Some theorems in the theory of microstretch thermopiezoelectricity

International Journal of Engineering Science 45 (2007) 1–16 www.elsevier.com/locate/ijengsci

Some theorems in the theory of microstretch thermopiezoelectricity D. Iesßan, R. Quintanilla

*

Department of Mathematics, ‘‘Al.I. Cuza’’ University, 700506 Iasßi, Romania Matematica Aplicada 2, ETSEIAT-UPC, 08222 Terrassa, Barcelona, Spain Received 29 September 2006; accepted 6 October 2006 Available online 8 December 2006

Abstract The electromagnetic theory of microstretch thermoelasticity is an adequate tool to describe the behaviour of porous bodies, animal bones and solids with deformable microstructures. In this paper we study the linear theory of microstretch thermopiezoelectricity. First, we establish a reciprocity relation which involves two processes at different instants. This relation forms the basis of a uniqueness result and a reciprocal theorem. Then, we study the continuous dependence of solutions upon initial data and body loads. A variational characterization of solutions is also presented. Finally, we investigate the effect of a concentrated heat supply and the effect of a concentrated volume charge density in an unbounded homogeneous and isotropic body.  2006 Elsevier Ltd. All rights reserved. Keywords: Microstretch elastic solids; Piezoelectricity; Thermoelasticity

1. Introduction The theory of nonpolar thermopiezoelectricity has been studied in various papers (see, e.g., [1–4]). In [5], Eringen established the electromagnetic theory of microstretch thermoelasticity. The constitutive equations for anisotropic materials display new physical phenomena. The material particles of the microstretch bodies can stretch and contract independently of their translations and rotations. The theory introduced in [5] involves interactions of electromagnetic fields and thermomechanical deformations for porous bodies like bones, solids with microcracks, foams and other synthetic materials. In this paper we study the linear theory of microstretch thermopiezoelectricity introduced by Eringen [5,6]. In Section 3, we establish a reciprocity relation which involves two processes at different instants. By using this relation we derive a uniqueness result and a reciprocal theorem. The uniqueness theorem is established without using definiteness assumptions on the elastic coefficients. The proof of the reciprocal theorem avoids the use of the Laplace transform. Section 4 is devoted to the investigation of the continuous dependence of solutions *

Corresponding author. Tel.: +34 937398162; fax: +34 937398101. E-mail address: [email protected] (R. Quintanilla).

0020-7225/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2006.10.001

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D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

upon initial data and body loads. In the Section 5 we present a variational characterization of solutions. Finally we study the effect of a concentrated heat source and the effect of a concentrated volume charge density in an isotropic and homogeneous body that occupies the entire three-dimensional euclidean space and is subject to steady vibrations. 2. Basic equations We consider a body that at some instant occupies the region B of the euclidean three-dimensional space and is bounded by the piecewise smooth surface oB. The motion of the body is referred to the reference configuration B and a fixed system of rectangular cartesian axes Oxi (i = 1, 2, 3). We denote by n the outward unit normal of oB. Boldface characters stand for tensors of an order p P 1, and if v has the order p, we write vij. . .k (p subscripts) for the components of v in the cartesian coordinate frame. We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (1, 2, 3), summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. In what follows we use a superposed dot to denote partial differentiation with respect to the time t. We consider the linear theory of microstretch thermopiezoelectricity. The basic equations of this theory consist of the equations of motion tji;j þ fi ¼ q0 € ui ; € j; mji;j þ eijk tjk þ gi ¼ I ij u

ð2:1Þ

€; pk;k  r þ G ¼ j0 u the energy equation T 0 g_ ¼ qi;i þ s;

ð2:2Þ

the equations of the electric fields Dj;j ¼ p;

Ek ¼ w;k ;

ð2:3Þ

the constitutive equations ð1Þ

tij ¼ Aijrs ers þ Bijrs jrs þ Dij u þ F ijk fk þ kijk Ek  aij T ; ð2Þ

mij ¼ Brsij ers þ C ijrs jrs þ Eij u þ Gijk fk þ kijk Ek  bij T ; r ¼ Dij eij þ Eij jij þ nu þ hk fk þ kið3Þ Ei  sT ; ð4Þ

pk ¼ F ijk eij þ Gijk jij þ hk u þ Akj fj þ kkj Ej  d k T ;

ð2:4Þ

g ¼ aij eij þ bij jij þ su þ d k fk þ ak Ek þ aT ; qi ¼ k ij T ;j ; ð1Þ

ð2Þ

ð3Þ

ð4Þ

Dk ¼ kijk eij  kijk jij  kk u  kjk fj þ vkj Ej þ ak T ; and the geometrical equations eij ¼ uj;i þ ejik uk ;

jij ¼ uj;i ;

fj ¼ u;j :

ð2:5Þ

In (2.1)–(2.5) we have used the following notations: tij is the stress tensor, fi is the body force, q0 is the reference mass density, ui is the displacement vector, mij is the couple stress tensor, eijk is the alternating symbol, gi is the body couple, Iij is the microinertia tensor, ui is the microrotation vector, pk is the microstretch stress vector, u is the microstretch function, r is the microstress function, G is the microstretch body force, j0 is the microstretch inertia, g is the entropy per unit volume, T0 is the constant absolute temperature of the reference configuration, qi is the heat flux vector, s is the heat supply, Dk is the dielectric displacement vector, p is the volume charge density, Ek is the electric field vector, w is the electrostatic potential, eij, jij and fk are kinematic strain measures, T is the temperature measured from T0, and Aijrs ; Bijrs ; C ijrs ; Dij ; Eij ; F ijk ; Gijk ; hi ;

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

3

ðaÞ

ð4Þ n; Aij ; kijk ; kð3Þ i ; kij ; vij ; aij ; bij ; s; d k ; ak ; a and kij are constitutive coefficients. We assume that the microinertia tensor and the constitutive coefficients satisfy the symmetry relations

I ij ¼ I ji ; Aijrs ¼ Arsij ; vij ¼ vji ; k ij ¼ k ji :

C ijrs ¼ C rsij ;

Aij ¼ Aji ;

ð2:6Þ

The second law of thermodynamics implies that k ij ni nj P 0

ð2:7Þ

for all nk :

The components of surface traction, the components of the surface moment, the surface microforce, the heat flux and the normal components of the electric displacement are given by ti ¼ tji nj ;

mi ¼ mji nj ;

p ¼ pk nk ;

q ¼ q k nk ;

D ¼ Dj nj ;

ð2:8Þ

respectively. Let Sr (r = 1, 2, . . . , 10), be subsets of oB so that S 1 [ S 2 ¼ S 3 [ S 4 ¼ S 5 [ S 6 ¼ S 7 [ S 8 ¼ S 9 [ S 10 ¼ oB; S 1 \ S 2 ¼ S 3 \ S 4 ¼ S 5 \ S 6 ¼ S 7 \ S 8 ¼ S 9 \ S 10 ¼ ;. We consider the following boundary conditions ui ¼ e u i on S 1  I; e on S 7  I; w¼w

e i on S 3  I; ui ¼ u

e on S 5  I; u¼u

T ¼ Te on S 9  I;

e i on S 4  I; mji nj ¼ m qj nj ¼ e q on S 10  I;

tji nj ¼ et i on S 2  I; e on S 8  I; pk n k ¼ e p on S 6  I; Dj nj ¼ D

ð2:9Þ

e Te ; et i ; m e and e e i; u e ; w; e i; e p; D q are prescribed functions, and I = (0, 1). where e ui; u The initial conditions are ui ðx; 0Þ ¼ u0i ðxÞ;

u_ i ðx; 0Þ ¼ v0i ðxÞ;

ui ðx; 0Þ ¼ u0i ðxÞ;

u_ i ðx; 0Þ ¼ m0i ðxÞ;

uðx; 0Þ ¼ u0 ðxÞ;

