NONLINEAR PARTIAL DIFFERENTIAL EQUATIONSIN ENGINEERING AND APPLIED SCIENCE

Proceedings of a Conference Sponsored By

Office of Naval Research

Held At

University of Rhode IslandKingston, Rhode Island

edited by

ROBERT L STERNBERGOffice 01 Naval Research

Boston. Massachusetts

ANTHONY J. KALINOWSKINaval Underwater Systems CenterNew London. Connecticut

JOHN S. PAPADAKISUniversity oi Rhode Island

I\ingston. Rhode Island

COPYRIGHT © 1980 by MARCEL DEKKER, INC.

MARCEL DEKKER, INC. New York and Basel

ON THE EXISTENCE OF HYDROSTATIC PRESSURE IN REGULAR

FINITE DEFORMATIONS OF INCOMPRESSIBLE

HYPERELASTIC SOLIDS

P. LeTallec and J. T. Oden

University of Texas at AustinAustin, Texas

§l. INTRODUCTION

In this article, we establish conditions for the exis-

tence of hydrostatic pressures in a class of hyperelastic

incompressible materials subjected to finite deformations.

Although our approach is quite general, we restrict ourselves

to materials for which the stored energy function is defined

on an appropriate Sobolev space (Wl,p(Q))n; these include,

for example, the Mooney-Rivlin materials. Among the implica-

tions of our results is that hydrostatic pressures may exist

only in a very weak sense if the minimizers u of the total

energy are irregular (e.g., u E (W1,p(n)n -- (W2,p(Q»)n), but

if ~ is sufficiently smooth (e.g., ~ E (W2,p(Q))n), then

pressures p exist which can be characterized so that (~,p) is

a solution of the weak equilibrium equations.

Theorems on the existence of minimizers of the total po-

tential energy of hyperelastic bodies were recently advanced

by Ball [1] who introduced the notion of polyconvex functions.

Owing to the complex structure of the spaces in which these

minimizers exist, Ball was unable to show that these mini-

mizers could be characterized as weak solutions of the equa-

tions of elastostatics. For incompressible materials, the

problem of characterization also requires the proof of the

existence of a sufficiently smooth hydrostatic pressure.

Related work for compressible hyperelastic materials has been

1

2 P. LeTallec and J. T. Oden

done by Oden and Kikuchi [5] who also provided conditions for

the characterization of minimizers as weak solutions of the

equilibrium equations. Rostamian [6] has employed the results

of Ball [1] in the development of a penalty method for treat-

ing certain constrained problems in nonlinear elasticity.

§2. SOME PRELIMINARIES

We consider the deformation of an incompressible hyper-

elastic body relative to a fixed reference configuration.

Equilibrium problems for such bodies are often approached as

problems of finding minima of the total potential energy

subject to the incompressibility constraint

detV~ = 1

(2.l)

(2.2)

Here ~ = ~(~) denotes the position of particle ~ in the cur-

rent (deformed) configuration, the particles are labeled by

their positions ~ = (xl' x2' ..., xn) relative to a fixed

rectangular Cartesian coordinate system in the reference con-

figuration which is the closure of a bounded open subset n of

Rn (with n typically l, 2, or 3), cris the strain energy func-

tion, and -f(~) is th~ potential energy of the external forces.

In (2.l), n(~) is the value of the total potential energy pro-

duced by a kinematically admissible motion ~; cris an objec-

tive, real-valued, measurable, differentiable function of the

material gradient Vu of u aVu)i = aui/ax 1 5 i, a ~ n).- - - a a

Typically, ~ takes on a prescribed value ~ on a portion anlof the boundary an of n, surface tractions ~, measured per

unit area in the reference configuration, are prescribed on

an2 (an = aITl U aQ2' anl n an2 = ~), and f is a linear func-

tional of the form

f (~) f Pof'~ dx + f E'~ dsn aQ2

(2.3)

where Po is the mass density in the reference configuration

and £ is the body force per unit mass. If the material is

HYDROSTATIC PRESSURE IN FINITE DEFORMATIONS 3

not homogeneous in its reference configuration then, crmay

also depend explicitly on ~.

We shall confine our attention here to Mooney-Rivlin

materials, and will give an analysis of more general cases in

a later paper. For Mooney-Rivlin materials, cr is of the form

where Cl and C2 are material constants,

I 2 Tv~1 = trace (v~ v~)~(~) = adj v~, I~(~) 12 = trace [(adj V~)T adj V~)

(2.4)

(2.5)

and adj V~ denotes the adjugate of V~ (i.e., the transpose of

the matrix of cofactors of V~).

