Extreme elastic deformations - Springer Link

Arch. Rational Mech. Anal. 115 (1991) 311-328. @ Springer-Verlag 1991

Extreme Elastic Deformations P. PODIO-GUIDUGLI & G. VERGARA CAFFARELLI Communicated by M. E. GURTIN Co~e~s 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . Extreme Deformations . . . . . . . . . . . . . . . . . . . . . . . . . Energy and Subenergies . . . . . . . . . . . . . . . . . . . . . . . . Energy-Growth Conditions in Terms of Subenergies . . . . . . . . . . . . Growth Conditions and Material Symmetry . . . . . . . . . . . . . . . . 6.1 Volume Preservation in a Symmetry Transformation . . . . . . . . . . 6.2 Growth Conditions and Fluidity . . . . . . . . . . . . . . . . . . . 6.3 Growth Conditions and Solidity . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 315 316 319 320 322 322 324 325 327

1. Introduction I n the absence o f b o d y forces the equilibrium o f a finitely deformed, h o m o geneous elastic b o d y is governed by a well-known nonlinear system o f partial differential equations: (1.1)

Div

(Db(Df(x))) = 0

for x C t2,

with s typically, art open, b o u n d e d region o f three-dimensional space. The calculus of variations associates with (1.1) the functional (1.2)

f v-->f ;r(Of(x)) dr(x). /2

Here ~, the (1.3)

stored energy density, is a m a p p i n g f~-> ~(F)

f r o m the collection of all orientation-preserving 3 • 3 matrices into the nonnegatire reals, such that at each F (1.4)

?r(RF) = ~r(F)

for all rotations R.

3l 2

P. PODIO-GUIDUGLI& G. VERGARACAFFARELLI

?r(DJ(x)) represents the energy stored per unit volume at a point x of the reference configuration -(2 when a deformation f with gradient D f occurs. The behavior of (1.1) depends crucially on the properties of b. Our main purpose here is to discuss the formulation of growth conditions on 3" for very large deformations. To simplify the discussion, we first introduce some notation. We write Lin for the set of all 3 • 3 matrices, Lin + :-- (FE Lin I det F > 0} for the orientationpreserving group of Lin, and Rot : = {R E Lin + I RRT = I}, with I the identity matrix, for the rotation subgroup of Lin +. For F E Lin +, (1.5)

F* = F*(F) : = (det F) F - r

denotes the cofactor o f F . We also w r i t e r for the reals, 1%+ :~-- ]0, -k ~ [ , N+ :--~

[0, + ~ [ . As an important example of a consistent and effective set of hypotheses on ~, we recall that J. M. BALL'S [1] existence results concerning minimizers of the functional (1.2) are obtained when ~ is polyconvex, polycoercive, and consistent with the growth condition: (A)

~(F)-+ -k oo

as det F - + 0 + .

Precisely, t) is polyconvex if there is a convex function (1.6) over L i n • 2 1 5

(X, Y, ~) ~ a(X, Y, 6) such that, for each FE Lin +,

a(F, F*, det F) - b(F).

(1.7)

is polycoercive if there are constants p, q, r, x, and 2, with

(B++)I

p>2, =

q > ~P =p--

1'

I"> 1,

such that, for all F E Lin +, (B++)2

gr(F) >~ z{llflI p ~- IIF* Iiq -~- (det/7)~} -k 2,

x> 0

(cf [1] and [2], Chapter 4)? Conditions (A) and (B++) imply that the stored energy mapping blows up as either (]IFll + liE* I1 + det F ) - + + ec or det F---~ 0 + ; beside serving the needs of mathematical analysis, these conditions reflect the physical expectation that very large deformations should involve very large energies. It is not an easy matter to give expectations of this sort unequivocal mathematical statements in the form of general growth conditions on the stored energy: not only has a careful notion of "very large deformation" to be developed, but it must be decided what form those conditions should have, and to what material 1 A typical assignment of ~ that agrees with BALL'S assumptions is ?~(F) ~ al(F) q- a2(F*) -t- aa (det F), with each ai convex, nonnegative, and coercivejust as required by (B++) (a~(F) ~ Y.t IIFllp, etc.), and with a3 satisfying (A) (cf. [1, 2, 3]).

