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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.