Trends and Applications of Pure Mathematics to Mechanics: Invited and Contributed Papers presented at a Symposium at Ecole Polytechnique, Palaiseau, France November 28 – December 2, 1983

427 Pages·1984·5.72 MB·English-French

by  J. M. Ball (auth.), Philippe G. Ciarlet, Maurice Roseau (eds.)

Lecture setoN
in scisyhP
detidE by .H Araki, ,otoyK J. Ehlers, ,nehcnOM K. ,ppeH Ziirich
.R ,nhahneppiK ,nehcnOM .H A. WeidenmOller, grebledieH
dna J. Zittartz, K6ln
591
Trends dna Applications
of eruP Mathematics
to Mechanics
Invited and Contributed Papers presented at a
Symposium at Ecole Polytechnique, Palaiseau, France
November 28 - December 2,1983
Edited by PG. Ciarlet and .M Roseau
Springer-Verlag
Berlin grebledieH New York oykoT 4891
Editors
eppilihP G. Ciarlet
Analyse Numerique, Tour 55
Maurice Roseau
Mecanique Theorique, Tour 66
Universite erreiP et Marie Curie
4, Place Jussieu, F-75005 Paris Cedex 50
AMS Subject Classifications (1980): 35Xx; 49Xx; 70xX; 73xX; XX67
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PREFACE
The "Fifth Symposium on sdnerT in Applications of Pure Mathematics to Mechanics"
saw held November 28-December 2, 1983, at 1'Ecole Polytechnique, Palaiseau, under
the auspices of the International Society for the Interaction of Mechanics and
Mathematics, in continuation of the meetings held ylevisseccus in ecceL (1975),
Kozubnik (1977), Edinburgh (1979), and Bratislava (1981).
The purpose of the Society is to promote and enhance the exchanges between
mathematics and mechanics and this symposium, sa the previous ones, saw a vivid
illustration of this purpose. Twenty-four srekaeps from eight different countries
delivered lectures which perfectly exemplified the interplay between the two
.secneics yehT derevoc the most recent secnavda in the mathematical analysis of
the equations of mechanics (bifurcation theory, compensated compactness, singu-
larities and nonlinearities, homogenization, the Schrodinger equation, the Boltz-
mann equation, Hamiltonian ,smetsys ).cte sa well sa the more mechanical stcepsa
(propagation of ,sevaw phase transformations, stability, composite materials, -ocsiv
elasticity, thermoelasticity, finite elasticity, ,).cte with a pervading emphasis
on nonlinearity.
It is a pleasure to sserpxe our warmest sknaht to all the invited srerutcel
whose inspiring communications made the sseccus of this symposium. The support
of the following contributing organizations saw also deeply appreciated: Association
Dniversitaire de Mecanique, Centre National de la Recherche Scientifique,
Commisariat a‘ 1'Energie Atomique, Ecole Polytechnique, Electricit de ,ecnarF
Institut National de Recherche en Informatique et en Automatique, Office National
d'Etudes et de sehcrehceR Aerospatiales. Last but not least, our deepest sknaht
are due to our colleagues of the Scientific Committee: srosseforP P. Germain,
J.L. Lions, and R. Temam.
This volume contains the stxet of all the lectures delivered at the symposium
tpecxe( the text yb K. Maurin).
Paris, yraunaJ 1984 P.G. Ciarlet, M. Roseau
Universiti! Pierre et Marie Curie
Lecture Notes in Physics
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Statistical .scisyhP Proceedings, 1981. Edited yb S. Albe- Mechanics. Proceedings, 1983. Edited yb PG. Ciarlet and
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1982.
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Vol. 175: Defect Complexes in Semiconductor Structures.
