## Abstract

In this paper, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear theory of thermopiezoelectricity of a body with inner structure. This theory permits propagation of thermal waves at finite speed. We establish a uniqueness result and a continuous dependence of the solutions upon initial data and body supplies. Some applications (the problem of a concentrated heat source, the problem of an impulsive body force, the deformation of a thick-walled spherical shell) are presented.

## 1. Introduction

There has been very much written in recent years on the subject of the theory of elastic bodies in which the deformation is described by not only the usual vector displacement field, but also by other vector or tensor fields. The origin of the modern theories of continua with microstructure goes back to the papers of Ericksen & Truesdell (1958), Eringen & Suhubi (1964), Green & Rivlin (1964) and Mindlin (1964). Much of the theoretical progress in the field is discussed in the books of Kunin (1983), Ciarletta & Ieşan (1993) and Eringen (1999). The interaction of electromagnetic fields with elastic continua has been the subject of many investigations (Toupin 1963; Eringen 1999; Yang 2006). A theory of elastic solids with inner structure, including electromagnetic and thermal interactions, has been established by Eringen (1999) by discussing restrictions on the constitutive equations with the help of the classical entropy production inequality. The constitutive equations display new physical phenomena. Eringen (1999) introduced a class of continua with microstructure called microstretch continua. The material particles of the microstretch bodies can stretch and contract independently of their translations and rotations. A theory of microstretch elastic bodies subjected to electromagnetic fields was derived by Eringen (2004). The intended applications of this theory are to porous bodies such as bones and ceramics, solids with microcracks and synthetic materials with microreinforcements. To show that the theory is applicable to porous bodies, we first note that Goodman & Cowin (1972) have presented a nonlinear theory for the flowing of granular materials. By using the concept of distributed body introduced by Goodman & Cowin (1972), Nunziato & Cowin (1979) have established a theory for the behaviour of porous solids in which the skeletal or matrix materials are elastic and the interstices are void of material. The intended applications of this theory are to geological materials such as rocks and soils and to manufactured porous materials such as ceramics and pressed powders. The linear theory of elastic materials with voids has been established by Cowin & Nunziato (1983). The theory of elastic materials with voids has been extensively studied (Nunziato & Walsh 1980; Bedford & Drumheller 1983; Cowin & Puri 1983; Batra & Yang 1995; Iovane & Sumbatyan 2005; Ciarletta & Straughan 2007 and references therein). It is important to note that if we neglect the microrotation vector field, then the linear equations that describe the behaviour of a microstretch elastic body coincide with the equations of an elastic material with voids established by Cowin & Nunziato (1983). If we take in the equations (2.29) of the present paper *φ*_{i}=0, *ψ*=0, *τ*=0, *g*_{i}=0, *f*=0, *S*=0, *κ*=0, *λ*_{2}=0, *β*_{0}=0, *c*_{0}=0, *ν*_{1}=0, *ν*_{3}=0, then these equations reduce to the equations of the linear theory of isotropic and homogeneous materials with voids. We conclude that the theory of microstretch continua is an adequate tool to describe the behaviour of porous materials. In particular, the theory presented in this paper is applicable to ceramics, which are piezoelectric porous materials.

Green & Rivlin (1964) introduced the theory of dipolar elastic bodies. A microstretch continuum is a dipolar continuum in which the dipolar displacement *u*_{ij} has the form *φδ*_{ij}+*ϵ*_{ijk}*φ*_{k}, where *δ*_{ij} is the Kronecker delta and *ϵ*_{ijk} is the alternating symbol. Here, *φ* is called the microstretch function (or porosity function) and *φ*_{k} is the microrotation vector. In a microstretch continuum, the microelements undergo a uniform microdilatation (a breathing motion) represented by *φ* and a rigid microrotation by *φ*_{j}. When *φ* is zero, we obtain the Cosserat model (micropolar continuum).

Lakes (1982, 1986) and Yang & Lakes (1982) published some experimental observations on the elastic properties of human bone. Lakes (1986) states that: ‘Human bone, a natural fibrous composite, displays size effects in torsion and bending which are consistent with Cosserat elasticity, rather than classical elasticity’. A physical exercise therapy programme designed for bone healing is based on deformations and motions of bones under the application of a mild amount of stress. Clinically, it is also known that an electromagnetic field applied to bones hastens the healing process (Satter *et al.* 1999). These processes involve interactions of electromagnetic fields and mechanical deformations of porous solids.