Extreme Elastic Deformations

3 13

classes they should apply. It is the purpose of this paper to undertake such a task, and to discuss a priori requirements such as (A), (B++), and other ones, milder than (B++), within the frarnework of the constitutive theory of finite elasticity. Such a discussion is hardly found in textbooks, with the notable exception of [2]. 2 After some preparatory algebraic results (Section 2), in Section 3 we recapitulate how the gradient F(x) : = Df(x) measures local changes in volume, area, and length accompanying a deformation f : the Jacobian det F(x) is the ratio of deformed to undeformed volume, whereas IF*(x) n ] and IF(x) e ] are the area and length ratios, respectively, for a material surface with normal n at x and a material curve with tangent e at x. We then define a deformation family to be extreme whenever an n-dimensional material element, with n = 1, 2, or 3, is deformed into another element whose n-dimensional measure tends to either zero or infinity. The following facts are easily established: (Proposition 3.1) If a deformation family is such as to blow a three-dimensional material element up to infinite volume, then some material surface and line elements should also explode (that is, both [IF*!I--> .4 00 and IIFII-~ .4 00 when d e t F - + + ec). (Proposition 3.2) If in a deformation family there is a family of material surface elements that explodes, then again both IIF*[[-> + 00 and liFl[--> + 00; if, instead, a family of material line elements explodes, then ]!FIt-+ @ 00. In addition, we show (Proposition 3.3) that when a deformation family is such as to shrink a threedimensional material element to zero volume, there are some material surface and line elements that are shrunk to zero area and length (that is, there are families of unit vectors n and e such that both IF*n I--->O+ and iFe}--~ 0.4). We also show (Proposition 3.4) that if a deformation family is such that there is a unit vector n (or e) for which IF*n I--> O+ (or IFel--> 0@) and, moreover, det F is bounded away from zero, then [IF*f1-~ 4 00 (or IIf[l--> .4 00). We conclude that a deformation family is extreme if and only if at least one of the following occurs: d e t F - > . 4 0 0 ; detF--->0.4; IIF*II-~ . 4 0 0 ; [Ifl[--> .400. The notion of extreme deformation is purely kinematical. However, it seems reasonable to consider the implications of assuming that extreme deformations of certain elastic materials involve infinite energy, more precisely, that the stored energy mapping definitively grow unbounded when an extreme deformation sequence is performed. In this way, equilibria involving extreme deformations of some body parts would be penalized in energy. In order to state mechanically sensible growth conditions, we should consider two issues. First, given a stored energy mapping ~, it should be possible to appreciate unambiguously the energetic cost that ~ assigns to changes in volume, area, and length: e.g., for harmonic solids defined by ~(F) = [IFE[z, it is not immediately obvious whether extreme volume changes would require infinite energy. Second, the consistency of growth conditions and material symmetry for a given stored energy mapping should be investigated: e.g., for elastic gases, rather than assuming

2 Vid. in particular Section 4.6, where reference to the path-breaking research of S. S. ANTMAN on the subject is made ([4, 5, 6, 7]). Some of the concepts used and the results reported here were anticipated in [8].

314


that the stored energy blows up to infinity with the volume, one would stipulate that (1.8)

~(F)--> 0 +

as

det F--+ + oo. 3

As to the first issue, we propose here a new way of stating growth conditions on the stored energy in terms of equivalent conditions on three associated mappings, the volume, area, and length subenergies. Each subenergy measures the energetic weight assigned asymptotically to a specific deformational effect by a given stored energy. Subenergies are always well-defined, even when the mathematical prescription for the stored energy does not explicitly exhibit the relative kinematical ratio (say, det F, in the case of harmonic materials). Subenergies are introduced in Section 4 and used to state growth conditions in Section 5. As an example of our reasoning there, let us fix attention on volume changes. In this case we decompose F as follows: F - = @(F)) ~ W(F),

with ),(F) : = det F and W(F) :=: (det F) -~ F

(so that, in particular, det W(F)= 1); we write

gro(~,, w ) : = ~(7' w ) and we define the volume subenergy % associated with ~ by %(~) •

inf

detW~l

~o(7~,W).

Then the growth condition (A) on (} requires that the volume subenergy %(7) tend to infinity as 7 tends to zero. More generally, we show that each growth condition on the stored energy is equivalent to an explicit growth condition on the appropriate subenergy (Proposition 5.3). Remarkably, the polycoercivity condition (B++) translates into separate and simultaneous conditions of qualified coercivity for each one of the volume, area, and length subenergies (Proposition 5.5). We also consider growth conditions weaker than polycoercivity, namely, unqualified coercivity conditions on the stored energy expressed so/ely in terms of one or another of the three basic deformational effects; we prove that the coercivity condition (B~), which penalizes changes in length: (Be)

~(F)-+ + ~

as IIF[l-+ + oo

not only implies all the others, but also is equivalent to the combined condition (B+)

~(F) ~ § oo

as (]IF[[ § [IF* 1[ § det F)--> § oo

(Propositions 5.1 and 5.2). Our final Section 6 is devoted to the second issue, consistency of growth conditions with material symmetry. We begin by showing that if a stored energy mapping obeys the growth condition (A), then the material symmetry transformations for b must necessarily 3 Indeed, growth conditions like (A) and (1.8) may be regarded, just as is done in Section 6, as part of the constitutive requirements defining an elastic gas [8].


