thermoelasticity of initially stressed bodies, asymptotic equipartition of energies

Pergamon

PII:

Int. Z Engng Sci. Vol. 36, No. 1, pp. 73~6, 1998 ~) 1997 Elsevier Science Limited. All rights reserved

Printed in Great Britain S0020.7225(97)00019.0 0020-7225/98 $19.00+0.00

THERMOELASTICITY OF INITIALLY STRESSED BODIES. A S Y M P T O T I C E Q U I P A R T I T I O N O F E N E R G I E S

M. MARIN and C. MARINESCU

Dept. of Mathematics, "Transylvania" Univ., Str. Iuliu Maniu, 50, 2200 Brasov, Romania

(Communicated by E. S. SUHUBi)

Abstract--The purpose of this work is to study the asymptotic partition of total energy for the solutions of the mixed initial boundary value problem within the context of the thermoelasticity of initially stressed bodies. © 1997 Elsevier Science Ltd.

1. INTRODUCTION

The asymptotic equipartition property is a familiar notion in the field of differential equations. In short, this means that the kinetic and the potential energy of a classical solution with finite energy becomes asymptotically equal in mean as time tends to infinity. We find such a property in various papers for physical systems governed by nondissipative hyperbolic partial differential equations or systems of such equations. The system of equations governing our problem consists of hyperbolic equations with dissipation and, therefore, does not belong to one of the categories considered previously in the literature on this topic. By using the dissipative mechanism of the system, we can prove that the equipartition occurs between the mean kinetic and strain energies. Instead of an abstract version of this question, we prefer to emphasize the technique itself on the thermoelasticity of initially stressed bodies. The theory of initially stressed bodies is derived from the theory of micropolar bodies which is introduced by Eringen and Suhubi (see also Refs [4] and [3]). The plan of the paper is as follows. We first write the mixed initial boundary value vroblem within the context of thermoelasticity of initially stressed bodies. Then we establish some Lagrange type identities and we also introduce the Cesaro means of various parts of the total energy associated to the solutions. Finally, we establish, based on previous estimations, the relations that describe the asymptotic behaviour of the mean energies. It should be noted that there are many papers which use the various refinements of the Lagrange identity such as Refs [8], [11], [9] and [6]. We also find many papers that use the Cesaro means, such as Refs [2], [8] and so on.

2. B A S I C E Q U A T I O N S

Let B be an open region of three-dimensional euclidian space occupied by the reference configuration of a homogeneous initially stressed body. We assume that B is regular and we denote the closure of B by / ) . The boundary dB of B is closed and bounded. We use a fixed system of rectangular cartesian axes and adopt cartesian tensor notation. Points in B are denoted by x s and t e [0,~) is time. Also, the spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. In the absence of the body force, body couple force and heat supply fields, the field for linear thermoelasticity of initially stressed bodies is

(~s + ~'~sL = ~ai, ]d'ijk,i "[- O'jk 31- Us,~M~k + ~ k i M j i - - ~Okr,iNijr = I~/bs~, (1)

- - qi,i---- pOoi#,(x,t) ~ B × [0,oo) (2)

Equations (1) are the motion equations and equation (2) is the energy equation.

73

74 M. MARIN and C. MARINESCU

In the above equations we have used the following notations: u/components of displacement, q~q components of dipolar displacement, %, o-q,/~qk components of stress, q/components of the heat conduction vector, r / the specific entropy, p the constant reference density, 00 the constant reference temperature, Iq coefficients of inertia.

A superposed dot denotes the differentiation with respect to time, t, and a subscript preceded by a comma denotes the differentiation with respect to the corresponding spatial coordinate. When the reference solid has a centre of symmetry at each point, but is otherwise non-isotropic, then the constitutive equations are

Ti] = Uj,kPki "Jr Cijmn ~, mn "~- amnij~/mn q- FmnrijZmnr -[- D q ( 0 + a 0) ,

o'i] = - ~OjkMik + ~Ojk,rNrik Jr Gqm.e.n. + nijmn'Ymn + Dq,..rZ.n.r + Ei]( 0 --b ol 0) ,

~i jk = ULrNirk -}- Fijkmn~'mn q- D,~.qkemn + AiimnrXmnr q- El]k( 0 "q- Ot 0) ,

qi = - OoKqO j,

On = a + dO + h O - D q e q - E q y q - FqkZi/k,(X,t) ~ B × [0,~) (3)

In the above equations we have used the following geometrical equations

EiJ = 2 (ui'] "~ uj'i)' "~ij = Uj,i -- ~Oj,i, Xijk = ~Ojk,i (4)

The tensor coefficients in equation (3) are constants subject to the symmetry conditions

Cqmn = Cmnij = Cjimn, nqmn = Bmnq, Gqmn = Gqnm,

Aijkmnr = Amnrijk, Fijkmn = Fijknm, g i j = g]i , e q = e j i (5)

In equations (1) and (3), Pq, Mij and N0k are prescribed functions which satisfy the equations

(Pji + Mii),j = O, Nqk.i + Mjk = 0

The density p, the coefficients of inertia, Iq, and temperature 00 are given constants which satisfy the conditions

O >0, 0o>0, Iq>0 (6)

From the entropy production inequality we obtain the following conditions

d a - h > 0, Kq~i~:j- 0, V~:~ (7)

In concordance with the conditions (7) we assume that Cqm., Bqm., Aqkmnr and K 0. are positive definite tensors, i.e.

