An Existence Theorem for Slightly Compressible Materials ...caa.epfl.ch/publications/17-Charrier-Dac-Hanouzet-Laborde1988.pdf · 1988 Society for Industrial andApplied Mathematics

SlAM J. MATH. ANAL.Vol. 19, No. 1, January 1988

1988 Society for Industrial and Applied Mathematics005

AN EXISTENCE THEOREM FOR SLIGHTLY COMPRESSIBLE MATERIALSIN NONLINEAR ELASTICITY*

P. CHARRIER, B. DACOROGNAt, B. HANOUZETt AND P. LABORDEf

Abstract. We show that if a hyperelastic material is slightly compressible, in this case the stored energyfunction is a function of the "modified invariants," then the existence results of Ball are still valid. We thenstudy the behavior of the solutions when the compressibility tends to zero.

Key words, nonlinear elasticity, slightly compressible materials, polyconvexity, coercivity

AMS(MOS) subject classifications. 49, 73

Introduction. In an important article Ball 1 has established an existence theoremfor nonlinear hyperelastic and compressible materials. Roughly speaking, it is shownthat the problem

(P) inf{I(u)=f. W(Vu(x))dx, u=uoonOll,UE WI’p(-; R3)}admits a solution; where ’C2R is the reference configuration, 0- is part of theboundary 012, u:12- and VuEM3+ (i.e., Vu is a 3x3 matrix with detVu>0),W: M3+ - is the stored energy function which is assumed to be coercive and polyconvex(for precise definitions see the next section).

In particular, if the material is isotropic then it is well known that W can bewritten as

(0.1) W(F) dP( i(FTF)where : (+)3.+ and i(F) denotes the principal invariants of F, i.e., i(F)=(il(F), i2(F), i3(F))= (tr F, tr (adj F), det F).

It is the aim of this article to show that the theorem of Ball still holds, under someextra hypotheses, if one replaces the principal invariants by the so-called "modifiedinvariants" i* (cf. [7], [8]) defined as

(0.2) i*(F) (i*l i* (ili- 1/3, i2i2/3).

Let us be more precise and first explain the importance ofthe "modified invariants."In practice it is a hard problem to determine experimentally the function W (or , ifthe material is isotropic) and for slightly compressible materials, one is led to proceedindirectly. One way of determining W may be as follows: first we make some experi-ments (such as simple traction, biaxial traction, simple shear, etc.) for which we canassume that the volume changes are negligible,

(0.3) W*(F) W(F)ldetF=or for isotropic material

*(il, i2) (il, ia, 1).

* Received by the editors November 4, 1985; accepted for publication (in revised form) December 15,1986.

" Universit6 de Bordeaux I, U.E.R. de Math6matiques et Informatique, 351, Cours de la Lib6ration,33405 Talence Cedex, France.

$ Ecole Polytechnique F6d6rale de Lausanne, D6partement de Math6matiques, MA (Ecublens) CH-1015Lausanne, Switzerland.

70

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

EXISTENCE THEOREM IN NONLINEAR ELASTICITY 71

However, since the material is actually slightly compressible, we assume that a

strong pressure p leads to change of volumes and postulates that

(0.4) p(Vu) g(det Vu).

It is then easy to show (cf. Proposition 1) that W must be in the following form:det F

(0.5) W(F) W*((det F)-I/3F)+ g(v) dvdl

or if the material is isotropic

(0.5’) (il, i2, i3) d*( i*l i’2) + g( v) dr.dl

However it is not obvious that such functions W (or ) remain coercive andpolyconvex (the two crucial hypotheses in Bali’s theorem). And indeed for the Mooney-Rivlin stored energy function W*(F) lFI= //31adj El, the polyconvexity is actuallylost.

In the second section of this article we show (Theorem 5) that for a large classof materials including some of the Ogden materials (but not Mooney-Rivlin materials),W is coercive and polyconvex. Then, using the same techniques as in Ball [1], weshow existence of minima when W satisfies (0.5).

In the last section we study the convergence ofthe minima when the compressibilitytends to zero and, in particular, we show convergence to the incompressible case.

Finally, we should mention that this kind of rheology has been used for engineeringpurposes and for numerical computation of rubberlike materials; for more details werefer to Charrier and Pouyot [2] and Pouyot [9].