315

be volume-preserving (Proposition 6.1). We then examine the consequences of coercivity conditions of type (B). First, for fluid materials, we note that all growth conditions may be expressed in terms of the volume subenergy ~0,; we distinguish liquids from gases according to whether % tends to infinity or zero when its argument tends to infinity. From this we argue that the behavior of any elastic material for very large volume changes is either liquid-like or gas-like, according to whether the uniform limit of the mapping ~,(y, W) for y --> q- oc is -k e~ or 0q- (i.e., whether (B,), introduced in Section 5, or (1.8) holds). Second, with a view to distinguishing solids from other material classes, we consider an example of a subfluid, and show by an explicit computation that such a material cannot obey any coercivity condition more stringent than (B,). Inspired by this example, we go on to prove that the condition (B~)

a(F)---> q- oo

as IIF*II--> + ~

implies that the material described by ~ must necessarily be solid (Proposition 6.2). Our last results illustrate the constitutive implications of growth assumptions that form the bases of current mathematical theories of elastic behavior, and may therefore help us to appreciate the applicability of those theories. More generally, in the spirit of [10, 4, 5, 6, 7], we believe that the discussion in this paper may help to give formulations to the basic analytical assumptions that are both physically significant and as mild as possible.

2. Some Algebraic Inequalities Let C be a symmetric, strictly positive matrix, and let t~(C) (i = I, 2, 3) denote the orthogonal invariants of C : As is well known,

(q)z >= 3ta,

(2.1)

(,z) z >= 3qt3,

with equality occurring if and only if C is spherical (i.e., C is a strictly positive multiple of the identity matrix: C = 721). To see this, let the generalized mean be the function from 1%\ {0} into 1%+ defined by / z ( p ) : = ( y ~ + Y~ + Y~)~. 3 where yi(C) (i = 1, 2, 3) are the proper numbers of C. One easily verifies that #(--1) = 3 ~-2-3 (the harmonic mean); g2

:(p) =

,3+

/~(1) = ~ tl

(the geometric mean); (the arithmetic mean);

lira #(p) = min {Yt, >'2, Ya};

p - ~ - - c~

lim /~(p) = max {71, 72, Ya}-

p - ~ + oo

4 q(C) := tr C, 2t2(C) : = (tr C) z -- tr C z, ~i(C) := det C.

316

P. PODIo-GuIDUGLI & G. VERGARACAFFARELLI

Inequalities (2.1) are a direct consequence of the fact that the generalized mean is a monotonically increasing function. In particular, let C -~ FrF for F E Lin +. Then (2.2)

q ( F r F ) = ]IF]]z,

t2(FrF) ~ ]IF* I1z,

t3(FrF) = (det F) z,

and (2.1) has the form (2.3)

Ilfj] z > 1/3- IIF*II,

IIF*I[ z > 1/2 I[FI[ (det F),

with equality occurring if and only if F is conformal (i.e., F -~ ~R, with 7 > 0 and R a rotation). It follows from (2.3) that (2.4)

]]FII3 ~ 3 1/3- (det F),

I]F*II 3 ~ 3 1/3- (det F) z.

In addition to the Euclidean n o r m II'(l implicitly defined by (2.2)~, we shall occasionally use the equivalent matrix n o r m IFI : = sup [Fe I.

(2.5)

[el=l

3. Extreme Deformations The deformation gradient F(x) :----Df(x) measures the changes in length, area, and volume at a point x E ,Q under a deformation f. Precisely, for e a unit vector, let

Oef(x) : = F(x) e;

(3.1)

then, for n ----- e 1 • e 2 with {e~, e2, n} a (positively oriented) orthogonal triplet of unit vectors, (3.2)

F*(x) n = ~,lf(x) • O~f(x),

det F(x) - F(x) n . F*(x) n;

IF(x) el, I F * ( x ) n l a n d d e t F ( x ) are the area, and volume ratios, respectively, for material surface with normal n at x, and about x. We now consider a given deformation Lin +. 5

local deformed-to-undeformed length, a material curve with tangent e at x, a a (three-dimensional) material element sequence {F (k)}; for each index k, F (k) E

Proposition 3.1. I f det F (k~~ + cx~, then there are sequences of unit vectors {e (k)} and {n (k)} such that

(3.3)

IF ~ e~l~ ! -+ +

and

!(F(~)* n(~l-+ + ~ .