Cijmn~ij~mn ~ ko~q(q, ko > O, V~ii = (ji,

Bijmn~i]~mn ~ kl~ij~ij, kl > O, V~q,

Aijkmnr(ijk~mnr ~ k2(i jk~ijk, k2 > O, V (ijk ,

KqJ~,~j >-- k3~:~(,, k3 > O, V~:, (8)

Moreover, according to Green and Lindsay [5], we can assume that

a>O,h>O, d a - h > 0 (9)

Now, we assume the following prescribed boundary conditions

u, = 0 on aB~ x [0,oo), (1"j~ + os-,)n j = 0 on aB1 x [0,oo),

~oq = 0 o n c~B 2 X [0 ,oo ) , tzqkni = 0 on aB~ X [0,oo),

0 = 0 on c?B 3 X [0,oo), q,n, = 0 on dB~ X [0,oo), (10)

where 3Ba, c~Bz, e)n3 and dB~, 8B~, c~B~ are subsets of 8B and their complements with respect to

Thermoelasticity of initially stressed bodies. Asymptotic equipartition of energies 75

aB, and ni are the components of the unit outward normal to c~B. Introducing equation (3) into equations (1) and (2), we obtain the following system

~Ui = [Uj,kPki-t- (Cjimn + Gfimn)F, mn+ (Gmn fi + njimn)~t mn + + (Fmnrj i + Ojimnr)Xmnr - q~ykMik

"~ ~Ojk,rNri k + (Dji + Eji)( 0 + ot ~))]j,

Iks~js = [Ui,rNirk + Fiikmn~mn + Omnijk~/mn + Aijkm.rZ,,,nr + Fqk( O + a 0)],,

-~- -- ~ j i~ l k i "q- ~Dji,rNrki "~ Gjkmn~'mn + ajkmn~/mn "~- Oik.n.rZm.r + Ejk( 0 + a 0) ( 1 1 )

hO = - dO + D~jk# + E#~'ij + FokZqk + Kifl,ij, (x,t) e B x [0,oo) (12)

Furthermore, we assume the following initial conditions

u,(x,O) = u°(x), i,,(x,O) = u°(x), ~#(x,O) = ~°-(x), ~#(x,O) = ~o°(x), O(x,O) = O°(x), O(x,O) = 0 % ) (13)

By a solution of the mixed initial boundary value problem of the thermoelasticity of initially stressed bodies in the cylinder flo = B × [0,~) we mean an ordered array (u~,~j,O) which satisfies equations (11) and (12) for all (x,0~f~o, the boundary conditions in equation (10) and the initial conditions in equation (13). We observe that if meas aB1 = 0 and meas 3B2 = 0, then there exists a family of rigid motions and null temperature which satisfy the equations (11), (12) and the boundary conditions (10). For this reason we decompose the initial data u °, ~o, /~o, ~o., as follows

• , 0 . 0 _ . , U 0 = U.* t .~_ U O, [,lo = u . . t .]_ ~fo, ~oo = ~oij .~_ f~ij, ~oij - ~ i j "~- + 0 ( 1 4 )

where u*, ~ , u*, ~0" * are determined so that Ui,° O0 ,o /_)~,, O,.j* satisfy

fB(oU° + I~'I'°~)dV=O, fB(OO° + I~*°~)dV=O (15)

If meas ~Ba = 0 and meas c~B2 # O, then we have the restriction

f a o U ° d V = O , fB o u ° d V = O

Finally, if meas cgB 3 = 0, then there exists a family of constant temperatures and null motion, which satisfy equations (11), (12) and the boundary conditions (10). Therefore, we decompose the initial data O °, 00 as follows

0 ° = 0* + T °, 0o = / ) , + ~r o (16)

where 0* and 0" are constants determined so that

fB T°d V = O, fa ~r°d V = 0 (17)

3. SPECIFIC NOTATIONS

We denote by Cm(B) the class of scalar fields possessing derivatives up to the m-th order in B which are continuous on B. For f ~ Cm(B) we define the norm

rn

11 f ]1 c'w)--- ~ ~ m a x ]f,im...ik I k=0 it,i2,...,i k B


By Cm(B) we denote the class of vector fields with twelve components cm(n). For w e Cm(B) we define the norm

12

II w II , ' - ( ~ --- ~ II w, II c-(~ i = 1

By WIn(B) we denote the Hilbert space obtained as the completion of Cm(B) by means of the norm II • II w~(~) induced by the inner product

(Lg)Wm(~) ~ ~, Li,...,~.,,...i~dV

By WIn(B) we denote the completion of Cm(B) by means of the norm [I • II w~(s) induced by the inner product

12

(w,o,)w.(~--- ~ (w,,~,i)w~(~ i = 1

We will use as norm in cartesian product of the normed spaces the sum of the norms of the factor spaces. Let us introduce the following notations

C ( B ) - { z E C t ( B ) : z = O o n c ~ B 3 ; i f m e a s c ~ B 3 = O , then f s z d V = O } ;

{~I(B) --= {(vi,~b0. ) ~ E l ( B ) : v i = 0 o n OB,, #'i~ = 0 on dB2

if meas c~B1 = meas aB2 = 0, then ~a(Ovi + Ik~$js) d V = 0 ; if meas ~B1 - 0 and m e a s aB 2 # 0, then fs ovidV=O}; lYVl(b) -=the completion of CI(B) by means of I I . II wl(~; WI(B) - t h e completion of CI(B) by means of II • II w , ( ~ .