Green & Naghi (1991*a*,*b*) developed a thermomechanical theory of deformable continua that relies on an entropy balance law rather than an entropy inequality. A theory of thermoelastic bodies based on the new entropy balance law has been derived by Green & Naghdi (1993). The linearized form of this theory does not sustain energy dissipation and permits the transmission of heat as thermal waves at finite speed. Moreover, the heat flux vector is determined by the same potential function that determines the stress. The Green–Naghdi theory has been studied in various papers (Chandrasekharaiah 1998; Hetnarski & Ignazack 1999; Quintanilla & Straughan 2000, 2002, 2004; Quintanilla 2001, 2002; Puri & Jordan 2004 and references therein).

In this paper, we derive a theory of thermopiezoelectricity of a continuum with microstructure that is capable of predicting a finite speed of heat propagation. In §2 we use the theory established by Green & Naghdi (1993) and the results of Eringen (1999, 2004) to obtain a linear theory of microstretch thermopiezoelectricity, which admits the possibility of ‘second sound’ and leads to a symmetric conductivity tensor. A uniqueness theorem in the theory of anisotropic bodies is presented in §3. In §4 we study the continuous dependence of solutions upon initial data and body loads. The continuous dependence theorem proves that, in the motion following any sufficient small change in the external data system, the solution of the problem is arbitrarily small in magnitude. Section 5 is concerned with the effects of a concentrated heat source in a body that occupies the entire three-dimensional Euclidean space. In contrast with the classical thermoelasticity, in the present theory the thermal waves propagate with finite speeds and the solution has no dissipative term. Section 6 is devoted to the problem of an impulsive body force that acts in an unbounded domain. We determine the temperature variation and the electric potential generated by this body force. Section 7 is concerned with the problem of a thick-walled spherical shell. We suppose that the surfaces of the shell are subjected to constant pressures. We determine the porosity function and the electric potential induced by the given tractions. The salient feature of the solution is that the displacement field and the electric potential contain new terms characterizing the influence of the material porosity and thermal field. The values of these functions are therefore modified from the values predicted by the classical theory.

## 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 surface ∂*B*. The motion of the body is referred to the reference configuration *B* and a fixed system of rectangular Cartesian axes *Ox*_{i} (*i*=1, 2, 3). We denote by ** n** the outward unit normal of ∂

*B*. Boldface characters stand for tensors of an order

*p*≥1 and if

**has the order**

*v**p*, we write

*v*

_{ij}

_{…}

_{k}(

*p*subscripts) for the components of

**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), whereas Greek subscripts are confined to the range (1, 2); summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate. In the following, we use a superposed dot to denote partial differentiation with respect to the time**

*v**t*.

We restrict our attention to the linear theory of microstretch continua. Let ** u** be the displacement field over

*B*. The local form of the conservation law of linear momentum may be written in the form(2.1)where

*t*

_{ij}is the stress tensor;

*f*

_{i}is the external body force per unit mass; and

*ρ*

_{0}is the density in the reference configuration. The local form of the conservation law of the moment of momentum is(2.2)where

*m*

_{ij}is the couple stress tensor;

*ϵ*

_{ijk}is the alternating symbol;

*g*

_{i}is the external body couple per unit mass;

*I*

_{ij}is the coefficient of inertia; and

*φ*

_{i}is the microrotation vector. The components of surface traction and the components of the surface moment are given respectively by(2.3)

Let *f* be the density of free charge, ** D** the electric displacement field, and

**the electric intensity. Then, Maxwell's equations for the quasi-static electric fields can be written in the form(2.4)where**

*E**ψ*is the electric potential.

