Bloch expansion in generalized thermoelasticity

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Int. J. Engng Sci. Vol. 27, No. 1, pp. 55-62, 1989 Printed in Great Britain. All rights reserved

Copyright @I 1989Pergamon Press plc

BLOCH EXPANSION

IN GENERALIZED

N. TURF& Laboratoire de Modelisation en Mkanique, Tour 66, Universitk Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France Abstract-The solution of the equations of linear thermoelasticity for a composite material with a periodic structure is expressed by means of Bloch expansion techniques. This allows a comparison with the homogenized method.

1. INTRODUCTION In the linear dynamical theory of the~oeleasti~ity, the evolution of displacement field is by a wave-type equation while the temperature field is subjected to a diffusion-type equation. This fact implied that a mechanical or thermal disturbance has an immediate effect at long distance points from the perturbation source and this leads to a notion of infinite velocity of propagation. To avoid this phenomena, physically inadmissible, a modified Fourier law was proposed by several authors [I]. From this law, a generalized thermoelasticity theory was constructed and used to the study of isotropic and, more recently anisotropic media [2,3]. It must be noted also that, in this theory, a lot of various problems are now solved 14, 51. This paper is a contribution, in the framework of generalized thermoelasticity, to the study of wave propagation in periodic media. First, we set the problem in an appropriate functional form from which existence and uniqueness of the solution are deduced. This solution is constructed by means of Bloch expansion techniques [6,7]. As an application of this expansion, an approximation of the solution is obtained when we assume that the data are slowly variable functions compared to the period of the material. This approximation is nothing but the solution of the homogenized problem, studied in classic theory by H. I. En6 [S] and 6. I. Pasa [91. governed

2.

FORMULATION

OF THE PROBLEM

2.1. Local equations We consider an unbounded thermoelastic material with a periodic structure (laminated media, media reinforced by fibers, etc.). More precisely, let Y be the set (IO, I[)” c iw3, We assume that all the characteristic constants of the medium: density pO(y), eiastic tensor Cijkl(y), strain-temperature tensor bij(Y), thermal conductivity tensor aij(y), are bounded functions on 2~rY (i.e. periodic functions of period 2n in each variable). We also assume that al1 the classic properties of symmetry and positivity of the coeflicients cijkl and aij hold [lo}: cijltn

=

%nij

=

cjilm9

V&ij,Eij= &ji,3&a > 0; Vcjy 3aC,>O;

aij

=

b, = bji

llji,

Cijfm Elm 5ij2 Qb&ijEq aij&tj

L

(1) (2)

(3)

culLL

a,,, LY~are constants and E denotes the complex conjugate of E; O
(4) u(y, t) and @ the temperature The equations of balance of

N.TURBl?

56

momentum and energy [l, lo] are

(5)

In eqn (6), C, and to are respectively the specific heat at constant strain and thermal relaxation time, introduced in the modified Fourier law. Note that the usual thermoelasticity theory holds for r. = 0 and that, in uncoupled theory, displacements terms of (6) are neglected.

Problems (5) and (6) can be expressed as an initial value problem in some Hilbert space. Let us introduce H = L2(R3), space of the square power summable functions. Also let V be the space H’(llP) = {f E L2(R3), $ E L2(lR’)) I

Vi and a for eqns (5) and (6) we have, after integration following equivalent form of eqns (5) and (6): By taking test functions

dul dVi

cijlm ay,

dy

by parts, the

VviEH’([W3)

_

ayj

This global form of the equation suggests the introduction of operators denoted by C, B and A. Weighted inner products with weights poCE and p. respectively, are defined on spaces H and H3 (resp.). Operator C is the associated operator (see [ll]) to the sesquilinear form u, v

v3,

E

c(u, v) =

I cijlm34 acdy ay,

lQ3

ayj

[i.e. c is linear (resp. antilinear) with respect to the first (resp. second) argument]. way, B is associated to the form b: a

E

v,

v

v3,

E

b(a, u)=~a3bijo~dy=~n,

Oi$(bija)dy

In the same

(8)

I

I

In the sequel, B associated to b(cv, v) will be used. Finally, A is associated to the form a: 8, cy E

The problem is now reformulated:

v,

a(e,

a) =

aeda

aij-dy aYj aYi

find (u, 0), function of t with values in H3 x H, such that d2u

z+Cu-BB=O

de dt+

u(0) = u”,

(9)

2 ro$+AB+

g (0) = ul,

(10)

T,fi

(11) e(o) = 80,

$ (0) = 81

(12)

Bloch expansion in generalized themoelasticity

51

We assume that u”, ul, 8’, f? are given functions such that u” E v3,

u1 E H3,

#E H

e” E v,

(13)

From (2) and (4) we deduce that the forms c and a in (7) and (9) are coercive on V3 and V (resp.). There exist positive constants PO and /I1 such that vu E v3,

c(u, u) 3 Bo llUll2,~

veev,

a(@ 0) z=B1lwl:

So, in the classic framework of semigroup theory (see for instance [ll]), the necessary conditions to define an operator that is a generator of contraction semigroup in an adequate space are fulfilled and, by the way of Lumer-Phillips theorem, we conclude to the existence and uniqueness of the solution of the problem (lo)-(13).

