A spatial decay in the linear theory of microstretch piezoelectricity

Mathematical and Computer Modelling 47 (2008) 1117–1124 www.elsevier.com/locate/mcm

A spatial decay in the linear theory of microstretch piezoelectricity R. Quintanilla ∗ Matematica Aplicada 2, ETSEIAT-UPC, 08222 Terrassa, Barcelona, Spain Received 1 March 2007; received in revised form 7 June 2007; accepted 22 June 2007

Abstract The electromagnetic theory of microstretch elasticity is an adequate tool to describe the behavior of porous bodies, animal bones and solids with deformable microstructures. In this paper we study the linear theory of microstretch piezoelectricity. First, we establish a spatial decay estimate. Then, we obtain an upper bound for the amplitude term. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Microstretch elastic solids; Piezoelectricity; Elasticity; Spatial decay estimates

1. Introduction The theory of nonpolar thermopiezoelectricity has been studied in various papers (see, e.g., [1–3]). Eringen [4] established the electromagnetic theory of microstretch thermoelasticity. The material particles of the microstretch bodies can stretch and contract independently of their translations and rotations. The theory introduced by Eringen [4] involves interactions of electromagnetic fields and thermomechanical deformations for porous bodies like bones, solids with microcracks, foams and other synthetic materials. Some recent results in this theory have been obtained [5–7]. In addition, interest has developed in the derivation of spatial decay estimates for static solutions to piezoelectric boundary value problems [8–12]. However, the dynamical case has not been studied yet. We believe that the main problem is that the usual arguments for dynamic problems cannot be used in this situation. It is worth recalling that what is usual is to bound the time derivative of a certain measure of the solutions by means of a combination of the spatial derivatives of the same function (see [13–15]). However, in piezoelectricity one does not know how to do it because of the ellipticity of the equation corresponding to the electrostatic potential. In fact, when we want to obtain a measure (see (3.4)), we will need to work with expressions which involve the time derivative of the electrostatic potential (see (3.11) and (3.12)). However we do not know how this term can be controlled by the measure. We propose an alternative argument which appears to be new and as far as we know has been used previously to study only dispersive equations [16] and quasistatic nonlinear viscoelasticity [17]. Specifically, a spatial decay estimate is derived for the solution to the problem of cylinder in motion subject to zero initial data and boundary data except for that prescribed on the base. The amplitude for this estimate is then bounded in terms of the base data. A disadvantage, however, of this alternative method is its slowness and unboundedness as the axial distance vanishes. ∗ Tel.: +34 937398162; fax: +34 937398101.

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.06.023

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In Section 2 we recall the basic equations which define the problem. The spatial decay estimate is obtained in Section 3. The upper bound for the amplitude in terms of the boundary conditions is obtained in Section 4. 2. Basic equations We consider a body that at some instant occupies a semi-infinite cylinder B = (0, ∞) × D, where D is a twodimensional bounded domain of the Euclidean space with smooth boundary. This means that the cross-section is independent of x1 and time. The motion of the body is referred to the reference configuration B and a fixed system of rectangular cartesian axes O xi , (i = 1, 2, 3). Throughout boldface characters stand for tensors of order p ≥ 1, and if v has the order p, we write vi j...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. We use a superposed dot to denote partial differentiation with respect to time t. The semi-infinite cylinder {(xi ) ∈ B, x1 > z} will be denoted by B(z), and the cross-section {(xi ) ∈ B, x1 = z} by D(z). We consider the linear theory of microstretch piezoelectricity. We recall that in this case the effect of the magnetic flux vector is ignored. The basic equations of this theory (see [7]) consist of the equations of motion t ji, j + f i = ρ0 u¨ i , m ji, j + εi jk t jk + gi = Ii j ϕ¨ j ,

(2.1)