The article is divided as follows: 1. Stored energy function of hyperelastic andslightly compressible materials; 2. An existence theorem; 3. Convergence to theincompressible case.

1. Stored energy function of hyperelastic and slightly compressible materials.1.1. Notation. We start by recalling the usual framework of nonlinear elasticity

(for references, see [10] or [3] for instance).We denote by M the set of 3 x 3 matrices and by

(1.1) M3+ {A M3: det A > 0}.

We endow the space M with the scalar product A. B =tr (ABT) and we denoteby IAI the associated norm. We also denote by i(A) the principal invariants of A M3,i.e.,

(1.2) i(A) (il(A), i2(A), i3(A)) (tr A, tr (adj A), det A)

where adj A denotes the transpose of the matrix of cofactors of A.Let f c 3 be the reference configuration (f is a bounded open set), u" 1)3 a

deformation of the body satisfying det V u > 0, i.e., F Vu M3+.Let T(u(x)) be the Cauchy stress tensor defined on the deformed configuration

u(f) and let S(x) be the first Piola-Kirchhoff stress tensor defined as follows:

(1.3) S(x) det (Vu(x))T(u(x))[(Vu(x))7"]-1.We also assume that the material under consideration is hyperelastic (and

homogeneous), i.e., there exists W: M3++ the stored energy function such that

(1.4) S= W’(Vu)

where W’ (0 W/OFia)l<_i,a<__3.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

72 CHARRIER, DACOROGNA, HANOUZET AND LABORDE

The pressure is then equal to

(1.5) p --1/2 tr T-- 1/2(det XTu)- W’(Vu) Vu.

If the material is also isotropic then W assumes the following form

(1.6) W(Vu)

where : (R/)3_/.As we mentioned in the Introduction, one way to construct the function (or

more generally W) is to determine at first

(1.7) *(il, i2) (il, i2, 1)

(or W*(F) W(F)ldetF=,).In the second step we postulate that pressure changes induce volume changes that

are related in the following way:

(1.8) p(Vu) g(det Vu)

where g: + +.In Proposition 1 we show that relations (1.7) and (1.8) determine completely the

function (resp. W), once * (resp. W*) and g are prescribed. And in fact we find

(1.9) W(F) W*((det F)-/3F)+ G(det F),

(1.9’) (i, i2, i3)--t*(i;1/3i, i;2/3i2)-FG(/i3)where G is a primitive of g, i.e.,

I(1.10) G(x)= g(z) dz.

The identity (1.9’) leads to the introduction of the so-called "modified invariants"(see Ogden [6], Penn [8])

(1.11) i* =- i/3il and i’2 i/3i2.1.2. Determination of the stored energy function. We can now state the proposition

which shows (1.9) and (1.9’).PROPOSITION 1o Part 1. Let W: M3+ + be differentiable and p: M3+ + be such

that

(1.12) W’(F). F= 3 det Fp(F).

Let g Loc(O, +oo). The following conditions are then equivalent"(i) For every F M3+,

(1.13) p(F) g(det F).

(ii) There exists W*" {F M3+; det F 1}-+ such thatdet F

(1.14) W(F) W*((det F)-I/3F)+ g(z) dz.

Part 2. Furthermore if satisfies (1.6) then (1.13) is equivalent to the fact thatthere exists *: x+ such that

(1.14’) @(i, i2, i3)= *(il*, i’2)+ g(z) dzdl

where i*l i* satisfy (1.11).

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Proof. Part 1. First suppose (ii). If W satisfies (1.14) then

dd--t W(tF)lt= 3 det Fg(det F)

and since (1.12) holds, we indeed have (1.13).Conversely, assume that (i) holds and let

det F

(.) h()= w()- (z) az.dl

We then have

h’(F). F= W’(F). F-(adj F)rFg(det F)

=3 det F(p(F)-g(det F)) 0.

Therefore h is homogeneous of degree 0. Let W* be the restriction of h on {FM3" det F 1}; then we immediately have (1.14). This completes Part 1.

Part 2. This is proved exactly in the same way. [3

1.3. Choice of a rheology. We first start with a definition introduced by Ball [1].DEFINITION. A function W: M3-+ is said to be polyconvex if there exists

W: M M [ -+ convex, such that

(1.16) W(F)- w(F, adj F, det F).

