EXISTENCE THEOREMS FOR A CLASS OF PROBLEMS IN

NONLINEAR ELASTICITY II. ANALYSIS OF A

MODEL PROBLEM OF FINITE PLANE STRAIN

J. T. Oden and C. T. Reddy

The University of Texas at Austin

i 1J

EXISTENCE THEOREMS FOR A CLASS OF PROBLEMS IN

NONLINEAR ELASTICITY II. ANALYSIS OF A

MODEL PROBLEN OF FINITE PLANE STRAIN

J. T. Oden and C. T. Reddy

1. Introduction

2. A Model Problem in Elastotatics

3. Some Properties of the Operator A

4. Existence Theorem

5. Singular Behavior and Local Invertibility

6. Implications of the Strong Ellipticity Condition

7. Concluding Comments

Acknowledgements

References

1. Introduction

In [1], several theorems establishing sufficient conditions for

the existence of solutions of a class of nonlinear equations on reflexive

Banach spaces were proved. It is also pointed out in that paper that

these results are applicable to certain nonlinear boundary-value problems

in elastostatics provided the operators involved are bounded, hemicontin-

uous, and coercive from some separable reflexive Banach space Uinta

1

2

its topological dual U~, and provided the operators satisfy a special type

of inequality referred to in [1] as a generalized G~rding inequality.

In the present paper, we present a collection of theorems which show

that the conditions established in the abstract theorems of [1] do indeed

hold for a class of boundary-value problems in plane elastostatics. In

particular, we consider as a model problem the plane deformation of an

isotropic, hyperelastic, compressible body, subjected to body forces and

prescribed boundary displacements, for which the strain energy function is

of the form

(1.1)

Here Eq' El' EZ' E3' and E4 are material constants and II' 12, and 13

are the principal invariants of the Green deformation tensor. The fact

that our model involves a hyperelastic material subjected to conservative

(dead) loads is incidental--we chose our model constitutive law so as to

resemble and generalize some which have been suggested by experiments and/

or kinetic theories for certain isotropic elastomers. However, our ap-

proach is applicable to bodies which do not possess a strain energy func-

tion and to bodies subjected to non-conservative loads.

Owing to the presence of the term -E In Ifo' 3 in (1.1), the strain

energy exhibits the natural singular behavior

as I -+ 03 (1.2)

provided EO, is positive. In Sections 2-4 of the paper, \~e ignore this

singular character and confine ourselves to cases in which E ::O. Weo

3

demonstrate that solutions do exist for the problem of place for the

model material, that for fixed choices of material constants and data

multiple solutions may exist; that these solutions need not be smooth--

they may exhibit wild oscillations and weak discontinuities in their first

partial derivatives. In Section 5 we list some sufficient conditions

for which the results can be amended so as to exhibit the singular be-

havior (1.2) and we give arguments which establish that solutions of our

model problem satisfy the local invertibility condition, 113 > 0 a.e.

in n

Regrettably, our analysis of this case involves an assumption con-

cerning the local regularity of solutions. We are able to show that, for

any \I satisfying 0 < \I < 1 there exist solutions to our model problem

in a nonempty closed convex set K ,\I

containing the origin, where K\I

is a subset of the non-convex set M\I = {~: II3(y)~ \I a.e. in n}. We

then assume that if solutions exist to our problem for given choices of

data, then all of them have a type of local regularity embodied in the

condition

where w is a solution and

no = {~(- n: 113(~T < 0, a < 0 < 1 ) .

Under this assumption, which certainly holds whenever II3(~(:)) is

continuous on certain interior sets no C Q, we show that there exists

a constant depending on the data, such that a.e.

in n. Finally, we choose such that and show that

solutions exist in K' C C K\If \If

Since these must include those with

and, hence, are solutions of the weak equilibrium condi-

4

the above regularity, these locally regular solutions must belong to the

interior of K'\If

tions. Without these regularity assumptions, we are only able to show

that solutions to a variational inequality related to our problem exist

in the convex set K for any choice of \I> 0 and, indeed, that solu-\I

tions also exist in v~O K\I .

A very important aspect of the proofs of the existence theorems

presented here is that they lead to conditions on the form of the strain

energy. Indeed, our theorem admits solutions for bounded hemicontinuous

operators whenever the requirements of coercivity and a generalized GRrding

inequality are satisfied. We show that these hold whenever the material

constants satisfy certain inequalities. Thus, our theory adds another

candidate to a long list of proposed solutions to TRUESDELL'S "main un-

solved problem in the theory of finite elasticity" [2] (see WANG and

TRUESDELL [3] for a survey and critique of various proposals).

We also mention that our theory does not make use of the assumption

of strong ellipticity. In the existence theorems of AN~~ (e.~. [4-6])

for elastic rods and in the recent work of BALL [7], the assumption of

strong ellipticity plays a crucial role in the analysis. The strong ellip-

ticity condition manifests itself in the inequality

V µ,A :f 0 (1. 3)

where µ,A <- lR3 and wi,a

are components of the displacement gradients.

(For various interpretations of (1.3), see WANG and TRUESDELL [3] or

HAYES [8].) However, KNOWLES and STERNBERG [9,10] have recently shown

5

that the strong ellipticity condition can be violated for physically rea-

sonable motions of isotropic, hyperelastic materials. He elaborate on

these points in Section 6; there we show that the assumption of strong

ellipticity does imply that the conditions on the material constants

(which need to be met in order that the requirements of our existence

theorem are satisfied) do hold. However, the converse does not hold.

2. A Model Problem in Elastostatics

We shall consider the problem of finite plane strain of a homo-

geneous, isotropic, hyperelastic body n subjected to body forces and

fixed along its boundary an. We wish to find the displacement field

I Qka(V~)Vk,a

n

where k,a = 1,2 ,

v v E u(n) (2.1)

U(Q) = space of admissible displacements

{u

~ = 0 on an, k ~ 1,2}

w. (X) are the cartesian components of displacement1 -

the response functional for the first Piola Kirchhoff

(pseudo) stress, given as a frame-indifferent fuction

of the cartesian components of the displacement gradi-

ents tensor (V~)ka = wk,a ::aWk/a~; a,k = 1,2

6

p (X) is the mass density of the body in the referenceo _

configuration

k kf = f (X) are the components of body force.

Throughout this study, we assume that n is regular and that its

00

boundary an is smooth; e.g. an { c

hold when an is only Lipschitzian.

however, nany of the results

Equation (2.1) describes a variational boundary-value problem; it

represents, in physical terms, a principle of virtual work. Frequently,

U(Q) is a separable, reflexive Banach space whose norm depends upon the

form of the constitutive law Qka (Vw) • If P(Q) = V(Q) x V(n) is dense

in U(n) , where V(Q) is the locally-convex topological space of test

functions, then (2.1) is equivalent to the distribution differential equa-

tion

o (2.2)

where a denotes a distributional derivative. Then (2.1) formally de-a

fines an operator A:V(n) -+ V(Q)~. The expression on the left side of

(2.2) is, thus, a distribution and its restriction to U(n)~ is the

functional appearing in the variational problem (2.1). See [1] for addi-

tional details on these interpretations.

\~e must be more specific about the space U(n) and, equivalently,

the form of the constitutive functional Qka(V~)

7

Let G (X) = (0 k + wk (X»(0 k + wk (X» denote the componentsa8 ~ a ,a - 8 ,8 -

of the Green deformation tensor and 11, 12, and 13 denote its principal

invariants. For plane deformations,

(2.3)

det (Ga8)

For an isotropic hyperelastic material, there exists a strain

We shall first consider the class of problems for which

the form

(2.4)

assumes

(2.5)

where El' E2, E3, and E4 are material constants.