πk,k − σ + G = j0 ϕ, ¨ the equations of the electric fields D j, j = p,

E k = −ψ,k ,

(2.2)

the constitutive equations (1)

ti j = Ai jr s er s + Bi jr s κr s + Di j ϕ + Fi jk ζk + λi jk E k , (2)

m i j = Br si j er s + Ci jr s κr s + E i j ϕ + G i jk ζk + λi jk E k , (3)

σ = Di j ei j + E i j κi j + ξ ϕ + h k ζk + λi E i ,

(2.3)

(4) πk = Fi jk ei j + G i jk κi j + h k ϕ + Ak j ζ j + λk j E j , (1) (2) (3) (4) Dk = −λi jk ei j − λi jk κi j − λk ϕ − λ jk ζ j + χk j E j ,

and the geometrical equations ei j = u j,i + ε jik ϕk ,

κi j = ϕ j,i ,

ζ j = ϕ, j .

(2.4)

In (2.1)–(2.4) the following notation is used: ti j is the stress tensor, f i is the body force, ρ0 is the reference mass density, u i is the displacement vector, m i j is the couple stress tensor, εi jk is the alternating symbol, gi is the body couple, Ii j is the microinertia tensor, ϕi is the microrotation vector, πk is the microstretch stress vector, ϕ is the microstretch function which gives a measure of the internal expansion or contraction of the microstructure, σ is the microstress function, G is the microstretch body force, j0 is the microstretch inertia, Dk is the dielectric displacement vector, p is the volume charge density, E k is the electric field vector, ψ is the electrostatic potential, ei j , κi j and ζk are (α) (3) (4) kinematic strain measures and Ai jr s , Bi jr s , Ci jr s , Di j , E i j , Fi jk , G i jk , h i , ξ, Ai j , λi jk , λi , λi j , χi j are constitutive coefficients. We consider the following boundary conditions ui = e ui ;

ϕi = e ϕi ;

u i = ϕi = ϕ = ψ = 0;

ϕ=e ϕ;

e ψ = ψ;

on {0} × D × I,

on (0, ∞) × ∂ D × I,

e 2 , x3 , t) are prescribed functions, and I = (0, ∞). where e u i (x2 , x3 , t), e ϕi (x2 , x3 , t), e ϕ (x2 , x3 , t), ψ(x

(2.5)

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The initial conditions are assumed to be homogeneous u i (x, 0) = u˙ i (x, 0) = ϕi (x, 0) = ϕ˙i (x, 0) = ϕ(x, 0) = ϕ(x, ˙ 0) = 0,

(2.6)

x ∈ B.

For materials with various symmetry groups, constitutive tensors in (2.3) have a simplified expression. For isotropic, centrosymmetric and homogeneous bodies the constitutive tensors become Aklmn = λδkl δmn + (µ + κ)δkm δln + µδkn δlm , Ci jr s = αδkl δmn + βδln δkm + γ δlm δkn , (4) λi j

= λ2 δi j ,

χi j = χ δi j ,

Di j = λ0 δi j ,

ξ = λ3 ,

Ii j = J δi j , (1)

(3)

Bklmn = Fi jk = E i j = h k = λi jk = λi

Akl = a0 δkl (2)

G i jk = b0 εi jk ,

λi jk = λ1 ε jik

= 0,

where δi j is the Kronecker delta and λ, µ, κ, λ0 , α, β, γ , b0 , λk and χ are constitutive constants. Then, the constitutive equations (2.3) reduce to ti j = λerr δi j + (µ + κ)ei j + µe ji + λ0 ϕδi j , m i j = ακrr δi j + βκ ji + γ κi j + b0 εi jk ζk + λ1 ε jik E k , σ = λ0 err + λ3 ϕ,

(2.7)

