (5.5)1'

By condition (5.8\, for an arbitrary positive number t > 0, there exists

z E K such that~€ '"

div z = tp"'€ €

where p is the part of the saddle point of the perturbed Lagrangian L C',')€ €

defined by (4.7):

ex> 1

By (5.8'L) for such z and p,...,d "'€ €

Then

29

This implies that

~(tllp€1I0,~ Ip€"0,2 - F(z,€)

? (t - ~C2t2) IIp€II~,2 - Ctll.~1I0,2I1p€1I0,2

By taking the constant t so that

122t - "2 µC t > 0

we can conclude that

(5.9)

(5.10)

1.e., the condition (5.3), 1.e., (5.4), is satisfied if (5.8) is true.

Results such as (5.8) were obtained by LADYSHENSKAYA [8], BABUSKA [2J,

and BREZZI [4]. An elegant proof has been given by TARTAR [14J. 0

It is noteworthy that in none of the results developed thus far in this

section was it necessary to assume differentiability of the functional (B (~) , q >JV"

If we now assume that condition (iii) of (4.20) holds, an alternative condition

sufficient to guarantee that lipII is bounded can be obtained.€cy To arrive at

this condition, let us start with the characterization (4.23) of a saddle point

We assume that the following conditions(x ,p ) of the perturbed Lagrangian~€ €

hold:

V €(i)

(ii)

L .€

x belongs to the interior of K~€ '"

The operator A: ~ ...~I of (4.20~

>0 1is bounded

(5.12)

30

independent of €, such that

In addition, assume

There exists a positive constant Cl'

(5.13)1J

I (C(~€)p€,~I)cul

ill-lieusup

yE.eu-(o}I.~ ""-I '"

When conditions (5.12) hold, fran (4.23)1'

(C (~Jp€' !)eu = (~- A (~) , r)eu'"

where C2 is a constant independent of €. Obviously, if (S.l3) also holds,

Hence:

Theorem 5.2. If conditions (4.l), (4.2), (4.20), (5.l2), and (5.13) hold,

then lipII is uniformly bounded in €. 0€o/

An existence theorem summarizing our main results established up to this

point now follows as a simple corollary of Theorems 4.3, 5.1, and 5.2.

Theorem 5.3. Let conditions (4.1) and (4.2) hold. Then there exists at

least one saddle point (x,p) E K 1l M of the functional L of (4.7) if either'"

of the following hold:

.-"

31

(D Conditions (4.3l)) (5.D, and (5.3)

(ii) Condi tions (4.2)), (5.l2), and (5.13)

Moreover, if (4.20) holds and x E int K and p E int M, then the saddle~ '"

point (~}p) is characterized by equations (4.24). [J

6. Existence Theorems for Incompressible Hyperelastic Bodies

In this section, we establish sufficient conditions for the existence of

solutions to an alternate variational formulation of the classical boundary-value

problem (2.13) of an incompressible hyperelastic body flJ subjected to body

force f and surface traction !o on its boundary ~. For simplicity, we con-

fine our attention to cases of dead loading; i.e., cases in which the magnitude

and direction of f and ~ remain unchanged during the deformation for each

particle X E flJ.

Let n = int ~ (:d) be a bounded open region in the n-dimensional

Euclidean space nR , where is the reference configuration. We assume

that the boundary an of n is sufficiently smooth. In applications of our

theory, it is sufficient to assume that an is Lipschitzian [9].

We denote by ~ a reflexive Banach space, the elements of which are'"

deformations y defined on the open bounded domain nc Rn for which V has~

meaning} where V is the total strain energy:

V(y)~ (6.1)

Clearly, the characterization of ~ will depend on the form of the strain-

energy function a(y). In the next section we consider a case in which a(y)~ ~

32

1 4is chosen such that ~ = ~' (~/:o(Sl

1 4Sobolev space of vector-valued functions on n with components in W' (~

and !1(0) is the space of constant vectors defined on n.

For purposes.which will be clear later, we require the reflexive Banach

space CU to satisfy an abstract "trace propertyll (see, e.g., AUBIN [ll): There

ex~ a pivot Hilbert space ~, a Banach space ~ of functions defined on d~-and a continuous linear operator y : CU -+ ~ such that

'" '" ,...,

(i) y maps CU onto ~'"

(ii) CU is contained in ~ with a stronger topology (6.2)

(iii) the kernel ~ of y is dense in ~'"

The operator J is called a trace operator. We also require the space !2(O) :=

n(D(O)) of vector-valued test functions wi th compact suppa rt con tained in 0

to be dense in ~. Hence, the following inclusions hold:

CU c '1f = 'If I C CUi'"

c <£(0))' }(6.3)

where the injections are continuous and dense.