_ 0Þ ¼ w0 ðxÞ; uðx;

0

T ðx; 0Þ ¼ T ðxÞ;

ð2:10Þ

x 2 B;

where u0i ; v0i ; u0i ; m0i ; u0 ; w0 and T0 are given. We assume that: (i) fi, gi, G, s and p are continuous on B  ½0; 1Þ; (ii) q0 ; I ij ; j0 ; u0i ; v0i ; u0i ; m0i ; u0 ; w0 and T0 are continuous on B; (iii) the constitutive coefficients and the microinertia tensor satisfy the symmetry relations e and Te are continuous e i; u e; w (2.5); (iv) the constitutive coefficients are continuous differentiable on B; (v) e ui; u e and e e i; e on S1 · I, S3 · I, S5 · I, S7 · I and S9 · I, respectively; (vi) et i ; m p; D q are continuous in time and piecewise regular on S2 · I, S4 · I, S6 · I, S8 · I and S10 · I, respectively. Let M and N be non-negative integers. We say that f is of class CM,N on B · I if f is continuous on B · I and the functions  n  om of ; m 2 f0; 1; 2; . . . Mg; n 2 f0; 1; 2; . . . ; N g; oxi oxj . . . oxp otn m þ n 6 maxfM; N g; exist and are continuous on B · I. We write CM for CM,M. By an admissible process P ¼ fui ; ui ; u; w; T ; eij ; jij ; fi ; Ek ; tij ; mij ; pi ; r; g; qi ; Di g we mean an ordered array of functions ui, ui, u, w, T, eij, jij, fi, Ej, tij, mij, pi, r, g, qi and Dj defined on B  ½0; 1Þ with the following properties: (i) ui, ui, u 2 C2, w 2 C2,0, T 2 C2,1, eij, jij, fk, Ek 2 C1,0, tij, mij, pk, qi, Dj 2 C1,0, r 2 C0, € i ; ui;j ; u; u; _ u € ; ui;j ; w; w;i ; T ; T ;j ; eij ; jij ; fk ; Ej ; tij ; tji;j ; mji ; mji;j ; pi ; pj;j ; g 2 C0,1 on B · I; (ii) ui ; u_ i ; € ui ; ui;j ; ui ; u_ i ; u _ qi and qj,j are continuous on B  ½0; 1Þ: r; Dk ; Dj;j ; g; g; By a solution of the mixed problem we mean an admissible process which satisfies Eqs. (2.1)–(2.5) on B · I, the boundary conditions (2.9) and the initial conditions (2.10). In the formulation of the constitutive equations it is often convenient to chose eij, jij, u, g, fi, Di and T,i as independent constitutive variables. Then, Eq. (2.4) are replaced by

4

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16 ð1Þ

ð1Þ

tij ¼ aijrs ers þ bijrs jrs þ d ij u þ fijk fk þ lijk Dk þ Bij g; ð2Þ

ð2Þ

mij ¼ brsij ers þ cijrs jrs þ eij u þ gijk fk þ lijk Dk þ Bij g; ð3Þ

r ¼ d ij eij þ eij jij þ mu þ vi ui þ li Di þ Bð3Þ g; ð4Þ

ð4Þ

ð2:11Þ

pi ¼ frsi ers þ grsi jrs þ vi u þ aij fj þ lij Dj þ Bi g; ð1Þ

ð2Þ

ð4Þ

ð5Þ

T ¼ Bij eij þ Bij jij þ Bð3Þ u þ Bi fi þ li Di þ bg; qi ¼ k ij T ;j ; ð1Þ

ð2Þ

ð3Þ

ð4Þ

ð5Þ

Ei ¼ lrsi ers þ lrsi jrs þ li u þ lji fj þ Rij Dj þ li g; ðpÞ

ðaÞ

ðsÞ

ð4Þ

ð4Þ

where aijrs ; bijrs ; d ij ; fijk ; lijk ; Bij ; cijrs ; eij ; gijk ; m; vi ; li ; lij ; Bð3Þ ; Bi ; aij, b, kij and Rij are constitutive coefficients. For isotropic and homogeneous bodies the constitutive Eq. (2.4) reduce to (see [5]) tij ¼ kerr dij þ ðl þ jÞeij þ leji þ k0 udij  b0 T dij ; mij ¼ ajrr dij þ bjji þ cjij þ b0 eijk fk þ k1 ejik Ek ; r ¼ k0 err þ k3 u  b1 T ; pi ¼ a0 fi þ k2 Ei þ b0 ersi jrs ; g ¼ b0 err þ b1 u þ aT ; qi ¼ kT ;i ;

ð2:12Þ

Di ¼ k1 ersi jsr  k2 fi þ vEi ; where dij is the Kronecker delta, and k, l, j, k0, b0, a, b, c, b0, kk, b1, a0, a, k and v are constitutive constants. It follows from (2.1)–(2.3), (4.1) and (2.5) that the field equations of the theory of homogenous and isotropic bodies can be expressed as ðl þ jÞDui þ ðk þ lÞuj;ij þ jeijk uk;j þ k0 u;i  b0 T ;i þ fi ¼ q0 €ui ; € i; cDui þ ða þ bÞuj;ji þ jeijk uk;j  2jui þ gi ¼ J u €; ða0 D  k3 Þu  k2 Dw  k0 uj;j þ b1 T þ G ¼ j0 u

ð2:13Þ

k2 Du þ vDw ¼ p; _  cT_ ¼ s; kDT  T 0 ðb0 u_ j;j þ b1 uÞ where we have used the notations c = aT0 and Iij = Jdij. 3. Reciprocal theorem and uniqueness This section is devoted to reciprocity and uniqueness results. We establish a reciprocity relation which involves two processes at different instants. This relation forms the basis of a uniqueness result and a reciprocal theorem. The proof of the reciprocal theorem avoids both the use of the Laplace transform and the incorporation of the initial conditions in the equations of motion. The uniqueness theorem is established without using definiteness assumptions on the elastic constitutive coefficients. Let u and v be scalar fields on B · I that are continuous in time. We denote by u * v the convolution of u and v, Z t ½u  vðx; tÞ ¼ uðx; t  sÞvðx; sÞ ds: ð3:1Þ 0

We introduce the functions gðtÞ ¼ t;

‘ðtÞ ¼ 1;

t 2 I:

If F is defined on B · I, then we write F for ‘ * F, that is Z t F ðx; tÞ ¼ ½‘  F ðx; tÞ ¼ F ðx; sÞ ds: 0

ð3:2Þ

ð3:3Þ

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

5

Following [7,11], Eq. (2.2) is equivalent to T 0 g ¼ qi;i þ W ;

ð3:4Þ

where W ¼ s þ T 0 gðx; 0Þ:

ð3:5Þ ðaÞ ðaÞ e ðaÞ ; et ðaÞ ; m e iðaÞ ; u e ðaÞ ; Te ðaÞ ; w e ei ; e ui ; u p ðaÞ ; i ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ ¼ fui ; ui ; uðaÞ ; w ; T ; eij ; jij ; fðaÞ i ; (a)

ðaÞ ðaÞ ffi ; gi ; GðaÞ ; sðaÞ ; pðaÞ ; ðaÞ

ðaÞ

We consider two external data systems L ¼ 0ðaÞ 0ðaÞ 0ðaÞ 0ðaÞ ðaÞ e D ;e q ðaÞ ; ui ; vi ; ui ; mi ; u0ðaÞ ; w0ðaÞ ; T 0ðaÞ g (a = 1, 2). Let P ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ Ek ; tij ; mij ; pi ; rðaÞ ; gðaÞ ; qi ; Di g be a solution of the mixed problem corresponding to L . We denote ðaÞ