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TABLE FO CONTENTS
J.M. BALL: Minimizers and the Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . 1
E. BUZANO, .G GEYMONAT: Geometrical Methods in Some Bifurcation Problems
of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
C.M. DAFERMOS: Conservation Laws Without Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
.J.R DIPERNA: Conservation Laws and Compensated Compactness . . . . . . . . . . . . . . . . . . . . 25
.G DUVAUT: Homogeneisation et Materiaux Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
.W FISZDON, M. LACHOWICZ, A. PALCZEWSKI: Existence Problems of the Non-linear
boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
R. GLOWINSKI: Numerical Simulation for Some Applied Problems Originating
from Continuum Mechanics . . . . . . . . . . . . . . . . . . . . ..*...................... 96
.G GRIOLI: Linear Problems Associated to the yroehT of Elastic Continua with
Finite Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
M.E. GURTIN: One-Dimensional Structured Phase Transitions on Finite Intervals . . 159
.J.W HRUSA, J.A. NOHEL: Global Existence and Asymptotics in One-Dimensional
Non-linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
R. ILLNER: Discrete Velocity Models and the Boltzmann Equation .... .......
.F :NHOJ Formation of Singularities in Elastic sevaW ..............
K. KIRCHGASSNER: Solitary sevaW Under External Forcing .............
.J LERAY: Sur les Solutions de 1'Equation de Schradinger Atomique et '1
Particulier de deux Electrons ............................... . 532
O.A. OLEINIK: nO Homogenization Problems .................................... 248
P.J. OLVER: Hamiltonian and Non-Hamiltonian Models for retaW sevaW ............. 273
P. PODIO-GUIDUGLI, .G VERGARA-CAFFARELLI: nO a Class of Live Traction Problems
in Elasticity .......................................................... 291
.G.W PRITCHARD: Some Viscous-Dominated Flows . . . . . . . . . . . . . . . . . . ............ 305
M. RENARDY: Initial Value Problems for Viscoelastic Liquids . . . . ............... 333
E. SANCHEZ-PALENCIA: Perturbation of Eigenvalues in Thermoelast iN ytic and
Vibration of smetsyS with Concentrated sessaM * . . . . . . . . . . . . ................ 346
.C.J SIMO, J.E. MARSDEN: ssertS ,srosneT Riemannian Metrics and the Alternative
Descriptions in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
L. TARTAR: Etude des Oscillations dans les Equations aux Derivees Partielles
Non Lin'eaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
.F VERHULST: Invariant Manifolds and Periodic Solutions of Three Degrees of
Freedom Hamiltonian smetsyS . . . . . . ..*............................... 413
188
......... 194
.......... 211
e saC
. . . . . . . . . .
MINIMIZERS AND THE EULER-LAGRANGE EQUATIONS
J. M. Ball t
Department of Mathematics
Heriot-Watt University
Edinburgh, EHI4 4AS
Scotland
Consider the problem of minimizing an integral of the form
I(u) = f(x,u(x),?u(x))dx
subject to ~iven boundary conditions, where ~ c ~m is a bounded open
set and the competing functions u : ~ ÷ ~n . Frequently it is possible
to use the direct method of the calculus of variations to establish the
existence of a minimizer u in an appropriate Sobolev space I~'P(~ ~n).
Then formally we expect that u satisfies the weak form of the Euler-
Lagrange equations
f 8 ~ i Sf,~ + -~ idx = 0 for all ~ 6 C~(~;~ )n )I(
~u l
but a search of the literature reveals that in general the theorems
guaranteeing this make stronger growth assumptions on f than are nec-
essary to prove existence. That this is not just a technical difficulty
can be seen from one-dimensional examples due to Mizel and myself that
are announced in 6. One of these examples concerns the problem of
mlnlmlzlng
I(u) = (x4-u6) 2 (u' + ~(u ) dx (2)
subject to u(-l)=-k, u(1)=k, where r > 14 is an integer, ~ > 0 and
d
0 < k ~ 1. (Here m=n=l and the prime denotes ).~--~ Note that the
integrand f(x,u,u') in )2( is smooth and regular (i.e., fu'u' > )0
so that the Euler-Lagrange equation can be reduced to the form u" =
= g(x,u,u'). Given k, let ~ > 0 be sufficiently small. Then I
attains an absolute minimum on the set 5~= {v 6 WI'I(-I,I) : v(±l) =
= ±k} and any minimizer u satisfies u(0)=0, u'(0)=+~. Furthermore
fu ~ L~oc(-l'l) and hence )i( does not hold. Also, we have that
inf I(v) > I(u) (the Lavrentiev phenomenon). )3(
v £ WI'~(-1,1)
v(±l) = ±k
I will now sketch the most important part of the proof, which estab-
lishes (3), that u(0)=0, and that if 0 ~ p < 1 then In(x) l~pklxl 2~
t Research supported by a U.K. Science & Engineering Research Council
Senior Fellowship.