3. BLOCH 3.1. Operators C(k), A(k),

EXPANSION

OF THE SOLUTION

B(k)

In the same way as in [6] and [7], the periodic structure of the material leads us to introduce operators acting on functions that are defined on the basic cell 23dY. Let Hi(2~rY) be the space of functions of H’(2nY) which take equal values on two opposite points of two opposite sides of the cell 2~rY. The space (~5*(2nY))~ being equipped with the weight inner product of weight po, C(k), with k E Y, are defined by u, v E (H;(~JEY))~,

(C(k)u,

‘)=l_~C~~,~(~+ik,)u,(~+iS)vidy

(14)

Space ,!,*(2~rY) is also equipped with a weighted inner product of weight poC,. A(k), are the operators

8, a E H;(~JGY),

(A(k)B,

a)=!

k E Y,

~ij(~+ik,)e($+iki)ndy 2nY

(15)

I

Properties (2) and (3) involve that C(k) and A(k) are positive, self-adjoint operators with compact resolvents, i.e. that exist constants y. and y1 such that (C(k) + yo)-’ and (A(k) + yl)-l are compact operators, with eigenvalues and eigenvectors. So, for each k E Y, there exist sequels of eigenvalues 0 < o;(k) S w:(k) - - -* +a (resp. 0 G L:(k) 6 A:(k) - - .+ +m) with corresponding eigenvectors QJ’(Y;k), $(y ;k) - - * (resp. W’(y; k), #(y; k) * - *) of operators C(k), (A(k) resp.), which form an orthonormal basis in (L*(~J~Y))~(L*(~J~Y) resp.). Furthermore, we introduce another family B(k), k E Y, 8 E H;(2nY),

u E (H;(~JTY))~,

(B(k)09

u)=InYbije(~+ikj)uidy I

(16)

3.2. Representation of the solution

Any function of (J~*(IW~))~,L2(lR3) respectively, with complex values, can be expanded means of eigenfunctions of the operators C(k), A(k) resp. [6, 71: u E (J~*([W~))~ u(y) = t&(k) =

IY

eik.y dk 5

I R’

Li,(k)cpm(y;

by

k)

m=O

e-‘“?&)@;“(x;

k)po(x) dx

(17)

where k . x = k,Xi; 8 E L2([w3)

8(y) = j- eik.y dk 5 Y

&(k)qP(y;

k)

n=O

8,,(k) = Ls emik.IwP(x;

k)po(x)C&)

dr

(18)

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58

The solution u(y, t), B(y, t) of problem (ll)-(13) has an expansion like (17), (18), with coeflicients 2, and 8, depending on t. These expressions are introduced into eqns (11) and (12) and it results, by “projections” on vectors 9pmand @“, that for each k E Y, C,(k, t) and &(k, t) are solutions of

[no summation on m in (19) and n in (20)]. The coefficients b,,(k) in (19) and (20) are

Initial conditions are deduced from (12), using (17) and (18):

li,(k, 0) =

I3e-ik%~(x)@;“(x;k)p&)

Analogous expressions hold for d&/dt

d.x

(22)

(k, 0), 8,(k, 0) and d&/dt (k, 0).

4. HOMOGENIZATION

4.1. Setting of the problem Let E be a small positive parameter. functions of y by means of E:

We assume that the initial data are slowly varying

U(Yt 0) = UO(&Y)>

$(Y,

O)=u’(v)

@(Y, 0) = @(EY),

$(Y,

0) = @‘(q)

(231

The period of the medium seems to be small compared to the data scale and it is natural to find, as an approximation of the solution, the solution of the homogenenized problem [ll]. In this new problem, two space variables appear: x = my slow variable of the initial data and y, fast variable of the period of the medium. Naturally, this problem is described with the variable x but, in order to use Section 3, we achieve the change x = my and the equations are (24)

(25)