If E2 and E = 0 , and if we add the term Eo ln li3 ' (2.5)3

reduces to a strain energy function proposed by BLATZ [II]for certain com-

pressible elastomers. If 13 = 1 (incompressible materials), the case

E2 = E3 = a corresponds to the classical neo-Hookean materials studied

by RIVLIN (e.g., [l2]); the case E2 = a corresponds to the well-known

Mooney-Rivlin material (see for example [13]); the form El,E2,E3:f a

was proposed by ISHIHARA, HASHITSUME and TATIBANA [14]for natural rubbers

8

on the basis of a kinetic theory. However, we shall not consider incom-

pressible materials in the present study.

By introducing (2.5) into (2.4) and making use of (2.3), we arrive

at the following constitutive equations for materials governed by (2.5):

(2.6)

)

=

=lJ

(2.7)

Now the space U(Q) of admissible functions should, in view of

(2.5), contain classes of functions whose generalized partial derivatives

2 I 2 1are such that, for example, Il(~) { L (Q). Since II(~) E: L (n) 9

a wi(X) ( L4(n) , we are led to the consideration of the Sobolev spacea -

4{u :u,a u E L (n), aa 1,2, u o on an} (2.8)

which can be equipped with the norm

lIul~,4(n) [In Ivu 14dUr4 (2.9)

This space is separable and reflexive, and its topological dual is denoted

9

(2.10)

For vector-valued functions with components in WI

,4(Q) , we use the no-

tations

(2.11)

(2.12)

We also use the notations

II~Ilif ' p (Q )

(2.13)

II ~IIJ,P (n)

With these preliminaries now completed, we can state in specific

terms the nature of our model problem of finite plane strain: Find the

displacement ~ = (wl

,w2) f U(Q) such that

(p f ,v)0_ _ (2.14)

where (.,.> denot es duality pairing on U (Q) ~ x U (Q) :

(A(w) ,v) (2.15)

(p f,v)0- -

(2.16)

Here are given by (2.6) and we take

1.0

(2.17)

Formally, A is given as the distributional operator

A(w) (2.18)

For additional discussion of this interpretation, see [I].

3. Basic Properties of the Operator A

In this section we prove a number of theorems which establish some

fundamental properties of the operator A defined in (2.15), with Qka

given by (2.6).

Theorem 3.1: Let A be the operator defined in (2.15). Then A

bounded sets in WI,4(n) into bounded sets in-0

is bounded as an op~rator from into -1,4/3(Q) ;W

-1,4/3(n)W

Le. A maps

I 4Proof: We note that V w ~ W ' (n) ,- _0

\IA(w) II -1,4/3(n)- - W

(3.1)

and

<A(w) , v) IQ

Qka(VW)Vk,adU

Next, we observe that if a(' ) and b(·) are given by (2.7), then

la(El,E2,E3'~) I2

)< 2[2+ 1El +E3 -4E21 +4IE2\lv~1 ]-

(3.2)

Ib(E3,E4'~) \2< 1E4 + 4E 31 + 21 E311 v~ I

where we use the notation

II

From the definition of Qka, we have for k,a = 1,2 :

(3.l:.)

In deriving (3.4), we used the fact that for

and p,q > 0 ,

a. E :R+, I < j < n ,] - -

Then,

(3.5)

(A(w) ,v) < J Cl(1+lv~13) Iv~ldun

< C111I+lvw1311 4/3 II1vvll14~ L (0) ~ L (0)

" C1 (Ia (1 + IV~13)4/3dVY/4(Ja (Iv~l) 4duf/4

12

< C1(320)1/4(1+ (mesQ)3/4) (1 + 1I~1I~,4(n) 1I~IIWI,4(r.)~o ~

C2(1 + II~II~,4(n) II~ "if,4 (n)~a ~

where

C2

= (320) l/ 4 (1 + (mesn) 3/4) CI

Therefore,

and

} (3.6)

(A("!!) ''!.>

II ~ I/~, 4 ill)~o

~ C2(I+II~II~l,4(Q)~o

Hence, in view of (3.1), if

oIIA(w)!1 -1,4/\n) .~ ~ W

II~II~,4 (n )_0

is bounded, so is

An operator A: U -+ U~, U a Banach space, is said to be hemi-

continuous if it maps line segments in U continuously into the dual

topology of U; i.e. A is hemicontinuous if, for e E (O,l),

lim+ (A(u + EN),w> = (A(u) ,w>6-i-Q

v u,v,w E U, where (.,.> denotes duality pairing on U~ xU.

(3.7)

Theorem 3.2: The operator A of (2.l5) is hemicontinuous from

wl,4(n) into w-l,4/3(n)_0

13

Proof: \~e examine a typical term in the sum

For example, observe that

where a(o) and b(o) are defined in (2.7).

2 3+ SA (u,v,w) + S B (u,v,w) + e c (u,v,w)a _ _ _ 0 _ _ _ a _ _ _

where

A (u,v,w)a _ _ _

B (u,v,w)o _ _ _

c (u,v,w)o _ _ _

14

cj>(u,v)

K(V)

Thus, QI1(~'8 + 8~'8)wI,1 is bounded by a measurable function

g = g(u,v,w) for all 8 f (0,1) . Therefore, it follows from the

Lebesgue dominated convergence theorem that

I lim Ql1(U'D +8~'8)wI 1 du8-+0 - µ ,

Q

Ill) .Q (~'13 Wl,l<lu

n

Similar arguments apply to the remaining terms in

Qka (u,D + 8v,D)Wk . Hence A is hemicontinuous. 0_ µ _ µ ,a

We recall that an operator A: U -+ U~ is coercive if and only

if

~-++<Xl as

We next have

Theorem 3.3: Let A denote the operator defined in (2.15) with*

A(O) a Then

*The requirement of a stress-free initial state, A(O) = 0, is con-venient but by no means essential to our arguments. Cases-in which ~(Q) I Qare easily handled using the techniques demonstrated by ODEN [I], at the ex-pense of greater algebraic complexity.

1. A is a coercive operator from ~,4(n)-0

into

IS

if

either of the following conditions hold:

E2 > IE31 or 1J

:(3.8)E2 > a and E3 ~ 0

II. If E2 = 0 , A is a coercive operator from Wl,2(Q) into- _0

W-1,2(Q) if

(3.9)

Proof: By noting that the stress free state corresponds to w = 0

(A(O) = 0) , we obtain, after considerable algebra,

= J (2(E1 +E3)IVwI2 + 2E3(I~(~) +1)(wI,lw2,2 -wl,2w2,1)Q

~ 2 2 12)+2E/I3(~)-1) +2E2(Il(~)-3) +2E2(II(~)-3)lv~ du

Noting that wI IW2 2 - wl 2w2 I" "

also that J canst (Wl,l +w2,2) du' = 0 ,

ncation, at the inequality

we arrive, after further simplifi-

(3.10)

<A(w) ,w) 2(E1 +E3) 11"1I~1.2W) + In [4E313(~) + 2E2(11 (~) - 3)2

+ 2E2(II(~) -3)lv~12 - 2E3Ii(~)(W1,1 +\012,2»)dU'

First let us consider the case when E2 > a and E3 ~ O. Then,

16since 13(~) > 0 ,

(A(w) ,w) (3.11)

where,

2E2 I (11 (~) - 3)2 + (11 (~) - 3) IV~12) du

11

J2(~) = I 2E3Ij(~)(Wl,1 +W2,2) du

n

(3.12)

Introducing the expressions for Il(~) and I~(~) given in (2.3), we get

Jl(~) = 2E2I{~(WI,1+W2,2)+IV~I~2 + ~(Wl,1+W2,2)+IV~I~IV~12)dUn

> 2E2 I ~(Wl,l +W2,2)IV~12 + 2IV~I~ duQ

4 I 3> 4E2 II w" 1 4 - 20E2 Iw /I 1 3~ 'i ' (n) ~ W ' (n)

= \2E3 I GW1,1 +W2,2)

(WI,lQ+W2,28dU I

< 2E3 f ~IV~12 + 1(lwI,113 + IW2,213) + 1IV~I~ dun

(2 3 2).5. 4E3 311wll 1 3 + Ilwll I 2

~ W ' (n) ~ W ' (n)

Therefore,

(A(w) ,w)

(3.13)

17

From which coercivity follows immediately.