ðaÞ

ðaÞ

ti ¼ tji nj ; ðaÞ

W

¼‘s

ðaÞ

ðaÞ

pðaÞ ¼ pj nj ;

ðaÞ

DðaÞ ¼ Dj nj ; ðaÞ

ðaÞ

mi ¼ mji nj ; qðaÞ ¼ qj nj ; ðaÞ

þ T 0 g ðx; 0Þ

ð3:6Þ ða ¼ 1; 2Þ:

Theorem 3.1. Assume that the symmetry relations (2.6) hold. Let K ab ðr; sÞ ¼

Z h i 1 ðaÞ ðbÞ ðaÞ ðbÞ ti ðrÞui ðsÞ þ mi ðrÞui ðsÞ þ pðaÞ ðrÞuðbÞ ðsÞ þ DðaÞ ðrÞwðbÞ ðsÞ  qðaÞ ðrÞT ðbÞ ðsÞ da T0 oB Z h i 1 ðaÞ ðbÞ ðaÞ ðbÞ þ fi ðrÞui ðsÞ þ gi ðrÞui ðsÞ þ GðaÞ ðrÞuðbÞ ðsÞ  pðaÞ ðrÞwðbÞ ðsÞ  W ðaÞ ðrÞT ðbÞ ðsÞ dv T0 ZB h i 1 ðaÞ ðbÞ ðbÞ € jðaÞ ðrÞuðbÞ € ðaÞ ðrÞuðbÞ ðsÞ  qðaÞ ð3:7Þ ui ðrÞui ðsÞ þ I ij u  q0 € i ðsÞ þ j0 u i ðrÞT ;i ðsÞ dv; T0 B

for all r, s 2 I. Then K ab ðr; sÞ ¼ K ba ðs; rÞ

ða; b ¼ 1; 2Þ;

ð3:8Þ

for all r, s 2 I. Proof. Let ðaÞ

ðbÞ

ðaÞ

ðbÞ

ðaÞ

ðaÞ ðbÞ J ab ðr; sÞ ¼ tij ðrÞeij þ mij ðrÞjij ðsÞ þ pi ðrÞfðbÞ i ðsÞ þ r ðrÞu ðsÞ ðaÞ

ðbÞ

 Di ðrÞEi ðsÞ  gðaÞ ðrÞT ðbÞ ðsÞ;

ð3:9Þ

were, for convenience, we have suppressed the argument x. It follows from (2.4) and (3.9) that ðbÞ

ðbÞ

ðaÞ ðbÞ ðaÞ ðaÞ ðbÞ ðrÞeij ðsÞ þ C ijmn jðaÞ J ab ðr; sÞ ¼ Aijmn emn mn ðrÞjij ðsÞ þ nu ðrÞu ðsÞ þ Aij fj ðrÞfi ðsÞ ðaÞ

ðbÞ

ðbÞ

ðaÞ

ðbÞ

ðaÞ ðaÞ  vij Ej ðrÞEi ðsÞ þ Bijmn ðjmn ðrÞeij ðsÞ þ eij ðrÞjðbÞ mn ðsÞÞ þ Dij ðeij ðsÞu ðrÞ ðaÞ

ðbÞ

ðaÞ

ðaÞ

ðbÞ

ð1Þ

ðbÞ

ðaÞ

þ eij ðrÞuðbÞ ðsÞÞ þ F ijk ðeij ðsÞfk ðrÞ þ eij ðrÞfk ðsÞÞ þ kijk ðeij ðsÞEk ðrÞ ðaÞ

ðbÞ

ðaÞ

ðbÞ

ðbÞ

ðaÞ

ðbÞ

ðaÞ

þ eij ðrÞEk ðsÞÞ þ Eij ðjij ðsÞuðaÞ ðrÞ þ jij ðrÞuðbÞ ðsÞÞ þ Gijk ðjij ðsÞfk ðrÞ ð2Þ

ðbÞ

ðaÞ

ðaÞ

ðbÞ

þ jij ðrÞfk ðsÞÞ þ kijk ðjij ðsÞEk ðrÞ þ jij ðrÞEk ðsÞÞ þ hi ðfiðaÞ ðrÞuðbÞ ðsÞ ðaÞ

ðbÞ

ðaÞ

ð3Þ ð4Þ ðbÞ ðaÞ ðbÞ ðaÞ þ fðbÞ i ðsÞu ðrÞÞ þ ki ðEi ðrÞu ðsÞ þ Ei ðsÞu ðrÞÞ þ kij ðE j ðrÞfi ðsÞ ðbÞ

ðbÞ

ðaÞ

ðbÞ

þ Ej ðsÞfiðaÞ ðrÞÞ  aij ðT ðaÞ ðrÞeij ðsÞ þ T ðbÞ ðsÞeij ðrÞÞ  bij ðT ðaÞ ðrÞjij ðsÞ ðaÞ

ðbÞ

þ T ðbÞ ðsÞjij ðrÞÞ  sðT ðaÞ ðrÞuðbÞ ðsÞ þ T ðbÞ ðsÞuðaÞ ðrÞÞ  d k ðT ðaÞ ðrÞfk ðsÞ ðaÞ

ðbÞ

ðaÞ

þ T ðbÞ ðsÞfk ðrÞÞ  ak ðT ðaÞ ðrÞEk ðsÞ þ T ðbÞ ðsÞEk ðrÞÞ  aT ðaÞ ðrÞT ðbÞ ðsÞ:

ð3:10Þ

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D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

In view of (2.6), we find that J ab ðr; sÞ ¼ J ba ðs; rÞ

ða; b ¼ 1; 2Þ;

ð3:11Þ

for all r, s 2 I. On the other hand, with the help of (2.5), (2.1)–(2.3) and (3.4), we obtain   1 ðaÞ ðaÞ ðbÞ ðaÞ ðbÞ ðaÞ ðaÞ ðbÞ ðbÞ ðbÞ J ab ¼ tji ðrÞui ðsÞ þ mji ðrÞui ðsÞ þ pj ðrÞu ðsÞ þ Dj ðrÞw ðsÞ  qj ðrÞT ðsÞ T0 ;j ðaÞ

ðbÞ

ðaÞ

ðbÞ

þ fi ðrÞui ðsÞ þ gi ðrÞui ðsÞ þ GðaÞ ðrÞuðbÞ ðsÞ  pðaÞ ðrÞwðbÞ ðsÞ  ðaÞ

ðbÞ

ðaÞ

ðbÞ

€ j ðrÞui ðsÞ  j0 u € ðaÞ ðrÞuðbÞ ðsÞ þ  q0 € ui ðrÞui ðsÞ  I ij u

1 ðaÞ W ðrÞT ðbÞ ðsÞ T0

1 ðaÞ ðbÞ q ðrÞT ;i ðsÞ: T0 i

ð3:12Þ

If we integrate (3.12) over B and use the divergence theorem, (3.6), (3.7) and (3.11), then we obtain the desired result. h The next theorem is a consequence of Theorem 3.1. Theorem 3.2. Assume that the relation (2.6) hold. Let P ¼ fui ; ui ; u; w; T ; eij ; jij ; fi ; Ei ; tij ; mij ; pi ; r; g; qi ; Di g be a e et i ; m e e e i; u e ; Te ; w; e i; e solution corresponding to the external data system ffi ; gi ; G; s; p; e ui; u p ; D; q ; u0i ; v0i ; u0i ; m0i ; 0 0 0 u ; w ; T g and let  Z  1 U ðr; sÞ ¼ fi ðrÞui ðsÞ þ gi ðrÞui ðsÞ þ GðrÞuðsÞ  pðrÞwðsÞ  W ðrÞT ðsÞ dv T0 B  Z  1 ð3:13Þ þ ti ðrÞui ðsÞ þ mi ðrÞui ðsÞ þ pðrÞuðsÞ þ DðrÞwðsÞ  qðrÞT ðsÞ da; T0 oB for all r, s 2 I. Then Z  Z d 1 ðq0 ui ui þ I ij uj ui þ j0 u2 Þ dv þ k ij T ;j T ;i dv dt ZB T0 B Z t ¼ ½U ðt  s; t þ sÞ  U ðt þ s; t  sÞ ds þ fq0 ½u_ i ð2tÞui ð0Þ þ u_ i ð0Þui ð2tÞg 0