πi = a0 ζi + λ2 E i + b0 εr si κr s , Di = −λ1 εr si κsr − λ2 ζi + χ E i . It follows from (2.1), (2.2), (2.7) and (2.4) that the field equations of the theory of homogeneous and isotropic bodies in the absence of supply terms can be expressed as (µ + κ)∆u i + (λ + µ)u j,i j + κεi jk ϕk, j + λ0 ϕ,i = ρ0 u¨ i , γ ∆ϕi + (α + β)ϕ j, ji + κεi jk u k, j − 2κϕi = J ϕ¨i ,

(2.8)

(a0 ∆ − λ3 )ϕ − λ2 ∆ψ − λ0 u j, j = j0 ϕ, ¨ λ2 ∆ϕ + χ ∆ψ = 0, where ∆ is the well-known Laplacian operator. 3. Decay estimate

We consider the linear theory of homogeneous and isotropic bodies. The basic equations of this theory are given by (2.8). It is convenient to have Eq. (2.8) rewritten in the non-dimensional form. Thus, we introduce the dimensionless variables xi0 =

xi , l0

t0 =

c1 t , l0

u i0 =

ui , l0

ϕi0 = ϕi ,

ϕ 0 = ϕ,

ψ=

ψ , ψ0

(3.1)

where l0 is a standard length, c1 = [(λ + 2µ + κ)/ρ0 ]1/2 , and ψ0 is a standard electrostatic potential. Introducing (3.1) into (2.8) and suppressing primes we find that ξ1 ∆u i + (1 − ξ1 )u j, ji + ρ1 εi jk ϕk, j + ρ2 ϕ,i = u¨ i , ξ2 ∆ϕi + ξ3 ϕ j, ji + ρ1 εi jk u k, j − 2ρ1 ϕi = J1 ϕ¨i ,

(3.2)

(ξ4 ∆ − ξ5 )ϕ − ξ6 ∆ψ − ρ2 u j, j = τ ϕ, ¨ ξ6 ∆ϕ + ν∆ψ = 0, where ξ1 =

µ+κ , ρ0 c12

ρ1 =

κ , ρ0 c12

ξ2 = ρ2 =

γ l 2ρ

2 0 c1

λ0 , ρ0 c12

,

ξ3 = J1 =

α+β , l02 ρ0 c12

J , l0 ρ0

a0 , 2 l0 ρ0 c12

ξ4 = ν=

χ ψ02

, 2

l02 ρ0 c1

τ=

ξ5 = j0 . l0 ρ0

λ3 , ρ0 c12

ξ6 =

λ2 ψ0 , l02 ρ0 c12

(3.3)

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Let us introduce the function Γ1 by Z Z   1 t Γ1 (z, t) = u˙ i u˙ i + J1 ϕ˙i ϕ˙i + τ (ϕ) ˙ 2 + 2Λ + νψ,i ψ,i dsdv, 2 0 B(z)

(3.4)

where 2Λ = (1 − 2ξ1 + ρ1 )err ess + ξ1 ei j ei j + (ξ1 − ρ1 )e ji ei j + 2ρ2 err ϕ + ϑγrr γss + (ξ3 − ϑ)γi j γ ji + ξ2 γi j γi j + 2B0 εi jk ϕ,k γi j + ξ4 ϕ,i ϕ,i + ξ5 ϕ 2 .

(3.5)

Here we use the notation ei j = u j,i + ε jik ϕk ,

γi j = ϕ j,i ,

ϑ=

α , l02 ρ0 c12

B0 =

b0 . l02 ρ0 c12

(3.6)

In what follows we assume that the elastic potential Λ is a positive definite quadratic form in the variables ei j , γi j , ϕ,i and ϕ. Thus, there exist two positive constants κ1 and κ2 such that κ1 (ei j ei j + γi j γi j + ϕ,i ϕ,i + ϕ 2 ) ≤ Λ ≤ κ2 (ei j ei j + γi j γi j + ϕ,i ϕ,i + ϕ 2 ),