In the Green–Naghdi theory of thermomechanics of continua, the entropy balance has the form(2.5)for every part *P* of *B* and every time, where *η* is the entropy per unit mass and unit time; *s* is the external rate of supply of entropy per unit mass; *ξ* is the internal rate of production of entropy per unit mass; and *G* is the internal flux of the entropy per unit mass. Following Green & Naghdi (1991*a*,*b*), from (2.5) we get(2.6)where *Φ*_{k} is the entropy flux vector and *n*_{j} is the outward unit normal at ∂*P*. In view of (2.6), the balance of entropy reduces to the local equation(2.7)

Let be an arbitrary material volume in the continuum, bounded by a surface ∂ at time *t*. We suppose that *P* is the corresponding region in the reference configuration *B*, bounded by a surface ∂*P*. Let *h* be the heat flux across the surface ∂ measured per unit area of ∂*P*. We denote by *q*_{j} the flux of heat associated with surfaces in the deformed body, which were originally coordinate planes perpendicular to the *x*_{j}-axes throughout the point ** x**. Then,(2.8)where

*θ*is the absolute temperature. With the help of (2.8), the equation (2.7) can be written in the form(2.9)

In the Green–Naghdi theory, the reduced energy equation is regarded as an identity for all thermodynamical processes and will place restrictions on the functional dependence of the constitutive equations.

Following Green & Naghdi (1991*a*) and Eringen (2004), we postulate an energy balance in the form(2.10)for all regions *P* of *B* and every time, where *φ* is the microstretch function; *j*_{0} is the microstretch inertia; *e* is the internal energy per unit mass; *ℓ* is the microstretch body force; and *p* is the microstress function. If we use the relations (2.1)–(2.3), (2.6) and the divergence theorem, then (2.10) reduces to(2.11)for all regions *P* of *B* and every time. With an argument similar to that used in obtaining (2.3), from (2.11) we find that(2.12)where *π*_{k} is the microstretch stress vector. By using (2.12) and the divergence theorem, from (2.11) we obtain the local form of energy balance(2.13)where(2.14)and

(2.15)

If we introduce the electric enthalpy *A* by(2.16)and take into account the relation (2.9), then equation (2.13) becomes(2.17)

Green & Naghdi (1991*a*, 1993) have introduced the thermal displacement *α* by We require constitutive equations for *A*, *t*_{ij}, *m*_{ij}, *π*_{i}, *σ*, *η*, *Φ*_{i}, *ξ* and *D*_{i}, and assume that these are functions of the set of variables *Π*=*e*_{ij}, *κ*_{ij}, *ζ*_{j}, *φ*, *θ*, *α*_{,i} and *E*_{j}. For simplicity, we regard the material to be homogeneous.

Introduction of constitutive equations of the forminto the energy equation (2.17), yields(2.18)where . Following the procedure of Green & Naghdi (1993), we find that the necessary and sufficient conditions for equation (2.18) to be satisfied under the above constitutive assumptions are(2.19)Clearly,(2.20)Following Green & Naghdi (1991*a*), we assume that there exists the reference time *t*_{0} such thatwhere *T*_{0} and *α*_{0} are constants. If we denote(2.21)then we have(2.22)

In the following, we restrict our attention to the linear theory. We assume that , , , and , where *ϵ* is a constant small enough for squares and higher powers to be neglected, and and are independent of *ϵ.* For a linearized theory, we assume that *U* is a quadratic function of the variables *e*_{ij}, *κ*_{ij}, *ζ*_{j}, *φ*, *T*, *τ*_{,j} and *E*_{k}, so that(2.23)where the constitutive coefficients satisfy(2.24)It follows from (2.19) and (2.22)–(2.24) that(2.25)In the context of the linear theory, from (2.8) and (2.25) we find that(2.26)The equation of entropy (2.9) reduces to(2.27)where *S*=*θs* is the external rate of supply of heat per unit mass.

The basic equations of the linear theory are the equations of motion (2.1), (2.2) and (2.15), the equation of entropy (2.27), the equations of the electric fields (2.4), the constitutive equations (2.25) and (2.26) and the geometrical equations (2.14). For isotropic and homogeneous bodies the constitutive equations reduce to(2.28)where *δ*_{ij} is the Kronecker delta and *λ*, *μ*, *κ*, *λ*_{0}, *β*_{0}, *α*, *β*, *γ*, *b*, *λ*_{1}, *ν*_{1}, *a*_{0}, *λ*_{2}, *ν*_{2}, *ξ*_{0}, *c*_{0}, *a*, *k*, *ν*_{3} and *Χ* are constitutive constants. It follows from (2.1), (2.2), (2.15), (2.27), (2.4), (2.28) and (2.14) that the field equations of the theory of homogeneous and isotropic bodies can be expressed as(2.29)where Δ is the Laplacian. In the classical theory of piezoelectricity for homogeneous and isotropic bodies, the constitutive equation for the electric displacement field is ** D**=

*Χ*

**, where**

*E**Χ*is a constant. Then, from (2.4) we obtain the equation

*Χ*Δ

*ψ*=−

*f*. The fourth equation of (2.29) is a generalization of the classical equation for the electric potential. The new terms represent the influence of porosity and thermal fields on the electric potential.