Consequently, (3.4)

[IF (k~1[-+ § cx~

and

[[F(~)* I[--> + c~.

5 The use of deformation families more general than sequences would not lead to results finer than those proved below.


317

Proof. For each fixed index k there is a unit vector e(k) such that [ ( F % e (k) l = IF(k)] 9

By (2.4), and the equivalence of the matrix norms introduced in Section 2, there is a constant ~ such that

IFr

I(F (k)) e(k)I = Z

> Ilfl[ >= 1/3-(det

F(k))II3;

from this, both (3.3), and (3.4), follow. An analogous argument demonstrates (3.3)2 and (3.4)2. [ ] Proposition 3.2. (i) Let {n (k)} be a sequence of unit vectors such that (3.5),

[(F(k)) * n(k) t-+ + co;

then

(3.6),

II(F(k))*ll-+ + o o

and

IIFCk)lI-+ + ~ .

(ii) Let {e(k)} be a sequence of unit vectors such that (3.5)z

IF (~) e(k) I -+ + oo ;

then

(3.6)2

]IF Ck)II -~ + ~ .

We omit the easy proof, which requires the same ingredients and similar reasonings as the proof of Proposition 1, and go on to consider deformation sequences {F (k)} such that the volume ratio vanishes in the limit for k - + oo. Proposition 3.3. I f det F (k) -~ 0 + , then there are sequences {e(k)} and {n (k)} of unit vectors such that (3.7)

IF (h) e(k)!--> O+

and

I(F(k)) * n(k)l -+ 0 + .

Proof. By the definition of the cofactor, (3.8)

FF *r = (det F) I.

On the other hand, for each fixed index k there is a unit vector a (k) such that (3.9)

i (F(k)),r a(k) I = ](V(k)),rl.

Let (3.10)

e (k) : = I(F(k)) *T 1-1 (f(k)) *T a (k).

From (3.8), (3.9), and (3.10) we have that (3.11)

IF(h) e(k)]

~

[(F(k))*T ]--I (det F(k));

but then, once again by the equivalence of the matrix norms introduced in Section 2 and (2.4)2, there is a constant z > 0 such that (3.12)

IF (k) e(k) [ =< z. (de{ ff(k))l/3.

318


Thus, (3.7)1 follows. The proof of (3.7)2, which is completely analogous, follows from the identity (3.13)

F * F r = (det F) I.

[]

This proposition means that when a material element is made to shrink to zero volume, there is always some material line (or surface) element shrinking to zero length (or area) as well. In this vein, one would regard a deformation sequence {F (k)} as extreme 6 if the limit of {detF (k)} is + c~ or 0 + , or else if there are unit vectors e, n such that one or the other of sequences {IF (k) e I} and {IF(k)* n I} tends to + ~ or 0 + . The fact that, with this notion of extremality, neither e nor n depends on the current index k entails no loss of generality. Indeed, given e and {e(k)}, there is a sequence of rotations {R(k)} such that e = R(k)e (k) for each index k; moreover, composition of F (k) with a rotation R (k) leaves unchanged both the quantities IlF(k)[] and ]I(F(k))*]l involved in our next proposition.

Proposition 3.4. Suppose that inf det F (k) > O. k

I f there is a unit vector e such that

lim IF(k) e I = 0 ,

(3.14)

k---~ eo

then

(3.15)

lim lIF(k)ll = + ~ .

k-+~

I f there is a unit vector n such that

(3.16)

lira I f (k)* n I = 0,

k---~ c~

then

(3.17)

lim IlF(k)*ll = + o.~.

k-->~

Proof. Suppose, to the contrary, that there is a constant no such that IlFCk)l[G no for infinitely many values of the index k. There would then be an extracted subsequence, which we continue to denote by {F(k)}, converging to some Fo E Lin. For such a subsequence, IF (k) e I --> ]Fo e ] - - 0 because of (3.14). Since le [ = 1, this would imply that det Fo = 0, which cannot be, since the sequence {det F (k)} has a positive limit equal to det Fo by hypothesis. Thus for each no, IIF(k) 11> ~o.

[]

As anticipated in the Introduction, we conclude that a deformation f a m i l y is extreme i f at least one o)C the following limits occurs: detF-+ +c~, 6 At a point x ~ ~; henceforth.

detF-+0+,

IIF*II-+ @cx~,

I[FlI---~+ o ~ .

for simplicity, the x-dependence will not be shown here and


319

Remark. In Section 4.6 of [2] it is proposed to regard a deformation as extreme when one of the proper numbers of the strain tensor C = FrF tends to either 0 + or -~-oc while the two other proper numbers are confined into a compact subinterval of ]0, + c~[. We observe that this notion of extreme deformation is more restrictive than ours: e.g., homothetic families F = 2/, for 2 that tends either to + c~ or to 0 + , would not be regarded as extreme. However, this notion turns out to be useful in the analysis of the extreme behavior of polyconvex energies [2, 3].