In these relations Wm(B) represent the familiar Sobolev space, (see Ref. [1]), and Win(B) - [WIn(B)] 12. We note that hypothesis (8) assures that the following Korn's inequality, (see also Ref. [7]), holds for all (v, ~b) ~ WI(B),

f B ( CijmnF, ij(1), ~.1) 'if'ran( U, ~l] ) + 2 Gmnijg. i]( V, ~1) ~/ mn( V, ~r l) "[- nijmn'Yij( V, ~1) ~/mn( t.}, ~.l )

+ 2Fm..#e #(v, g/)Z,,,.r(V, ~b) + + 2Di/m.r~/ij(v, ¢)Zm.~(v, ~b) + Pk~Vj,kV~,i -- 2MikVj,,~qk

+ NrikVj,i~bjk,~ + Aijkm.rZi~k(V, ~b)Zm.~ (v, ~b) d V >- ml fB (vivi + vijvij + ¢ijd/i~ + ~ij,k~ii,k) d V (18)

where ml > 0, ~ii(v,~b ) = 1/2(vj, i + vi~), yij(uAb) = vj, i - ~b#, Xijk(V,'P ) = ¢~i. Moreover, under the hypothesis (8), for all Z e ff'l(B) the following Poincare's inequality

holds

f a K i j z , ~ , j d V > - m 2 f B z 2 d V , m2>O (19)

If meas aB1 = 0 and meas 0B2 = 0, then we shall find it is a convenient practice to decompose the solution (u~,q~o-,0) to the form

= = * " * ( 2 0 ) U i U~ + tu* + v~, ~ ~pi~ + tq~q + ~bu, 0 = Z

where ((Vl, ~bli),X) ~ l~gl(B) × 17¢1(B) represents the solution of equations (1), (2) and (10) with the initial conditions

11 i = V O, i) i = ~fO ~ij = I~)O, ~lij -- " 0 - qb#,X = 0 ° , 2 = 0° ,onB, a t t = 0


Let us now consider that 3B3 = 0. Then we use the equations (16), (17) and (2) in order to decompose the solution ((u~,~o),O) to the form

hi ( ,,t)]o,_,_ U i ~- Wi, ~0 O ~- I/I/j, 0 = O* "1- - ~ 1 - exp - ~ - Z (21)

where ((vi, Oo),Z) e ~/I(B) × l~rl(B) represents the solution of equations (1), (2) and (10) with the initial conditions

17i = uO, JJi = ~lO, ~1i] = ~ oO, /~t//- = ~0 O, ,~ = T °, j( = ~r °, on B, at t = 0

4. SOME PRELIMINARY IDENTITIES

In this section we shall establish some evolutionary integral identities which are essential in proving the relations that express the asymptotic partition of energy. The first theorem presents a conservation law of total energy. THEOREM 4.1. l_x'.t ((U~,~%),0) be a solution of the initial boundary value problem defined by equations (11), (12) and (10). We assume that

o o . o . o W o ( B ) , 0 ° O ° W o ( B )

Then the following energy conservation law holds

E(t) -~ - f [ Q~!i(t) iti(t) + Ikr~Ojr(t)~Ojk(t ) + CijmnSij(t ) 8rnn(t) + 2aijmnSij(t)Ymn(t)

+ B0.,.y0(0 ym,(t) + 2Fmnrijeij(t)Zmnr(t) + 2DomnryO(t)Zmnr(t) + Aokm~rZiSk(t)Zm,r(t)

+ PkiUs,g(t)us,i(t ) -- 2MikULi(t ) ~Olk(t ) + NrikULi(t ) ~Ojk,r(t ) + agooa(t ) Oj(t) + d 02(0

+ ah~?2(t) + 2hO(t)O(t) + [KoO,i(s)Oj(s ) + - h)O (s)ldVds = E(O) (22)

PROOF. In view of equation (11) we get

2 ds d

- - - - [ el'li l'li "Jr 6r~jr~Ojk] = [/~j( ~'ji "~- O'ij) + ~Ojk~ijk], i- C j i m n ~ m n ~ j i - Gmnji( ~trnn~ji + ~tmn~'ji)

-- Fmnrj i (Xmnr~j i q- X, mnr~,ji) -- g f imnYmn~j i

-- Djimnr(~,mnr~ji-]-X, mnr~ji) -- A j i kmnr~ , j i kXmn , - PkiUj.ki~j.i

+ Mik(Ui , i~ jk -]- llj,i@jk ) -- Nrik(Uj,i@jk,r + Uj,i~Ojk,r)

-- Dj i ( 0 q- ol O)~ji - E f t ( 0 + ot O)~/ji - F#k( O + a O)}(S,k (23)

Conversely, by using the energy equation (12), we obtain

1 d (Oji~]i 27 Eji~/]i -]- Fj ik)( j ik)( 0 n t- Of O) = -- [ g i j o j ( 0 -~- ol 0)1,i -~- -~- d-s- [d 02 + aKiiO,iO d + ah 02

+ 2001 + KoO,O J + (dot - h~O (24)

such that from equations (23) and (24) by integrating over B × (0,t) and by using the boundary conditions (10), and the initial conditions (13), we arrive at the desired result, equation (22).