for all x £ -i,i, provided s > 0 is sufficiently small. The ar-
gument is an adaptation of Mania 9 (cf. Cesari 8,p.514). Further
details can be found in Ball and Mizel 7. Let V be any element of
~. Then v(x0) = 0 for some x 0 6 (-i,i) and by symmetry we can sup-
pose that x 0 ~ 0. Suppose further either that x 0 ~ 0 or x 0 = 0
and 0 < v(x) < ~kx 2/3 for some x 6 (0,i). Let ~ < ~ < i. In eith-
er case there exists an interval (Xl,X2) , 0 < x I < x 2 < i, on which
ukx2/3 v(x) ~ ~kx 2/3 and such that v(x) = UkXl2/3 , v(x2)=vkx22/3.
On this interval (x4-v6) 2 > xS(l-(~k)3)2,~and hence
I(v) ~ (1-(vk) )3 2 ° I x2 x8 (v,) 2rdx.
J x 1
0 2r-9
Putting y = x , where 8 = ~-~, we get, using Jensen s inequality
xS(v,)2rdx = @2r-l 2(dv~2r
~xJ \~/ yd
1 2/3 _ ~x2/3)2r
> @2r-lk~r(~x2 i def
_ = h (Xl,X 2 ) •
2r-i
• ~ ~
(x 2 - x )I
it is easily verified that if r ~ 14 then 0<xi~2< 1 h(Xl,X )2 > 0,
and it follows that I(v) ~ N > 0 for all v as above, ~ being in-
dependent of .s Now let {(x) = Ixl2/3sign x for lxI ! k3/2, v(x)=
= k for x > k 3/2 ~(x) = -k for x < -k 3/2 Then ~ 65~ and
k3/2
I(~) = 2~ ~: (~x -I/3)~ 2dx ,
J0
which is less than ~ if s is sufficiently small. Thus u(0) = 0,
lu(x) l ~ ~klxl 2/3 for any minimizer u, and )3( holds. As far as we
are aware the examples in 6,7 are the first in which the singular set
in Tonelli's partial regularity theorem i0, p. 359 has been shown to
be nonempty.
I now turn to nonlinear elastostatics, which in fact motivated the
work in 6,7. Consider a simple mixed boundary value problem in which
it is required to minimize
I(u) = W(Vu(x))dx
J
on the set ~ = {u £ WI'I(~;~ n) : I(u) < ~, ulS~l = u0 in the sense
of trace}. Here ~ c ~n is a strongly Lipschitz bounded open set,
8~i c 8~ has positive n-i dimensional measure, and W : M n×n ÷ ~+
U {+~} is the stored-energy function of a homogeneous material. We
suppose that W 6 cl(M~xn) , where _M+n ×n = {A 6 Mnxn : det A > 0}, that
W(A) = +~ if det A < 0, W(A) ÷ +~ as det A ÷ 0+, and that for some
e 0 > 0
3
~(CA)A T ,< const. (W(A) + )i )4(
for all A,C 6 M+n×n with Ic-iI < s .0 Let u minimize I in -~; ex-
tra hypotheses guaranteeing the existence of a minimizer can be found
in 1,5. Let v : ~n + ~n be C 1 with Vv uniformly bounded and
vou 0 = 0. Define for e > 0
~i
us(x) = u(x) + ~v(u(x)).