4. Energy and Subenergies We have seen that changes in length, area, and volume at a point x, when a deformation f with gradient F = Dr(x) occurs, are connected with three real numbers:

(4.1)

0~(f) : = llFI[,

H(F):--IIF*(F)II,

7(F) : = det F.

We note that the mapping F~+ F* defined by (1.5) is a diffeomorphism of Lin +. F o r each F 6 Lin + we introduce three multiplicative decompositions of F, F* and again F that involve 0~(F), fl(F) and 9,(F), respectively; these are (4.2)~

F----- ~x(F) U(F),

U(F) : =

IIFII-~ F;

(4.2)2

F* = fl(F) V(F),

V(F) : =

Ilr*{l -x F*;

(4.2)~

F = (~,(F)) ~

W(F), W(F) : = (det F) -~ F.

We define

(4.2,)

Rad + : = {FC Lin + I IIFII = 1},

(4.4)

Uni : = {FE Lin l det F = 1},

and note that Lin + is a diffeomorph of both R+ • Rad + and ~ + • Uni (so that, in particular, for each FC Lin +, U(F) and V(F) belong to Rad +, the "radial" subset of Lin +, whereas W(F) belongs to Uni, the "unimodular" subgroup of Lin. To each stored energy mapping (4.5)

~ : Lin + -~ a 0 ~,

= a(F)

we associate three mappings (4.6)1

~rt: N + • Rad + -+ R~-,

~rl = ~1(oc, U),

(4.6)2

~ :R+•

% ---=~(fi, V),

(4.6)3

by : E+ • Uni -~ ~ + ,

+,

~v = ~o(;~, W),

such that (4.7)

~ = b(F) = at(~(F), U(F)) = 8a(fl(F), V(F)) = av(~(F), W(F)).

329

P. PODIO-GuIDUGLI & G. V E R G A R A CAFFARELLI

The mapping ~, has a direct link with the volume subenergy associated with b, 7 i.e., the function % from R+ into ~ o defined by (4.8)

7 ~%(7)

:=

inf b(F).

detF=~,

Indeed, it is clear that % ( 7 ) - - inf ~(7, W).

(4.9)

W~Uni

In a completely analogous manner we introduce the area and the length subenergies ~o~ and ~0l: (4.10)

~0,([3) : =

(4.11)

q~(c~) : :

inf ~ ( F ) =

flF*][ ~/3

inf A(F) =

IIF][:~

inf ~,(/3, V);

V~Rad +

inf

U~Rad +

b,(0r U). -

Given a hyperelastic material described by a stored energy ~, the mapping % associated with ~ by (4.8) bounds from below the energy stored as a consequence of changes in volume: e.g., if ~ ( F ) = I]F]]z, then k ( f ) ~ 3 (det F) 2/3 = ~ov (det F) since (4.1)3, (4.2)3, and (2.4)~ imply that IIF}]z = (7(F)) el3 I1W(F)It 2 ~ 3(7(F)) 2/3. Similarly, as we shall see in the next section, given an extreme deformation family, ~v, and q~t bound from below the energy stored as a consequence of changes in area and length as IIF*II-+-? oo and I]FII-+ + 0% respectively. More generally, given a specific deformationaI effect, a subenergy may be defined that estimates the energy stored as a consequence of that effect. It is precisely in terms of such a subenergy that "natural" growth conditions might be expressed, as we shall demonstrate in the case of volume, area, and length subenergies.

5. Energy-Growth Conditions in Terms of Subenergies We recall from Section 3 that a deformation family is reckoned as extreme if det F tends to zero, or else if at least one of det F, IF* I, or IF I tends to infinity. Accordingly, we consider the following growth conditions for the stored energy: (A)

b(F)--+-~-e~

as d e t F - + 0 + ;

(B~)

&(F) -+ + cx~

as det F - > + ~ ;

(Ba)

t~(F)---> + oo

as IIF*II ~ + co;

(BI)

~(F)---> ~- 00

as IIFI[--~ + oc.

