Invariance principle on Banach space for generalized thermoelastic stability analysi

Infante, E. F.

Slemrod, M.

English

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CDS TECHNICAL. REPORT 69-3
NL SLFAIROD AND E. F. INFANTE
AN INVARIANCE. PRINCIPLE FOR DYNAMICAL
SYSTEMS ON BANACH SPACE: APPI,ICA"!ON
TO THE GENERAL PROBLEM OF THERMO ' ASTIC
STABILITY
JUNE: 17, 1969
316j' 69-
(ACCESSION NUMBER,
(PAE5,
(NASA CR OR TMX OR AD NUMBER,
CENTER FOR DYNAMICAL SYSTEMS
(VARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS ON BANACH SPACE:
.CATION TO THE GENERAL PROBLEM OF THERMOELASTIC STABILITY+
by
M. Slemrod
and
E. F. Infante
Center for Dynamical Systems
Division of Applied Mathematics
Brown University
Providence, Rhode Island
+This research was supported in part by the National Aeronautics and Space
Administration.  under grant NGL-40-002-015 $ and iqpart by the United
States Army, Durham,, under contract DA-31-124-ARO 	 70.
AN INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS ON BANACH SPACE:
APPLICATION TO THE GENERAL PROBLEM OF THERMOELASTIC STABILITY
M. Slemrod and E. F. Infante
1. Introduction
Elastic stability is usually discussed from strictly mechanical con-
siderations. Recently.  however.  attempts have been made to analyze the
influence of the usually neglected thermodynamic properties of elastic
materials. More specifically ,  one may ask what effects the second law of
thermodynamics has on the asymptotic stability of equilibrium of thermo-
elastic materials.
KOITER [1] has studied the general nonlinear thermoelastic problem.,
and for materials with internal friction he obtains asymptotic stability of
the equilibrium solutions. ERICKSEN [2] has posed the question as to the
asymptotic stability of the equilibrium solutions of elastic materials with-
out imposing the assumption of internal friction. DAFERMOS [37 answered this
question to some degree by obtaining a description of the states that the
material approaches as t - ► -.
This same question is studied here in a more general setting than was
done by SLEMROD [4], but in the same spirit: it is shown that the results of
[3] can be obtained as a simple application of an invariance principle for
abstract dynamical systems [4,5].
2. Mathematical Preliminaries
The principal analytical tool to be used is a generalization due to
HALE [5] for abstract dynamical systems of the well known invariance
principle of LASALLE [6] for ordinary differential equations. The following
-2-
brief presentation of this tool emphasizes notation and concepts to be used
in studying the problem of thermoelastic stability.
Let R+ = [O. m) and let -4 be a Banach space with norm ^^m^^ for
e -0. Then,
Definition 2.1. u is a dynamical system on a Banach space 	 if u
is a function u: R+ x.0 -i.V	 such that u is continuous, 1(0
., ,P ) = mP
u(t+T, m) = u(t.,u(T.,O)) for all t . T ? 0 and all m in ,4. The positive
orbit 0+ (0) through 0 in -4 is defined as 0+ (m) = U u(t.,O). A point
t?0
in -4 is an equilibrium point if 0+ (*) = *.
This set of definitions simply generalize familiar notions from the
theory of differential equations to dynamical systems.
Definition 2.2. A set	 M	 in .^ is a positively invariant set of
the dynamical system u if for each 4) in M., e (0) C M.
Definition 2.3. A set M in
	 is an invariant set of the dynamical
system u if for each (V in M there exists a function U(s^ ^)^ U(O ., O) = ¢
defined and in M for a e (-^^w) and such that u(t..U(s^m)) = U(t+s,O)) for
all teR.
Definition 2.2 is well known. The second definition is used to extend
backward in time those solutions of the dynamical system which lie in an in-
variant set. It is clear that if a set M is invariant it is positively
invariant but the converse is
,,
 in general., false.
-3-
Definition 2.4, If u is a dynamical system on .4 and V is 4
continuous scalar functional on .q X define the functional
^ 1 [V(u(t,m)) - qt)].
Then
Definition 2.5. V:	 -► R is said to be a Liapunov functional on a
set G in -4 if V is continuous on G, the closure of G, and if
V(0) s 0 for 0 in G. Furthermore, denote by S the set S = (0 in
GjV(m) = 0) and let M be the largest invariant set in S for the dynamical
system u. With these definitions it is then possible to prove:
Theorem 2.1 (HALE [5]). Let u be a dynamical system on .V. If V
is a Liapunov functional on G and a positive orbit 0+ ( 40) belongs to G and
is in a compact set of .q then u(t^ 40) -* M as t a*.
It is self-evident that in applications to the problem of asymptotic sta-
bility of an equilibrium point * it is necessary to show that M 	 More-
over, it should be emphasized that the usefulness of this theorem in applications
depends on the very relaxed assumptions imposed on the Liapunov functional V
and its derivative V. These conditions should be compared to the much stronger
conditions imposed by standard asymptotic stability theorems ( see
., 
for example,
PARKS [71).
3. Constitutive Equations of Linear Thermoelasticity
A material point is identified by x = (x l
,x2i x3 ) in its state of
-4-
equilibrium (no stresses, constant temperature - yo). The displacement field
at some time t following an initial disturbance at time t - 0 is given by
u(j,t) and the temperature deviation by T(jp t); P(x) denotes the density
at x. in the equilibrium state.
Let a be an open, bounded, connected set in E 3 which is properly
regular [8]; let 262 denote the boundary of a. The constitutive equations
of thermoelasticity are taken then in the form
PBi - (CijkJuk
, d , j - ( miJT),jj.	 (3.1)
PCDI + mijroui,j - ( KiiT.4 J) .0i; 	(3.2)
where body forces and heat sources have been excluded. In these equations
Ci j kl = Cj iki w Ckji j' mi3 = m j i , Ki3 - Ka i and CD, P,, Ci.,j k.8' mi j and
Kii are assumed to be smooth functions of x.
Let now to
 > 0. By a classical solution of the mixed initial-boundary
value problem in A X (O,to) we mean a pair (u,T) satisfying equations
(3.1) and (3.2) together with the boundary conditions
u - 0 on W X (O,to) (closed boundary),	 (3.3)
T - 0 on 262 X (0,to) (constant temperature);
	 (3.4)
and with initial conditions
(u(x,o))A(jO), T(x,O)) - ( 2jo(E), uo(x.), To( x )),
	