POG (

System (24)-(25), associated to initial conditions (23), has the same form as system (5)-(6), with 2nY periodic coefficients. So, the Bloch expansion theory of Section 3 can be applied. The solution of (24)-(25) has an expansion like (17)-(18) and, from these expressions, we look for the limit of the solution when E tends to zero. Previous study [ll] suggest the change k = EK. More, the behaviour of the solution is governed by the functions

2 &z(EK, t, E)#%‘;

E3ii= E3ii(EK, y, t, E) =

EK)

(26)

EK)

(27)

m=O

E’6

=

E3&&, y, t, E) =

2

&(EK,

n=O

t, E)$J”(y;

Bloch expansion in

59

generalizedthermoelasticity

These functions EMU’ E (L.‘(~J~Y))~ and ~~8 E L*(~J~Y) are solutions of the system (see [7]):

2 (E%)+ $ C(&K)E31i- ‘,B(&K)E38= 0 -$s36)+ r,,$(F’~)i$A(~K)~~~+~~(eK)($(e~fi)+

(28) ~&(e~fi))

=O

(29)

with initial conditions deduced from (17) and (18). For instance, &K)JR3e -““?4p(&x)@(X;

&K)p,(X)&3 dx

A(eK),

We must establish

&34=o = .ZO V(Y; In (28)-(29) properties.

appear

the operators

4.2. Properties of the operators

B(eK),

C(eK).

(30) again their

A(EK),B(EK),C(EK)

In [6]and 171, the following behaviour was settled. For m # 0, the eigenvalue simple or multiple, it is possible to find a vector @“(y; 0) such that Oi(&K) = o;(o)

+ O(E);

V(Y

; EK) =

w:(O), being

V(y ; 0) + O(E)

So we have C(&OPm(Y;

EK) = &l(0)~“(Y;

0) + O(E)

For m = 0, w;(O) = 0 is an eigenvalue associated to any arbitrary constant vector. This eigenvalue produces three holomorphic branches of eigenvalues and 09(&K) and ~‘(y; &K) admit the expansions Oi(&K) = &%22(K) + 0(E3) v’(Y; &K) = (P’(Y; 0) + ET:~& + ~2~~pqKPKq+ O(E~) where the constant vector q”(y ; 0), which depends on K, is one of the three eigenvectors Q(‘)(K) (r = 1, 2, 3), orthonormed vectors in (L’(~J~Y))~, associated with the eigenvalues S&(K). In [7] we also proved that qvP is function of ‘p”(y ; 0) and W’, vectors which are introduced in the homogenization theory of elasticity problems [ll]. It results that C(eK)cp’(y;

The operator A(eK)

EK) = e29$,(K)@“‘(K)

has same properties N&K)V”(y

;

+ 0(e3)

(r = 1, 2, 3)

for small E: for n # 0, it comes [6] EK) = %(O)w”(y; 0) + O(E)

For n = 0, we have A(EK)I,~‘(~;

An estimation of the term (B(eK)$P(y; (B(&K)V(y;&K), (B(eK)q”(y;

EK) = e2A2(K)Y

+ 0(e3)

EK), rp”(y; EK)) also is needed:

cP”(y; &K))=l~~br*“(y;O)~(y;O)dy

EK), q”(y; EK)) = ej-

b,qF(y;

2nY

+O(~);mf0

O)[F

KP + iK/DI”] dy + O(E~)

(31)

(32)

I

written in the sequel (B(&K)V”(y; (B(eK)qP’(y;

&K), V(Y;

&K)) = kz, + O(E)

EK), q”(y; eK)) = ~b:) + O(e)

4.3. Behaviour of the functions e3zi and ~“6

First, we study the initial conditions. In (30) and analogous, we adopt the data scale and we change EX into X. From the following property: if h(x, y) is smooth on R3 x 2nY and has

N. TURB6

60

compact support in x, then

it comes &“ti(O)= (2n)““~&@~‘(K)W(K)

+ O(E)

.s3; C(O) = (2Jr)“‘*&a:&,I’)(K)@~)(K)

+ O(E)

&“e(o) = (2n)3’2(p,C,)-~o~ll,

+ O(E)

Egg (0) = (23G)“‘(p&)-@+r/J

+ O(E)

(33)

where f is the mean value, on 2nY, of the function f and p is the Fourier transform of J Now, for any t > 0, we replace expressions (26) and (27) into eqns (28) and (29). Equation (28) is projected on q”(y; EK) (m = 0, 1, . . .) and we identify the successive powers of E. We shall denote &J&K, 1, E) = liO,(K, t) + Efi;(K, &+K,

t, E) = &K,

t) + E&K,

t) + O(E2)

(34)

t) + O(E2)