7 In continuum mechanics the concept of volume subenergy has been introduced by ERICtCSZN [11], who elaborated an idea of FLORV [12], to model certain types of phase transitions within the framework of thermoelasticity. More recently the concept has been used by CmeoT & KINDERLEHRER[13] and FONSECA[14] in the variational study of crystal equilibria, and by POOIo-GtnDU6LI [15], in order to state a yielding criterion for finite elasto-plasticity that generalizes the classical criterion of distortional energy due to VON M I S E S .


321

Just as (A), the mild coercivity conditions of type (B) are physically well motivated; they are all implied by (B +)

~(F) --> + c~

as (1[FIl + I[F* 11+ det F) -+ + ~x~

(and afortiori by the polycoercivity (B++). We observe that (B 3 is a multidimensional version of the familiar weak coercivity condition in the calculus of variations. Although it is perhaps not completely evident, (Bt) implies (B+), as we shall prove shortly. In the light of definitions (4.6) and (4.7), conditions (A) and (B~) may be alternatively written in terms of the mapping bo, while conditions (B~) and (Bz) may be written in terms of mappings oa and ~'t, respectively: (uniformly for WC Uni);

~'-->0 +

(B~)

lim ~(7, W) : + c~

(uniformly for W C Uni);

(Ba)

lim ~(fl, V ) :

+ cx~

(uniformly for V E Rad +);

lira bz(a:, U ) = + o~

(uniformly for U E Rad+).

(B3

or

+ oo

With these alternative statements, all growth conditions acquire an explicit and rigorous form; moreover, comparison of the various coercivity conditions introduced so far is made possible.

Proposition 5.1. (Bt) ~ (Ba) ~ (B~). Proof. We prove only the first implication; the second is proved analogously. By hypothesis, for each ~ > 0 there is #k > 0 such that ~(F) > z whenever IIFII >/~k. Then, if we choose ~k = 3 ~#k, it follows from (2.3)~ that ]IFII > #k whenever ]IF*II> vk, and the desired conclusion obtains. [] Proposition 5.2.

(i)

(B+) ~ ((B3 & (B~) & (B,)); (B ~-)(Bz).

(ii)

Our next step is to express growth conditions in terms of subenergies; once again, the uniformity in the limit processes appearing in (A), (B~), (Ba), and (Bt) will turn out to be of the essence. Consider the growth conditions: (a)

% (det F) -+ + ~

as det F---~ 0 + ;

(b,)

% (det F ) - + q-e~

as det F - + + e c ;

(b,)

r

as IIF*ll-+ + c o ;

(b3

F* I]) --> + ~ 9~(llFl[)-+ + ~

as ]IFII-+ + o 0 .

322

P. PODIO-GUIDOGLI& G. VERGARACAFFARELU

Proposition 5.3. (A) (a); (B~) 0,

(Pa(~) ~ 74a~q -}- ha,

~a > O,

~0l(0r ~ ~ J

~t > 0.

+ 21,

In the light of this proposition the coercivity condition (B++)2 is interpreted as an assumption on the asymptotic behavior of the subenergies linked to the three main kinematical effects when det F, IIF*I1, llF]t tend to § ~ .

6. Growth Conditions and Material Symmetry 6.1. Volume Preservation in a Symmetry Transformation In the formulation of NOEL [16, 17] a stored energy mapping has the material symmetry described by the group (6.1)

X : = {HE Uni 16(FH) = 6(F) for all FE Lin+};

the elements of 5f are visualized as those changes in local reference configuration that are undetectable on the basis of experiments on the energy. An easy, but relevant, consequence of (6.1) is that (6.2)

6 ( n ) = 6(1)

for all H E ~ .

One can motivate [18, 19, 3] the following notion of an extended material symmetry group : (6.3)

~ : = {H E Lin+ I ;r(FH) = det H ( b ( F ) - 6(1)) + 6(H) for all F E Lin+}.

As an example when the inclusion of 5f in ~f is proper, we consider the stored energy mapping 6(F) = a(l]Fll, [[F* lI, det F), with the nonnegative function (x, y, z) ~ a(x, y, z) defined in the positive orthant of R 3 such that a(o;x, o~2y, o;3z) = a.3a(x, y, z)

for each o~> 0.

In this case, for any conformal matrix H, one verifies that b(H) -- (det H) b(1) = 0,


323

and, more generally, ;~(FH) = (det H) b(F)

for all F E Lin+,

so that, in view of definition (6.3), H E ~ . With the above prescription of b we have that ~ ( ~ I ) = e3b(I), so that gr(el)--+O+ as e - + 0 § thus, b cannot satisfy the growth condition (A), according to which the stored energy has to blow up when volumes shrink to nothing. Remarkably, as we now proceed to prove, the extended material symmetry group shrinks down to the usual group if (A) prevails. Proposition 6.1. (A) ~ (~" ~-~ ~). Proof. We prove that if (A) holds, then det H = 1 and b(H) : ~(I) for each

Note that for each H E ~ , (6.4)

b(tt) -- (det 1t) b(I) = --(det H) b(H -~) § 5(I),

and thus, for each integer n, (6.5)

5(1)

-

1 1 " " 1 + det H n~r(H ) ~ 1 + det H - "

5(H-").