The classical problem then reduces to finding minimizers

of the functional

(2.6)

n being a constant, subject to the constraint (2.2). Theoscalar-valued function p which appears as a Lagrange multi-

plier associated with the constraint is, physically, the hy-

drostatic pressure. A knowledge of the pressure is necessary

to determine the stress in the body. Indeed, the Piola-

Kirchhoff stress tensor 9 is given by

(2.7)

Thus, the mere determination of a minimizer of n is insuffi-

cient insofar as the stress analysis of the body is concerned;

the pressure p must also be determined.

§3. EXISTENCE OF MINIMIZERS

For completeness, we review briefly the ideas underlying

the proof of the existence of minimizers of n. Although our

approach is slightly different, the principal ideas follow

those of Ball [l]. We use the well-known generalized

4 P. LeTallec and J. T. Oden

Weierstrass theorem (cf. Vainberg [8]):

Theorem If n : K + R is a proper, coercive, weakly lower

semicontinuous functional defined on a nonempty weakly sequen-tially closed subset K of a reflexive Banach space, then n isbounded below on K and attains its minimum on K.

In the present case, we introduce the product space

(3.l)

equipped with the norm

(3.2)

where

1I~IIL2 = In1V~12 dx, IItlll~,2 = In trace j:!T!:Jdx (3.3)

We assume mes(ani) > 0, i = l, 2. The space 2 is a reflexiveBanach space when endowed with the norm (3.2).

Let

K = {(~,B) E g : B = adj V~, det V~in n, ~ = ~ on anl}

1 a.e.

(3.4)

and consider the functional n K + :R defined by

+ no (3.5)

no being a constant. Clearly, for Cl, C2 > 0,

and

Thus TI is coercive and weakly lower semicontinuous on all of U.The verification of the remaining conditions of the

theorem for TI rests on the following lemma due to Ball [1]:

HYDROSTATIC PRESSURE IN FINITE DEFOID1ATIONS

Lemma 3.l

(i) If u E ~1,2(Q), then ~(~) E ~1(n) and

i+2 i+l )(u u 'a+1 'a+2

i+2 Hl )- (u u 'a+2 'a+1

5

(3.6)

(ii)

in V' W), 1 ~ i, a ~ 3

If u E Wl,2(Q) and A(u) E L2(n), then- - 1 - - -

det Vu E L (0) and

det VI:!3 1 1I (u (A (u)) ), in V I (Q)

a=l - - a a(3.7)

ax ;Here commas denote differentiation with respect toi i Clu, = au lax .a

We need only show that K is weakly sequentially closed.

Let {(u ,H )} be a sequence from K converging weakly to-n -n(~,~) E Q. By (i) of Lemma 3.1 and the definition of K,H = adj Vu and for ~ E V(Q),-n -n

H2 i+l )- (un un,a+l'~'a+2 V

< i+2 i+l )+ un un,a+2'~'a+l V

where <','>V denotes quality pa1r1ng on V' (n) x V(n). Thus,

taking the limit as n + ~, we have

f i+2 i+l i+2 i+ldx = - n(U u'a+l~'a+2 - u u'a+2~'a+l) dx

= «~(~))~,~>v = fn(~(~))~~ dx

Hence, ~ = adj V~ and, therefore, adj Vu E L2(Q).

Next, using (ii) of Lemma 3.1,

Jn~ dx

6 P. LeTallec and J. T. Oden

-.

for all ~ E V(Q), so that again using Lemma 3.l (ii), we have

In det V~n~ dx + (det v~,~)v = IQ

det v~~ dx = IQ~dx

Hence, det V~ = 1 in Ll(Q).

According to the theorem, we have established the exis-

tence of a point (u ,H ) E U such that-0 -0

Let

Then

li(u ,II ) ~ n(u,H)~o -0 - -v (~,!!) E K

n(u )-0

li(u ,H ) ~ n(u,H)-0 -0 - -

v ~ E K

where K is the set equivalent to K defined by

K = {~ E wl,2(n) : adj V~ E t-2(Q)

det V'~ = 1 a.e. in n, ~ = ~ on aQl}

Thus, u is a minimizer of n on K.-0

§4. SOME ASSUMPTIONS AND PRELIMINARY RESULTS

(3.8)

Having established the existence of a minimizer ~ of the

total potential energy, we will now introduce some assumptions

concerning its regularity which, we will show, are sufficient

to prove the existence of a hydrostatic pressure p. We recall

that the motion ~ is effectively a map of the initial config-

uration of the body into its current configuration. Let

n c ~3 denote the image of U under the motion~. Let ~ be a

minimizer of the functional n of (2.6) on the set K of (3.8).