(For m = 0, @, represents one of the B,,,, r = 1, 2, 3.) At order c-* for m # 0 and n f 0, we have

&(K,

liO,(K,t) = 0;

t) = 0

according to the result (33). At order c-l, it comes CL&(O)& - 6,@

= 0;

n:(o) e: = 0

(35)

(no sum on m and n). As a consequence, for E+ 0, the significant terms of the solution (u, 6) are uo = uo(x, t) =

13~ =

t) =

0,(x,

I

(36)

eiK.I dK@Y IlR3

(37)

eiK.xdK i &,,(K, t)@“(K) r=l r%s

where fi,,, and 6: are solutions of equations obtained by the projections and (29) on I+!J’(Y;EK). So at order co, we have after the use of (35) d* 2 ti,, + B$,(K)fi,,,

dt 1

+

T

0

2 lboml*de: I m=l

4.4. Homogenized

w;(O)

>( dt

t

’

- l@(K)@

d*@+A*(K)&+ -) dt*

of (28) on

q”(y

= 0

Torgl @+(K)(F+

;

EK)

(38) ~~3)

=0

(39)

constitutive law

From (38) and (39), we prove that the approximation (uo, 0,) is solution of the homogenized equations. This property results from several remarks. The derivative of uo, or eo, with respect to xi is equivalent to the product by iKj of the expansion of uo, or Bo. Also, we must state the dependence of Q$, and A* on K. In [7] we proved (40)

61

Bloch expansion in generalized thermoelasticity

where ct,,,, is the classical homogenized

coefficients [ll]

c;,rn = Eiilm+ (cijkhz)-

(41)

The vectors W”” being defined by W’” E (Hi(2JrY))3 Cijkh

aWArndOi - dy = aYk

$ I 2nY

aYj

cij,,,,ijidy

v?J

E

(2$(2xY))3

Yi

(42)

Also we have a;KiKj; .?

A2(K) =&-

(43)

with functions Wj which satisfy Wk E H32nY)

awk ati!

a..--dy ”

Va E H>(2?rY)

=

3Yj aYi

In (38) and (39), the term b$‘(K) defined by (32) has to be transformed.

(44)

From [8],

@;) = i@$,@ Let us introduce the vector z E HL(~~GY) of [8] or [9]:

az1a”

cijJm ay,

dy

vu E (H;(2JrY))3

=

ayi

(45)

Then, taking in account (45) with v = Wkp and (42) with v = z, we have br’(K) = -iY\

(cijkpz Kp6g’ + bljKjQp’) dy 2nY I

Let us define like in [8] or [9] (46)

Then b$‘(K)@r’(K)

= -iKj’Pz

(47)

Finally, we must study the coefficient bo,. From (31), using (45) with v = @“(y; 0), we have born = -w;(O)(z,

V(Y;

O)P’

It results that

2 PornI = -5TPoCJ m=l dz@) where, like in [8] or [9] (48)

The equations satisfied by (uO, 0,) are easily deduced from all these remarks: (49) (50)

62

N. TURBfi

Equations (49)-(50) are the macroscopic equations, obtained for z. = 0, by EnC [8]. Pasa [9] proved that the asymptotic process E+ 0 (with to = 0) is convergent. In the case z. # 0, we can expect that this property holds. It also would be of great interest to analyze the effects of temperature on the relaxation time and coupling coefficients and to study the behaviour of the solutions in all different cases.

REFERENCES [l] H. W. LORD and Y. SHULMAN, J. Mech. Phys. Solids 19, 299 (1967). [2] J. N. SHARMA, ht. J. Engng Sci. 25, 463 (1987). [3] J. N. SHARMA and R. S. SIDHU, Znt. J. Engng Sci. 24, 1511 (1986). (41 Y. ARGANAL, 1. Elasticity 8, 171 (1978). [5] C. V. MASSALAS, Acta me&. 65, 51 (1986). [6] A. BENSOUSSAN, J. L. LIONS and G. PAPANICOLAOU, Asympfotic Analysis for Periodic Structures. North-Holtand, Amsterdam (1978). [7] N. TURBF, Math. Meth. appl. Sci. 4, 433 (1982). [8] H. I. ENE, Int. J. Engng Sci. 21,443 (1983). [9] G. I. PASA, ht. J. Engng Sci. 21, 1313 (1983). [lo] P. GERMAIN, Cows de Mhxznique. Masson, Paris (1986). [ll] E. SANCHEZ-PALENCIA, Non Homogeneous Media and Vibration Theory. Springer, Berlin (1980). (Received 29 October 1987; received for publication 5 May 1988)