To the basic equations we must adjoin boundary and initial conditions. Let *S*_{m}(*m*=1, 2, …, 10) be subsets of ∂*B* such that , *S*_{1}∩*S*_{2}=*S*_{3}∩*S*_{4}=*S*_{5}∩*S*_{6}=*S*_{7}∩*S*_{8}=*S*_{9}∩*S*_{10}=∅. In the case of the mixed boundary-value problem, we consider the boundary conditions(2.30)where and are prescribed functions, and .

The initial conditions are(2.31)where and are given.

## 3. Uniqueness

In this section we establish a uniqueness result for the mixed boundary-initial-value problem. We introduce the notations(3.1)

*Assume that*

*ρ*_{0}*and j*_{0}*are strictly positive*,*I*_{rs}*is a positive definite symmetric tensor*,*the constitutive coefficients satisfy the relations*(*2.24*),*Λ is a positive semi-definite form, and**Π is a positive definite quadratic form*.

*Let* *be the difference of any two solutions of the mixed problem. Then*,(3.2)*Moreover, if S*_{7} *is non-empty, then the mixed problem has at most one solution*.

With the help of the constitutive equations (2.25), we obtain(3.3)

On the other hand, by (2.14), (2.4), (2.1), (2.2), (2.15) and (2.27) we find that(3.4)Thus, (3.3) and (3.4) imply(3.5)By using the divergence theorem, from (3.5) we get(3.6)Suppose that there are two solutions. Then their difference ^{*} corresponds to null data. Thus, from (3.6) we obtain(3.7)where ^{*} is the function defined by (3.1) and associated to the solution ^{*}. The initial conditions imply that ^{*}(0)=0 so that we conclude that ^{*}=0 on . With the help of the hypotheses of the theorem, we find that , , and on *B*×. Since and *τ*^{*} vanish initially we conclude that (3.2) holds. Clearly, if *S*_{7} is non-empty, then we obtain *ψ*^{*}=0. ▪

This uniqueness result can be extended to unbounded domains. A uniqueness result can be derived for the equilibrium theory and for the case of internal concentrated loads.

## 4. A continuous dependence result

This section is concerned with the linear theory of homogeneous and isotropic solids. We establish the continuous dependence of solutions upon initial data and body supplies. It is convenient to have equations (2.29) rewritten in non-dimensional form. We consider the dimensionless variables(4.1)where *ℓ*_{0} is a standard length; ; and *ψ*_{0} is a standard electrostatic potential. Introducing (4.1) into (2.29) and suppressing primes, we find the equations(4.2)where(4.3)We restrict our attention to the boundary-initial-value problem characterized by the equations (4.2), the initial conditions (2.31) and the following boundary conditions:(4.4)where and are given functions, and *t*_{1} is a prescribed positive constant.

Let us introduce the notations(4.5)where ; ; ; ; and . It is a simple matter to see that equation (4.2) can be written in the form(4.6)

Let us consider two solutions , (*α*=1, 2), corresponding to the external data systems . If we define , , and , then is a solution of the problem corresponding to the system where . We denote this problem by () and introduce the function *Φ* on [0,*t*_{1}] by(4.7)where(4.8)(4.9)It is easy to see that in view of the boundary conditions, the terms with coefficients *B*_{1} and *B*_{3} have no influence on *Φ*.

We assume that is a positive definite quadratic form in the variables *e*_{ij}, *γ*_{ij}, and *ψ*,_{k}. Thus, there exist the positive constants *k*_{1} and *k*_{2} such that(4.10)for all the variables and any *t*∈[0,*t*_{1}].