If E3 is negative, we need to consider the term I 4E313 du

n

<

41E31 f (l+WI,l +W2,2+Wl,lW2,2 -Wl,2W2,1)2 du

n

(3.14 )

We note that, for every

f ~"l,l du = a andQ

We shall prove that

I W2,2 duQ

a

To prove this assertion, we recall that00

c (n)o is densely imbedded in

Then, integration by parts gives

o

InnWI 1Wz 2 d u, ,

n

18

Thus,

I {Gl,l + (W~,I-wl,lnt2,2+ (w~,2-w2,2nn

- t!,2 + (w~,2 -Wl,2J] t2,1 +(w~,l -\"2,1)]} du

and, using Holder's inequality, we have

Next, we take the limit as n + ~, and get

o 2 I (Wl ~w2 2 - wI 2w2 I) du, ...!.., "

n

By applying the equality with signs reversed and following the same pro-

cedure, \"e get

a 2 - J (wI,lw2,2 - wI,2w2,1) du ,n

which proves (3.IS).

We now return to the inequality (3.14). In view of (3.14), we now

can write

19

f 4E2I3(w)du ..::41E31rmesQ + 211wll2l 2 +~31IwI13l 3 +llwll\ 4 JL - W ' (Q) - W '(n) - W ' (n)n

Then, combining this result with (3.13) and demanding that E3 < 0, we get

(3.16)

From above we see that A is coercive whenever E > IE31- 2

Also, when E2 = 0 , \oleobserve that A is then a coercive operator

from ~,2(0) -+ W-1,2(0) if E = 0 and EI > 0 . 0_0 _ 3

We next derive an intermediate result which establishes an import-

ant property of the operator A which should be useful in approximation

theories.

Theorem 3.4: Let A be the operator defined in (2.15). Then, for

w, W, z E Wl,4(Q) , there exists a positive constant C3 independent of_0

w, w, z such that

I (A(w) -A(w),z) I ~ c3g(w,w) Ilw-wll 1 4 11:11 I 4 '- - - - W ' (0) W' (Q)

where

(3.17)

(l+ IIwll2l 4 + Ilwll\ 4 )- W '(n) - W ' (n)

(3.18)

20

In particular, (3.17) holds if we take

1/

C3 = 152 (1 + 8/6)( I E21 + I E31 )(1 + (mes n)'2)

Proof: By direct expansion, we get

(3.19)

<~(-:)-~(~).~>~ L{~(EI +E3)(wI•I -wI•I) + 4E2(II (-:) - 3)(I+wI•I)

-(II (~) - 3) (1 +Wl,l») + E4 (w2,2 - w2,2)

- (II(~) -3)(1+W2,2») + E4(wI,1 -wI,I)

I;(~)(1 +W2.2»)}2.2} du(3.20)

21

Applying Holder's inequality to (3.19), we get

where

(3.21)

4E2 Ie {~Il (~)-1)(1+"1,1) - (Ii (~) -l)(1+'\,l~Zl,l

+ GIl (~) -1)wl,2 - (II (~) -1)wl,JZ1,2

+ GIl (~) -1)W2,1 - (II (~) -1)W2,JZ2,1

+ G'l(~)-l)(1+"2,2) - (Il(~)-1)(1+;;2'2~Z2'2}dU

(3. 22a)

- I J {r:~ ~- - :lG2(~'~':) = 2E3 n t:,(~)(l+wI,I) + 13(~)(1+wI,1)JzI,1

r:~ ~- - 1- L:3(~)w2,1 - 13(~)w2,llZl,2

~~ ~- - :l }+ t3(I+w2,2) - 13(~)(I+w2,2)Jz2,2 du

Noting that for every a,b,c,d f ~ ,

ab - cd = t Ga+C)(b-d) + (a-C)(b+d~

we reduce equations (3.22) to the identities

(3. 22b)

(3.23)

22

+ (w2,1 +w2,1)z2,1 + (2+W2,2-W2,2)Z2,J} du I(3. 24a)

and

- (wI 2 +w1 2)z2 I + (2 +w2 2 +w2 2)z2 J} du I, " t"

(3.24b)

In the above two equations, we recognize that

(3.25)and, by again using the identity (3.23), we have

- (w2,1 +w2,1)(wI,2-wI,2) - (wI,2+wl,2)(w2,1 -W2,1~

(3.26)

23

Introducing (3.25) and (3.26) into (3.24) and again applying Holder's in-

equality, we obtain

- -) II - II II z 11__] 4< Cr-gl ('::"'::' ~ - ~ WI,4(Q) - W-' en)

(3.27)

where Cl and C2

are positive constants and

(3.28)

Upon simplifying the above expressions, we easily obtain

C (1 +llwll\ 4 + IIwll\ 4 J- W '(n) - W ' (Q)

(3.29)

where C is a positive constant. We next reinforce inequalities (3.27)

by introducing (3.29) and incorporate the resulting inequality into (3.21).

The result is then (3.17), as asserted.

By retracing the steps in our proof, it can be shown that it is

sufficient to choose the constant C3

in (3.19) in order that (3.17) hold.

We omit these purely computational details. 0

24

Lenuna 3.1: 2For every U,V E L (Q) and v :f 0, for every E > 0 , there

00

exists on infinitely differentiable function $ E C (n) such that

I $v dun

(3.30)

Proof: For every u, V€ L2 (n) , we have the following obvious

inequality.

In (sgnv) lui v du

The function (sgnv) lui is in L2(n). Since the space of test functions

2V(Q) is everywhere dense in L (Q), there exists a sequence {$ } ofn

infinitely differentiable functions in V(Q), converging strongly to

(sgnv) lui. Moreover,

limn-+<x> J $ v du = J (sgnv) lui v dun n n

Thus, for every E > 0 , there exists an integer n > 0 such that foro

Then,

We now choose <I> <l>n· 0

26

II * II 2Ca (p)

Ill/JII 2a L2(n) L (n)

Proof; By definition,

-2 J~ = P Q wp(~-¥) l/J(y)dy

Then,

(

~ -2 J aw (x-y) l/J(l) dyax= p p ~

et Q axa

~ II = f -4 [f ow (x-y) 12

ax P aP l/J(y)dy dx~ X ~)

L2(n) n Q·a

Using the Holder's inequality on the second integral, we have

(3.33)

~axa

<

Hence,

Since the second integral is defined only on B (x), we can setp ~

-2 J [ow (~-y) )2w (x) = P P aXa J dy

P, a ~Q

~aXa

L2(n)

(3.34)

25

Let us examine the function $ in Lemma 3.1 in more detail using

the notion of mollifiers on star-shaped domains, let uE:LPcn).