B

_ _ þ I ij ½u_ j ð2tÞui ð0Þ þ u_ j ð0Þui ð2tÞ þ j0 ½uð2tÞuð0Þ þ uð0Þuð2tÞ dv:

ð3:14Þ

Proof. In view of (3.8), Z 0

t

K 11 ðt þ s; t  sÞ ds ¼

Z

t

K 11 ðt  s; t þ sÞ ds:

ð3:15Þ

0

Let us apply this relation to the process Pð1Þ ¼ P: From (3.7) and (3.13) we obtain Z t Z t Z tZ 1 K 11 ðt þ s; t  sÞ ds ¼ U ðt þ s; t  sÞ ds þ k ij T ;j ðt þ sÞT_ ;i ðt  sÞ ds dv T0 0 B 0 0 Z tZ € j ðt þ sÞui ðt  sÞ þ j0 u € ðt þ sÞuðt  sÞ dv ds: ½q0 €  ui ðt þ sÞui ðt  sÞ þ I ij u 0

B

ð3:16Þ Similarly, Z t Z tZ Z t 1 K 11 ðt  s; t þ sÞ ds ¼ U ðt  s; t þ sÞ ds þ k ij T ;j ðt  sÞT_ ;i ðt þ sÞ dv ds T 0 B 0 0 0 Z tZ € j ðt  sÞui ðt þ sÞ þ j0 u € ðt  sÞuðt þ sÞ dv ds:  ½q0 € ui ðt  sÞui ðt þ sÞ þ I ij u 0

B

ð3:17Þ

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

7

With the aid of the relations Z t Z t _  sÞf_ ðt þ sÞ ds; f€ ðt þ sÞhðt  sÞ ds ¼ f_ ð2tÞhð0Þ  f_ ðtÞhðtÞ þ hðt 0 0 Z t Z t € _ _ _  sÞf_ ðt þ sÞ ds; hðt  sÞf ðt þ sÞ ds ¼ hðtÞf ðtÞ  hð0Þf ð2tÞ þ hðt 0 0 Z t Z t _  sÞ ds ¼ hð0Þf ð2tÞ þ f ðtÞhðtÞ þ f ðt þ sÞhðt f_ ðt þ sÞhðt  sÞ ds; 0

0

from (2.6), (3.14)–(3.17) we obtain the desired result.

h

We now use Theorem 3.2 to establish the following uniqueness result. Theorem 3.3. Theorem 3.3. Assume that (i) (ii) (iii) (iv) (v)

q0 and j0 are strictly positive; Iij is positive definite; the symmetry relations (2.6) hold; the constitutive equations can be presented in the form (2.11); kij and Rij are positive definite.

Let P ¼ fui ; ui ; u ; w ; T  ; eij ; jij ; fi ; Ek ; tij ; mij ; pi ; r ; g ; qi ; Di g be the difference of any two solutions of the mixed problem. Then ui ¼ 0;

ui ¼ 0;

u ¼ 0;

T  ¼ 0;

w;i ¼ 0

on B  I:

ð3:18Þ

Moreover, if S7 is nonempty, then the mixed problem has at most one solution. Proof. The difference of any two solutions corresponds to null data. Thus, from (3.14) we obtain Z  Z tZ d 1 ½q0 ui ui þ I ij uj ui þ j0 ðu Þ2  dv þ k ij T ;j T ;i dv dt ¼ 0; 0 6 t < 1: dt B T0 0 B Since ui ; ui and u* vanish initially, the above relation implies that Z Z tZ 1 ½q0 ui ui þ I ij uj ui þ j0 ðu Þ2  dv þ k ij T ;j T ;i dv dt ¼ 0: T 0 0 B B

ð3:19Þ

ð3:20Þ

In view of hypotheses (i), (ii) and (v), from (3.20) we find ui ¼ 0;

ui ¼ 0;

u ¼ 0;

T ;i ¼ 0

on B  I:

ð3:21Þ

It follows from (2.4) and (3.21) that qi ¼ 0; so that Eq. (2.2) implies that g_  ¼ 0. Since g* vanishes initially, we have g ¼ 0

on B  I:

ð3:22Þ

If we use the constitutive Eq. (2.11) and the relations (3.21) and (3.22), then we obtain ð5Þ

T  ¼ li Di ;

Ei ¼ Rij Dj :

ð3:23Þ

By using the divergence theorem and (2.3), we get Z Z Z     Ei Di dv ¼  w Di ni da þ w Dj;j dv: B

oB

From (3.23) and (3.24) we find that Z Rij Dj Di dv ¼ 0: B

B

ð3:24Þ

8

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

Since Rij is positive definite, this relation implies that Di ¼ 0

on B  I:

ð3:25Þ

It follows from (3.23) and (3.25) that T* = 0 and w;j ¼ 0 on B · I. Obviously, if S7 is nonempty, then w* = 0. h Uniqueness results in classical piezoelectricity have been established in various papers (see, e.g., [2,3]). Theorem 3.1 forms the basis of the following reciprocal theorem. Theorem 3.4. Assume that the symmetry relations (2.6) hold. Let PðaÞ be a solution corresponding to the external data system L(a) (a = 1, 2). Then   Z 1 ð1Þ ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ ð2Þ g  ti  ui þ mi  ui þ p  u þ D  w  ‘  q  T da T0 oB  Z  1 ð1Þ ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ ð2Þ F i  ui þ Gi  ui þ C  u  g  p  w  g  W  T þ dv T0 B   Z 1 ð2Þ ð1Þ ð2Þ ð1Þ g  ti  ui þ mi  ui þ pð2Þ  uð1Þ þ Dð2Þ  wð1Þ  ‘  qð2Þ  T ð1Þ da ¼ T0 oB  Z  1 ð2Þ ð1Þ ð2Þ ð1Þ F i  ui þ Gi  ui þ Cð2Þ  uð1Þ  g  pð2Þ  wð1Þ  g  W ð2Þ  T ð1Þ dv; þ ð3:26Þ T0 B where ðaÞ

ðaÞ

ðaÞ Gi

ðaÞ gi

F i ¼ g  fi ¼g

0ðaÞ

0ðaÞ

þ q0 ðtvi þ

þ ui

0ðaÞ I ij ðtmj

þ

Þ;

0ðaÞ uj Þ;