For the present discussion, we consider a bounded domain n inn - -R with a smooth boundary aQ = anl U aQ2' aUl n aU2 = ~,

mes an2 > 0, and a motion H belonging to the set of admissible

motions

HYDROSTATIC PRESSURE IN FINITE DEFORMATIONS 7

K = {~ E (Wl,r(n))n : adj V~ E (Lr(n))nxn,

~ = ~ on ani' det v~ = 1 a.e. in n} (4.l)

and an energy function n : K + R of the form (2.6). Our prin-

cipal assumption is

H.l There exists a function ~ which minimizes n on K

such that

(i)

(ii)

(iii)

~ is a bijection from n onto QQ is a domain in Rn with boundary an of class CS

,

s a nonnegative integer

u E (Ws,p(n))n, s > ~ + 1p

Under these assumptions, it is possible to construct a

tangent set at K in~. Then ~ can be characterized as a solu-

tion of a weak equation of equilibrium. Physically, our tech-

nique can be interpreted as establishing the existence of a

pressure in an Eulerian frame in the current configuration,

wherein the incompressibility condition is then the linear

constraint div y = 0, followed by application of results of

Temam [7] for the analysis of this constraint, after which we-l -then apply the smooth map ~ from n to n so as to make con-

clusions on the existence of a pressure defined on n.

The essential tools in our analysis are summarized in

the following lemmas:

Lemma 4.l (cf. Cantor [2]) Let hypothesis H.l hold.

Then(i)

(ii)

(iii)

For r ~ sand s > ~, the map $ : ~,p(Q) x ws,p(n)p

~ ~,p(n) given by the product

is continuous.

~ is a Cl-diffeomorphisn

The map $ (f) = f 0 u is continuous from~ - - -(Wr,p(n))n into (~,p(n))n, r ~ s

or from

(Wl,q(n)n into (Wl,q(n»)n, 1 < q < 00

8 P. LeTallec and J. T. Oden

(iii) is an-l s P - n~ E (W ' (Q)) and the map

isomorphism whose inverse is

(iv) lji ofu

lji--l'u

We will also need the following result due to Temam [7].

Lemma 4.2 Let

H = {y E (Ws,p(i1))n : y = a on anl}o - -

Then the divergence operator div is a surjection with split-- s-l p -ting kernel from Ha onto W ' (0). Moreover, if

- - s-l p -Hal = {~ E Ho : ~~ = Vq, q E W ' (n), ~ = A~l on an}

where ~l is a function in (Ws-l/p,p(aQ))n such that Pl = a on

an1 and fanPl'~ ds = l, then

HO = HOl $ Ker (div)

These preliminaries allow us to go directly to the con-

struction of a tangent space in ~ at K in (Ws,p(n))n. We

first introduce by definition the notation

[y,~] §ef trace [(adj V~)Vy]

and

The next result establishes that any ¥ in Ha with the property

that [~,~] = 0 belongs to the tangent set in u at K.

Lemma 4.3 Let H.l hold. Then, for every ¥ E HO such

that [~,¥] = 0 there exists a vector-valued function ~ from a

neighborhood (-a,a) of the origin in R into HO such that

1jJ(0) = 0, 1jJ is continuous, and- -~ + t(¥ + ~(t)) E K V t E (-a,a).

Space limitations do not permit us to give the lengthy

proofs of these results here, so we plan to present complete

details in a companion paper (LeTallec and Oden [4]). How-

ever, we remark that Lemma 4.3 is crucial to our analysis and

that our proof employs the implicit function theorem and the

techniques of Crandall and Rabinowitz [3].

HYDROSTATIC PRESSURE IN FINITE DEFORMATIONS

§5. THE HYDROSTATIC PRESSURE FIELD P

In addition to H.l, we now assume

H.2 The total potential energy functional n is Frechetdifferentiable on (Wl,q(n))n, q > l.

Let

H = {l E (Wl,q(n))n : ~ = Q on anl

)

Then HO is densely and continuously embedded in H. We have:

Theorem 5.l Let H.l and H.2 hold. Then, for everyy E H such that [u,y] = 0 we have- -

Dn (!:!) • ~ = 0

where Dn(u) • y is the Frechet derivative at the minimizer u- -in direction l'

9

ProofLemma 4.3.

Let ~ E HO satisfy [~,~) = 0 and ~(t) be as inThen ~ + tel + ~(t)) E K for t E (-a,a) for some

a. Since u minimizes n, where t ~ 0,

-lt [n(~ + tel + ~(t))) - n(~)] ~ 0

By definition of the Frechet derivative, this leads to

Vn(~) • (l + ~(t)) + e(t)lI~+ !(t))lIl,q ~ 0

limlle(t)lIl = 0t+O ,q

Letting t ~ 0 so that IIIj1(t)1I and, hence, IIIj1(t)lIl~ 0- s,p _,qgives

Dn (u) • l ~0

Replacing y with -y gives the result in H. But since H is- - 0 0

dense in Hand Dn is continuous on H, the property extends to

any l in H such that [~'l] = O.