In the incompressible case a class of rheologies (i.e. stored energy functions) forisotropic materials is that of Ogden [6], where

M N

(1.17) W*(F)= Y a, tr(C,/-)+ Y, bitr ((adj C) fli/2)i=1 i=1

where C FTF. If ai, b>0 and a,fl >- 1 then W* defined as in (1.17) is polyconvex(cf. Ball [1]).

One can choose the coefficients in (1.17) such that W* interpolate, with a reason-ably good approximation, the experimental measurements.

We will also, marginally, consider W* satisfyingM N

(1.17’) W*(F)= Y a, lF[,+ biladj FI,i=1 i=1

with a, b >- 0 and a, fl => 1, which is obviously polyconvex.Note also that if w*" M3M / in (1.17) or (1.17’) is such that

(1.18) W*(F) w*(F, adj F)

then we have the following coercivity condition

(1.19) w*(F,) K(IFI /lalwhere K >0 is a constant and a max __< i_<M {ai} and/3 maxl_j_<v {/3j}.

We now discuss the compressibility law. As we mentioned, we postulate that forslightly compressible materials the pressure p satisfies

(1.20) p(Vu) g(det Vu).

We assume that if

(1.21) G(x) g(z) dz

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


then G satisfies

G" (0, +oo) [0, +oo) is convex,

lim G(x)= +oo,x-0

(1.22)G(x)>-Cx’forsome C>0 and y>l, x large enough,

G(x)=0 if and only ifx=l.

A good example of compressibility law is, for every F e M+,

(1.23) p(F) g(det F)=e-e det F-det--which satisfies (1.22). In particular, if det F is close to 1 we have

l(detF 1)=1(oo 1)P"’e e

where Po is the initial density and p is the density of the deformed body. This is usuallyconsidered as reasonable for numerous slightly compressible materials.

Therefore, consider a rheology of the form

(1.24) W(F) W*(F) + G(det F)

where W* and G satisfy (1.17) and (1.22). We could also consider more general W*(cf. (2.6) below). We seek deformations u such that u Uo on 0f/1 (where 0f/1 = 0f)and which minimize the potential energy of the system (we suppose for simplicity thatthe material is homogeneous and that there are no external forces), i.e.,

(1.25) I(v) f, W(Vv(x)) dx

among all kinematically admissible deformations v which belong to

A,r= {v Wl"(O, R3), adj Vv (L(f))9, det Vv LV(f/)

det Vv > 0 a.e. and u Uo on 0f/l}.

The methods and results of Ball [1] can then be applied as follows.TrEOREM (Ball). Let ce>3/2, /3>1, 1/c+1//3<4/3 and y>l. If there exists

At such that I() < oo then there exists u At so that

I(u) inf {I(v)" v Av}.Remark 2. In the definition ofAtr, if we assume c -> 2, then there is no ambiguity

in the definition of adj Vu since it is an L function. However if 3/2 < a <2, adj Vuis extended as a distribution by continuity denoted Adj 7u by Ball [1]. A similarremark is applicable to det 7 u.

It is the aim ofthis article to extend Ball’s result to the case ofa slightly compressiblematerial which satisfy (of. Proposition 1)

(1.26) W(F) W*((det F)-I/3F)+ G(det F)

where W* and G satisfy (1.17) and (1.22).The theorem of Ball cannot be applied directly since, even if W* is polyconvex

and coercive when det F 1, it is not, in general, the case for W satisfying (1.27). Thiswill be investigated in the next section.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


We end this section with some remarks.Remark 3. (i) A simple example of stored energy function is the so-called

Mooney-Rivlin rheology, i.e.,

(1.27) W*(F) a tr C + b tr (adj C)= alFl-/ bladj El=.We shall see in the next section that, even though (1.27) is polyconvex the function

(1.28) W(F)=a(det F)-:/3lFl-+b(det F)-4/3ladj F[2+ G(det F)

is not polyconvex.(ii) It is known (cf. Gurtin [4]) that in an isotropic material the constraint in the

reference configuration is a pressure. The choice of a law of compressibility satisfyingG’(1) g(1) 0 implies that the reference configuration is free of constraint and thusis a "natural" configuration.