(3.7)

for all the variables ei j , γi j , ϕ,k , ϕ. We also define the function Z Z   1 t Γ2 (z, t) = u¨ i u¨ i + J1 ϕ¨i ϕ¨i + τ (ϕ) ¨ 2 + 2Λ∗ + ν ψ˙ ,i ψ˙ ,i dsdv, 2 0 B(z) where 2Λ∗ = (1 − 2ξ1 + ρ1 )e˙rr e˙ss + ξ1 e˙i j e˙i j + (ξ1 − ρ1 )e˙ ji e˙i j + 2ρ2 e˙rr ϕ˙ + ϑ γ˙rr γ˙ss + (ξ3 − ϑ)γ˙i j γ˙ ji + ξ2 γ˙i j γ˙i j + 2B0 εi jk ϕ˙,k γ˙i j + ξ4 ϕ˙,i ϕ˙,i + ξ5 (ϕ) ˙ 2.

(3.8)

We introduce the notations s ji = (1 − 2ξ1 + ρ1 )err δi j + ξ1 e ji + (ξ1 − ρ1 )ei j + ρ2 ϕδi j , µ ji = ϑγrr δi j + (ξ3 − ϑ)γi j + ξ2 γ ji + B0 ε jik ϕ,k − Lεi jk ψ,k , h = ρ2 u j, j + ξ5 ϕ,

(3.9)

σi = ξ4 ϕ,i − ξ6 ψ,i + B0 εr si γr s , d j = −Lεr s j γsr − ξ6 ϕ, j − νψ, j , where L = λ1 /(l02 ρ0 c12 ). Eq. (3.2) imply that s ji, j = u¨ i , µ ji, j + εi jk s jk = J1 ϕ¨i , σi,i − h = τ ϕ, ¨ d j, j = 0.

(3.10)

In view of (3.9) and (3.5) we obtain ˙ + νψ,i ψ˙ ,i . si j e˙i j + µi j γ˙i j + σi ϕ˙,i + h ϕ˙ − d˙ j ψ, j = Λ

(3.11)

On the other hand, taking into account (3.10), we find that si j e˙i j + µi j γ˙i j + σ j ϕ˙, j + h ϕ˙ − d˙ j ψ, j = (s ji u˙ i + µ ji ϕ˙i + σ j ϕ˙ − d˙ j ψ), j − u¨ i u˙ i − J1 ϕ¨i ϕ˙i − τ ϕ¨ ϕ. ˙

(3.12)

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We define the functions Φi , i = 1, 2 by Z tZ
(t − s) s1i u˙ i + µ1i ϕ˙i + σ1 ϕ˙ − d˙1 ψ dads, Φ1 (z, t) = − 0

Φ2 (z, t) = −

D(z)

Z tZ 0


(t − s) s˙1i u¨ i + µ˙ 1i ϕ¨i + σ˙ 1 ϕ¨ − d¨1 ψ˙ dads. D(z)

As we are interested in the spatial asymptotic behavior, we are going to assume that the solutions satisfy the spatial asymptotic condition as x1 → ∞, u i, j , ϕi, j , ϕ,i , ψ,i , u˙ i, j , ϕ˙i, j , ϕ˙,i , ψ˙ ,i , u¨ i, j , ϕ¨i, j , ϕ¨,i , ψ¨ ,i → 0,

(3.13)

uniformly in x1 for every t, x2 , x3 . We note that if a solution satisfies the initial and boundary conditions and the asymptotic condition (3.13), then Φ1 (z, t) = Γ1 (z, t)

and

Φ2 (z, t) = Γ2 (z, t).

Theorem 3.1. Assume that J1 , τ, ν are strictly positive and that Λ and Λ∗ are positive definite forms. Let {u i , ϕi , ϕ, ψ} be a solution to the problem. Then, there exist two computable positive constants C1 and C2 such that Φ1 (z, t) ≤ tC1 (Φ1 (0, t) + C2 Φ2 (0, t))z −1 ,

t > 0, z > 0.