THEOREM 4.2. Let ((Ui, ~Di]), O) be a solution of the initial boundary value problem defined by equations (11), (12), (10) and (13). We assume that

o o W , ( B ) , . o . o W o ( ~ ) , 0 ° ~ w , ( B ) , 6 ° Wo(B).

Then the following identity holds

2 f [eui(t)ui(t) + lk~#(t)~(t)]dV + 2 f [(da - h)O2(t) + Ko( f~ O,i(')d')( f~ oJ(¢)d')

(fo ) fo'f. + 2aKqOi(t) Oj(C)d# d E = 2 [~ l i (S ) [ . l i (S ) + I k , ~ # ( S ) ~ k ( S ) - Cijmn,~,i j(s)~mn(S )

- Bqm.Yi j (S)ym.(S) - 2Gmni~eq(S) ym.(S) - 2Fmnrqeij(S)Zm.(S) - 2Oqm~rYi~(S)Zmn(S)

- - A i j k m n r Z i j k ( S ) X m n r ( S ) -- ekiUj,k(S)l l j , i (S ) "b 2MikUj,i(S ) ~ jk (S ) -- 2NrikUj, i (S) ~Ojk,r(S )

- dO2(s) - 2hO(s) O(s) - ahO2(s ) - aKqOi ( s ) O , i ( s ) l d V d s + 2 f o [ e u ° i t ° + I i j (a°ip°]dV o

fof + ( d e - h ) ( O ° ) 2 ( t ) d V - 2 (a - e~7°)[0(s)+ a O ( s ) ] d V d s (25)

where c q ° = a + dO ° + hO ° Dqe o. _ o o - E q y i j - FijkXijk

PROOF. First, by using the equations (11), we obtain

d

ds - - [ Q U i i t i + Igr~Ojr~jg] = [lgi("i'ji q- O'ji) "4- ~ j k / J ~ i j k ) , i - C j i m n S m n S j i - 2GmnjiYmneji - 2Fm,,diXmnreji

- - B j i m n ~ / m n ~ / j i - 2Ojimnr,,~mnr,Yji- AjikmnrXmnrXjik- PkiUj,kUj,i

+ 2Mi~uj,i~j~ - 2N.~uj,i~j~,. - D#(O + ~ O) ~# - E#(O + ,~ O) Yji - Fjig(O + a 6)Zjik + pit iui + Ikr~j~(O~k (26)

However, by using the energy equation (12), we obtain

fo [ + K,j(O; + o~ 6 9 Oj(~)d~ - 1%(

f o + ( d a - h ) O O + K i j O , i O j ( ( ) d ( + d 0 2 + c ~ h 6 Z + 2 h O O + ( a - p ' q ° ) ( O + a0) (27)

From equation (26) and equation (27), it results

d d---~ [ ~ u i [li -4- I k r ~ j r ~ j k ] = [Ui( "l"ji -~ or ji ) -~- ~Djk[Zijk),i - Cjimn~,mnS ji - - 2Gmnji'Ymn13 ji - n j imnrmn~] i

- 2Fm,,rjiZm,,reji - 2Djim,,rXmnP/~i - A#km,,rZm,rZiik - PkiUj,kUj,i + 2MikUy,i~jk -- 2NrikUy,i~Ojk,r

[ fo ] + p ~ f i ~ i + I q ~ d a i + K q ( O + a O ) 0j(~)d~: (a - tgr /° ) (0+ a 6 ) + a K q O , O j

[Io ] i - o~Ko 0 i 0 j ( ~ ' ) d ~ + 0 , 0 ; - KoO, 0~(~ ' )d~ ' - dO 2 - o~h62 - 2bOO - (ao~ - h)O0 (28 )

An integration of the identity (28) over B x (o,t), followed by the use of the boundary


conditions (10)), the initial conditions (13) and the symmetry relations (5), lead to the identity (25) and the proof of Theorem 4.2 is complete.