We denote by OY a strictly convex reflexive Banach space, whose dual is

also strictly convex, the elements of which are hydrostatic pressures q defined

on 0, and. require that the range J(CU) of the operator y -+ J(y) = 'ily defined'" I"V '" ,..,."

on CU is contained in the dual space ~I of 4r. Here the material gradient v

is interpreted in the generalized sense. We require the space D(O) of scalar-

valued test functions with compact support contained in 0 to be dense in ~ and

that cy is contained in a pivot space 'If which has a weaker topology. Hence, we

have the following inclusions:

33

V (li) c:: OY c: 1e = 1t' c: 0)" c: (D (ll)) I

where the injections are continuous and dense.

(6.4)

The trace property of the reflexive Banach space ~ gives meaning to

traction boundary conditions in the present abstract setting. Throughout this

and the next section, we assume that

(6.5)

so that the potential energy of the external loads -W has meaning, where

W(y)'"

(f y-x >. =, '" '" qj,

r Po f· (y-X) dvO

+ f tn· y(y-X) dsOn '" '" '" dn ".IV '" '" '"

(6.6)

With these preliminaries now established, we want to determine saddle

points of the functional of modified total potential energy defined by

L(y) = U(y) + (B(y), q)"'-J ~ l'V'y

(6.7)

where U is the total potential energy defined by

(6.8)

and the operator B ~~ ~' is defined by'"

= 1 - J(y), (B(y),q) = f q(l-J(y))dvO'" '" 0/ n '"(6.9)

ff

34

The constraint of incompressibility is

B(y) = 0'"

or ( 6.10)

Throughout this analysis, it is always understood that the strain-

energy function is such that

a = a(F) is differentiable with respect to F'" ~

( 6.11)

Then it can be easily verified that the total potential energy U: CU...:R is,...

Gateaux differentiable on CU and its G~teaux derivative OU/oz is such that'"

(6.l2)

where the operator A: ~ -. ~I is defined by

(6. 13)

and the linear functional f E cut is defined by,...

J pof·z dvO +J t,,'y(z)dsOn ,...~ on ".IV ,... ~

(6.14)

Lermna "6.1. Let q be a fixed element of ~. Assume that qy z ism,M r,R

Lebesgue integrable over n for any '/./~ E~. Then the functional y -. (B(y),q~

of (6.9)2 is Gateaux differentiable on Moreover,

35

= - f q adj vyT: 9z dvv ..., 0n

~. Since, almost everywhere in n,

we have

(B(X-H}~),q).y - (B(l),q~ = -f q(J(l-H}~) - J(X))dvOn

= - 8 f q(adj 9l) T: 9~dvOfl

from which we obtain

(6.l5)

( 6.16)

1(C(y) q, Z )G1' :: lim+ '8

'" '" ~ 9-+ 0o

The following lemma is a direct application of the results in Section 4

to nonlinear incompressible elasticity problems.

Lemma6.2. Let ~ and CY be real reflexive Banach spaces. Let

L: CUJc Oy -+ R be the functional defined by (6.8) and let the following conditions'"

hold:

•

36

(i) The operator y ~ J(y) = det vy is a weakly sequentially con-'" '" '"

tinuous operator from CU into OV'

(ii) The total potential energy U defined by (5.9) is weakly

lower semicontinuous on CU and

(6.17)

lim U(y) = +00IIXllu ~ 00 'V

Then, for every € > 0, the perturbed Lagrangian

has at leastone saddle point (x, p) in cu t <:Y."'€ €

Proof. It suffices to show that, for any q E~, the functional

(B(·),q) defined by (6.10) is weakly lower semicontinuous on CU. Indeed, for0/

any q E CY,

'"gq: CY' - ~ •

(B(.),q) defined by (6.10) can be regarded as a functional~

Recalling that the range of the operator J is contained in ~',

it is obvious that g is linear, G~teaux differentiable with respect toq

J E ~I, and that

Hence, it follows from (6.17) and Theorem 3.1 that, for any q E1v,

(B (.) , q) : CU ~ R is weakly lower semicontinuous. 0"'Y

We now arrive at the main existence theorem for weak solutions of the

boundary-value problem (2.13).