(3.14)

Proof. Our first step is to estimate the function Φ1 in terms of the spatial derivatives of −Φ1 and −Φ2 . We have Φ1 (z, t) ≤

8 t X J ∗, 2 i=1 i

where J1∗ = J3∗ = J7∗

Z tZ s1i s1i dads, 0

D

Z tZ 0

µ1i µ1i dads, D

Z tZ

d˙1

= 0

D


2

dads,

J2∗ =

Z tZ u˙ i u˙ i dads, 0

D

Z tZ J4∗ = ϕ˙i ϕ˙i dads, 0 D Z tZ J8∗ = ψ 2 dads.

J5∗ =

Z tZ 0

D

σ12 dads,

J6∗ =

Z tZ 0

˙ 2 dads, (ϕ) D

0 D ∗ J1 , J3∗

In view of the definition of Λ, we see that and J5∗ can be estimated in terms of the spatial derivative of −Φ1 . ∗ ∗ ∗ The terms J2 , J4 and J6 can be also estimated in terms of the spatial derivative of −Φ1 . To do the same with J8∗ we need to use first the Poincar´e inequality. To estimate J7∗ we can use the spatial derivative of −Φ2 . Thus we obtain   ∂Φ1 (z, t) ∂Φ2 (z, t) Φ1 (z, t) ≤ −tC1 + C2 . ∂z ∂z Notice that C1 and C2 are calculable in terms of the constitutive coefficients and the Poincar´e constant for the domain D. It follows that the inequality Z z 1 L(z, t) + Φ1 (s)ds ≤ L(0, t), tC1 0 is satisfied, where L(z, t) = Φ1 (z, t) + C2 Φ2 (z, t). On the other hand, we know that the spatial derivative of Φ1 (z, t) is always less than or equal to zero. Thus ∂ ∂Φ1 (zΦ1 (z, t)) = Φ1 (z, t) + z ≤ Φ1 (z, t). ∂z ∂z

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A quadrature implies that Z z Φ1 ( p, t)d p ≤ tC1 L(0, t), zΦ1 (z, t) ≤ 0

which implies the polynomial decay Φ1 (z, t) ≤

C1 t L(0, t). z

The last inequality concludes the proof.



Remark. I. We point out that the analysis can be extended to the thermal conduction case (see [7]) and also to the anisotropic case. II. Right hand side of (3.14) becomes unbounded when z tends to zero. This fact could be saved. If we take A an arbitrary positive constant, we can obtain the inequality: ∂ ∂Φ1 ((z + A)Φ1 (z, t)) = Φ1 (z, t) + (z + A) ≤ Φ1 (z, t). ∂z ∂z This implies the estimate Φ1 (z, t) ≤

C1 t A Φ1 (0, t) + L(0, t). (z + A) (z + A)

(3.15)

Right hand side of this inequality does not become unbounded when z vanishes. III. One suspects that estimates such as (3.14) or (3.15) could be improved to obtain some kind of exponential decay. However, as we pointed out in the introduction the usual arguments for the dynamical cases which take to this kind of estimates do not seem easy to be used in this situation. Though estimate (3.14) seems slow, we believe that it is the first one obtained for this theory. 4. Upper bound for the amplitude term To have a complete description of the estimate (3.14), we need to obtain an upper bound for Φi (0, t), i = 1, 2 in terms of the boundary conditions (2.5). Let us consider the functions vi (x, t), ηi (x, t), η(x, t), ζ (x, t), defined on B × I such that they agree with the boundary conditions (2.5) and tend uniformly to zero as x1 goes to ∞. We also assume the same asymptotic behavior for their time derivatives. In view of the relation si j (v˙ j,i +  jik η˙ k ) + µ ji η˙ i, j + σ j η˙ , j + h η˙ + d j ζ˙, j = (s ji v˙i + µ ji η˙ i + σ j η˙ − d˙ j ζ ), j − u¨ i v˙i − J1 ϕ¨i η˙ i − τ ϕ¨ η, ˙ where si j , µi j , h, σi and di are defined as functions of the variables u i , ϕi , ϕ and ψ, we have that Z tZ
Φ1 (0, t) = − (t − s) s1i v˙i + µ1i η˙ i + σ1 η˙ − d˙1 ζ dads 0