In order to construct a mollifier of u, we introduce a special

function w (x) which for any choice of a real number p > 0 , has 'thep -

following properties.

(i). w (x) has continuous derivatives of all orders in Rn; i.e.p -

w (x) E C<» (Rn)p

~i). wp (~) = 0 for lei ~ p and wp (e) > 0 for Ixl < p : i.e.

w (x) and its partial derivatives vanish outside a ball B (0) of radiusp p -

<»p centered at ~ = 0 , such that w (x)~ C (B (0») and within B (0),

- P - 0 P - p -

w (x) is positive.p -

(iii),

w (x) dX = 1p -

(3.31)

Now let u be an arbitrary function locally integrable over Q.

The mollifier operator J is defined for each p > a byp -

J u(x)p p-n IQ u(y) w (x - y) dy- p-

(3.32)

For a detailed account of mollifiers, see [18]. We are immediately led

to the following lemma.

2Lemma 3.2: Let QCR be open, bounded and star-shaped. Let ~ be

a function in L2(n) and let $ = Jp~' Then, for arbitrary p > 0 , there

exists a constant C (p) dependent on p such thata

27

We can define a constant C (p) such thata

C (p)a ]

1/2(x) dx

(3.35)

Then, substituting (3.35) in (3.34), we get (3.33). 0We next establish a fundamental property of A

Theorem 3.5: Let A be the operator defined in (2.15) and let

Then, for every pair of displacements w,W E B (0) , 0 < µ E:R ,µ -

B (0)II _ IIu Ii 1 4 < ll}W ' (Q)

(3.36)

and for arbitrary

that

E E~, there exists a positive constant y (E,~)a such

(A(w)-A(w), w-w) ~ (Ez-E)llw-wIl414- - W ' (Q)

+ 2(El +E3 - 4EZ) 11~-~II~,2(Q)

- Yo(E,µ)II~-wI14/3- L\Q)

(3.37)

Proof: By direct expansion of the left side of (3.37) we get

Z8

<~(~) -~(~),~-~> = f {~(I+WI,I) - a(l+wI,I~ (wl,l -wI,I)n

+ ~(l +wZ,2) - 1>(1+~Z,Z~ (wl,l- ~l,l) + (awl, 2- aWl,2) (w1,2- wl,2)

If" - 2 - Z - 2~2(EI+E3)0wl,I-W1,1) + (w2,Z-w2,Z) + (wI,Z-wI,Z)

n

+ (wZ,1 -W2,1)J + ZE4Qw1,1 -wl,l) ("'Z,2 -wZ,2)

+ 4EZ[(11- 3)w2 1- <11- 3)w ](w2 I - w2 1), 2,1"

29

where, for compactness, we have denoted a = a(EI,E2,E3'~)'

a = a(EI,E2,E3'~)' b = b(E3,E4'~)' b = b(E3,E4'~)' 13 = 13(~)' 13 = 13(~)'

etc. (recall (2.7).

Let11 = W-\07

Since n t Wl,4(n) , we note from (3.15) that_0

J (nl,ln2,Z-nl,2n2,1)du = 0Q

Applying the equation (3.15) with ~ replaced by n = w - W

we arrive, after considerable effort, at the equality

<~(~) -~(~),~-~> = J {2(El+E3)IV(~-~)12 + ZE;(I3('::)_I;(~»2Q

- Z It - ~ I - 2)+ ZEZ (II(~) - II (~» + ZE2 0II (~) - 3) + (11(~) - 3)J V (~ - ::)I du

(3.38)

we have

<~(~) -~(~),~-~> >

wherein

2(EI +EZ) 11~-~11~1,2(n)

(3.39 )

(3.40 )

A

We easily derive a lower bound on JI(~'~):

30

Ezil ~ - ~":1,4(n) 8E211~ - ~11~l,z(Q)

Next, we consider the term j 2 (::r'~) in (3.36). By direct expan--

sian, and denoting (w+ w) = a we get

We will now show that,

J(VQ+ ~ detV~) detVndu

n 2

4/34

< (£ +£ ) II ~ 1\' I 4 +C(µ, £ ) I~II (3 .43)1 Z W ' (Q) 2 - 4

L (n)

where £l > 0 , £2 > a are arbitrary, C(µ, £2)-+00 as £2 -+ 0 , and µ is

a bound on 1~111 4 and II ~ II Wl,4(Q)tr' (n) -

31

We first apply Lemma 3.1 and write

where replaces £ in (3.30) and 4> -+ sgn (detVr!)[V·e+detVe]n - - -

as n+=. The value of n can be determined for fixed first term. Then,

integrating by parts,

J (V·~+ ~etVe)detVlldU 2 £1111l1/\ 4 + J (4) 11ll 2-$ 21lll)llZdu- - - - W ' (Q) ""

Q n

We next construct the mollifier of sgl'(detVn)[V·e+detV8] and make use of

Lemma 3.Z to obtain

(3.1.4)

.£.LII < C (p) IIax - aaLZ(n)

1sgn detVll (V·e+ ¥etVe) II 'l

- - L'-(Q)

c (p) II v·e + k2 etVe 112a - - L (0)

We note that e = w + wand v·e = el I + e2 Z ' detV8 = el I 82 2-el 2eZ I '- , , - , , "

We also recall the following fundamental inequality

p p-l p P(al + a2 + ... + a) < n (al + ... + a ), p > In - n -

for every a such that a. > a I < j < n. Using (3.45) repeatedly, weJ - -

obtain

(3.45)

32

1 1/2 2 zII v·e+ 7dZ etVellz ~ [l08Q..+mesn)] (1+11w " 1 4 +11w II 1 4 ) (3.46)

- - L (n) - w' eQ) - W' (n)

Since w, w E B (0) c Wl,4 (n), we may setµ -

2C(µ) = (108 (1+mesn»1/2 (1 + 2µ ) (3.47)

Then, by applying Holder's inequality to (3.44), we get

4

\J (V· e+detVe}detVndu I < Ell n III 4 + (II <I> IIIZ II nl zll 4Q - - - - - w' (n) 'L (Q) 'L (Q)

+ II 4> 211 z II n II 4 ) II nzll 4, L (Q) l,l L (Q) L (n)

Taking C _ (p) = max C (p), and applying (3.45) and (3.46) to the abovea a a.

inequality, we get

fore, setting C(µ) = Cn(P)C(µ) , we get

4which is bounded by µ • There-

J4 -

(v·e+ -kzetVe)detVndu ~ EIII nil 1 4 +C(µ)llnlll 4 II nl14Q - - - - w' (n) - W ' (Q) - L (n)

(3.48)

33

Now we make use of the Young's inequality in the form, (see [IS]),

ab < e: al/Z + (u!e:)u/(l-a) (I-a) bl/(l-a) uE: [O,l]

and get,

where,

and C_(p) = max C (p) , C (p) is defined by (3.35).a a u a

Thus (3.43) is proved, and we have

(3.49)

(3.50)

ZE3C(µ,e: ) 11111r/3Z _

L4(Q)

(3.51)

Setting E = ZE3(EI+E2) and substituting (3.41) and (3.51) into (3.35)

gives

<A(w) - A(w),w - w) >

- 2 (I -114/3+ 2(EI +E3-4EZ)llw-wILl Z - Yo e:,µ) Iw-w 4'W-' (Q) L (n)

34

which is precisely (3.16). Here £ > 0 is arbitrary and

(3.5Z)o

Corollary 3.5.1: Let A be the operator defined in (2.15) with-EZ = 0 . Then for every pair of displacements "! ' ~ E Bµ (~) . o < 11 E .R

B (0) = {u ( WI,Z (n) : II u II if- 2 < µ} (3.53 )µ - _ _0 , (Q)

and for arbitrary positive £ E ~, there exists a positive constant

(A(w) - A(w) ,w - w> .:: (2El + 2E3 - £) Ilw - wll \ Z- - W ' (Q)

(3.54,)

Proof: The proof of this theorem follows lines very similar to that

of Theorem 3.5 and we will not repeat the details. Note that if EZ = 0 ,

we use, instead of (3.40), the familiar inequality

ah < 2 + 1 b2£a 4£ £ > 0

in obtaining (3.44). In this case, we find

(3.55)o

35

4. Existence Theorem

We have now reached a point in our study at which we can prove the

existence of solutions to our model problem (Z.14). We first lay down a

general theorem on the existence of solutions to equations defined on

Banach spaces.