THEOREM 4.3. Let ((u~,~0~/), 0) be a solution of the initial boundary value problem defined by equations (11), (12), (10) and (13), corresponding to the initial data

(Ui,~Dij)0 0 ~. W I ( B ) , (Ui,~Oi]).0. 0 ~ Wo(B), oo ~ WI(B) , 0o ~ Wo(n )

Then the following identity holds

2 fB [Oui(t)izi(t) + lkr~/r(t)~jk(t)]dV + fB [ [(da- h)OZ(t) + K~/( f/ o,'(~)d~)( f~ o,J(~)dg )

+ fB [(d°t-h)O°O(2t)+aKijO°(f to'j(')d/d)] dV+ fof. (a-co°)(O(t+s) - O(t - s ) + a [ O ( t + s) - O(t - s ) ] ) d V d s . (29)

PROOF. Let f~(x,s) and f~(x,s) be twice continuously differentiable functions with respect to time variable s. It is easy to see that

d

d s - - [ O (fi(s) g,i(s) - f i (s)gi(s))] = O [ f i (s)gi(s) - f i (s)gi(s)]

such that by integrating over B × (O,t), it results

fB'?[fi(t)gi(t)-f;(t)g;(t)]dV=f~fso[f;(s)gi(s)-fi(s)g;(s)]dVds• /

+ fB O[fi(0)g/(0) - f i ( O ) g i ( O ~ d V (30) /

/

By setting f / (x , r )=ui (x , t - O , g i ( x , 7") = U i ( X , t + 7"), 7" ~ [0,t], t ~ (0, °°) the relation (30) becomes

fof ' 2 O u ; ( t ) i t i ( t ) d V = e[u°•,(2t) + i t ° u i ( 2 t ) ] d V + O[u,(t + S)//,(t - s)

.J / i

- u;(t - s)ii;(t + s)]dVds,t E (0,oo) I (31)

Similarly, we have ~-~

fof 2 Ik,~O/,( t )~o/k(t)dV = Ik,[~oO,~o/k(2t) + ~o°,~O/k(2t)]dV + + Ik,[~o#(t + S)~Ojk(t -- S)

-- ~Oj,(t -- S)ib/k(t + s ) ] d V d s , t ~ (0,~) (32)

We now eliminate the inertial terms in the last integral of relations (31) and (32). In view of equation (11) and with aid of the symmetry relations (5), we get

e [ u i ( t + s) t i i ( t - s) - u i ( t - s)[,ii(t d- s)] -b Ikr[ qVjr(t -F S)(Pjk( t -- S) -- ~ j r ( t -- S ) ¢ j k ( t q- S)]

= [u/(t + s)( ' r j i ( t - s) + o'ji(t - s ) ) + u;( t - s ) ( ' r j i ( t + s) + o~ji(t + s))],;

+ [~ jk ( t + s)I~j ik( t -- S) -- ~ /k ( t -- S)I~jik(t + S)],i + [O#ej i ( t - s) + Eji3~ji(t - s)

+ FjigZjik(t -- s ) ][O( t + s) + a O(t + s)] -- Dyieji( t + s) + Eji?/yi(t + s)

+ F/ikZjik(t + s )][O( t -- S) + a O(t -- S)] (33)


However, in view of equation (12) we obtain

[Djisji(t - s) + EjiYji(t - s) + FiikZjik(t - s)][O(t + s) + ot O(t + s)] - Ojieji(t + s) + EiiYji(t -+- s)

+ Fjik,~jik(t + S)][O(t -- S) + a O(t -- S)] = (a - pr/°)[O(t - $) - O(t + s) + a(O(t - s)

[ (fo -s - O ( t + s ) ) ] + ( d a - h ) [ O ( t - s ) O ( t + s ) - O ( t + s ) O ( t - s ) ] + K i j O,i(t+s) Oj(~:) d~

- O,,(t - s) ( ff+~ O,j( ~)d~) ] + ~Kij[ O,,(t + s) ( f/-" O,j( e)d~) - O,,(t - s)O j(t + s) ]

+ ,~gij[ O,i(t - s) ( f/+~ Oj(,)d,) -O,,(t + s)Oj(t - s) ]

( fo ]fo ) +K~j [O(t - s) + ~O(t - s)] Oj(~:)d~:- [O(t+s) + a O ( t + s ) Oj(~:)d~: (34) ' i

We now substitute equation (34) into equation (33) and we use the boundary conditions (10), in order to obtain the following identity

2 fR [OUi(t)iti(t)+ 'krq~'r(t)ip'k(t)]dV= f . [o(uOiti(2t)+ fiOui(2t))+ IiJ(q~O~ik(2t) + iP°~°'k(2t))]dV

fof + (a - O'O°)[O(t+s) - O(t - s) + oL(O(t+s) - O(t - s ) ) ] d V d s

+fof. [(cl~-h) d (°(t+s)°(t-s))+~s fo °'(')a' fo o~(

fo + agqo i ( t + s) O,j(~:)d~ + aKifl,i(t - s) O j ( ~ ) d ~ ] d V d s (35)

If we use the initial conditions, equation (13), in equation (35) then we arrive at the desired result, equation (29), and the Theorem 4.3 is proved.