2. An existence theorem.2.1. Statement of the result. We consider a stored energy function W in the form

(2.1) W(F)= w(F, adj F, det F)= W*((det F)-I/3F)+ G(det F)

where

(2.2)

and w* satisfies

W*(F) w*(F, adj F)

M N

(2.3) w*(F, H)= Y a, tr (C,/2)+ Y bj tr (D/2)i=1 j=l

with a, > O, b > O, C FrF, D HrH and

ai>= i=l," ",M,(2.4)

/3j _-> 3, j=l,’" .,N.

We also assume that

(2.5) G satisfies 1.22).

Finally, if we let a max { ai, 1 <_- -< M} and/3 max {flj, 1 <_-j =< N} we then assume

3cy 3>-p--

c+3y 2’

3/3’,/,(/3 ->_ 3 and y > 1 imply q > 1),(2.6) q

2/3+3y

1 1 1 4--+--p q a fl y 3

For instance (2.6) is verified if a 2, fl 3, y 2.Remark 4. The same results hold if instead of (2.3) w* is of the form

M N

(2.3’) w*(F, G)= E ailFI’+ Y blGI t’,i=1 j=l

and if we assume all the other hypotheses (2.1)-(2.6).Recall that we want to minimize

(2.7) I(v) =- f W(Vv(x)) dx

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


over the space

Apq3,-- {t) WI’P(-’ 3), adj Vv (Lq(’))9, det Vv LV(l),(2.8)

det V v > 0 a.e. and v Uo on 01}.

The theorem can then be stated as follows.Tno 5 (Existence Theorem). Let Wsatisfy (2.1)-(2.6) and suppose that there

exists v Apv such that I v < c, then there exists Apq so that

I(a) inf{I(v): v Aq}.

In order to prove the above theorem, we proceed according to Ball [1] and wedivide our proof in three steps. In the first one we discuss the polyconvexity of W, inthe second one we study the coercivity of I in the space Apq and last, we pass to thelimit on the minimizing sequence.

2.2. Polyconvexity of W.PROPOSITION 6. Let a > O, fl > O.(i) Let

(2.9) W(F) ([F[(det F)-1/3);

then W is polyconvex if and only if a >- 3/2.(ii) Let

(2.10) W(F) (ladj F](det F)-2/3)’;

then W is polyconvex if and only if fl >-3.(iii) Let Vl, rE, v3 denote the principal stretches, i.e., the eigenvalues of (FTF) 1/2

and let

(2.11) W(F)= a v-/3 + _, b v+v+2-/3

i=1 k=l j=l j=l

where det F (with the notation v4 v and vs v) and a, b > 0. Furthermore if>- 3/2 and >- 3 for every i, j chert W is polyconvex.Remark. It is interesting to note that, following the above proposition, the

Mooney-Rivlin stored energy function

W*(F)- alF]2+ bladj FI2

does not lead to polyconvex W:

W(F) a]Fl(det F)-/3 + bladj F](det F)-4/3

since a 2 and/3 2.Proof (i) Step 1. Let h: + x+ be defined as

(2.12) h(x, ‘3 =- (x6-1/3) ’*.

It is then easy to show that h is convex if and only if a >_-3/2. Let us remark now that

(2.13) W(F) w(F, det F), where w(F, 6)= h([F[, ‘3).

If a->_ 3/2, polyconvexity of W is deduced from the convexity of h and the fact thatx h(x, ,3) is a nondecreasing function.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Step 2. It now remains to show that a >_-3/2 is also necessary for W to bepolyconvex. To prove this is slightly more involved. We first define for F M3+, a, b 3such that F + ta (R) b M3+ for every E+ (where a (R) b (aibj) l__<i,j_-<3 M3)"

(2.14) q(t)=- W(F+ ta(R)b)=(lF+ ta(R)b[(det (F+ ta(R)b))-l/3).It is then easy to show (cf. Ball [1]) that

(2.15) W polyconvex implies q convex

(a function W having the property that the associated q is convex is called rank oneconvex).

In order to simplify the notation we let hi,. , h5 be defined as

[F+ta(R)b]2 ht2+h2t+h23,

(2.16)det (F+ ta(R)b)= ,4t-F A5

for every t/ (observe that det (F+ ta(R)b) is linear in t).Since W C2, so is p and an elementary computation leads to

qg"(t) (h2t2 -(/3[ () 3+h2t+h)a/2)-2(h4t+As) )-2 hh24t4 (2a-3) -FO(t3)

thus

and consequently

p"(t) >_- 0 for every + implies

W polyconvex implies a _-> .(ii) The second part of the proposition is proved in exactly the same way.(iii) In order to prove the third part we set

W(F) b(v,, v2, v3, v2v3, v, v3, v, v2, vv2v3)

and use a result of Ball [1] which asserts that, if p: (R+)7-R is convex and increasingin each of the first six variables, then W* is polyconvex. It is clear that if ai >- 3/2 and

fl-> 3 then @ has the required properties (as in the first part). E

2.3. Coercivity of the energy functional I. We now want to show that I definedin (2.7) is coercive over the space Apq introduced in (2.8).