Theorem 4.1: Let U and V be separable reflexive Banach spaces,

with U continuously embedded in V, with the injection i: U -+ V com-

pact. Let A be an operator mapping U into its topological dual U~ ,

and let A be

i) bounded,

ii) hemicontinuous,

iii) coercive, and

iv) let there exist a continuous, non-negative valued function

+ + + + .h:lR x lR -+ lR , where lR = [0,(0), tY1ththe property

1lim "6 h(x,ey)6-+0

= o +V x, y E lR , e > a (4.1)

such that for every u and v in the ball

the following inequality holds:

(A(u) -A(v), u-v) ~ - h(µ,llu-vllv)

(4.2)

(4.3)

Then A is surjective; i.e. for every f E U~ there exists at least

one u E U such that

(A(u),v) = (f,v) v v E U (4,4)

o

36

The proof of this theorem is given in [1]. We remark that the

existence theorems in [I] apply to a more general class of operators than

those considered here; in [1], A is assumed to be factorable into the

form A(u) = A(u,u) where A(. ,v) satisfies (4.3) for any v E U and

A(v,') is, in essence, weakly continuous for any v E U.

Returning now to our model problem, we have

Theorem 4.2: Let A be the operator defined in (Z.14), let (3.17)

hold, and let the material constants satisfy either the inequalities

E > 0Z

(4.5)

or

Then,for any data Pof E W-l,p~(n) ,

(4.6)

there exists at least one solution

(A(w) ,v) < P f,v)0- -

(4.7)

where (.,.) denotes duality pairing on

p~ = p/(p-l) , and p = 4 if E2 > 0, p = Z if EZ = 0 .

Proof: The proof amounts to a simple application of Theorem 4.1.

By Theorems 3.1 and 3.2, A is bounded and hemicontinuous. If E2 > 0

and A is coercive from into If, in

37

addition, EI+E3~4E2' (3.33) reveals that A satisfies (4.3) for

Here e:where

Since

y (e:,µ)aWl,4(rl)_0

is given by (3.42).

is compact in L4(rl)

yo(e:,µ) Ilw-wll4/3

- - L4(Q)

is chosen so that E2 - £ > 0 .

the theorem follows for cases in

which (4.5) hold.

If EZ = 0, E3 = 0, and El > 0, A is bounded, hemicontinuous,

and A is coercive from Wl,2(n) into its dual. From (3.45), we observe

-0

that Yl(e:,µ)= 0 when E = 03 and hence the operator is monotone. Thus

we get unique solutions from the theory of monotone operators. 0

5. Singular Behavior and Local Invertibility

The theory we have developed up to this point has two shortcomings

from the viewpoint of general finite elasticity. First, we have ignored

the local invertibility condition,

J(w) > 0 v w <: U(n) (5.1)

where J(w) = II3(~) , this condition holding a.e. in rl. Hence, the

deformation X(X) = X + w(X) may not be locally invertible and orientation

preserving at a.e. X E n Second, the form of the strain energy function

chosen in (2.5) does not exhibit the following singular behavior required

of physically reasonable constitutive equations for compressible elastic

solids:

38

as J -+ a

"aa -+

aWk,a- 00 as J -+ 0 k = a 1,Z .

The present section is devoted to a study of these two conditions. Our

treatment of the local invertibility condition has, unfortunately, one

defect: We are forced to make an (apparently mild) assumption of local

regularity of the solution in order to show that these solutions are, in

fact, weak solutions of the model elastostatics problem. Without this

regularity, we can only show that solutions belong to a certain family

of closed convex sets.

Again, we confine ourselves to a model problem in which all of the

assumptions laid down in Section 2 are in force, except that now we con-

sider a material for which the strain energy function is of the form

a (I ,J)

+ o(I,J) ,

J = 0

J > 0

(5.3)

where I = II' O(I,J)

fined in (Z.5), and

a (J) = - E In Ja aE > ao (5.4)

E being a material constant. The corresponding constitutive functionala

-ka ~Q = aa/awk is now of the form,a

(5.5)

39

where Qka (Vw) is defined in (2.6) and

r~ J(w) ~ 0 a.e. in Q,Qka (Vw) -= (5.6)o _

-1- EoJ (~)(aJ(~)/awk,a) , J (w) > 0 a.e. in n

Clearly, the form (5.3) exhibits the singular behavior described in (5.2).

We observe that instead of the operator A of (Z.15), our model

problem now involves the operator

A = A +A_0

where, V v E U(n) ,

J ka(A (w), v) = Q (Vw)vk du_0 _ _ 0 ,an

(5.7)

(5.8)

Thus, we are now concerned with the problem of finding a w ~ U(Q) such

that

(p f ,v)o<A(w) , v)

J(w) > a

Let us introduce the set

v v E U(n)

a.e. in n } (5.9)

M = {u E U(n) :J(u) ~ \I > 0 a.e. in n}\I - _(5.10)

If the operator A of (5.9) where hemicontinuous, coercive, and pseudo-

monotone from M into U~ for each \I > 0, we could reformulate (5.9)\I

with J(w) > a replaced by J(w) ~ \I > 0, as a variational inequality

and use a theorem of BREZIS [15] to show that solutions do exist, if M\I

40

were convex. Unfortunately, the set M\I

is not convex. However, we will

show that (an apparently large) subset K of M can be identified which\I \I

is nonempty, closed, convex, and which contains the origin u = (0,0) .

Let

f 1 Z I 4\X(x ,x ) E Wo' (Q) : l+XI,1 > 0 a.e. in

(5.11)

and let

J (x,y) (5.IZ)

~We introduce two sets Kl C Kl and KZ C K2 such that for every

we have

J(X,y) > \I a.e. in n .

Then we define a set K\I

by

Lemma 5.1: For every \I satisfying 1 > \I > 0 , the set K\I

of (5.13) is a closed, convex, non-empty subset of

the origin.

containing

Proof: First we shall show that K\I

is convex. Let (X,y), (x,y) E K\I

and let e E [0,1] . Then

41

= (6(1+X'1) + (1-6)(1+x, 1) (6(I+y'2) + (1-6) (l+Y,Z)

\I

Hence, the convexity of K\I

is proved.

A

The set is clearly non-empty, for if we choose a set Kl C Kl

such that for every x ( KI , l+x'l>a>o and X'Z = 0 a.e. in nA A \I

and a set KZ C K2 such that for every y (; K2 ' I + y, 2 >a > 0 andA A A

Y'l = 0 a.e. in n , we produce a set K\J = KI x KZ C K\I which is

clearly non-empty.