Z tZ = 0

D(0)

(t − s)(s ji (v˙ j,i +  jik η˙ k ) + µ ji η˙ i, j + σ j η˙ , j + h η˙ + d j ζ˙, j − u˙ i v¨i − J1 ϕ˙i η¨ i − ν ϕ˙ η)dvds ¨

B tZ

Z − 0

B


d j ζ, j − u˙ i v˙i − J1 ϕ˙i η˙ i − ν ϕ˙ η˙ dvds.

After applying A-G inequality with suitable weights, we can find a positive function N (t) such that Z tZ 1 (v¨i v¨i + η¨ i η¨ i + (η) ¨ 2 + v˙i, j v˙i, j + η˙ i, j η˙ i, j Φ1 (0, t) ≤ Φ1 (0, t) + N (t) 2 0 B + η˙ , j η˙ , j + ζ˙, j ζ˙, j + ζ, j ζ, j )dvds.

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In a similar way, we can obtain Φ2 (0, t) ≤

Z tZ ... ... ... 1 ... ... Φ2 (0, t) + N (t) ( v i v i + η i η i + ( η )2 + v¨i, j v¨i, j + η¨ i, j η¨ i, j 2 0 B + η¨ , j η¨ , j + ζ¨, j ζ¨, j + ζ˙, j ζ˙, j )dvds.

It is worth noting that the function N (t) becomes unbounded as t tends to infinite. Let us select vi (x, t) = u˜ i (x2 , x3 , t) exp(−α1 x1 ), ηi (x, t) = ϕ˜i (x2 , x3 , t) exp(−α1 x1 ), ˜ 2 , x3 , t) exp(−α1 x1 ), η(x, t) = ϕ(x ˜ 2 , x3 , t) exp(−α1 x1 ), ψ(x, t) = ψ(x where α1 is an arbitrary positive constant. We have Z tZ Z tZ 1 v˙i, j v˙i, j dvds = (u˙˜ i,β u˙˜ i,β + α12 u˙˜ i u˙˜ i )dads. 2α 1 0 0 B D Here β can take only the values 2 and 3. We can estimate in a similar way the integrals Z tZ Z tZ Z tZ v¨i, j v¨i, j dvds, ϕ˙i, j ϕ˙i, j dvds, ϕ¨i, j ϕ¨i, j dvds, . . . . 0

0

B

B

0

B

We can obtain Φ1 (0, t) ≤

N (t) (I1 + I2 + H1 ), α1

where Z tZ   I1 = u˙˜ i,β u˙˜ i,β + ϕ˙˜ i,β ϕ˙˜ i,β + ϕ˙˜ ,β ϕ˙˜ ,β + ψ˙˜ ,β ψ˙˜ ,β + ψ˜ ,β ψ˜ ,β dads, 0 D Z tZ   ˙˜ 2 + (ψ) ˙˜ 2 + (ψ) ˜ 2 dads, u˙˜ i u˙˜ i + ϕ˙˜ i ϕ˙˜ i + (ϕ) I2 = α12 0 D Z tZ   ¨˜ 2 dads. H1 = u¨˜ i u¨˜ i + ϕ¨˜ i ϕ¨˜ i + (ϕ) 0