ð3:27Þ

CðaÞ ¼ g  GðaÞ þ j0 ðtw0ðaÞ þ u0ðaÞ Þ: Proof. We take in (3.8), r = s and s = t  s and integrate from 0 to t. With the aid of (3.7) we find that  Z  1 ð1Þ ð2Þ ð1Þ ð2Þ ti  ui þ mi  ui þ pð1Þ  uð2Þ þ Dð1Þ  wð2Þ  ‘  qð1Þ  T ð2Þ da T0 oB  Z  1 ð1Þ ð2Þ ð1Þ ð2Þ fi  ui þ gi  ui þ Gð1Þ  uð2Þ  pð1Þ  wð2Þ  W ð1Þ  T ð2Þ dv þ T0 B  Z  1 ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ ð2Þ € j  u i þ j0 u €  u  ‘  k ij T ;j  T ;i dv q0 € ui  ui þ I ij u  T0 B  Z  1 ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ti  ui þ mi  ui þ p  u þ D  w  ‘  q  T ¼ da T0 oB   Z 1 ð2Þ ð1Þ ð2Þ ð1Þ fi  ui þ gi  ui þ Gð2Þ  uð1Þ  pð2Þ  wð1Þ  W ð2Þ  T ð1Þ dv þ T0 B  Z  1 ð2Þ ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ € ð2Þ € q0 €  u þ j  u  ‘  k T  T ui  ui þ I ij u  u dv: ð3:28Þ ij ;j j i ;i 0 T0 B It is a simply matter to see that ðaÞ

ðaÞ

0ðaÞ

g€ ui ¼ ui  tvi

0ðaÞ

 ui

:

ð3:29Þ

Taking the convolution of the relation (3.28) with g, we conclude with the aid of relations (3.29) and (2.6) that (3.26) holds. h The method to obtain the reciprocal relation has been established in [8].

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

9

4. A continuous dependence result In this section we study the continuous dependence of solutions upon initial data and body loads. We consider the linear theory of homogeneous and isotropic bodies. The basic equations of this theory are given by (2.13). It is convenient to have Eq. (2.13) rewritten in nondimensional form. We introduce the dimensionless variables x0i ¼

xi ; l0

t0 ¼

c1 t ; l0

u0i ¼

ui ; l0

u0i ¼ ui ;

u0 ¼ u;

T0 ¼

T ; T0

w¼

w ; w0

ð4:1Þ

where l0 is a standard length, c1 = [(k + 2l + j)/q0]1/2, and w0 is a standard electrostatic potential. Introducing (4.1) into (2.13) and suppressing primes we find n1 Dui þ ð1  n1 Þuj;ji þ q1 eijk uk;j þ q2 u;i  k 1 T ;i þ F i ¼ €ui ; € i; n2 Dui þ n3 uj;ji þ q1 eijk uk;j  2q1 ui þ Gi ¼ J 1 u ð4:2Þ

ðn4 D  n5 Þu  n6 Dw  q2 uj;j þ k 2 T þ H ¼ s€ u; n6 Du þ mDw ¼ P ; KDT  k 1 u_ j;j  k 2 u_  fT_ ¼ S; where lþj c aþb a0 k3 ; n2 ¼ 2 2 ; n3 ¼ 2 2 ; n4 ¼ 2 2 ; n5 ¼ ; 2 q 0 c1 q0 c21 l0 q0 c1 l q0 c1 l0 q0 c1 k2 w j k0 J kT 0 ; q2 ¼ ; J1 ¼ ; K¼ ; n6 ¼ 2 02 ; q1 ¼ 2 2 q0 c1 l0 q0 q0 c1 l0 q0 c31 l0 q 0 c 1 n1 ¼

b T0 k1 ¼ 0 2 ; q0 c1 l0 fi ; Fi ¼ q0 c21

b T0 vw2 cT 0 j k2 ¼ 1 2 ; m ¼ 2 0 2 ; f ¼ ; s¼ 0 ; q 0 c1 q0 c21 l0 q0 l0 q0 c1 g G wp l0 s Gi ¼ i 2 ; H ¼ ; P ¼ 02; S ¼ : q0 c21 q0 c1 q0 c1 q0 c31

ð4:3Þ

To Eq. (4.2) we add the initial conditions (2.10) and the following boundary conditions ui ¼ ~ ui ;

~ i; ui ¼ u

~; u¼u

~ w ¼ w;

T ¼ T~

on oB  ½0; t1 ;

ð4:4Þ

~ and T~ are prescribed functions and t1 is a given positive constant. ~ i; u ~; w where ~ ui ; u ð1Þ ð1Þ ð2Þ ð2Þ Let us consider the solutions fui ; ui ; uð1Þ ; wð1Þ ; T ð1Þ g and fui ; ui ; uð2Þ ; wð2Þ ; T ð2Þ g corresponding to the ð1Þ ð1Þ 0ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ~ T~ ; u ; v0ð1Þ ; u0ð1Þ ; m0ð1Þ ; u0ð1Þ ; w0ð1Þ ; T 0ð1Þ g and ~ i; u ~ ; w; external data systems J ¼ fF i ; Gi ; H ; P ; S ; ~ui ; u i i i i ð2Þ ð2Þ ~ T~ ; u0ð2Þ ; v0ð2Þ ; u0ð2Þ ; m0ð2Þ ; u0ð2Þ ; w0ð2Þ ; T 0ð2Þ g, respectively. If we define ~ i; u ~ ; w; ui ; u Jð2Þ ¼ fF i ; Gi ; H ð2Þ ; P ð2Þ ; S ð2Þ ; ~ i i i i ð1Þ ð2Þ ð1Þ ð2Þ ui ¼ ui  ui ; ui ¼ ui  ui ; u ¼ uð1Þ  uð2Þ ; w ¼ wð1Þ  wð2Þ ; T ¼ T ð1Þ  T ð2Þ , then {ui, ui, u, w, T} is a solution of the problem corresponding to the external data system J ¼ fF i ; Gi ; H ; P ; S; 0; 0; 0; 0; 0; u0i ; v0i ; ð1Þ ð2Þ ð1Þ ð2Þ u0i ; m0i ; u0 ; w0 ; T 0 g, where F i ¼ F i  F i ; Gi ¼ Gi  Gi ; H ¼ H ð1Þ  H ð2Þ , P ¼ P ð1Þ  P ð2Þ ; S ¼ S ð1Þ  S ð2Þ ; 0ð1Þ 0ð2Þ 0ð1Þ 0ð2Þ 0ð1Þ 0ð2Þ 0ð1Þ 0ð2Þ u0i ¼ ui  ui ; v0i ¼ vi  vi ; u0i ¼ ui  ui ; m0i ¼ mi  mi ; u0 ¼ u0ð1Þ  u0ð2Þ ; w0 ¼ w0ð1Þ  w0ð2Þ ; and T0 = T0(1)  T0(2). We denote this problem by (A). Let us introduce the function C on [0, t1] by  Z  Z t 1 _ 2 þ 2K þ fT 2 þ mw;i w;i þ 2 C¼ KT ;i T ;i ds dv; ð4:5Þ u_ i u_ i þ J 1 u_ i u_ i þ sðuÞ 2 B 0 where 2K ¼ ð1  2n1 þ q1 Þerr ess þ n1 eij eij þ ðn1  q1 Þeji eij þ 2q2 err u þ #crr css þ ðn3  #Þcij cji þ n2 cij cij þ 2B0 eijk u;k cij þ n4 u;i u;i þ n5 u2 :

ð4:6Þ

10

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

Here we have used the notation eij ¼ uj;i þ ejik uk ;

cij ¼ uj;i ;

a

#¼

l20 q0 c21

;

B0 ¼

b0 2 l0 q0 c21

ð4:7Þ

:

In what follows we assume that the elastic potential K is a positive definite quadratic form in the variables eij, cij, u,i and u. Thus, there exist the positive constants j1 and j2 such that j1 ðeij eij þ cij cij þ u;i u;i þ u2 Þ 6 K 6 j2 ðeij eij þ cij cij þ u;i u;i þ u2 Þ;