37

Theorem 6.1. Let the conditions of Lemma 6.2 hold and let the pair

(x,p) be a saddle point of the perturbed Lagrangian L (y,q) defined by"'€ € € '"

(6.18), for any € > O. In addition, suppose that either of the following

conditions hold:

(i) There exists an element z E ~ such that--B(z) = j (tq)

'"in 0/1

for some positive t and any q E ov, where j is the duality

map from ay onto ayl with respect to the gauge function

~(.), and B and V are defined by (6.9) and (6.1), re-

spectively.

(ii) The operator A defined by (6.13) is bounded and there

exists a positive constant Cl, independent of €, such

that

(6. l~)

(6.l9)

supyE~-(O) ,..,

Then the sequence ((x ,p )) has a subsequence which converges weakly"'€ €

to a saddle point (x,p) E ~ x :y of the Lagrangian L defined by (5.8). More-'" "-

over, (x,p) satisfies the equations'"

(A (:9 "l )~+ (c (~)p, l )~= (f ,l )~'" '" '"

(B(x),q) = 0'" l}

(6.20)

38

This theorem follows directly from Lemma 6.2. and Theorem 5.3. Recall-

ing the definitions (6.9), (6.l3), (6.14), and (6.15) for B, A, f, and C, we

rewrite (6.20) in the following expanded form:

(6.21)

J q(J(;s) - 1) dVO = 0n

'V q E'V

Equation (6.21)1 is, of course, a statement of the principle of virtual work;

(6.21)2 is a weak form of the incompressibility condition, J(x) = 1 holding

a.e. in n. The variational problem (6.2l) is equivalent to the boundary-value

problem (2.13) provided the terms appearing in (2.13) are interpreted in the

sense of distributions. This equivalence follows from (6.3), (6.4), and standard

arguments given in, e.g., AUBIN [lJ.

Remark 6.1. For the class of elasticity problems considered here, condi-

tion (6.18)1 assumes the form

J (z) = 1 -+ tq'"

(6.22)

where J(z) = det ~z, q E or, and t >0. By a simple change of variables'" '"

(~ = ~<p),. this equation reduces to the well-known Monge-Ampere equation of dif-

ferentia1 geometry. We will discuss this aspect of our theory in more detail in

the next section. For linear incompressible elasticity problems, (6.22) reduces

to

div z = tq'"

(6.23)

in analogy with the auxiliary problem of LADYZHENSKAYA [8] defined in (5.~)

(recall Remark 5.1).

39

IEquation (6.23) amounts to a linearization (Frechet

derivative) of (6.22) evaluated at the origin.

The Fr~chet derivative of the operator ~ ~ qJ(r) is

(qJ' (x) , y)'" '"

Hence, roughly speaking, the alternate condition (6.19) in the statement of

Theorem 6.l is a derivative of condition (6.18). Satisfaction of either

guarantees unifonn boundedness of lipII .€ Gy'

Note that these two conditions are apparently indistinguishable in the

linear theory, since the Frechet derivative of z ~ div z is again div( ). []

Remark 6.2. In (6.22), it is sufficient to take q = p. We discuss€

this possibility in Section 7. []

7. An Application to a Model Boundary-value Problem

In this section we describe an application of the general existence

theorem, Theorem 6.1, to a specific model boundary-value problem. We will consider

the existence of equilibrium configurations in R2 of an incompressible hyper-

o elastic body characterized by a strain-energy function a of the form

(7 .1)

where El, E2, and E3 are material constants. Our analysis will also provide

sufficient conditions on the material constants in order that solutions of the

corresponding boundary-value problems exist. This special material was suggested

by ISIHARA, HASHITSUME and TATlBANA [7 J.

First,

40

we identify precisely the space ~ of deformations y and the'" '"

space OY of hydrostatic pressures q for this specific problem. The total

strain energy of the incompressible hyperelastic body characterized by the

strain-energy function (7.1) is defined by

(7.2)

where l(V = VI : VI and lI(l) = adj VI : adj ~.