5. ASYMPTOTIC E Q U I P A R T I T I O N OF E N E R G Y

In this section we shall use the identities (22), (25) and (29) such that in conjunction with the hypotheses made in Section 2 we establish the asymptotic partition of total energy. First, we introduce the Cesaro means of energies contained in (22). Thus, we define

~c( t ) ---- ~ [oiti(s)iti(s) + Ik,~Ojr(S)~Ojk(S)]dVds,

c(t) -- ~ [PkiUj,k(S)Uj,i(S)

+ CijmnSij(S)8ran(S) + 2Gm.qeii(s)ym.(s) + 2Fm.~ijeii(s)x=.~(s)Bijm.Yii(s)"Ymn(S) +

+ 2Dijm.ryij(S)Xm.r(s) + aokmnrZqk(S)Zm..(S )

-- 2MikULi(S) q~jk(S) + 2NrikUj,i(S) ~0jk,r(S)] d V ds,


~ c( t) - - ~ . otKiyO.i(s) O j(s) d V ds,

1roy " ~ c ( t ) - - ~ . dO2( s )dVds ,

'Iof. 5grc(t)------ -~- a h O 2 ( s ) d V d s ,

lfo'fo'fB ~ c ( t ) =-- ~-~ [KqO,~(~:) + ( d a - h ) O 2 ( ~ ) ] d V d ~ d s (36)

We are now in a position to state and proof the main result of our study.

THEOREM 5.1. We assume that the hypotheses of the Section 2 hold. Then, for all choices of initial data

(ui,~oij)° o E Wl(O), (ui,~olj)'° o E Wo(B), 0 ° E WI(B), 0 ° E Wo(B)

we have

lim ~e(t) = O, lim Zae(t) = 0 (37) t---->oo t---->ec

Moreover, the following assertions hold i) If m e a s o~B3 e-~ 0, then

lim ~g~(t) = 0 (38) t....>oo

ii) If meas 0B2 := 0, then

t--,= -~ (dO* + h O * ) d V lira 5g~(t) = -~-

iii) If m e a s t~B 1 ~ 0 and m e a s 69B 1 ~ 0, then

lim ~ ¢ ( t ) = l im ~ ( t )

(39)

(40)

lim ~ ( t ) = E(0) - 2 lim ~ ( t ) = E(0) - 2 lim ~ ( t ) t - . . - ~ t --~.oo t " -~o°

(41)

iv) If meas dB1 = 0 and meas OB1 = O, then

1 fB " * ' * " * ' * lim ~ ( t ) = lim ~ ( t ) + -~- [ • U i U i dr- Ikr~Ojr~Ojk] d V t...~e~ t - ~ o o

(42)


-- 4- I kr~O yr~O ] k ] d V lim ~e(t) = E(0) 2 lim ~e(t) + ~ [OUi U i t---~co l . . . . ~

= E(0) - 2 lim ~ ( t ) - ~- [ Q/t*/~* + Ikr~Oj*r~OTk ] d V t____~¢~

(43)

PROOF. We use the energy conservation law, equation (22), in conjunction with the hypotheses of Section 2 in order to prove the relation (37). Thus by hypotheses (9), we have

1 h dO2(t) + ah 02 + 2h O(t)O(t) = --ff (dO(t) + h 0(0) 2 + ~ (da - h)02(0 = h (0(t) + ot 0(t)) 2 Ol

1 + - - (as - h)O2(t) >_ 0 (44)

O~

Now, with aid of the equations (36) and (22), we get

1 ~E~(t) <- -f~ ha(dt~ - h)-lE(0) (45)

t ~

~¢(t) -< -~- E(0). (46)

Letting t ~ ~ into equations (45) and (46), we deduce the relation (37). i) Suppose that meas 0B3 ~ 0. It is easy to prove that OEWI(B). Then by using the Poincare's

inequality (19), and the identity (22), we get

fo fBdO2(s)dVds <- _ _ d d

t o Kifl,i(s) Oj(s) d Vds <- E(O) (47) m2 m2

From relation (47) and by means of equation (36), we obtain the conclusion (38). ii) We now suppose the meas 0B3 = 0. We use the decomposition, equation (21), and the fact

that ZEff'I(B) in order to obtain the following identity

h 0 , ) 2 d V + h h 0* e x p ( - - ~ - - fBo2(t)dV= fB (O*+ -~ fB2"2(t) dV- -2 fB -~ (O*+ -~ )0* dt )dV

h2 " 2 (48)

In view of (48) and (36) we get

-2-1 fn la 2-tl fo fn 1 [ - e x p ( d ) ] ~ ( / ) = -:(dO* +hb*)2dV + d2~(s)dVds- -- 1 - t t -h

h 2 , 1 1 - e x p - 2 d t (/)*)zdV X -~O*(dO*+hO*)dV+ a t - ~ (49)


The next estimate follows with the aid of the Poincare's inequality (19), the identity (22) and the fact that z~Wt(B)

fof "fof 1 dz2(s )dVds <_ _ _ Kijx,i(s)zj(s)dVds -~" 2tin2

d KijO,i(s ) Oj(s) d V ds <- - - E(0) (50) 2trn2 2tin2

Letting t---> ~ into equation (49), we are led to equation (40), taking into account equations (9) and (50). w e now use the relation (45), the energy conservation law (22), and the hypotheses of Section 2 in order to obtain the following estimates

fB 1 E(0), t ~ [0,~) (51) oZ(t)dV <- 2a d a - h

fB [QEli(t){li(t) + Ikr~jr(t)~jk(t)]dV < 2E(0), t E [0,oo) (52)

fo' fD KijO,i(r) O j(r) d V dr <- E(O), t ~ [0, oo ) (53)