Assume now, with no loss of generality, that det H < (6.6)

1. Then from (6.5),

1 gr(I) > . . . . ,, ?r(H"). = 1 + act/1

By (A), the right side of (6.5) diverges with n, a contradiction. Thus, ~ C Uni. In particular, (6.4) reduces to (6.7)

a(H) -- ~(I) = - - a ( H -1) q- 5(1).

In order to complete the proof, we observe that in view of (6.3), (6.8)

gr(FH) -- 5(F) = gr(H) -- 5(I)

for each FE 5~. Writing (6.8) for F = H, H z . . . . , H "+1, and adding the resulting relations, we obtain (6.9)

5(H,+ l) _ ~(H) = n(~(H) -- gr(I)).

From (6.9) and the fact that (} is bounded below it follows that (6.10)

~(H) -- b(I) >= 0

for all H E

~.

Conditions (6.10) and (6.7) together imply the desired result.

[]

More precisely, if (A) is to prevail, on the one hand we may dispense with one of the standard items in NOLL'S notion of a material symmetry transformation ([10, 11]); on the other hand, there is no point in introducing notions more general than NOLL'S.

324


An elastic material is a .fluid if Y" = Uni, a solid if .U Q Rot (in particular, a solid is isotropic if 5~~= Rot). The growth condition (A) is considered appropriate for all (compressible) fluids and for all solids. We now proceed to examine the implications of growth conditions of type (B), and show that significant differences emerge between different fluids, and between fluids and solids.

6.2. Growth Conditions and Fhddity For a compressible fluid, the stored energy has the equivalent representations (6.11)

?r(F)=~r(detF)=?r,~(detF, W) =: ~v,~(det F)

for all W 6 U n i .

Thus, ~ and the volume subenergy % coincide. A weak coercivity condition such as (b,) is appropriate for liquid fluids only; indeed, for % a convex function agreeing with (A), i.e., such that (a)

lira q0~(7) - - q- cx~,

~'~0+

it has been shown in [9] how to resolve fluids into liquids and gases by inspection of the solutions of certain elementary equilibrium problems, according to whether

(by) or (c)

tim q)o(y)

=

0+

prevails. For a typical compressible fluid with stored energy k and volume subenergy %, ~ controls the length subenergy ~0zas follows" (6.12)

0 ~ ~ot(~) --

inf %(y) ~ % 0 1 (if, instead, we considered the case 9(H) < 1, a completely analogous reasoning would lead us to the same conclusion). Then, by (6.17), there is at least one proper number of H whose absolute value is greater than 1, and at least one whose absolute value is less than 1. Thus n

~(H -1) = ~lim jfN]H-'I] = 90 > 1. -o- oo Since (~o -- eo)" < [IH-'[I

for eo -- eo > 1

and for sufficiently large values of n, and since IIH-~J[ =

II(H*)nIl,

it follows that (6.18)

lim I](H*)"ll = + ~ .

n--~- o o

On the other hand, (6.2) and the definition (4.10) of area subenergy imply that (6.19)

b(I) = a ( H ") ~

~.(II(H:")~II)

for each integer n,

so that (6.18), (6.19) aod (b~) are contradictory. Necessarily, then, if (B~) prevails, all material symmetry transformations must have unit spectral radius: (6.20)

2t(H) = ~2(H) -/- flZ(H) = 1

for all H E ~c.

We look for elements H C ~ , if any, that are not similar to any rotation and that satisfy (6.20). As this combined requirement excludes all solutions of (6.20) with distinct proper numbers, we may replace (6.20) by (6.21)

2,(H) = 1,

z z ( n ) - 2a(H) -- ~ 1 .