(3.5)
where io(x), uo(x) and To(x) are given functions on n.
4. The Thermoelastic Problem as a D.vnamical Svstem
In this section we show, by recalling some results of DAF'ERMDS [31,
that the generalized solutions of the mixed initial boundary value problem
described above can be viewed on an appropriate Banach space as a dynamical
-S-
system (ZUBOV [91). Once this is done, the application of Theorem 2.1 per-
mits us to draw immediate conclusions on the asymptotic behavior of the solu-
tions of our problem.
Consider the Sobolev spaces W2k)(SZ) and W2o ) ( n) , k = 1,2,...
(see, for instance SOBOLEV [10, 11], AGNDN [121). Assume that
ess inf p(x) > 0, ess inf CD(x) > 0 0	 (4.1)S2	 i2
Kjj tit i ? Cititp C1 > 0 constant,	 (4.2)
(a reformulation of the Clausius-Duhem inequality, LANDAU and LIFSHITZ [13])
and for all vi E W20 ) (S2)
fnC i j ki i,j rk' Id—' C2f n'i, jvi, Jdx, C2 > 0 constant( 4.3)
a general property of the tensor of the elastic modulii for infinitesimal elas-
ticity (TRUESDELL and NOLL [14]).
Define now the spaces Ho	
P
01) W& ) (a) X L2(S2) X L2(n) with norm
2
I (v i w i ^ R)I o = fn[ pw iw i + Ci jkjvi'j vk' j + ^,^t2]
0 onto
dx
 and H(n) = W20
W	
)(0) X
20) (n) X W20) (n).	 Define the map P: HoM -^ Hl(il) sending (vi ^wi'R) E
Ho(n) onto (ui^vi^T) E H1(n) C H(St) where (ui,T) E W20 )(n ) X W20) (a ) is
defined by the solution of'the system
f S2 Ci j k1ukP I9i' J dx. _ -f n[ Pwiei + mi j T8i13 ] dx
fnKiiT'?'idx = fD[ pCDR + mij rovi' j ]D dx,
for every DjPei E W^0) (n). The mapping P is linear, well defined on Ho(n)
and one to one. Hence, defining Pm=P°P°...° P let Hm(n) denote the range
of the map Pm. It is clear that ml exists and maps Hm(tl) onto Ho(n).
Let * e Hm(n) and define ( *I m = ( m11y 1 o, Thent
L' 4.11 (DAFEWS [33_). Hm is a Hanach space with norm I'Im.
H0(n) :) H( n) :) ... :D Hm(a) algebraically and topologically. Furthermore,
Hm(') is dense in H, (n) for m 3 I and the imbedding Is Hm( ') -+ Him
is compact.
Let us now define appropriately a generalized solution of our prob-
lems
Definition 4.1. ( ui tLi T) will be calleu a generalized solution
of (3.1) - (3.5) on A x (O,t0) if for all smooth test functions (vi,R) with
compact support on 0 and vanishing on it X 0.
to
fo fn( ( t-to)1P1hiVi Oijkl
 K,lvi, j + mirip j +
• PD TR .+ mi jui ^
 jR] + PuiIi + P CD TR +
TO	 0	 ( 4.4)
• mi jui, jR i f o(Ki i) jT dt) dxdt •,
o P
_ 
-tof J P60ivil t*0 + ro o I ts0 + mi juoi 
jR ( t„0]dx
With this definition it follows thats
Theorem 4.1 (DAFERmc)s [31). Under assumptions (4.1) - (4.3) the
_ F
triple ( ui fis T) describes a dynamical system on Hm(S2), m = 0,1,2,...,s	 #
where ( u t! T) is the generalized solution to the equations of linear
-
thermoelastieity satisfying equation (4.4). Furthermore, for t in (O,t0)	 -^
I ( uiPAi ^T ) ( t )I m + of tfnKi jT (i) T(^) dxdr .
op 0..
where T(m) (x,t) denotes the generalized mth derivative in time of T(x,t).^
5. Stability Analysis
The problem of thermoelastic stability has now been put in a setting
l
	 I
-7-
appropriate for the application of Theorem 2.1, which allows us to obtain
stability results in a simple and direct manner.
For this purpose, fix m it 1. ' Then, by Theorem 4.1 and (4.5) it
follows that for any initial data in Hm(n) the trajectory (uipai.T)(t)
will lie in a bounded set of Hm(n) for any t > 0. Hence, by Lemma 4.1 the
trajectory remains in a compact set G of HA(n), A < m. But then all the hy-
pothesis of Theorem 2.1 are met with -4 = Hi(n). For simplicity let I = 0
and V J(ui ui T)1 2 From (4.2) and ( 4.5) it immediately follows that
s
V : _ JngijT iT jdx s -c3^IT11L2, c3 > 0; therefore the set S is the set
	