PROPOSITION 7. Let W satisfy (2.1)-(2.6); then there exists K >0 such that forevery v e Apq

I(v) f W(Vv(x)) dx

(2.17) It w(V v, adj V v, det V v) dx

-> K(- + IlVvll,5 + [[adj Vvl[ , + [[det Vvl[[).

Remark. The result is still valid if instead of (2.3) we have (2.3’).Proof. It is easy to see that (2.3) implies that there exists K > 0 such that

(2.18) w*(F, H) >- g(lFI +IHI’)where a =max {ai’l =<i -<M} and fl =max {/3’1 =<j-< N}.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Let 6 det F and use (2.5) to get that there exists K > 0 (we write K genericallyfor constants)"

(2.19) G(6) >- K(6- 1);

hence W and w defined by (2.1) satisfy the following coercivity condition"

(2.20) w(F, H, 6)>-_ K(I6-’/F] + ]6-2/HI +- 1).

We now apply Young’s inequality to

<2.21) [FIp I-l/3FIp p/3.

So for every e >0 and m > 1, 1/m+l/m’= 1 we have

Em --m’(2.22) IF] p --Ia-1/3FIp + apm’/3,

m m

Choosing p as in (2.6), i.e.,

p-a+3y

and m such that pm a, we indeed obtain pro’= 3y and hence

(2.23) IFI =--e<p /P 16-’I3F[ +--P 8--3y/Pa y.

Similarly if q 3fly/(2 + 3y) we get

< qe/ql6_/3 i2q -3v/2q(2.24) ]HI q H

Combining (2.20), (2.23) and (2.24) we have indeed obtained

and thus proved the proposition.Remark 8. Observe that in the above proposition we have indeed used the coer-

civity of w(F, H, a) (i.e. where the arguments (F, H, 6) are independent) as in (2.25),and not only the condition W(F)=w(F, adjF, detF)gK(IFI"+IadjFIq+(detF)V-1),FeM which is, in general, weaker. In fact, the coercivity ofw(F, adj F, det F) does not seem to be sufficient to ensure the coercivity of I(v) inApqv if 3/2 <p<2, since we do not know whether adj Vv (which is defined only as adistribution in Lq) coincides with the almost everywhere definition. However, if p g 2and qp/(p-1), the coercivity of w(F, adjF, detF) is sufficient to obtain that ofl(v) on Apqv (see } 2.5, Theorem 5’).

2.4. Proaf of Theorem 5. We first state a lemma proved by Ball 1].LEMMA 9. Let be a bounded open set of 3 and let

u uW*(a; ).

(i) Ifp > , then

adj VUk adj Vu in (@,(fl))9.

(ii) Furthermore, ifp > 1 and

adj Vu---- adj Vu in (Lq())9, q>l

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


with 1/p) + (1/q) < then

det VUk-- det Vu in 9’(12).

Remark. Recall that in the lemma we have used the notation of Remark 2.

ProofofTheorem 5. We may now proceed exactly as in the proof of Ball’s Theorem.Let Uk be a minimizing sequence of I(u) on Apqv. A version of Poincare’s inequalityensures that

(2.26) fa ]uk(x)lP<--K[fa ]VUk(X)lP-l"(ffoa, [uO] ds) p]where K is a constant. Therefore combining (2.26) and Proposition 7, we find that(up to an extraction of a subsequence)

Uk U

(2.27) adj V Uk H

det V Uk 6

in WI’p(-’ 3),

in (Lq(f))9,

in LV(f).

The above lemma ensures that H- adj V fi and 6 det V tT. Using Proposition 6 we get

(2.28) lim inf I(Uk) >= I().kcx

Moreover, a=Uo on 01-11 and detVfi>0 a.e. Since, by (2.28), I()<.+, so Apqv.Thus is a minimizer of the energy.