Then it is not hard to show that u s 65~ and that
d{--d I(ug) I = ! ~l (Vu) uJ v,ji (u(x))dx = 0. )5(
Under further hypotheses (c.f. 2) u is invertible and )5( can then
be recognized as a weak form of the Cauchy equilibrium equations
. T~ = 0,
~u z
where T~ 1 is the Cauchy stress tensor. If instead we define for vl~=0,
us(x) = u(z) , x = z + ~v(z),
and make an analogous hypothesis to (4), we obtain the weak form of the
equation
e u i ~W)
~x~ (W~ - '~ ~u~ = .0 )6(
Details of these results will appear in 3. To obtain the weak form
dx = for ~ 6 C0(~;~n) )7(
T--T- ~,~
~U ~
of the equilibrium equations one would need to show that I(u6) is dif-
ferentiable with respect to ,£ with the obvious derivative, for a large
class of variations us(x) = u(x) + s~p(x), and it is not clear how to
do this under any realistic hypotheses on W. The one-dimensional ex-
amples suggest that infinite values of ?u(x) or ?u(x) -I may occur
in minimizers; this could be the source of the difficulty, and may also
be relevant to the onset of fracture.
Finally I remark that the Lavrentiev phenomenon severely restricts
the class of numerical methods capable of detecting singular minimizers;
see 4.
References
i J.M. Ball, Convexity conditions and existence theorems in non-
linear elasticity , Arch. Rat. Mech. Anal. 63(1977), 337-403.
2 J.M. Ball, Global invertibility of Sobolev functions and the in-
terpenetration of matter, Prec. Roy. Soc. Edinburgh 88A(1981), 315-
328.
3 J.M. Ball, in preparation.
4 J.M. Ball & G. Knowles, forthcoming.
5 J M. Ball & F. Murat, wl,p-quasiconvexity and variational prob-
lems for multiple integrals, to appear.
6 J.M. Ball & V. J. Mizel, Singular minimizers for regular one-
dimensional problmes in the calculus of variations, to appear.
7 J.M. Ball & V. J. Mizel, in preparation.
8 L. Cesari, 'Optimization - Theory and Applications', Springer-
Verlag, New York-Heidelberg-Berlin, 1983.
9 B. Mania, Sopra un esempio di Lavrentieff, Boll. Un. Mat. Ital.
13 (1934), 147-153.
10 L. Tonelli, 'Fondamenti di Calcolo delle Variazioni', Vol. 2,
Zanichelli, Bologna, 1923.
GEOMETRICAL METHODS IN SOME BIFURCATION PROBLEMS OF ELASTICITY
E.Buzano G.Geymonat
Dipartimento di Matematica Dipartimento di Matematica
Universit& di Torino Politecnico di Torino
Via Carlo Alberto i0 C.so Duca degli Abruzzi 24
1-10123 TORINO - Italy I-i0129 TORINO - Italy
i. INTRODUCTION
i.I. The general theory of buckling and post-buckling behavior of elastic structures
was enunciated by Koiter in 1945 and subsequently there has been a considerable
amount of research in this field from theoretical, numerical and experimental point
of view.
From a mathematical point of view the buckling corresponds to a bifurcationand
so much interest has been devoted to abstract bifurcation theory. In this paper we
shall consider the situation where the linearized problem has an eigenvalue of
finite dimension. It is interesting to remark that the classical results of Cran -
dall-Rabinowitz and Rabinowitz do not apply if the eigenvalue is of even dimension.
Tipieal examples of that situation are the following.
Example :i Mode Jumping in the Buckling of a Rectangular Plate 17. The undeformed
plate ~ = 0, ~x0,~ is subjected to a load ~ applied at the ends z = 0 and
1
z i = £z. The boundary conditions considered in 17 are )i( clamped on the ends
and simply supported on the sides z = 0 and z = Z or (ii) simply supported all
2 2
around. The yon Karman equations for w, the z -deflection of the plate, are the
3
Euler equations of the even functional
fCw,~) = -7- 1 llAwll 2 - -7-Itw X lr 5 ÷ --~- i 11 a N -1 rw,wl II 2
1
lr
where N denotes the L2-norm, A -I is the inverse of the Laplacian with Neumann
N
2
boundary conditions, w,w = 2 (w w - w ); the functional is defined on the
zlz I z2z 2 ZlZ 2
2
subspace H of H )~( of functions satisfying the stable boundary conditions, i.e.
7o w = ~ all around and 71w = 0 on the ends in the case )i( and 7oW = 0 all
around in the case (ii). The smallest eigenvalue of the linearized Euler equation
is double if and only if Z =/ k(k+2) in case )i( and £ =/ k(k+l) in case (ii).Then