o	 ' '
S ((ui ui T) a H0(a)	 11 T11
L2
 = 0). Le;
 now M+ be the largest positively
	
^	 $
invariant set in S. By the definition of generalized solution (4.4) it
follows that M+ _ (( ui iui T) a Ho(n) I JIT(- ,t)ll
L2
 = 0 for t ? 0). Choosing
now vi = 0 in (4.4) it follows that for (u i ui T) € e it is necessary
that
JoofSl dt (t-to)R)mijui,jdxdt = -toJnmijuo R dx
i,j
for every test function R. Choosing this function as R(x,t) _ ^ th(^)
t-t0
where q(x) is an arbitrary test function and cult) is the C°D "bump"
function of Serrin (15J, it follows that
Jnmlj('—`)ui,j(^ t)dx = feij (x)uoi ^ 3(x)d! t i 0
for (ui
 iii T) in M+. Hence, Theorem 2.1 applied to this context yields the
desired results
Theorem 5.1: For any initial data ( u ,	 ,T) in Hm, m it 1, and
°i 
ii 
°i
under assumptions (4.1) - (4.3) ( ui,iii,T) ( t) approaches asymptotically the set
11--i 11 {(vi,i►i,R) € W20 	X L2(n) X L2(n)1jnmij(2)Syi
,j (x,t)dx = jm ( X)u	 (X)dj*ij	 oi^j
t 0, R = 0} in the norm of the space W20
 X L^(n) X L2(n).
-8-
References
1. XDITER, W. T.s Report No. 3 60, Laboratory of Engineering Mechanics, De-
partment of Mechanical Engineering, Technological University Delft.
April, 1967.
2. ERICKSEN, J. L.s Internat. J. Solids and Structures 2, (1966) P. 573-580.
3. DAFERNDS, C. M.: Arch. for Rat. Mech. and Analysis 2^, (1968) p. 241-271.
4. SLEMROD, M.: Ph.D. Dissertation, Division of App. Math., Brown Univ.,
June, 1969.
5. HALE, J. K.: J. of Math. Anal. and Appl. 26, (1969) P. 39-59•
6. LASALLE, J. P.: J. of Diff. Eqns. 4, (1968) P. 57-65.
7. PARKS, P. C.s Intl Symp. on Diff. Eqns. and Dynamical Systems, ed. J. K.
Hale and J. P. LaSalle, New Yorks Academic Press 1967, p. 287-298.
8. FICHERA, G.s Lect. on Elliptic Boundary Di:rferential Systems and Eigen-
value P+.-oblems, Berlin-Gottingen-Heidelberg: Springer 1965, p. 21.
9. ZUBOV, V. I.s Methods of A. M. Liapunov and their Application, translations
Groniugene P. Noordhoff, Ltd. 1964.
1U. SOBCLEV, S. L.s Applications of Functional Analysis in Mathematical
Physics, Leningrad, 1950, English translation by F. Browder, Amer. Math.
loc., Providence, R. I. 1963.
11. SOBOLPV, S. L.s Sur les Equations aux Derivees Partielles Hyperboliques
Non-lineaires, Raines Edizioni Cremonese 1961.
12. AGMDN, S.: Lectures on Elliptic Boundary Value Problems, Princeton:
Van Nostrand, 1965.
13. LANDAU, L. D. and E. M. LIFSHITZ: Theory of Elasticity, Reading, Masss
Addison-Wesley 1959, p. 121.
14. TRUESDELL, C. and W. NOLL: The Nan-linear Field Theories of Mechanics,
Berlin - Gottingen - Heidelberg: Springer 1965.
15. SERRIN, J.s Proe. Symp. on Nonlinear Prob., ed. R. E. Langer, Madisons
Univ. of Wisc. 1963, p. 79.


An invariance principle for dynamical systems on Banach space - Application to the general problem of thermoelastic stability

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