In the next section we show that the coercivity condition may be slightly improved;and in the last section of the second part of this article we will also show that one canconsider more general stored energy functions.

2.5. Optimal coercivity of the energy. Recall that

W(F) w(F, adj F, det F)= W*((det F)-I/3F)+ G(det F),M N

W*(F)= w*(F, adjF)= Y aitr(C,/-)+ Y bjtr(adj C)6/2,i=1 j=l

(2.29)where C FF,

G(6) >= C6 ’ for some C > 0 and y > 1, 6 large enough.

In Proposition 7 we have shown that

w(F, H, 6) _>-- K (IFI p + IHIq + 6 1)(2.30)

provided

3ay 3fly(2.31) p_--< q_--<

a+3y’ 2/3+37

where a max {ai: 1 <-_ <- M} and/3 max {/3j 1 _-__j _-< N}.In fact, we have shown that for every (x, y, 6) (/)3

(2.32) (6-1/3x)aW(6-2/3y)t3+6Y>--g(xPWyq+63’-l).

The question is then,whether we can improve p and q by assuming that x IFI,y ladj F and 6 det F. This can indeed be done and we have the following optimalresult.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


PROPOSITION 10. Let W be as above, then the following two statements areequivalent:

(i) There exists K > 0 such that

W(F) >- K(IF[ p +ladj F[ q +(Oct F)- 1)

for every F M3+;

(ii) p <max q <max ya+33/ fl+6y] \ 2a

Proof Let us suppose (ii) and prove (i).We first show the following two inequalities:

(2.33) tr (adj A) <= tr A for every A M3,

(2.34) tr A/2 < tr adj A for every symmetric positive matrix A with det A 1.

Let ak be the eigenvalues of A. It is obvious that

tr (adj A) aa3 + aa3 + aa _-< al2 + a2 + a3 tr A2,

and hence (2.33). Similarly,

(a:a3)-l/2 / (alaa)-l/Z / (ala2)-1/2<_ a-l+ a-l/ a-1

and since det A ala2a3 we deduce that

tr A1/2 al/2 + a/ .4. a/ <- aEa3 + ala3 + ala2 tr adj A

and thus (2.34).Let us define

(2.35)C. (det F)-Z/3FTF (det F)-2/3C,

c=max a, /3=max /3,

Then using (2.33) and (2.34) we have

(2.36) tr (C/2) >= tr (adj C/4),and thus

tr (adj C,/) >- tr C/4)

(2.37) W(F) -> K (tr Cg/+ tr (adj C,/2) + (det F) v 1 ).

We now proceed exactly as in Proposition 7, where we have replaced a and/3 by c7

and/3, and thus we get (i). Conversely, we now assume that W satisfies the coercivity(i) and we want to show that p, q and r are as in (ii). The condition on r is immediatelydeduced from (i) by choosing F AI and letting A o0.

Furthermore, by letting vl, v2, v3 be the principal stretches of F and 6 vv2v3we get from (i)

(2.38)

(Vi-l/3) "4- 2 (ViVi+l-2/3)t "4- Ti=1 i=1

-> K v .4- V Vi+i--1 i=1

q/2

where 1)4 V

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Step 1. We let si be such that

(2.39) vi e s, and Sl s2 S3o

Then (2.38) is equivalent to (considering only the leading term)

(2.40)

Hence letting

e(Sl-(1/3)(sl+s2+s3))__

e(Sl+S2-(2/3)(Sl+S2+S3))fl + e(sl+sz+s3)v

>- K (eqp + e(sl+s2)q nt- e(sl+s2+s3)r- 1).

sl=s, s (1- 0-1)s, $3 (1 0-1 0-2)S

we must have from (2.40)

max (20-1 + 0-) 3; + 20-) (3 20-1(2.41)

=> max {p, (2 0-1) q, (3 2o-1- 0-2) r}

for every O" O"2 O.Step 2. In particular

p-<p-= min { max 1i(0-1,0-2) }a,o-2_>0 1<i<3(2.42)

where

(2.43) 11 (20-1 + 0-2)7, 12 (0-1 + 20"2).5

It now remains to show that

3ay 3fly }maxa+3r /3+63,

3aya+3y"

Case 2. a _<-/3/2, then 12--> 11 and therefore

/ min {max {12, 13}}.