For the plane problem, using (2.5)2' we have

For convenience, we decompose the total strain energy V into two parts:

V = Vi + V2

" 1 2Vl = V1 (I) ="2 ~ ((El+ E~ (1- 3) + E) (1-3) ) dvO

~ 1 2V2 = V2(J) = '2 J E2(J - 1) dvOn

(7.3)

(7.4)

Let

defined on

14 14 2W ' (0) = (W , (0)) be the Sobolev space of vector-valued functions'"

2 1 4 Io c ~ with components in W' (n). Let ·Il 4 be the Sobolev,norm for ~l,4 (n) defined by

(7.5)

We denote by the seminorm of the Sobolev space wi,4 (n), where'"

(7.6)

41

Then, by direct expansion, we obtain

2J I (!) dvO = IIr111,2o

(7 • 7)

We note that the total potential energy is defined as the functional

2U = V - W, where W is given by (6.6) for 0 c~ .

vThe following lemma is proved in, e.g., NECAS [7]:

Lemma 7.1. Let the boundary aO of the domain 0 c R2 be Lipshitzian

and J; l ~llao be the trace operator. Then, there exists a positive constant

N = N (0) such thaty y

(7.8)

where

(7.9)

o1 cl IWe no te that the space (~' (l1»)

*II· 111 4 is the dua1 norm defined by,

1 4is the dual space of W' (0), and that'"

.."

y .; 0'" '"

(7.10)

42

Lemma 7.2. For the total strain energy V defined by (7.2), the

estimate

holds. If the boundary dn of the domain n is Lipshitzian, the estimate

for -w

*-W(~) ~ (1Ipo!1I1,4+ Nyl.5>"_3/4,4/3,dtt cll,ll,4 + 1~ll,4) (7.l2)

holds. Moreover, if PO! E ~1,4(n)' and ~ E W-3/4,4/3 (dn), the total

1 4potential energy U = V - W is well-defined in the Sobo1ev space w' (m.~

Proof.

pansion (7.7).

we obtain

From (7.3), the estimate (7.10) can be obtained using the ex-

"Using Holder's inequality, J the triangle inequality, and (7.8)

(7.13)

The estimate (7.12) now follows by (7.13).

From (7.l1) and (7.12), we see that the functional U is meaningful

1 4when defined on the space W' (m. 0'"

1 4Then, the space ~ discussed in Section 6 has to be contained in W' (m.~ '"

"

43

For the strain-energy function a given by (7.l), the operator

1 4 1 4 ,A: W ' (rn -+ (W' (n)) defined by (6.13) can be reduced to

'" '"

(7.14)

1 4 , 1 4where C·'·)l 4 denotes duality pairing on (W' (n)) X W' (n)., '" '"

1 4 1 4 .The boundedness of the operator A : W ' (n) ~ (W ' (n))' def~ned by'" '"

(7.l4) is shown in the following lemma:

1 4Lemma 7.3. The operator A: ~' (Q)

bounded.

defined by (7.14) is

Proof: By direct expansion, we obtain

It follows ft'(Xl\ (7.14) and (7 .l5) ~h£t

(7. l5)

(7.16)

1 4 14,It is obvious from (7.16) that the operator A: W ' (n) -+ (W ' (Q) defined~ '"

1 4by (7.6) is bounded, i.e., it maps bounded set in ~' (n) into bounded set

in (Wl,4(q,) I. 0'"

44

The functional l - Vl<l)' defined by (7.4)2' is Gateaux differentiable on

1 4 "-W' <m and its Gateaux derivative is such that~

<OVl ('!) ~

(Al (X) ,~) 1,4 = o~' 71,4

(7.17)

Lemma 7.4. If

then; for every X, ~ E ~1,4(0),

(7.l8)

(7.19)

i.e., the Gateaux derivative Al of the functional V1 defined by (7.4)2 is

1 l~a monotone operator on ~' (n).

Proof: Denote x = y - z. Then~ '"

( (El +E2- 6E3) ~ + 2E2 (I (!) ~ - I (~) V'.~)) : ~ = (El +E2-4E3) ~ : ~

+ 2E2 ((VX: V'!) ~ - (V'~: V'~) V'~): ~

However,-

((V'y : 'ily) 'ily - (V'z : V'z) V'z\ : Vx"-I ~ I"V ...... "'" ""J

1 0 ((V'(~~~): (V'(~~~») V(~tB~) T= f 01\7 ) , d9V'x: V'xo \V~ '" 'V

= b 1(2'V(Z.jQ~) V(z.jQ~ T + V(!'.+G~): V(~+G~)) dG~: ~

(7.20)

(7.2l)

45

In the last step, we have used the fact that, for every x,y E R,

1 2 1 2J (x + By) de ~ l2 Yo

Substituting (7.2l) into (7.20) and taking integral of the resulting inequality

over 0 'gives (7.19). 0

The following lemma is an extension of some results of RESHETNYAK [9,10];

a proof has been given by BALL [3] and a detailed proof has been given by ODEN

and KIKUCHI [8 J •

Lemma 7.5. The operator

1 4continuous from W' (0) into'"

y ~ J(y) = det vy..... ~ '"

L2(O). 0is weakly sequentially

Thus, the Lagrange multiplier q, which corresponds to the constraint

2of incompressibility, must belong to the space L (m, since J maps of

into L2(m . In other words, we can take

where ~ is the space appearing in condition (i) of (6.17) and in the generalized

existence theorem Theorem 6.1.