*Let {u*_{i}, *φ*_{i}, *φ*, *ψ*, *τ**} be a solution of the problem* (). *Then*,(4.11)

In view of (4.5) and (4.8), we find that(4.12)On the other hand, by using (4.9) and (4.6), and the relation which holds in the quasi-static theory, we obtain(4.13)If we integrate (4.13) over *B* and use the divergence theorem, the boundary conditions and (4.12), then we obtain the desired result. ▪

We define the functions *F* and *M* on [0,*t*_{1}] by(4.14)

*Assume that J*_{1} *and J*_{2} *are strictly positive constants and that* *is a positive definite quadratic form. Let {u*_{i}, *φ*_{i}, *φ*, *ψ*, *τ} be a solution of the problem* (). *Then*, *there exist the positive constants ρ*_{1} *and ρ*_{2} *such that*(4.15)

With the help of the Schwartz inequality, from (4.11) we find that(4.16)From (4.14) and (4.16), we obtainwhich implies the inequality(4.17)In view of (4.7) and (4.10), we get(4.18)whereThe relations (4.17) and (4.18) imply that(4.19)whereIn view of the Gronwall inequality, from (4.19) we obtain the desired result (4.15). ▪

## 5. The effect of a concentrated heat source

In this section, we consider a special case of the theory of thermopiezoelectricity by neglecting the contraction of the microelements. In this case, the field equation (2.29) for a homogeneous and isotropic body reduces to(5.1)In the following, we study the effect of a concentrated heat source in an unbounded body. In classical thermoelasticity this problem has been intensively studied (Hetnarski 1964; Ieşan 2004 and references therein).

Throughout this section we consider a body that occupies the entire three-dimensional Euclidean space and assume that the body loads have the form(5.2)where ; ** y** is a fixed point; and

*W*

_{0}is a prescribed function. We consider the initial conditions(5.3)and the following conditions at infinity:(5.4)In the following, we assume that the forms

*Λ*and

*Π*, defined by (3.1), are positive definite. This fact implies that(5.5)It is natural to seek the solution in the form(5.6)where

*U*,

*F*and

*G*are unknown functions that depend only on the variables

*r*and

*t*. From (5.6), it follows that the displacement vector is collinear with

**−**

*x***.**

*y*The field equations (5.1) are satisfied if the functions *U*, *F* and *G* satisfy the equations(5.7)We introduce the notations(5.8)The equations of (5.7) reduce to(5.9)and(5.10)We define *Γ* by(5.11)It is a simple matter to find that if we take(5.12)where the function *g* of class *C*^{4} satisfies the equation(5.13)then the functions *U* and *G* satisfy (5.9). The initial conditions for the function *g* are(5.14)These conditions imply the initial conditions of (5.3).

We denote by the Laplace transform with respect to *t* of the function *f*,By using (5.14), from (5.12) we find that(5.15)The function *g* satisfies the equation(5.16)This equation can be written in the form(5.17)where(5.18)Let us consider the functions and that satisfy the equations(5.19)It is a simple matter to find that the solution of the equation (5.17) can be written in the form(5.20)We assume that(5.21)where *δ*(.) is the Dirac delta; *W*^{*} is a given constant; and *Φ* is a prescribed function. By using the conditions at infinity, from (5.19) and (5.21) we find that(5.22)where(5.23)In view of (5.6), (5.12) and (5.20), we obtain(5.24)We now assume that(5.25)where *H* is the Heaviside unit step function, i.e. *H*(*t*)=0 for *t*≤0 and *H*(*t*)=1 for *t*>0. In this case, from (5.22) we find that(5.26)whereIf we use the relationsthen, from (5.24) and (5.26), we find thatFrom (5.6), (5.10) and (5.24) we getThus,where *u***v* is the convolution of *u* and *v*.

We note that the discontinuities in the solution can occur only at *t*=*κ*_{1}*r* and *t*=*κ*_{2}*r*. In contrast with classical thermoelasticity, the thermal waves propagate with finite speeds and the solution has no dissipative term.

In the classical thermoelasticity (Hetnarski 1964; Sneddon 1974), the calculation of the inverse Laplace transforms is very complicated. The level of difficulty of these problems can be ascertained by the fact that, even in the case of a single spatial variable, the solutions are usually based on a few terms of the power series expansion with respect to the coupling parameter or on asymptotic expansions governing short- and long-time responses.