We also observe that (o,O)E K , since J(O,o) = 1 > \I, \I

Now, to show that K is closed, we consider the sequence {(xn,yn)}\I

such that x -+ X and y -+ y in WI, 4 (n) . Then, for given o > 0 ,n n

there exists a positive integer nosuch that whenever n > n- a

< <5 ,II x n - X II WI, 4 (Q) _

Consider the difference,

42

where we have used the identity (3.Z3). Taking the L2-norm, using

Holder's inequality, and simplifying, we get

IIJ(xn,l) -J(x,y)11 2

L (n)

< C(l+llxn+xll 1 4 + Ilyn+y II I 4 )llxn-xll 1,4( )/Iyn-Y11wl,4( )W ' ( Q) H' , (n) W n Q

Thus,

lim·.IIJ(Xn

.yn) - J(x,y) II 2 = 0n~ L (Q)

nk nkIt follows that we can extract a subsequence J(x ,y ) converging to

J(x,y) almost everywhere in Q (see e.g. SMIRNOV [17]). Therefore,

J(x,Y) =n n

(J(X k,y k) _ J(x,y»

>

Now taking the limit as nk

-+ CD we obtain

J (x, y) > \I a.e. in n . 0

We next prepare for future reference a list of some properties of

A and

Theorem 5.1: Let A be the operator defined in (Z.15) and let

be defined in (6.8). Moreover, let \I be a fixed real number satisfying

1 4I > \I > 0 and let K denote a nonempty closed convex subset of W' (Q) ,

\I _0

containing the origin, such that

K C M\I \I

(5.14)

43

where M is defined in (5.10). Then\I

(1) A :~-+ w-l,4/3(n) is bounded_0 -(ii) A : K\I-+ w-1,4/3(Q) is hemicontinuous

_0

(iii) is coercive provided EZ > IE3'

(iv) there exists a positive constant Ca such tha-C, for every

C

I<A (w) -A (;),z)1 ~ 20,g(~,~) 1I~-~1I __1,4 ,11:11 1,4( )-0 - -0 - - \I W- (OJ w n

where g(w,w) is defined by (3.18).

(5.15)

(v) 1 4for every w, w ~ B (0) n K c. W ' (Q) , and every e:>0 ,µ - \1_0

there exists a positive constant Yz = y2(e:,µ,\I) such that

+ 2(El + Ef 4EZ) II~- ~II~,z(n)

- Y2(e:,µ,\I)llw-wI14/3

- - L4(Q)

Proof: (i) Boundedness follows easily from the relation

(5.16)

I(A (w), v>1_0 _ _ If Q~ (V~)Vk,adVIQ

If Eo aJ(~)- J (w) . aw vk cx:d\ll

Q _ k,a'

44

where C > 0

< C Eo Ii vii 1 4 (1 + IIwll 1 4 )'J - W ' (n) - W ' (Q)

(11) Consider a typical term in kaQ (Vu+a(Vv-Vu», aE(O,l):o _ __

EI -. T ( II~ (V-II ») (1 + uZ, Z + a (v z , 2 - u Z, 2» w1,11

E< \10 IW1,11 (II+uZ,zl + alvZ,2-uZ,zl)

Hence, by the Lebesgue dominated convergence theorem,

I11lim Q (vu+a(vv-vu»wlldu

a-+O a - - - ,Q

III= Q (vu)wl 1 dua _ ,

Q

Similar arguments apply to the remaining terms.

(iii) Since for some C~ > 0

E 2)C~-.£. (1+ 11,-:::11 1,4(n)~ \I W

(A(w) ,w) >E Z)

f(w,w) - C~ \10 (1+11~11~,4(Q)

where f(w,w) is the right side of inequality (3.33), the growth of

(A(w) ,w) as II wll 1 4 -+ 00 is dominated by the positive term- W ' (Q)

44E211 ~IIWl,4(Q) Hence, -A = A + A is coercive.

_0

(iv) By direct expansion, we have:

45

E; I J(W)lJ(w)[IJ(~) +J(~~ GW2,2 -WZ,Z)ZI,lQ - -

+ (Wl I-WI l)zZ Z+(w2 l-wZ l)zl 2+(wl Z-wl 2)z2 II, , , , , , , , ,li

< I E02 I{ IJ(~) +J(w) Ilv(w-w) Ilv~1Z\I - -

+ il v (~- ~) IIV: I (IZ + wI, 1 + wI, II + Iz + w2, Z + w2, zi

In the steps above, we have used the identity (3.Z3). Now using Holder's

inequality and simplifying the terms, we arrive at (5.IS).

(v) To obtain (S.l6), we observe that

46

where, as before, n w-w Employing again Lemma 3.1 with v = det Vn

and u = (sgn v) . 1 , we introduce a function00

~ E C (n) such that

2E J(~o (':)- ~o (~),~ -~) ~ - \Ia ~ det V~ du

n

Finally, using (3.18) and noting that

(5.ln

<A (w) - A (w),w - w)+(A(w) - A(w) ,w- w)_0 _ _0 _ _ _ _ _ _ _ _ _ (5.18)

we introduce (3.17) and (5.17) into (5.18) and obtain (5.16) with

We remark that similar results hold whenever E2 = 0, E3 = 0,

14412 ZEI > 0 with W' (n) and L (n) replaced by W' (Q) and L (Q),

respectively. For brevity, we shall consider only the cases covered by

the hypotheses of Theorem 5.1.

We also remark that this theorem holds for any nonempty closed

convex set satisfying (5.14) and that set K\I

appearing in this theorem

need not be the set defined in (5.l3). Lemma 5.1 merely shows that sub-

sets K with the desired properties do exist.\I

The following result is due to BREZIS [16].

Theorem 5.Z. Let U be a reflexive Banach space, K a nonempty

closed convex subset of U containing the origin, and let A be a

coercive, hemicontinuous, pseudomonotone operator from K into U~.

Then, for every f t U~, there i!';at least one u E K such that

(A(u),u-v) ~ (f,u-v) V v ('K (5. ZO)

oMoreover, if u ~ K then (A(u), v) = (f, v) for every v E K . 0

47

Returning to our model problem for the operator A of (5.7)

defined on a convex set K eM ,v>O\I \I

In particular, we wish to

find a displacement field w (; K C u(n)\I

such that

<A(w) ,w - v) > <"p f,w-v"-"' 0 ",- - - v v E K

\I(5.21)

We next establish that so long as the material constants can be selected

so as to satisfy certain inequalities, solutions to (5.ZI) always exist.