Next we show that the material constants can be chosen so that the total

potential energy, U = V - W, is weakly lower semicontinuous on ~1,4(0).

Lemma 7.6. The total potential energy U

continuous on wl,4 (0) if'"

V - W is weakly lower semi-

(7.22)

46

Proof: The functional Vl1 4

is weakly lower semicontinuous on W~' (~~

since its G~teaux derivative is monotone. The potential energy of the external

loads -W defined by (6.6) is weakly lower semicontinuous on Wl,4(~ since'"

it is convex and lower semicontinuous on Wl,4(~ (indeed, it is linear and'"

continuous on Wl,4(~). The only property remaining to be proven is the weak'"

lower semicontinuity of V2 defined by (7.4)3. We observe that

(7.23)V2(r)~ 1 2 1V2(J(r)) ="2 E2 f J (r) dvO - "2 E2mes (n)

nA 2 ,.

If E2 ~ 0, then V2: L (~ ~~ is convex and Gateaux differentiable. Hence

1 4is weakly lower semicontinuous on W' (~~ by Theorem 3.1 and Lemma 7.5.

This completes the proof. []

Up to this point, we have shown that whenever inequalities (7.2~ hold,

the total potential energy U = V - W is a well-defined, Gateaux differentiable,

.14weakly lower sem~continuous functional on the Sobolev space ~' (~. In

1 4addition, the boundedness and monotonicity of the operator A of ~' (~

into its dual have been established. Thus, the remaining property of the

total potential energy U to be established is coerciveness. Toward this end,

we first obtain an a priori estimate on U.

Le nIna 7. 7 . If

and (7.24)

then the total potential energy U v - W satisfies the estimate

(7.25)

47

Proof: Using (7.7) and (7.12), the estimate (7.25) can be easily ob-

tained. 0

Thus, the total potential energy U

1 4Indeed, for every yEW' (~ such that'" '"

is not coercive in Wl,4 (n) .'"

a.e. in n

we observe that U(x) cannot approach co as IIX1l1,4 -. co. The problem, of

course, is that an arbitrary rigid translation of the body will produce no

changes in energy. To overcome this difficulty, it is necessary to recast

1 4the problem in the quotient space of ~' (n) modulo the space :!O (n) of all

rigid translations of the body n.

cu = ~l, 4(n) /~ (n)

If i E ~ is the coset (an equivalence class in ~, corresponding to

1 4X E ~' (n) with respect to ~ (n), then for any !: E!,O (n) ,

(7.26)

In other words, in the space cu one element can be distinguished from another

only to within a rigid translation of ~.

It is well known that the norm of the quotient space cu can be iden-'"

tified w1th the seminorm of the Sobolev space

is a representative of theis equivalent to the seminorm

se t y E CU.'"

IIXI11,4' where

l,4 h~ (n), i.e., t e norm

y'"

IlllI~

Continuing, we observe that the energy functionals V, W, and U on the

quotient space CU must be invariant under a rigid translation, i.e.,

48

and W(y) = W(y + r)'" '" '"

for every ~ E ~ (n).. Since v (y) = V (y + r)'" ........ obviously holds for every

.~ E "0(0), it suffices to consider W. Since W(·) is linear, we must have

.....That is,

W(r) = 0'"

(7.27)

This implies that

f Po f . r dvo + f t,,' r dso '" '" dO "-V ....

f POf dvO + f t" ds = 00.... dO "'V

(7.28)

(7.29)

if 1PO! E L (0)

satis fies (7.2)

and .!:o ELl(dO). Therefore, if the given data (po!'.!:o)

or (7.9), the functional U, V, and W can be considered in

the quotient space ~. We easily verify that (7.29) is also a necessary con-

dition on the data in our problem. We note that the condition (7.28) or (7.29)

are merely statements of the global equilibrium of the external forces. We

conclude that when (7.29) holds, the total potential energy U = V - W is

well-defined, Gateaux differentiable, and weakly lower semicontinuous on the

space ~. Moreover, the coerciveness condition

u (i) -+ +00 as I~I~-+ +00

now follows from Lemma 7.7.