## 6. The problem of an impulsive body force

We consider now the problem of a concentrated body force, acting in an unbounded material. We restrict our attention to the basic equations from §5. We suppose that the force is applied impulsively at the origin, parallel with the *x*_{3}-axis. In this section, we study equations (5.1) when the body loads are given by(6.1)where *f*^{*} is a given constant. Following Green & Naghdi (1993) we express the field equations in terms of the unknown *u*_{i}, *φ*_{i}, *ψ* and *T*. Thus, we obtain(6.2)and , where and are given by (5.8). We assume that the initial data are zero and consider the conditions (5.4) at infinity. First, we study the system (6.2). The function *ψ* can be found after the displacement, microrotation and temperature fields have been determined. In view of the initial data, any solution of the system (6.2) satisfies the equations (5.1). We introduce the notations(6.3)where(6.4)

*Let*(6.5)*where the function Q of class C*^{8} *satisfies the equation*(6.6)*Then u*_{i}, *φ and T satisfy the equations* (*6.2*).

Clearly,(6.7)If we substitute *u*_{k}, *φ*_{k} and *T* given by (6.5) into system (6.1), then in view of (6.6) and (6.7), we find that ▪

In the following, we use theorem 6.1 to study the solution of the system (6.2) when the body force is given by (6.1).

Let us consider the equation(6.8)where *h* is a given function. The Laplace transform of this equation can be written in the form(6.9)where(6.10)and *κ*_{1} and *κ*_{2} are given by (5.18). If the functions satisfy the equation(6.11)then the solution of equation (6.9) can be written in the form(6.12)where(6.13)In view of (6.1), (6.6) and (6.8), the solution of the problem is given by (6.5) where the function *Q* satisfies the equationOn the basis of (6.9), (6.11) and (6.12) we find that(6.14)It follows from (6.5) and (6.14) that we know the Laplace transform of the functions *u*_{j}, *φ*_{α} and *T*. Let us determine the temperature induced by the loads (6.1). We note that (6.11), (6.13) and (6.14) imply thatFrom (6.5) and (6.10) we obtainSimilarly, we can investigate the functions *u*_{j} and *φ*_{α}. The electric potential corresponding to the impulsive body force is given byThe method can be used to study the effects of a concentrated body couple.

## 7. The problem of a thick-walled spherical shell

In this section, we consider the general theory by taking into account the contraction of the microelements. We study the field equations (2.29) that describe the behaviour of homogeneous and isotropic bodies. We assume that *B* is a thick-walled shell, , *α*_{1}>0 and *α*_{2}>0. The body is in equilibrium in the absence of body loads. We suppose that the surfaces of the shell are subjected to constant pressures. We assume that the other boundary data are of Dirichlet type. Thus, we consider the following boundary conditions:(7.1)where , and are prescribed constants. In the theory of equilibrium, the basic equations reduce to(7.2)We seek the solution in the form(7.3)where *V*, φ, *ψ* and *τ* are unknown functions of *r*. The substitution of (7.3) into (7.2) yields the equations(7.4)where _{1} is an unknown constant and(7.5)The first two equations of (7.4) are equivalent to(7.6)where(7.7)and _{2}, _{1} and _{2} are arbitrary constants. The boundary conditions for tractions can be written in the form(7.8)It follows from the constitutive equations and (7.3) that(7.9)where(7.10)By (7.7) and the boundary conditions we obtain(7.11)where(7.12)Thus, we obtain(7.13)where(7.14)We conclude that(7.15)where(7.16)The boundary conditions (7.8) reduce to(7.17)Thus, we have determined the functions *V* and *φ*.

The first equation from (7.4) leads to(7.18)where _{α} are arbitrary constants. From the boundary conditions we obtain(7.19)From the last equation of (7.4), we find that the electric potential generated by the pressures *P*_{1} and *P*_{2} is given bywhere _{α} are arbitrary constants. From the boundary conditions for the electric potential we getThe porosity function *φ* has the form (7.6), where *φ*_{0} is given by (7.7) and the constants _{1}, _{2} and _{1} are defined in (7.11) and (7.17).

The results presented here can be used to obtain the solution in the case of the elastic space with a spherical cavity.

## Acknowledgments

I express my gratitude to the referees for their helpful suggestions.

## Footnotes

- Received October 11, 2007.
- Accepted November 22, 2007.

- © 2007 The Royal Society

## References

## Notice of correction

The references are now present in their correct form. 3 January 2008