ð4:8Þ

for all the variables eij, cij, u,k, u and any t 2 [0, t1]. Lemma 4.1. Let {ui, ui, u, w, T} be a solution of the problem (A). Then C_ ¼

Z

ðF i u_ i þ Gi u_ i þ H u_ þ ST Þ dv:

ð4:9Þ

B

Proof. We introduce the notations sji ¼ ð1  2n1 þ q1 Þerr dij þ n1 eji þ ðn1  q1 Þeij þ q2 udij  k 1 T dij ; lji ¼ #crr dij þ ðn3  #Þcij þ n2 cji þ B0 ejik u;k  Leijk w;k ; h ¼ q2 uj;j þ n5 u  k 2 T ;

ri ¼ n4 u;i  n6 w;i þ B0 ersi crs ;

P ¼ k 1 uj;j þ k 2 u þ fT ;

Qi ¼ KT ;i ;

ð4:10Þ

d j ¼ Lersj csr  n6 u;j  mw;j ; where L ¼ k1 =ðl20 q0 c21 Þ. The Eq. (4.2) imply that sji;j þ F i ¼ € ui ;

€ i; lji;j þ eijk sjk þ Gi ¼ J 1 u

ri;i  h þ H ¼ s€ u;

ð4:11Þ

_ Qj;j þ S ¼ P:

d j;j ¼ P ;

In view of (4.10) and (4.6) we obtain _  d_ j w;j ¼ K_ þ fT T_ þ mw;i w_ ;i : sij e_ ij þ lij c_ ij þ ri u_ ;i þ hu_ þ PT

ð4:12Þ

On the other hand, taking into account (4.11), (4.7) and the fact that for piezoelectric bodies we have P_ ¼ 0, we find that _  d_ j w;j ¼ ðsji u_ i þ lji u_ i þ rj u_  d_ j w þ Qj T Þ þ F i u_ i þ Gi u_ i sij e_ ij þ lij c_ ij þ rj u_ ;j þ hu_ þ PT ;j € i u_ i  s€ þ H u_ þ ST  €ui u_ i  J 1 u uu_  Qj T ;j :

ð4:13Þ

If we integrate (4.13) over B and use the divergence theorem, the boundary conditions and (4.12), then we obtain the desired result. h We define the functions U and V on [0, t1] by   Z  Z t _ 2 þ T 2 þ w;j w;j þ eij eij þ cij cij þ u;j u;j þu2 þ T ;i T ;i ds dv ; U¼ u_ i u_ i þ u_ i u_ i þ ðuÞ B

V ¼

Z

2

2

ðF i F i þ Gi Gi þ H þ S Þdv

0

1=2

ð4:14Þ

:

B

Theorem 4.1. Assume that J1, s, f, m and K are strictly positive and that K is a positive definite form. Let {ui, ui, u, w, T} be a solution of the problem (A). Then, there exist two positive constants c1 and c2 such that UðtÞ 6 c1 Uð0Þ þ c2

Z 0

t

V ðsÞ ds; t 2 ½0; t1 :

ð4:15Þ

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

Proof. By using Schwartz inequality, from Eq. (4.9) we get Z h i 1=2 2 2 _C 6 V _ þ T dv : u_ i u_ i þ u_ i u_ i þ ðuÞ

11

ð4:16Þ

B

It follows from (4.14) and (4.16) that C_ 6 V U:

ð4:17Þ

This inequality implies that Z t CðtÞ 6 Cð0Þ þ V ðsÞUðsÞ ds;

t 2 ½0; t1 :

ð4:18Þ

0

We note that on the basis of (4.5) and (4.8) we obtain CðtÞ P x1 U2 ðtÞ;

Cð0Þ 6 x2 U2 ð0Þ;

t 2 ½0; t1 ;

ð4:19Þ

where 1 1 minð1; J 1 ; s; 2j1 ; f; m; 2KÞ; x2 ¼ maxð1; J 1 ; s; 2j1 ; f; m; 2KÞ: 2 2 The relations (4.18) and (4.19) imply the inequality Z t U2 ðtÞ 6 c21 U2 ð0Þ þ 2c2 V ðsÞUðsÞ ds; t 2 ½0; t1 ; x1 ¼

ð4:20Þ

0

where c1 ¼ ðx2 =x1 Þ

1=2

;

c2 ¼ 1=ð2x1 Þ:

By using the Gronwall inequality, from (4.20) we obtain (4.15).

h

A similar continuous dependence result can be established for the case of Neumann boundary conditions. 5. A variational characterization of solutions In the first part of this section we present an alternative characterization of solutions by equations which incorporate the initial conditions. Then, we establish a variational theorem of Gurtin type. We restrict our attention to the case of homogeneous and isotropic bodies and assume that q0, J, j0, k and a are strictly positive, and that I = (0, 1). Let us introduce the notations P i ¼ g  fi þ q0 ðtv0i þ u0i Þ;

M i ¼ g  gi þ J ðtm0i þ u0i Þ;

N ¼ g  G þ j0 ðtw0 þ u0 Þ;

Q ¼ l  s þ T 0 g0 ;

ð5:1Þ

where g and l are defined in (3.1) and g0 ¼ b0 u0j;j þ b1 u0 þ aT 0 . Lemma 5.1. The functions tij, mij, pk, qj 2 C1,0, g 2 C0,1, r 2 C0, ui, ui, u 2 C0,2 satisfy Eqs. (2.1) and (2.2) and the initial conditions if and only if g  tji;j þ P i ¼ q0 ui ;

g  ðmji;j þ eijk tjk Þ þ M i ¼ J ui ;

g  ðpj;j  rÞ þ N ¼ j0 u;

T 0 g ¼ l  qj;j þ Q

on B  ½0; 1Þ:

ð5:2Þ

Proof. The proof of this lemma is similar to the proof of the corresponding lemma in the classical theory of elasticity [11]. In what follows we consider the boundary-initial-value problem characterized by Eqs. (2.1)–(2.3), (2.12), (2.5) on B · I, the initial conditions (2.10) and the homogeneous boundary conditions ui ¼ 0;

ui ¼ 0;

u ¼ 0;

w ¼ 0;

T ¼0

on oB  ð0; 1Þ:

ð5:3Þ

Lemma 5.1 allows us to give a characterization of the solution by equations which incorporate the initial conditions. h

12

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

In view of (2.5) and (2.12), the basic equations can be expressed in terms of the functions ui, ui, u, w and T. Thus, we obtain the following equations:  q0 ui  g  ðl þ jÞDui þ ðk þ lÞuj;ji þ jeijk uk;j þ k0 u;i  b0 T ;i ¼ P i ;  J ui  g  cDui þ ða þ bÞuj;ji þ jeijk uk;j  2jui ¼ M i ;  j0 u  g  ða0 D  k3 Þu  k2 Dw  k0 uj;j þ b1 T ¼ N ; T 0 ðb0 uj;j þ b1 u þ aT Þ  l  kDT ¼ Q;

ð5:4Þ

g  ðk2 Du þ vDwÞ ¼ R;

on B · [0, 1), where we have used the notation R = g * p. We denote by ðPÞ the problem which consists in the finding of the functions ui, ui, u, w and T which satisfy Eq. (5.4) and the boundary conditions (5.3). Let K be the set of all nine-dimensional vectors u = (ui, ui, u, w, T) with the following properties: (i) ui, ui, u 2 C2, w 2 C2.0, T 2 C2,1 on B · [0, 1); (ii) ui,   ½0; 1Þ: (iii) ui, ui, u, w and T satisfy the boundary conditions (5.3). ui, u, w and T are continuous on B By a solution of the problem ðPÞ we mean a vector u 2 K which satisfies Eq. (5.4). In what follows, the functions eij and jij associated to the vector u ¼ ðui ; ui ; u; w; T Þ 2 K will be denoted by eij(u) and jij(u), respectively, i.e. eij ðuÞ ¼ uj;i þ ejik uk ;

jij ðuÞ ¼ uj;i :