\.Je summarize these results in the following lemma.

(7.30)

" .'

49

Lemma 7.8. If the compatibility condition for the data (PO!'~)' (7.28)

or (7.29), is satisfied, then the functional U = V - W is coercive on the

1 4quotient space cµ = !:: ' (n) /!1 (n), whenever the conditions in Lemma 7.7 hold. 0

Remark 7.1. For the plane problem, the set of all rigid body motions

of the body n is given by

{ 1 4 1 2~ == ~ = (rl,r~ E W' (n) : rl=al +X b2-X bl,

r2=a2+xlbl +X

2b2, al,bl,b2 ER, Ibil~l, i = 1,2,

f)

1 a.e. in n} (7 .31)

Then, for!: E ~,

1!:11,4 -+ +00 if and only if II!!."0 ,4 -+ +00

where a = (a1,a~. Indeed, for r E :R,

44411,~111,4== 2(bl + b~ mes (n) < +00

(7.32)

Thus, in order that the norm of 1 4yEW' (!l) approach infinity for pure rigid-"" '"

body motions it is necessary to subject the body to rigid translations a

such that 1I~llo,4-+ 00. 0

With all of the preliminaries now in hand, we can now establish the fol10w-

ing existence theorem:

50

Theorem 7.1. Suppose that

(i) The data po! E (~1,4 (0))' and .fu E W-3/4, 4/3 (do.) and

(ii)

(iii)

El + E2 - 6E3 ? 0, E2? 0, E3 > 0, and

1 4there exists an element z E W' (0) and constants

'" IV

Cl,C2> 0 such that

2det 'Vz - 1 = q in L (0.)

(7.33)

for every q E L2(0.)

2Then there exists at least one saddle point (x, p) E ttl xL (0.), ttl being'" '" '"

defined in (7.30), of the functional L defined by

L(r, q) = U (y) + (B (y) , q) 2 ''" IV L U~

(7.34)

(B(!), q) L2(n) = J (1 - det 'Vy)q dvOIV,...n(7.35)

where U = V - W is the total potential energy, V is given by (7.3), and

W(y) = J ROf'(v-X) dvO + J to' (y-X) dsO'" n IV 'Iv IV dnIV'" IV

(7.36)

Moreover, the saddle point (x,p) E ttlxL2(n) can be characterized as a so-IV '"

lution of the variational boundary value problem

51

\J 1 4::: J POf'y dVO + J tf'l' y(y)dsO v yEW' (0.) /Pf'I(~0. '" IV d0. 'V\J '" '" '" "-'V

J q(J(~)-l)dVO :::0 V q E 12(0.)

0.

(7.37)

Proof. We will verify that the conditions of Theorem 6.l,particularly

(6.18), hold whenever the hypotheses are true. Condition (6.17) (i) follows

from Lemma 7.5. That is, y ~ J(y) :::det '\]v is weakly sequentially continuous'" '" ~

142from ~' (0.) into L (0.). The condition (6.17) (it) follows from LelJ'lIla7.6

and Lemma 7.8 under assumption (i) of (7.33) .

We now show the condition (6.l8) is satisfied whenever assumption (ii~)

of (7.33) holds. Taking q::: d} in (iii) of (7.33) gives

By (7.ll), Lemma 7.2,

for proper positive constants C3' C4' and Cs' where, in this case, ~(x) :::x,

x > O. Then

Taking

? (t - ciC3t~ IlQIl;,2 - (2ClC2C + ClC~ Ilqllo,2 - (C~C3 + C2C4 + C5)

2 2t so that t - C1C3t > 0, we obtain

"'.

52

Therefore, all of the conditions of Theorem 6.l are satisfied. This

completes the proof. []

An alternative result can be obtained by establishing sufficient condi-

tions·for property (ii) of Theorem 6.l to hold rather than property (i):

Theorem 7.2. Suppose that the conditions (i) and (ii) of (7.33) in

Theorem 7.l hold. Further, suppose that

(iii) I

'·1

C lll~"l, ~ ~ lip € 110 , 2

for some positive constant Cl independent of €, (7.38)

being a saddle point of the func-

tional L defined by€

ex > 1

Then there exists at least one saddle point (x,p) E ttlxL2(o.) of L, and'" '"

;' (x,p) satisfies (7.37).'"