327

Modulo trivial reorderings of proper numbers, there are two simple solutions t~ of (6.21), namely, the identity I and the rotation J : = 2a | a -- I of ~ about the proper vector a associated with the proper number 2~, but of course our combined requirement excludes those also. There are, however, nonsimple solutions of (6.21) that are similar to no rotations; their real canonical Jordan forms (cf. Section 6.7 of [22]) are: I+a|

I+a|174

J+b|

But, as exemplified for the transformations listed in (6.15), the iterated norms of each one of these putative members of 4" diverge (as well as the iterated norms of their inverses) and the previous contradictory situation is reproduced when the area subenergy is both uniformly bounded above by ~(I) and is forced to blow up to infinity by (Ba). [] In particular, then, the growth condition (B ++) is too much to ask of either a fluid or a subfluid. While the issue of growth conditions is sufficiently well understood for fluids, what coercivity assumptions are appropriate for subfluids remain to be seen. Also, it would be desirable to have an existence theory for elastic solids that employs coercivity assumptions as mild as (Ba); the recent work of GIAQUI~TA, MODmA, & SOUrEr: [23] may serve as a first step in that direction.

References 1. J. ~/[. BALL, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 1977, pp. 337-403. 2. P. G. CIARLE%Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. NorthHolland, 1988. 3. P. PODIO-GuIDUGLI,Polyconvex energies and symmetry requirements. To appear in J. Elasticity. 4. S. S. ANTMAN, Fundamental mathematical problems in the theory of nonlinear elasticity, pp. 35-54 of Numerical Solution of Partial Differential Equations, IIL North-Holland, 1976. 5. S. S. ANTMAN, Ordinary differential equations of nonAinear elasticity, I & II. Arch. Rational Mech. Anal. 61, 1976, pp. 307-393. 6. S. S. ANTraAN,Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders. Arch. Rational Mech. Anal. 83, 1983, pp. 1-52. Corrigenda, ibid. 95, 1986, pp. 391-393. 7. S. S. ANTMAN, The influence of elasticity on analysis: modern developments. Bull. Amer. Math. Soc. 9, 1983, pp. 267-291. 8. P. PODIo-GuDUGLI, On energy-growth conditions in elasticity. To appear in Rend. Sere. Mat. Torino, 1990. 9. P. POD~o-GuIDU~LI, G. VEROARACAVFARELH, & E. G. VIRGA, Cavitation and phase transition of hyperelastic fluids. Arch. Rational Mech. Anal. 92, 1986, pp. 121-136. 10. C. TRUESDELL, Some challenges offered to analysis by rational thermomechanics. Lecture 1 : Comments on elasticity, pp. 495-540 of Contemporary Developments in lo A linear transformation is called simple if the algebraic and the geometric multiplicity of all its proper numbers coincide ([22], Section 4.12).

328

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

P. PODIo-GuIDUGLI& G. VERGARACAFFARELLI Continuum Mechanics and Partial Differential Equations, G. M. DE LA PENHA(~; L. A. MEOEmOS Eds., North-Holland, 1978. J. L. ERICKSEN, Some simpler cases of the Gibbs problem for thermoelastic solids. J. Thermal Stresses 4, 1981, pp. 13-30. P. G. FLORV, Thermodynamics relations for high elastic materials. Trans. Faraday Soc. 57, 1961, pp. 829-838. M. CmPOT & D. KINDERLEHRER, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103, 1988, pp. 237-277. I. FONSECA,The lower quasi-convex envelope of stored energy function for an elastic crystal. J. Math. Pures Appl. 67, 1988, pp. 175-195. P. PODIo-Gun~UGL~, I1 criterio di snervamento della sottoenergia, pp. 559-568 of G. Ceradini's Anniversary Volume, Tip. Esagrafica: Roma, 1988. W. NOLL, On the foundations of the mechanics of continuous media. Carnegie Inst. Tech. Dept. Math. Rep. No. 17, June 1957. W. NOLL,A mathematical theory of the behavior of continuous media. Arch. Rational Mech. Anal. 2, 1958, pp. 197-226. C. TRUESDELL,A theorem on the isotropy groups of a hyperelastic material. Proe. Nat. Acad. Sci. U.S.A. 52, 1964, pp. 1081-1083. M. E. GURTIN, An Introduction to Continuum Mechanics. Academic Press, 1981. C. TRUESDELL & W. NOLL, T,~e Non-Linear FieM Theories of Mechanics. Handbuch der Physik III/3, S. FLO~GE Ed. Springer-Verlag, 1965. P. R. HALMOS, Finite-Dimensional Vector Spaces. Van Nostrand, 1958. P. LANCASTER& M. TtSMENETSKV, The Theory of Matrices. Academic Press, 1985. M. GIAQUINTA,G. MODICA& J. SOUeEK, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 106, 1989, pp. 97-159. Erratum and addendum, ibid. 109, 1990, pp. 385-392. Dipartimento di Ingegneria Civile Universith di Roma "Tor Vergata" 00173 Roma and Dipartimento di Metodi e Modetli Matematici per le Scienze Applicate Universit~t di Roma "La Sapienza" 00161 Roma (Received December 19, 1990)