Repeating the above argument we find

3flyp-/3+67"

Observe that the line 11 13 divides the cone 0"1 => 0, 0"2 0 into two regions and thatmax {11,/3} tends to +m if 0-1 or 0-2 does. Therefore, the minimum of max {11,/3} over

0-1 >= 0 and 0"2 => 0 is attained at one of the extremal points. Comparing the differentvalues of max {11,/3} at these points we find

min {max {11,/3}}"0"1,0"2=>0

13=(3--20"1--0"2) %

to obtain (ii).We need to consider three cases.Case 1. a >= 2/3; then 11--> 12 and hence

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


Case 3. fl/2 <-_ a <-_ 2/3, one shows in a similar manner that max {ll, 12, 13} attainsits minimum over oq _>- 0, tr2 -> 0 exactly at the point where ll 12 13 and this minimumis then

3a3’a+33’

hence the result.Step 3. It now remains to show that

(2.44) q_<-max 2/3+33,’2a3’We again use (2.41) to get that

(2.45) 2q<_-min /max{/l(r,r2)}--20O’1_0, O’20 1--<i=<3

where l’i li + qo’l.

A procedure similar to that of step 2 shows that t] is exactly max (33"/(2fl +3y), 3ay/(2a +63’)). And this concludes the proof of the proposition, fi

Combining Proposition 10 with Proposition 6, we obtain the following (exactlyin the same way as Theorem 5).

THEOREM 5’ (Existence Theorem). Hypotheses (2.1)-(2.5) are supposed to be

satisfied and (instead of (2.6))

maxka +37’ )/2’

(2.6’)( 3/33’

q-- max \2/;3’3a3’ )’2a3’ / 1

holds. Assume also that there exists v Ao such that I v < o; then there existsso that

I(fi)=inf {I(v): vAo,}.2.6. Polyconvexity of a more general stored energy function. Up to now we have

only considered stored energy functions constructed from the Ogden model (i.e., W*satisfying (2.1)-(2.3)). We now consider the more general case

(2.46) W(F) *(Vk*, Vk*+l, Vk*+2) + G()where the Vk* are the principal stretches of (det F)-I/aF and 8 =det F, and (Vk, Wk)stands for p(Vl, v2, v3, Wl, w2, w3).

For notational purposes we pose Wk--Dk+lDk+2. We also make the followinghypotheses on *:

if*: ([0, +))6+ is increasing,

(2.47) *: (Vk, Wk) *(V/3, W/3) is a convex function on [0, +)6,

*(v, w) e K v7 + E wf 1i= j=

and G satisfies (2.5). Then if

3a3’ 3 3/33’ 1 1 4(2.48) p= >- q-=> 1, -+-<- 3’> 1

a+33" 2’ 2/3+33’ p q 3’we have the following.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


THEOREM 5". If there exists v Apqy such that I(v) < o then there exists Apqysuch that

I(tT) inf{I(u): u Apqv}.

Proof. The only thing that remains to be checked is the polyconvexity of W. It isthen sufficient (see Ball 1]) to show that

(2.49) I[l( Vk, Wk, "1")--lilt(T-I/ark, 7"-2/3Wk) + G(T)

is increasing in the first six variables and is convex. The first property follows fromthe fact that q* is increasing. It thus remains to show the convexity of q, but this isobvious since we have from (2.47) and (2.49)

(2.50) q,(v, w, )= *((-/v)/(-/w))+ G().

Combining Proposition 6, the convexity of q* and G, we have indeed proved thetheorem. [3

Examples. (i) Ogden materials, considered in the preceding sections, represent aparticular case of the above theorem.

(ii) However there are examples, not of Ogden type, satisfying (2.47), for instance,

* V, W) E aijv 7’ l) d- biwi w+ cijv 7’wi,j=l

with aij, bi, co -> 0, ci ->_ 3/2 and/3 -> 3. If we also assume that the quadratic form on 6

f(x, y) Y (aijxxj + b,yiyj + c,jx,yj)i,j=l

is positive definite. Therefore, the convexity off and the fact that f is increasing implythat * is so.

3. Convergence to the incompressible case. We conclude this article by showingthat if in the above analysis we let the compressibility tend to zero we recover theincompressible case.

Let e > 0 and

(3.1) /(v) W*((det TV)--I/3vV) dx +- G(det Vv) dx

where W* and G satisfy (2.1)-(2.6).Let us consider the problem

(P) inf {I v v e Apq,}

and define

(3.2) Io(v) Jn W*((det dx.