P.roo£. Since all other conditions of Theorem 6.1 have been shown to be

true in the proof of Theorem 7.l, it suffices to verify the condition (6.19)

of Theorem 6.1. From (6.15), we know that, for two-dimensional cases,

53

(C(~€)P€,Z\,4

Hence, if ~ is such that (iii)' holds,

sup

l EWl,4 (q) IfrJ (0) - (~)

I (C(~€)p€'~\,41

IIIIll,4

which is precisely (6.19). 0

Finally, we address the auxiliary problem given in condition (iii) of

(7.33). As shown in the proof of Theorem 7.1, all requirements for the existence

of solutions to our problem are met if the data and material constants are

chosen so as to satisfy (i) and (iii) of (7.33) and if a z E ~1,4(0) can be

found such that

"det 'Vz :::1 + tq

(7 .39)

where Cl and C2 are positive constants,

small po.sitivenumber.

q E L2(0.), and t is a sufficiently

As noted earlier (recall Remark 6.1) a remarkable aspect of problem (7.39)

is that, through a simple change of the dependent variable, it can be reduced

2 4to the well-known Monge-Ampere equation. Let ~ E W' (0.) and set

~._J

54

Then (7.39) 1 becomes

M(~)A::: 1 + tq (7.40)

2where M is the Monge-Ampere operator in R,

2 i j _M(~) ::: det(d ~/dx dx) ::: ~'ll~'22 ~'l2~'21

Special cases of (7.40) have been studied by several authors.

example, for the case in which

20. is an open, bounded, strictly convex C -domain

22-in R, ~Ido.:::gldo. with g E C (0.)

CHANG and YAU [5] have recently proved the following result:

For

(7.41)

(7.42)

Lemma 7.9. If (7.41)holds and f E Ck(o.), k > 3, there exists a unique

convex solution ~ to the equation M(~) :::f

13 E (0,1). []

"in the Holder space

Our present analysis of (7.33) rests on three ideas:

1. To show uniform boundedness of lip€ 110,2' it is sufficient that

(7.39) hold for the choiceA

q ::: p€.

2. The perturbed Lagrangian L can be constructed in such a way that€

the mul tipliers are smooth for € > O.

.....

3. If (7.42) holds, Lemma 7.9 can be used to verify that condition (7.39)

holds.

We can then conclude from Theorem 7.1 that solutions to our model problem always

exis t for smooth domains 0. of the type in (7.42) .

55

1. The first step is not difficult. Suppose (7.39) is satisfied_for

the choice,.q :::p , where p

€ €is the Lagrange multiplier in the saddle point

1 + tp ). Then, by retracing the€

and (ii) of Theorem 7.1 hold has already been established.

tion of .(7.39) for q = p (i.e., det \7z.' 1 € ov€

Let z be a solu-"'€

steps in the proof of Theorem 7.1, we arrive at the inequality

where Ci, 1 ~ i ~ 5, are positive constants, independent of €, CI depends

on t, which is chosen sufficiently small, and C4' depends upon the data

PO! and .fu' Clearly, this inequality cannot hold if Ilp€110,2 -+ 00. Hence

lip II is bounded and, in fact, this bound is independent of €.€ 0,2

2. Since Ck(Q), k? 3, is dense in L2(n) and L2(0) is separable, we

can identify a countable class ofkC -func tions (v1J V2' ... J Vn' ... J everywhere

dense in L2(o.). Let €::: lin and denote

space spanned by (vl,v2, ...,vnJ. Clearly

U '\I is everywhere dense in L2(0.).&+D€

by 4v the finite-dimensional sub-€

>y1 c -1--2::::... ::::'Yn ::::... and

Next, we introduce the perturbed Lagrangian,

L : ttl x OV -+E.,€ ov €

1 12::: L(y, q) - ~llqlo 2'" ,

where qz is defined in (7.26), L(~, q) ::: U(~) + !q(J(l,) - 1) dVO' and U is the0.

total potential energy functional (recall (7.34». Since CY is reflexive and€

strictly convex, there exists at least one saddle point (x ,p ) E ttl x "y for allov€ € '" €

€ > 0, provided conditions (i) and (ii) of Theorem 7.1 hold.

, .

Suppose

a subsequence

56

IIp€110,2 is uniformly bounded in €:::

(p I} converging weakly to an element€

lin.

p

Then there exists

is the weak limit of (x I} in ttl, then (x,p) is a saddle point of L inrv€ I"V f"'V

2and a solution of the weak equilibrium equations (7.38). Conve rsely,ttlxL (0.)

if (7.39) holds for the choice .. k( then lip€1I0,2 is uniformlyq=p EC 0.),€

bounded in €.