02(r )dV -< - - E(0), t ~ [0,oo) (54) d a - h

However, the identities (25) and (29) imply

[ Oai(S)ai(S)-F lkr~Oir(S)~jk(S) -- PkiUj,(S)Uj,i(S) - C i j m n S i j ( s ) , ~ . m n ( S ) - 2emnq~O(S)'Ymn(S)

- 2Fm,,,qeij(~)Zm,,,(s) - O i j m n ~ / i ] ( s ) ~ / m n ( S ) - - 2 O i j m n r ~ i j ( S ) Z m n ( S ) - - AijkmnrXi.ik(S),~mnr(S)

- PkiUj,k(S)U/,i(S) + 2M~kUj,~(S) q~jk(S) -- 2Nr~kUy,,(S) q~jk,r(S)] d Vd s

fof 1 [dOE(s) + 2hO(s)O(s) - ah/~2(s) - aK~jO~(s)O,~(s)]dVds 2t

1 fn[ouOitO + o.o 1 fa __ fo fa (a_or lO)(O(s ) Ikr~Ojr~jk]dV (da - h)(O°)ZdV + 1 ' 2t 4t 2t

1 f(da_h)OOO(2t)+aKijOoff + ott~(s))dVds + 4 t OJ(sC)d~:]

d 1 (a - OTq°)[O(t+s) - O( t - s) + a---~s(O(t+s ) + O( t - s ) )]dVds +4--7 (55)

In view of notations (36) and by using equation (13), from the identity (55), it results

~e(t) - ~ ( t ) = -~- [(dc~ - h)(O ° + a(a - Cq°)][0(2t) - 0°]dV


l f)'f~ 'fo'f~ l f~ + 4t aK'iO°Oj(s)dVds + - - hO(s)O(s)dVds - - - [Ou°it ° + Ikr~O°,~O°kldV " t 2t

'L + YE~¢(t) - ~g(t) + --~ [Q(ft°ui(2t) + u°iti(2t)) + Ik,(~oFpjk(2t) + ~°r~jk(2t))]dV

lfo'fB + ~ ( t ) + --~ (a - ~l°)[O(t+s) + O(s)]dVds (56)

Now we shall use the Schwarz and Cauchy inequalities on the right side of equation (56). We also use the relations (45)-(47), (51)-(54) and thus we get

1

1 f~ [QuOitO+iar~O~Ok]dV 1 fa - - < d- Ikr ( ~jr~Ojk "l I- ~OOr~]k)] d V; 57 - ¥ [~ (u°u° + i ' ° i ' ° ) o o

L [ ( d a - h)(O°+ a ( a - On°)][O(2t) - O ° ] d V ~ -~- [ [ ( d a - h)(O°+ a ( a - gO°)] 2

+ 2(O°)2]dV + E(O); 2 t ( d a - h)

2t 1 2t .~ fo f, O,,,:,,o°o,,~,,~v,:,s <--~(fo f, O,,,:,,o°o°,-,v,:,s)' < 1 ) : ; >'( fo f,, o,,,:,,o,~,~:,o,,,,:,,:,v,:,, )'_ ~ (~o:, f,,,,,,,,,oooo~v,:,, ~

+ fo fahO(s )O(s ) d V d s < - + ( f o f a h O 2 ( s ) d V d s ) k X + ( f o f a h O 2 ( s ) d V d s ) ~

2 a ) ~ h E(0); <- --i-- d a - h

-~ fa [@(il°iui(2t)+u°iiti(2t))+ Ik~(iPO/'p'k(2t)+ ~.fip}k(2t)]dV ] if,, o o 1 <- - [ Qu °u° + Ikr~j,~ik] d V + - - E(O) (57) 8t 4t

iii) Assume that meas 0B1 ~ 0 and meas aB2 ~ 0. Since (ui,#ij)eVC~(B), from equations (7), (18) and (22) it results

L kL [eui(l")ui(~') + Ikr~jr(~')~pjk(~')]dV <- _ _ [Cijmnl?.ij(T)Smn('l ") + 2Gmnijeij(r) ymn(r) ml

+ 2Fmnrij~ ij(7) Xmn('7") "~ B#mn'~ij(T) Tmn(T) "q- 2Oi]mnrTq( 7")Xmn(I-) + mijkmnrZijk ( T)Zmnr(T)

2k + ekiUj,k(7")ULi(1" ) - - 2Mikuj.i(r)~jk(r) + 2Nr,kUi.,(O~oj~.~(r)]dgds <- E(O),¢ e [0,0o)

ml (58)

such that we obtain


1 f [p(itOiui(2t ) + uOiti(2t)) + ikr(ipoAOik(2t ) + ~O0~Ojk(2t))]d v -g J~

<--- [Qi~°i~° + Ik,ip°~O°k]dV + - - E ( O ) (59) 8t 4tmx

If we suppose that m e a s c~B 3 ~ 0, then we have

~ ¢ ( t ) + ~-~ (a -CO°)[O(t+s)+ O(s)]dVds <-YE~(t)