We define the admissible set for the incompressible case as

(3.3) K={ve wl’(;3),adjVv(L())9, detVv=l and V=UoOnOl)l}.

We will assume that K . Finally, let

(Po) inf { Io( v)" v Kt}.

Remark. Note that p < a and q </3.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


We now can state our last theorem.THEOREM 11. Under hypotheses (2.1)-(2.6) and thefurther assumption that G(1)

G’(1) -0 and G"(t)>- Co> O, then every sequence u Apq ofsolutions of (P) converges(up to the extraction ofa subsequence) to a solution Ko of (Po) in thefollowing sense:

u in Wl’p(-’ 3),adj Vu adj V in (Lq(f))9,det Vu -- 1 in L2(f) strongly,

(3.4)lim Io(u Io(tT),e--0

lim -1 f G(det Vu) dx O.e0 E Jn

Remark. Similar results have been obtained by Pouyot [9] and Le Dret [5] forrheologies of the form Wo(F)+(1/e)G(det F).

Proof. The existence of solutions u of (P) is obtained using Theorem 5; theexistence of solutions u of (Po) is a consequence of a theorem from Ball [1] forincompressible materials.

Let w K; then

(3.5) I(u) Io(u)+1 1 G(det Vu) dx <- I(w)= Io(w).E

Therefore u is bounded in Apq.y and, after a possible extraction of a subsequence, wehave

Vu-- Vt in Lp,(3.6) adj Vu adj V t7 in Lq,

det Vu det V in Lv.Furthermore, taking into account the fact that G(1) G’(1) -0 and G"(t) >= Co> O, wededuce that Co((t-1)2/2) <- G(t). Hence in combination with (3.6) we get

(3.7) det Vu- 1 in L2 strongly.

It remains to show that t7 is a solution of (Po) and the last two equalities of (3.4). Leteo> 0 be fixed and 0 < e < eo, we then have for w K(3.8) Io(Uo <-_ Io(U) <- I(u) <-_ I(w) Io(w);

thus

(3.9) Io(tT) leo(a) =<lim inf Io(U) =<lim sup Io(U)<= Io(w).e0 e0

This implies that Io(fi)< +c and hence t7 K and is a solution of (Po). Replacingin (3.9) w by t7 we have immediately

(3.10) lim Io(U)= Io()e--0

and returning to (3.8) we have indeed proved the theorem.

REFERENCES

1] J. M. BALL, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech.Anal., 3 (1977), pp. 337-407.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


[2] P. CHARRIER AND J. M. POUYOT, On the computation ofrubber like materials in severe configurations,in preparation.

[3] P. G. CIARLET, Lecture on three dimensional elasticity, Tata Inst. of Fundamental Research, Bombay,India, 1983.

[4] M. E. GURTIN, An Introduction to Continuum Mechanics, Academic Press, New York, 1981.[5] H. LE DRET, Incompressible behaviour of slightly compressible nonlinear elastic materials, to appear.[6] R. W. OGDEN, Large deformations isotropic elasticity. On the correlation of theory and experiment for

compressible rubberlike solids, Proc. Roy. Soc. London Set. A, 328 (1972), pp. 567-583.[7] , Volume changes associated with the deformation of rubberlike solids, J. Mech. Phys. Solids, 26

(1978).[8] R. W. PENN, Volume changes accompanying the extension of rubber, Trans. Soc. Rheology, 14 (1970).[9] J. M. POUYOT, Etudes numdriques de problmes d’dlasticitd non lindaire: application au calcul d’las-

tomb.res dans des configurations sdvbres, Thse 3me cycle, Universit6 de Bordeaux I, Bordeaux,France, 1984.

[10] C. TRUESDELL AND W. NOEL, The nonlinearfield theories ofmechanics, in Handbuch der Physik, Vo|.

III/3, S. Fliigge, ed., Springer-Verlag, Berlin-New York, 1965.

Dow

nloa

ded

05/0

7/13

to 1

28.1

78.6

9.10

1. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

Documents

An Existence Theorem for Slightly Compressible Materials ...caa.epfl.ch/publications/17-Charrier-Dac-Hanouzet-Laborde1988.pdf · 1988 Society for Industrial andApplied Mathematics