We finally come to the proof of the existence of apressure:

Theorem 5.2 Let H.l, H.2 hold and let 1jI* denote theutranspose of the map IjI defined in Lemma 4.l (iii). Then,u

lO P. LeTallec and J. T. Oden

for every y E H, there exists a uniquely defined function

P E Lq' (n)(l/q + llq' = l) such that

1jI*(Dn(u))~ - '!.= P div '!. (5.l)

Proof The map 1jI is an isomorphism from H onto H. Since- u

Du(u) E H', 1jI*(Dn(u))-EH' and~

1jI*(Vn (y)) • y = On (u) . you~ - - - -

Thus, according to Theorem 5.l, if '!.E Hand div '!.

have

[~, ¥ 0 ~] = (div ¥) 0 ~ = 0

0, we

which implies that On(u)

Next, let'!.0 u lj/*(Vn(u))

u - '!. o.

div ¥ = 9 for

where Hl denotes the H-closure of HOl defined in Lemma 4.2,

and denote

F(g) w* (Vn (u)) • y~ - -

According to Lemma 4.2 (which is applicable for s = landp = q in this case), the map g + y is an isomorphism and so

F is linear and continuous on Lq(~). Hence, there is a

unique p E Lq' (n) such that, for every g E Lq(Q), F(g) =

IQ pg dx. By the construction of F,

1jI*(Vn(u»)• y = F(div y) = f- P div Y dx Ii '!.E Hlu - - - n -

Let ¥ E H. According to Lemma 4.2, ¥ can be expressed

in the form

¥='ll+Y2; 'll E Hl div '!.2= 0

HYDROSTATIC PRESSURE IN FINITE DEFORMATIONS

Then

11

lj/*(Vn (u))u -

y = lj/*(Vn (u)) . ¥l + "'~(V1T(~)) . Y_ u - _2

lj/*(Vn(u)) • y + 0U - -l

1- p div Yl dx = 1- P div Y dxQ - 0 -

Theorem 5.3 Let the conditions of Theorem 5.2 hold.

Then the hydrostatic pressure p defined by

p = -p a u E L q' (n)

exists and satisfies

in H'

This follows immediately from (5.l) after applying (lj/*)-l.~

Our final remarks concern the characterization of p.Suppose V1T(~) is of the form

Vn(~) = Al + A2

q' nAl E (L W) , A2

E (Wl-l/q',q'(a02))n (5.2)

Then we easily can show that for y E H,

f P div ~ dx = Al . ~ + A2 . y, A.~-l

A. 0 u~

Taking ~ E (V(Q))n and integrating by parts twice yields

- - 3 - q' - nVp = -Al

in (V'(n) ; hence, Vp E (L (n»)

- 1 q' - 3 -Thus, P E (W' (0)) so that we can also take ~ E H for an

integration by parts. Then we obtain

J p yan2

so that p

¥ ds

l2

Summarizing, we have

P. LeTallec and J. T. Oden

Theorem 5.4 In addition to the hypotheses of Theorem

5.2, let (5.2) hold. Then the hydrostatic pressure p is

characterized by

p = -p 0 ~-l, P E wl,q' (n), 6pp • n = A on an

- 2 2

ACKNOWLEDGMENT

-div Al,

(5.3)

The work reported here was developed during the course

of a research project supported by Grant NSF-ENG-75-07846

from the National Science Foundation. We also wish to thank

Professor M. Cantor for advice and useful suggestions.

REFERENCES

1. Ball, J. M., Convexity Conditions and Existence Theorems

in Nonlinear Elasticity, Archive for Rational Mechanics

and Analysis, Vol. 63, No.4, 337-403 (l977).

2. Cantor, M., Perfect Fluid Flows over Rn with Asymptotic

Conditions, Journal of Functional Analysis, Vol. l8,

73-84 (l975).

3. Crandall, M. C., and Rabinowitz, P. H., Bifurcation from

a Simple Eigenvalue, Journal of Functional Analysis, Vol.

8, 321-340 (l971).

4. LeTallec, P., and Oden, J. T., On the Characterization of

Minimizers of the Energy in Incompressible Finite Elas-

ticity (in preparation).

5. aden, J. T., and Kikuchi, N., Existence Theory for a Class

of Problems in Nonlinear Elasticity: Finite Plane Strain

of a Compressible Hyperelastic Body, TICOM Report 78,

Austin, 1978.

6. Rostamian, R., Internal Constraints in Boundary-Value

Problems of Continuum Mechanics, Indiana University Mathe-

matics Journal, Vol. 27, No.4, 637-656 (l978).

• A

HYDROSTATIC PRESSURE IN FINITE DEFORMATIONS

7. Temam, R., Navier Stokes Equations, North Holland,

Amsterdam, 1978.

13

8. Vainberg, M. M., Variational Method and Method of Mono-

tone Operators in the Theory of Nonlinear Equations,

John Wiley and Sons, New York, 1973.