Theorem 5.1. For each t 2 I, let Kt(Æ) be the functional defined on K by Z  Kt ðuÞ ¼ fg  kerr ðuÞ  ess ðuÞ þ ðl þ jÞeij ðuÞ  eij ðuÞ þ leji ðuÞ  eij ðuÞþajrr ðuÞ  jss ðuÞ B

þbjji ðuÞ  jij ðuÞ þ cjij ðuÞ  jij ðuÞ þ a0 u;j  u;j þ k3 u  uvw;j  w;j  aT  T þ 2k0 err ðuÞ  u 2b0 err ðuÞ  T  2b1 u  T  2k2 u;j  w;j þ 2b0 eijk u;k  jij ðuÞ  2k1 ersi jsr ðuÞ  w;i þ q0 ui  ui þ J ui  ui þ j0 u  u  kT 1 0 l  T ;i  T ;i  2ðP i  ui þ M i  ui þ N  u  T 1 0 Q  T  R  wÞg dv; t 2 I;

ð5:5Þ

for any u ¼ ðui ; ui ; u; w; T Þ 2 K. Then dKt ðuÞ ¼ 0;

t 2 I;

ð5:6Þ

at u 2 K if and only if u is a solution of the problem ðPÞ. Proof. Let u = (ui, ui, u, w, T) and u0 ¼ ðu0i ; u0i ; u0 ; w0 ; T 0 Þ be nine-dimensional vectors such that u þ yu0 2 K

for every scalar y:

ð5:7Þ

We define du0 Kt ðuÞ ¼

d Kt ðu þ yu0 Þjy¼0 : dy

If du0 Kt ðuÞ exists and equals zero for every choice of u 0 consistent with (5.7) then we write dKt(u) = 0. It follows from (5.7) that the functions u0i ; u0i ; u0 ; w0 and T 0 satisfy the boundary conditions (5.3). We shall denote the functions tij, mij, pk, r, g and Dj associated to the vector u by tij(u), mij(u), pk(u), r(u), g(u) and Dj(u), respectively. In view of (5.5) we obtain Z  h du0 Kt ðuÞ ¼ 2 g  tij ðuÞ  eij ðu0 Þ þ mij ðuÞ  jij ðu0 Þ þ pj ðuÞ  u;j þ rðuÞ  u0 :gðuÞ  T 0 þ Di ðuÞ  w0;i B i 0 1 0  kT 1 l  T  T þ T Q  T þ q0 ui  u0i þ J ui  u0i þ j0 u  u0  P i  u0i  M i  u0i ;i 0 ;i 0   N  u0 þ R  w0 dv:

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

13

If we use the divergence theorem and the boundary conditions, then we find that Z

 fq0 ui  g  ðl þ jÞDui þ ðk þ lÞuj;ji þ jeijk uk;j þ k0 u;i  b0 T ;i  P i g  u0i du0 Kt ðuÞ ¼ 2 B

  þ J ui  g  cDui þ ða þ bÞuj;ji þ jeijk uk;j  2jui  M i  u0i  þ fj0 u  g  ða0 D  k3 Þu  k2 Dw  k0 uj;j þ b1 T  N g  u0 0 þ T 1 0 g  ½l  kDT þQ  T 0 ðb0 uj;j þ b1 u þ aT Þ  T þg  ðk2 Du þ vDw  pÞ  w0 Þ dv:

ð5:8Þ

If u is a solution of the problem ðPÞ, then in view of (5.4) and (5.8) we find that du0 Kt ðuÞ ¼ 0 for any u 0 . This fact implies that (5.6) holds. Conversely, by using the lemmas established by Gurtin [11, Section 65] we can prove that if du0 Kt ðuÞ ¼ 0 for any u0 2 K, then u is a solution of the problem ðPÞ. h As in [12], we can establish variational theorems for other kind of boundary conditions. 6. Effect of concentrated loads In this section we study the effect of a concentrated heat supply and the effect of a concentrated charge density in an unbounded homogenous and isotropic body. In the case of the classical thermoelasticity the problem of concentrated loads has been intensively studied (see, e.g., [9,10]). Throughout this section we consider a body that occupies the entire three-dimensional euclidean space and restrict out attention to the theory of steady vibrations. We assume that the body loads have the form fj ¼ 0;

gj ¼ 0;

G ¼ 0;

p ¼ Re½P ðrÞ expðxtÞ;

s ¼ Re½SðrÞ expðixtÞ;

ð6:1Þ

where Re½f  is the real part of f, P and S are prescribed functions, r = jx  yj, y being a fixed point, x is the frequency of vibration, and i = (1)1/2. In this case we seek for solution to (2.13) of the form u ¼ Re½grad U ðrÞ expðixtÞ; u ¼ Re½UðrÞ expðixtÞ;

u ¼ 0;

w ¼ Re½WðrÞ expðixtÞ;

T ¼ Re½V ðrÞ expðixtÞ;

ð6:2Þ

where U, U, W and V are unknown functions. The field Eq. (2.13) are satisfied if U, U, W and V satisfy the equations ða11 D þ a011 ÞU þ a012 U þ a014 V ¼ 0; a21 DU þ ða22 D þ a022 ÞU þ a23 DW þ a024 V ¼ 0;

ð6:3Þ

a32 DU þ a33 DW ¼ P ; a41 DU þ a042 U þ ða44 D þ a044 ÞV ¼ S; where a11 ¼ k þ 2l þ j; a21 ¼ k0 ; a32 ¼ k2 ;

a011 ¼ q0 x2 ;

a22 ¼ 0;

a012 ¼ k0 ;

a022 ¼ j0 x2  k3 ;

a014 ¼ b0 ;

a23 ¼ k2 ; a024 ¼ b1 ;

a33 ¼ v;

a41 ¼ ixT 0 b0 ;

a042 ¼ ixT 0 b1 ;

a44 ¼ k;

a044 ¼ ix:

ð6:4Þ

14

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

Let us introduce the notations j1 ¼ a012 a44 ;

j0 ¼ a012 a044  a014 a042 ;

k 1 ¼ a014 ða23 a32  a22 a33 Þ;

k 0 ¼ a33 ða012 a024  a022 a014 Þ;

n1 ¼ a11 a044 þ a44 a011  a41 a014 ;

n2 ¼ a11 a44 ;

s1 ¼ a11 a024  a21 a014 ;

s0 ¼ a011 a024 ;

a1 ¼ a22 a044 þ a44 a022 ;

a2 ¼ a22 a44 ;

n0 ¼ a011 a044 ;

f1 ¼ a012 a21 a44  a014 a22 a44 ; b1 ¼ a11 a042  a41 a012 ;

a0 ¼ a022 a044  a024 a042 ;

f0 ¼ a012 a024 a41  a012 a044 a21 þ a014 ða042 a21  a024 a41 Þ;

ð6:5Þ

b0 ¼ a011 a042 ;

g2 ¼ a11 ða22 a33  a23 a32 Þ; c3 ¼ a11 a2  a23 a32 n2 ;

g1 ¼ a33 ða11 a022 þ a22 a011  a21 a012 Þ  a32 a011 a23 ;

c2 ¼ a32 n1 þ a33 ða2 a011 þ a1 a11 þ f1 Þ;

c1 ¼ a33 ða11 a0 þ f0 þ a011 a1 Þ;

c0 ¼ a33 a011 a0 :