D

We can also obtain Φ2 (0, t) ≤

N (t) (I3 + I4 + H2 ), α1

where Z tZ   u¨˜ i,β u¨˜ i,β + ϕ¨˜ i,β ϕ¨˜ i,β + ϕ¨˜ ,β ϕ¨˜ ,β + ψ¨˜ ,β ψ¨˜ ,β + ψ˙˜ ,β ψ˙˜ ,β dads, 0 D Z tZ   ¨˜ 2 + (ψ) ˙˜ 2 dads, 2 ¨˜ 2 + (ψ) I4 = α1 u¨˜ i u¨˜ i + ϕ¨˜ i ϕ¨˜ i + (ϕ) 0 D Z t Z ... ... ... ... ...  H2 = u˜ i u˜ i + ϕ˜ i ϕ˜ i + ( ϕ˜ )2 dads. I3 =

0

D

Thus the amplitude term will be controlled by N (t) ((I1 + I2 + H1 ) + C2 (I3 + I4 + H2 )) , α1 where N (t) and C2 can be computed and α1 is an arbitrary positive constant. It is worth noting that I2 and I4 depend on the variable α1 .

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Acknowledgments This work was partially supported by the project “Qualitative study of thermomechanical problems” (MTM200603706) supported by the Education and Science Spanish Ministry and the European Regional Development Fund. The author thanks the anonymous referee for his/her useful comments which helped in the correction and improvement of the manuscript. References [1] F. Heidary, M.R. Esalmi, Pyroelectric effect on dynamic response of coupled distributed piezothermoelastic composite plate, J. Thermal Stresses 28 (2005) 285–300. [2] W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses 1 (1978) 171–182. [3] R.C. Batra, J.S. Yang, Saint-Venant’s principle in linear piezoelectricity, J. Elasticity 38 (1995) 209–218. [4] A.C. Eringen, Electromagnetic theory of microstretch elasticity and bone modeling, Internat. J. Engrg. Sci. 42 (2004) 231–242. [5] A.C. Eringen, Microcontinuum Field Theories I. Foundations and Solids, Springer-Verlag, New York, 1999. [6] D. Ies¸an, On the microstretch piezoelectricity, Internat. J. Engrg. Sci. 44 (2006) 819–829. [7] D. Ies¸an, R. Quintanilla, Some theorems in the theory of microstretch thermopiezoelectricity, Internat. J. Engrg. Sci. 45 (2007) 1–16. [8] R.C. Batra, X. Zhong, Saint-Venant’s principle for a helical piezoelectricity body, J. Elasticity 43 (1996) 69–79. [9] A. Borrelli, C.O. Horgan, M.C. Patria, Saint-Venant end effects in anti-plane shear deformations of linear piezoelectric materials, J. Elasticity 64 (2001) 217–236. [10] A. Borrelli, C.O. Horgan, M.C. Patria, Saint-Venant’s principle for anti-plane shear deformations of linear piezoelectric materials, SIAM J. Appl. Math. 62 (2002) 2027–2044. [11] A. Borrelli, C.O. Horgan, M.C. Patria, End effects for pre-stressed and pre-polarized piezoelectric solids in anti-plane shear, J. Appl. Math. Phys. (ZAMP) 54 (2003) 797–806. [12] A. Borrelli, C.O. Horgan, M.C. Patria, Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 1193–1212. [13] C.O. Horgan, L.E. Payne, L.T. Wheeler, Spatial decay estimates in transient heat conduction, Quart. Appl. Math. 42 (1984) 119–127. [14] J.N. Flavin, R.J. Knops, L.E. Payne, Energy bounds in dynamical problems for a semi-infinite elastic beam, in: Elasticity: Mathematical Methods and Applications, Ellis-Horwood, Chichester, 1989, pp. 101–111. [15] R. Quintanilla, End effects in thermoelasticity, Math. Methods Appl. Sci. 24 (2001) 93–102. [16] R. Quintanilla, G. Saccomandi, Spatial behaviour for a fourth-order dispersive equation, Quart. Appl. Math. 64 (2006) 547–560. [17] R. Quintanilla, G. Saccomandi, Quasistatic anti-plane motions in the simplest theory of nonlinear elasticity, Nonlinear Anal. R. W. A. (2007) (in press), doi: 10.106/j.nonrwa.2007.03.020.