+ 4t . fB(a-Prl°)ZdVds)½(f~ fB [O(t + S) + O(s)]2dVds )½ <-- ~Ee(t)

+ (a - CO°)2dV ~¢(2t)] ~ (60)

Letting t--~ ~ in equation (56) and taking into account the estimates (57), (59), (60) and the relations (37) and (38), we conclude that the relation (40) holds. Next we suppose that meas t~B 3 = 0. If we use the decompositions (16) and (21), the relations (17), (49) and the expression of 7 ° (from Theorem 4.2), we conclude that the following identity holds

1 fo f~ (a-p'q°)[O(t+s)+ O(s)]dVds- 4tl fB ...~h2 0,(dO, + 3 h O , )

× exp - 2-h- - 1 ] d V + --t -~T O*(dO* + h 0 * ) [ e x p - ~ - 1]dV

1 dzZ(s)dVds + [Diieo + o + --2t --4t Eijy#- dT ° - hT"°][Z(t + s) + Z(s)]dVds (61)

Now, by using the Schwarz and Cauchy inequalities in (61) and taking into account the relation (50), we have

lim ~ ¢ ( t ) + ~ (a-~'q°)[O(t+s)+ O(s)]dVds = 0 (62) t---.~oo

It is easy to see that the use of relations (37), (57), (59) and 62) in (56) implies again the conclusion (40). Also, it is a simple matter to see that the relation (41) is obtained from (22) by taking the Cesaro mean and by using the relations (37), (38) and (40).

iv) If meas c~B1 = 0 and meas ~B2 = 0, then we use the decomposition (20), the relations (14), (15) and the fact that (u~,q~i)~Wl(B) in order to obtain

1 [ouOiti(2t, ) + IkAOO~ojk(2t)]dV = --~ [Ou~,it. + ik,q~O~k]d V -4?

+ -21 [pit*i~* + IkfiPj*iPj*k]dV + --4t [P/'~/°vi(2t) + IkrdP°~OJk(2t)]dV (63)


Also, since (v~,~bo)~W~(B), the Korn's inequality (18) leads to the relation

--~ [el'li(7-)l)i(T) + Ikr~bj~(7-)Ojk(7-)] d V <-- [Ci]mn~i](7")~mn(7") + 2Gmmjeq(7") ymn(7-) ma

+ 2Fm.,Oeij( T)f(mn(7") "~ Oiimn ~/0(7-) ~mn("1") -]- 2Dijm.,~/O( 7")2m.(7") + Aokm.r21i ( 7")2ran(T)

+ Pk~aj,k( T)i~],i(7") - - 2M~kaj,,(~') (Ojk(~') + 2Nr~kaj,i(7-) (ojk,,(7-)] d V

k fB [PkiUj'k(T)Uj'i(T) -]- CijmnSi](T)Smn(T) "~- 2Gmnijeo(7-)Y'"(7-) + 2Fmnr#eii(7")Zmn(q') ml

+ nijmn~/ij(T)~/mn(T ) -}- 2Oijmnr~/ij(7-)Xmn(7 ) + Aqkmnr)(,ij(T)Xmn(7- ) -- 2M~kUjs(r)~Ojk(r )

2k + 2N,~kUj,i(7-) ~ojk,r(r)] d V -< - - E(0) (64)

m l

1 where 7-~[0,~), ~/i = ~-(vj,~ + vij,~ij)= vjs-¢ij,~,Ok = ~jkS" Letting t ~ in equation (56) and

by means of relations (37), (57), (60), (63) and (64) we arrive at the conclusion (42). Finally, (43) is proved on the basis of the relation (22) by taking the Cesaro mean and by

using the relations (37), (38), (42) and (57). The proof of Theorem 5.1 is complete. At last we remark that the relations (40) and (42), restricted to the class of initial data for

which t~*-q~0-"* = 0, prove the asymptotic equipartition in mean of the kinetic and strain energies.

REFERENCES

1. Adams, R. A., Sobolev spaces, Academic Press, New York, 1975. 2. Day, W. A., Arch. Rational Mech. Anal 1980, 73, 243-256. 3. Eringen, A. C., Int. J. Engng. Sci. 1990, 28, 1291-1306. 4. Eringen, A. C. and Suhubi, E. S., Int. J. Engng. Sci. 1964, 2, 389-411. 5. Green, A. E. and Lindsay, K. A., J. of Elasticity 1972, 2, 1-7. 6. Gurtin, M. E., Arch. Rational Mech. Anal 1994, 4, 305-335. 7. Hlavacek, I. and Necas, J., Arch. Rational Mech. Anal 1970, 36, 305-334. 8. Levine, H. A., J. Diff. Eqns. 1977, 24, 197-210. 9. Matin, M., Int. J. Engng. Sci. 1994, 8, 1229-1240.

10. Marin, M., C.R. Acad. Sci. Paris, t. 321, Serie II b, 1995, 475-480. 11. Rionero, S. and Chirita, S., Int. J. Engng. Sci. 1987, 25, 935-946.

(Received 27 June 1996; revised 20 September 1996)

thermoelasticity of initially stressed bodies, asymptotic equipartition of energies

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