We consider the operators K1 ¼ a23 Dðj1 D þ j0 Þ;

D2 ¼ Dðk 1 D þ k 0 Þ;

2

C1 ¼ a23 Dðn2 D þ n1 D þ n0 Þ; P1 ¼ ða11 D þ

a011 Þða2 D2

P2 ¼ a32 Dðs1 D þ s0 Þ;

C2 ¼ a33 Dðs1 D þ s0 Þ;

þ a1 D þ a0 Þ þ Dðf1 D þ f0 Þ;

ð6:6Þ

X1 ¼ a23 Dðb1 D þ b0 Þ;

2

X2 ¼ Dðg2 D þ g1 D þ g0 Þ: Theorem 6.1. Let U ¼ K1 F þ K2 G;

U ¼ C1 F þ C2 G;

W ¼ P1 F þ P2 G;

V ¼ X1 F þ X2 G;

ð6:7Þ

where F and G are functions of class C1 which satisfy the equations Dðc3 D3 þ c2 D2 þ c1 D þ c0 ÞF ¼ P ; Dðc3 D3 þ c2 D2 þ c1 D þ c0 ÞG ¼ S:

ð6:8Þ

Then, U, U, W and V satisfy the equations (6.3). Proof. Let us substitute the functions U, U, W and V given by (6.7) into the left-hand side of (6.3). For the last equations from (6.3) we obtain a41 DU þ a042 U þ ða44 D þ a044 ÞV ¼ ½a41 DK1 þ a042 C1 þ X1 ða44 D þ a044 ÞF þ ½a41 DK2 þ a042 C2 þ X2 ða44 D þ a044 ÞG:

ð6:9Þ

In view of (6.5) and (6.6) we find that a41 DK1 þ a042 C1 þ Xða44 D þ a044 Þ ¼ 0; a41 DK2 þ a042 C2 þ þX2 ða44 D þ a044 Þ ¼ Dðc3 D3 þ c2 D2 þ c1 D þ c0 Þ:

ð6:10Þ

It follows from (6.9), (6.10) and (6.8) that Eq. (6.3)4 is satisfied. Similarly, we can show that the other equations of (6.3) are satisfied. h Let us investigate the effect of a concentrated charge density in an infinite space. We assume that P ¼ dðx  yÞ;

S ¼ 0;

ð6:11Þ

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

15

where d(Æ) is the Dirac delta and y is a fixed point. In this case we take K = H and G = 0, where H satisfies the equation Dðc3 D3 þ c2 D2 þ c1 D þ c0 ÞH ¼ dðx  yÞ:

ð6:12Þ

From (6.7) we obtain the following solution of Eq. (6.3), U 1 ¼ K1 H ; W1 ¼ P1 H ;

U1 ¼ C1 H ; V 1 ¼ X1 H :

ð6:13Þ

Let us determine the function H. The Eq. (6.12) can be written in the form c3 DðD þ r21 ÞðD þ r22 ÞðD þ r23 ÞH ¼ dðx  yÞ; where r21 ; r22 c3 r6 

and

r23

ð6:14Þ

are the roots of the equation

4

c2 r þ c1 r2  c0 ¼ 0:

ð6:15Þ

In what follows we denote by rj (j = 1, 2, 3), the roots with positive real parts and assume that r1, r2 and r3 are distinct. Let us consider the functions hr (r = 1, 2, 3, 4), which satisfy the equations ðD þ r2j Þhj ¼ Q;

ðno sum; j ¼ 1; 2; 3Þ;

Dh4 ¼ Q:

ð6:16Þ

It is a simple matter to see that the solution of the equation DðD þ r21 ÞðD þ r22 ÞðD þ r23 Þh ¼ Q; can be expressed in the form h ¼ c1 h1 þ c2 h2 þ c3 h3 þ c4 h4 ; where

ð6:17Þ 1

c1 ¼ ðp1 r41  p2 r21 þ p3 Þ½ðr21  r22 Þðr21  r23 Þ ; 1

c2 ¼ ðp2 r22  p1 r42 þ p3 Þ½ðr21  r22 Þðr22  r23 Þ ; c3 ¼ ðp1 r43  p2 r23 þ p3 Þ½ðr21  r23 Þðr22  r23 Þ1 ; p1 ¼ p3 ¼

ð6:18Þ

2

ðr1 r2 r3 Þ ; p2 ¼ ðr21 þ r22 þ r23 Þ2 ðr1 r2 r3 Þ2 ; 2 ðr21 r22 þ r22 r23 þ r23 r21 Þðr1 r2 r3 Þ :

If Q¼

1 dðx  yÞ; c3

then from (6.17) we obtain 1 expðirj rÞ; hj ¼  4pc3 r

h4 ¼ 

1 : 4pc3 r

Thus, the function H, which satisfies (6.14), is given by " # 3 X 1 H ¼ c4 þ cj expðirj rÞ : 4pc3 r j¼1

ð6:19Þ

ð6:20Þ

If we substitute the function H, given by (6.20) into (6.13), then we obtain the solution of Eq. (6.3) corresponding to the loads (6.11). In a similar way we can study the effect of a concentrated heat supply, i.e., P ¼ 0; S ¼ dðx  yÞ: In this case we obtain the following solution of Eq. (6.3) corresponding to the sources (6.21), U 2 ¼ K2 H ; W2 ¼ P2 H ;

U2 ¼ C2 H ; V 2 ¼ X2 H ;

where H is given by (6.20).

ð6:21Þ

16

D. Iesßan, R. Quintanilla / International Journal of Engineering Science 45 (2007) 1–16

Acknowledgments This work is supported by the project ‘‘Estudio Cualitativo de Problemas Termomeca´nicos’’ (MTM200603706) of the MEC. References [1] R.D. Mindlin, Equations of high frequency, vibrations of thermopiezoelectric crystal plates, Int. J. Solids Struct. 10 (1974) 625–637. [2] W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses 1 (1978) 171–182. [3] W. Nowacki, Mathematical models of phenomenological piezoelectricity, in: O. Brulin, R. Hsieh (Eds.), New Problems in Mechanics of Continua, University of Waterloo Press, Ontario, 1983, pp. 30–50. [4] F. Heidary, M.R. Esalmi, Pyroelectric effect on dynamic response of coupled distributed piezothermoelastic composite plate, J. Thermal Stresses 28 (2005) 285–300. [5] A.C. Eringen, Electromagnetic theory of microstretch elasticity and bone modeling, Int. J. Engng. Sci. 42 (2004) 231–242. [6] A.C. Eringen, Microcontinuum Field Theories I. Foundations and Solids, Springer-Verlag, New York, 1999. [7] J. Ignaczak, A completeness problem for stress equations of motion in the linear elasticity theory, Arch. Mech. Stosow. 15 (1963) 225–235. [8] D. Iesßan, On some theorems in thermoelastodynamics, Rev. Roum. Sci. Tech. Ser. Me´c. Appl. 34 (1989) 101–111. [9] R.B. Hetnarski, The fundamental solutions of the coupled thermoelastic problem for small times, Arch. Mech. Stosow. 16 (1964) 23–31. [10] W. Nowacki, Theory of Asymmetric Elasticity, Polish Scientific Publishers and Pergamon Press, Warszawa and Oxford, New York, Paris, 1986. [11] M.E. Gurtin, The linear theory of elasticity, in: C. Truesdell (Ed.), Handbuch der Physik, vol. VIa/2, Springer, Berlin, 1972. [12] D.E. Carlson, Linear thermoelasticity, in: C. Truesdell (Ed.), Handbuch der Physik, vol. VIa/2, Springer, Berlin, 1972.