Theorem 5.3: Let A be the operator defined in (5.7) and let

contained in the set M of (5.10). Further, let\I

E2 > I E31 > 0 or EZ > 0 and E3 ~ 0

} (5.Z2)

EI + E3 - 4E2 > 0

and

K, \I a fixed positive constant, be a nonempty, closed convex subset of\I

w~,4(n)

Then, for any choice of the data-I 4/3p f (; W' (n), there exists at0_ _

least one solution w to problem (5.21). Any such solution, therefore,

has the property that J(w) ~ \I I a.e. in Q

<A (w),v> = (p f,v> V v f- K_ _ _ 0_ _ _ \I

aIf w E K

\Ithen

Proof: Under conditions (5.2Z), the properties listed in

Theorem 5.1 determine that A is bounded, hemicontinuous, coercive, and

satisfies a generalized G~rding inequality of the type studied in [1]

(specifically, (5.16». Hence, under these assumptions, A is coercive,

hemicontinuous, and pseu~omonotone from into its dual,

W-l,4/3(n). Hence, the assertion of the theorem immediately follows

from Theorem 5.2. a

constants E

48

Let us now return to the general problem (5.9). Let us tempor-

arily assume that a solution w(X) exists to (5.9) for which the

deformation is orientation preserving and locally invertible almost

everywhere in n. The solution will obviously depend upon the elastic

(Eo,El,EZ,EZ,E4)' the mass density po(~) , and the

data f(X) ; i.e. there will exist a function W with values in

wl,4(Q) such that_0

w(X) W(p ,f, E ,X)_ 0 _ _ _

Since w ~ Wl,4(n) , the Sobolev embedding theorem guarantees that w_ _0

will be continuous on Q for each choice of Po' f, and E ; it will not,

however, depend continuously on f , in general.

We observe that if A is the singular part of the operator de-_0

fined in (5.8), then

<A (w) ,w) =_0 _ _

=

so that if w is any solution of (5.8) we must have

fn

Z +V'w

J(w) duI

Ea(5.Z3)

where V'w and A is the regular part of our operator

(1.e. A = A - A is defined in (2.15»._0

Since (2+V'w)/J(w) > 0 a.e.

in n , we must have

where

Jn

2 +V'wJ(w) du

49

(5.24)

G(p ,f,E) = <p f,W(p ,f,E,·» -(A(W(p ,f,E,·),a _ _ 0_ _ 0 _ _ _ _ 0 _ _

W (p , f, E,·» + E mesn_ a _ _ a

and G(p ,f,E) is necessarily positive.0__

(5.25)

Theorem 5.4: Let \I be a positive constant such that J(w) ~ \I

a.e. in n for any solution w of (5.9). Then it is necessary that

\I <2E mesnaG(P ,f,E)

0__

(5.26)

Proof: From (5.23) we have

JQ

2 +V'W- du

\I= 2 mesn

\I> Jn

2+V 'WJ (W) du = o

We need to identify a lower bound on the constants \I of this

theorem for each choice of data and material constants.

We will describe one approach to this problem. Let us assume

that solutions w = W(p ,f,E,X) exist to (5.9) for given data p , f,_0___ 0_

and E. Let 0 be a positive number satisfying 0 < 0 < I and let

{x Ei- Q : w = W(p ,f,E,X) ; J(w) < 0 a.e. in Q} •_ _ 0 _ _ _ _-

Let us consider the following condition on all solutions w

lim.!.J (2+ V·w) du = 00

0-+0 0 no

(5.27)

(5.28)

50

i.e., while Qo decreases as 0 decreases, the solutions are such that

f1-£(2 +'V'w) du = 0(0 ),

Q -o£ > 0 .

Theorem 5.5: Let (5.28) hold for all solutions w = W(p ,f,E,X)_ _ 0 _ _ _

of (5.9) for given data p , f, and E. Then there exists a 0* > 0 sucho

that J(w) "::0*a.e. in Q.

Proof: We prove this by contradiction. Let no be as defined in

(5.27). Then from (5.24), we have

-1E G(p ,f,E)00_ _ (5.29)

when the condition (5.28) holds, by decreasing the value of 0 we can

violate (5.29), a contradiction. Therefore, there exists a 0* such that

*J (w) < 0 is not possible in Q. 0

As a slight improvement over (5.28), we remark that the conclusion

Theorem 5.5 also holds when (5.28) is modified to the condition

<Xl > lim.!. J (2 +'V-w) du >0+0 0 Qo -

-1E (p ,f,E)o 0 _ _ (5.30)

The arguments needed to establish the existence of solutions to

(5.9), subject to the assumption (5.28) are now clear: If all solutions

for given data, if they exist, have property (5.28) (or (5.30)), then we

*0< vf < 0 •where~onsider the question of existence in convex sets KvfObviously, by Theorem 5.3, the solutions satisfying (5.28) or (5.29) do

indeed exist and they must be on the interior of Kvf

solutions satisfy the weak equilibrium equations (5.9).

Hence, such

51

We remark that other regularity assmnptions lead to similar conclu-

sions. For example, suppose that solutions ware in W2,4(Q)(\~,4(n) .- _0

Then the derivatives wk are continuous on Q and J(w(X)) is a con-,a

tinuous function of X. Being continuous on a compact set Q, and being

strictly positive on n, J must assume its infimum v > 0f on Q; Le.

there must exist a positive constant vf' depending on E and the data

po!' such that J(~(po'~'~'~)) ~ vf a.e. in Q.

~IDreover,we can significantly weaken this regularity condition.

Recall that there exists a Banach scale of reflextive spaces Ws,p(Q) ,

p > 1, such that if,4(Q) () Wl,4(n) C Ws,4(Q) n ~,4(Q) C wl,4(Q) ,_ _0 _ _0 _0

1 < s < 2, the inclusions being dense and continuous. By the Sobolev

. s 4 0 2embedd~ng theorem, W' (n) C C (n) , Q C 1R, whenever s > 1/2. Thus,

3/2 + £ n nl 4for any £ > 0 and any w ( W (Q) W-' (n) , J (w) is continuous_ _ _0

on n. However, we do not need to have this degree of smoothness globally.

Then, if a solution w is globally in but

(5.31)

then J(w) is continuous on n~ Consequently, there

must then exist a fixed positive constant Vo such that

J(~(Po'!'~'~)IQ) ~ vao

a. e. in Q.

6. Implications of the Strong Ellipticity Condition

Theorems 4.2 and 5.3 demonstrate that among the conditions suffi-

cient to guarantee the existence of a solution to our model problem are

inequalities involving the material constants Eo' El' ..., E4' In

particular, we imposed conditions

52

or (6.1)

In this section, we investigate the role of the strong ellipticity con-

dition in providing a physical basis for inequalities such as (6.1).

We show that, in addition to providing conditions necessary for our

existence theorems to apply to our model problem, the strong ellipticity

condition imposes restrictions on the deformation not necessary for

physically reasonable solutions, As noted earlier, KNOWLES and STERN-

BERG [10] have shown recently that the strong ellipticity condition can

be violated for physically reasonable deformations. We arrive at simi-

lar conclusions by following a quite different approach.

If the strain energy function 0(11,12,13) is twice different-

'Q'8iable with respect to the displacement gradients and if A~ J denotes

the components of the fourth-order tensor

(6.2)

then the strong ellipticity condition demands that V w E U(n) and a.e.

X E Q

Vµ,A:fO (6.3)

For dynamical problems, this requirement is a necessary and sufficient

condition that second- and higher-order discontinuities travel at finite

speeds in the body. Statical interpretations of this condition have

been given by HAYES [8] and WANG and TRUESDELL [3].

53

A

In our model problem of finite plane strain for which a is given

by (Z.5), the components of the tensor AiajB = AiajS(V'W)

given by

of (6.Z) are

Allll Z Z+ lZEZ(l+wl,l) + Z(E3+ZEZ)(1+wZ,Z)

Z Z+ 4EZ(wl,Z +WZ,l)

Al2lZ = Z(El +E3-4EZ) + 4EZEl+WI,1)Z + (l+WZ,Z/]

Z Z+ lZEZWl,Z + Z(E3+ZEZ)WZ,1

AZllZ = AlZZl

AZlZl

= Z(El + E3-4EZ) + 4EZG1+Wl,1)Z + (l+WZ,Z)~

Z Z+ Z(E3 + ZEZ)wl,Z + lZEZwZ,l

54

A2l22

A1l22 , A2212 A 2221 A2l22

A2222 2 22(El +E3-4E2) + 2(E3+2E2)(1+wl,1) + l2E2(1+w2,2)

2 2+ 4E2(wl,Z+w2,1)

(6.4)

where, for simplicity in expression, we have not displayed the arguments

of the functional AiajB('ilw) . Similar expressions are obtained using

the more general energy functional (6.3), but, for our present purposes,

it suffices to study the algebraically simpler inequalities arising from

(2.5).