3 . Sup pas e t hat (7.42) h old s wit h g = go > 0, and let

p E Ck(o.) be constructed as in step 2 above. Then, by Lemma 7.9, there€

exists a unique convex ~€ E Ck+l,13(o.) such that

M(~ ) :::1 + tp€ €

Let k > 1 so that2 Suppose that p E C (0.)~ E C (0.) .

€ €

is a positive number y > 0 such that

Ip€ (:9 I :s y for every x € 0.

Then, if t is sufficiently small, for example, t < l/y,

M(~ ) = 1 + tp (x) > 0€ €'"

for every x E n. Moreover, if

so that there

(7.43)

(7.44)

a. E lR, i = 1, ... , 4,1

there exists a positive constant µ > 0 such that

~1 ~ inf (ala4-a2a3) I(ai + ... +a~qi ER

1=1, ... ,4

By (7.44), for every ~ E 0.,

(7.45)

# I· ..

57

Taking

µo::: mi~ µ(x)xE.o. '"

yields

Then

(7.46)

4 2lp€, ij S (l+tpJ ' 1.e.,

where I' 12 4 is the seminorm of the Sobolev space w2,4 (n). Then, taking,Z ::: 'V~ yields"'€ €

(7.47)

where Cl::: 1/µ0 and C2 = mes(O)/µO' Thus, condition (iii) of Theorem 7.1

is satisfied.

Summing up, we have proved:

Theorem 7.3. The conditions of Theorem 7.1 hold if (7.42) and conditions

(1) and (11) of Theorem 7.1 hold. 0

...

58

8. Concluding Comments

The general existence theorems described here, particularly Theorem 5.3,

cover a wide range of problems in nonlinear continuum mechanics are not limited

to problems in nonlinear elasticity. For example, in Remark 5.1 we demonstrated

that tha:conditions of Theorem 5.3 are met in the study of special classes of

incompressible non-Newtonian fluids. Moreover, the constraint on the motion

can take a variety of forms, incompressibility being only a representative

example. For instance, the extension of our results to problems with constraint

conditions involving weakly sequentially continuous operators or convex constraints

is straightforward. Along these lines, we could also treat nonlinear Signorini

problems describing contact of uncompressible hyperelastic bodies with lubri-

cated (frictionless) rigid foundations with convex c~ntours. Some applications

of this type will be discussed in future work.

In applications of the general theory to specific cases, a major step is

to verify that the generalized Ladyzhenskaya-Babuska-Brezzi condition (4.27)

holds, and this generally involves the analysis of an appropriate auxiliary

problem of the type discussed in Section 7. Since a variety of forms of the

operator B in (4.4) can be selected for a given constraint, the character of such

auxiliary problems can vary widely for each specific constraint.

There are, of course, important classes of problems in nonlinear elas-

ticity to which our results do not apply without modification. When the space

oy is not reflexive, our compactness arguments must be amended. This is not

a major difficulty in cases in which Alaglou's Theorem holds and compactness

arguments using the weak*-topoligies can be made. For example, if E3 = 0 in

(6.1), we obtain the strain energy function for a ~looney-Rivlin material.

" ., r.

For2

0. =R , we then have

59

and the addition of the constraint

J q(J(x) - 1) dvO = 0 places the hydrostatic pressure0. rv

that minimizers of the total energy exist in the set

p

K :::

00in L (0.). The proof

1 2{l E ~' (0.) 1£0 (0.) :

J(x) :::1 a.e. in 0. ) is not difficult and follows arguments essentially the~

same as .thosegiven in Section 4. However, our techniques for handling the

uniform boundedness of hydrostatic pressures are not applicable in this case

00because L (m is not reflexive. We hope to address such problems in future

work. We also note that the proof of the weak lower semicontinuity of the

functional V2 of (6.l~ is based on the property of J given in Lemma 6.3

and on the convexigy of J ~ V2(J). Methods for treating cases in which V2(J)

is not convex do not appear to be known.

Finally, we note that the conditions of our theory suggest conditions

on the form of the strain energy function. In particular, our results on the

model problem described in Section 7 are based on the assumption that the

material constants satisfy inequalities (6.19).

Acknowledgement: The results communicated in this article were developed over

several years. Early phases of the work were done under the support of the

U. S. National Science Foundation under Grant NSF ENG 75-07846. The completion

of the work reported here was done under the support of the U. S. Air Force

Office of Scientific Research under Contract F-49620-78-c-0083.

.-- ., ...

60

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