'a'SThe following conditions on the functions A~ J can be obtained

directly from the strong ellipticity condition (6.3):

Allll('ilw) > o , A2222 ('ilw) > 0

Al2l2 ('ilw) > o , A2l2l('ilw) > 0-

(Al112('ilw»2 < Alill (Vw)A12l2 ('ilw)- -(A2l22('ilw»2 < AZ12l ('ilw)A2222('ilw)- -

(A1222 ('ilw»2 < A12l2 ('ilw)A2222('ilw)- -(All2l ('ilw))2 < Alill ('ilw)A2l2l('ilw)

v w E U(n) (6.5)

55

These conditions are necessary but not sufficient for the strong

ellipticity condition to be satisfied. The first four of the above in-

equalities yield the following conditions on the constants:

E > 02-

(6.6)

The remaining four inequalities in (6.5) need not hold if (6.6) do not

hold, but they are automatically satisfied whenever the constants satisfy

them. We observe that not all of EZ' El + E3 - 4EZ' E3 + 2EZ can vanish

simultaneously.

In addition to the inequalities (6.5), we can also easily derive

the following sufficient condition for the strong ellipticity condition

from (6.3):

V vector A:f 0 (6.7)

We will study some implications of (6.7). For this purpose, we

will appropriately denote the coefficients of

56

>.; as Pl('il~),P2('il~),P3('il~),P4('il~),and P5('il~),respectively. Then

(6.7) can be rewritten in the form

P (i) (6.8)

where p. = p, ('ilW), 1 < j < 5. We next make use of Young I s inequalityJ J - - -

(6.9)

where £ > 0 is arbitrary, and write (6.8) as

p (>') >

V >. I- 0

for arbitrary £1'£2 > O. We choose £1 and £z so that

(6.10)

(i)

(it)

PI >Ip21

E:l -Z

>IP4L

P5 £z-Z

or 1 1 I I 2(b) (P3 -2£1 pzl -iE P4) <1 2

(6.11)

Condition (iii) is then equivalent to

2 2

1(iii) (a)P2 P4

P3 > -+-- 4Pl 4PS

or (b) o < < J

57

(6.lZ)

Clearly, inequalities (6.12) describe restrictions on the deforma-

tion, since Pj , (1~j~5), are functions of the components of 'ilw •

We may, therefore, easily concieve of physically reasonable solutions to

our model problem which do not satisfy (6.12) and, therefore, violate the

strong ellipticity condition. This observation suggests that the appli-

cation of the strong ellipticity condition as a constitutive inequality

for static problems of nonlinear elasticity rules out families of solu-

tions which have important physical significance.

7. Concluding Comments

The theory that we have developed here demonstrates that the

theory of pseudomonotone operators is applicable to a wider class of

nonlinear boundary-value problems than was heretofore believed. Be-

cause of its constructive nature, we have also shown that the theory

can be used as a basis for determining restrictions on the form of the

strain energy function in order that physically reasonable solutions

exist. Importantly, these restrictions do not arise from the imposi-

tion of the strong ellipticity condition, which we show can be violated

by solutions of our model problem. However, if we impose the strong

ellipticity condition, then we obtain restrictions on the form of the

energy which are sufficient to guarantee the existence of solutions.

We emphasize that multi-valued solutions are possible in our

theory. Indeed, the form of the G~rding inequality does shed some

58

light on primary bifurcation phenomena. For example, under reasonable

conditions on the material constants, a typical G~rding type inequality

will assume the form:

<A(w) -A(w),w-w) >

where C,Y(}.J)> 0, p..:::q, and p'"= p/(p-l). Obviously, the

parameter Y depends on the bound µ of the vectors wand w

which, in turn, will be bounded above by some appropriate bound on

Whenever the termthe data,

p'"

term Y(µ)II,:,-~IIIl(n) ,

dominates the

the operator A will be strictly monotone

and unique solutions are guaranteed. We can expect this to be the case

under "small" deformations; e.g., perturbations of the solution about

the origin. However, it is clear that as the magnitude of the loads are

increased one can encounter cases in which the righthand side of the above

inequality is negative. The operator is then no longer monotone, and

multi-valued solutions may exist.

The possibility of determining conditions on the data which will

produce this critical case is intriguing and is obviously associated with

primary bifurcation of the solution. To complete such an analysis, it

would be necessary to obtain an a priori estimate on the bound µ and

this is tantamount to the development of a regularity theory. Since, to

date, very little progress has been made toward establishing a regularity

theory for nonlinear boundary-value problems of any type, the development

of such a theory poses a challenging prospect for future research.

59

We also point out that our results are restricted to the Dirichlet

problem for the nonlinear operator ~. The Neumann problem poses a

challenging and physically interesting class of problems. To extend our

results to the Neumann problem, which we hope to do in the near future,

we must make use of trace theorems defined on reflexive Banach spaces.

Some of these theorems do exist for Sobolev spaces, and a study of methods

of putting them to use in determining more general forms of the Ggrding

inequality is underway. It is clear that for the Neumann problem it is

necessary to add boundary integrals into the variational formulation

(2.15), and that the data appearing in these integrals will likely effect

the parameter µ in the Ggrding inequality. An interesting class of

Neumann problems which we hope to consider will involve cases in which

the boundary tractions will be given in terms of a load parameter which

can supposedly be varied until the operator lo.ses its monotonicity, as

described above. This, then, would, again, be an indication of primary

bifurcation.

We note that all of our results apply to compressible hyper-elas-

tic materials. Extension of our method to elastic materials not charac-

terized by a strain energy function is trivial, since we made little use

of the fact that the operator A was derived from a potential. However,

the extension of our result to incompressible materials involves a number

of additional complications, not the least of which is the problem of

accommodating a hydrostatic pressure in the analysis. Obviously, the

trial space will, in this case, be a product space in which the solution

iA a pair of fields--a displacement field and a pressure field. We still

feel that essentially the same ingredients that we have developed here

60

will be needed for a complete existence theory for incompressible ma-

terials. However, the problem of establishing specific properties of the

operators should be more involved.

Finally, we again note that the conditions assumed in our existence

theorems effectively guarantee that the operator in our model problem is

pseudomonotone. This is an important property, because it immediately

makes available to us a wide collection of results that have already been

developed for pseudomonotone operators. In particular, once the Neumann

problem is successfully formulated, it should not be difficult to extend

our theory to unilateral problems, i.e., contact problems, in finite

elasticity. Indeed, variants of the theory of pseudomonotone operators

may also be instrumental in handling incompressible materials.

Acknowledgment

We wish to express our gratitude to our colleagues, Professors

R. E. Showalter and P. G, Ciarlet and Dr. N. Kikuchi, for many helpful

discussions of the subject reported herein; in particular, Professor

Ciarlet showed us the way to develop the simple proof given for

Lemma 3.1. The results communicated in this paper were obtained in

the course of an investigation supported by the National Science Founda-

tion under Grant NSF-ENG 75-07846.

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