## Abstract

We analyse electric field gradient (or quadrupole) effects in the anti-plane problem of a small, circular inclusion in polarized ceramics. An exact solution is obtained. The solutions show that, different from the classical inclusion solution from the linear theory of piezoelectricity, the electric field in the inclusion is no longer uniform. This has implications in field concentration and strength considerations and the prediction of effective material properties of composites.

## 1. Introduction

For elastic dielectrics, either the electric polarization vector (Toupin 1956) or the electric field vector (Tiersten 1971) can be used as the independent electric constitutive variable. Mindlin (1968) extended the polarization formulation by allowing the stored energy density to depend on the polarization gradient, in addition to the polarization vector and the strain tensor. Mindlin's polarization gradient theory has support from lattice dynamics (Mindlin 1969; Askar *et al*. 1970). Gradient theories can be considered as theories of weak non-local effects (Maugin 1979). These theories are closer to lattice dynamics than classical continuum theories (Mindlin 1972; Eringen & Kim 1977). Gradient theories are still applicable when the characteristic length of a problem is so small that classical continuum theories begin to fail. Size effect of small-scale problems is a standard application of gradient theories. Gradient theories also have important consequences in problems with singularities, e.g. concentrated sources and defects, etc. In dynamic problems, different from classical theories, gradient theories predict that short plane acoustic waves are dispersive, which agrees with lattice dynamics.

For dielectrics, it is known that the electric field gradient can also be used as constitutive variables (Landau & Lifshitz 1984). The resulting theory is called dielectrics with spatial dispersion and is equivalent to the theory of dielectrics with electric quadrupoles (Kafadar 1971) because electric quadrupole is the thermodynamic conjugate of the electric field gradient. Theories for elastic dielectrics with electric quadrupoles were also developed (Demiray & Eringen 1973; Maugin 1979; Maugin 1980; Eringen & Maugin 1990; Kalpakides & Agiasofitou 2002), which provide results similar to Mindlin's polarization gradient theory in problems with singularities or scale effects. Recently, in Yang *et al*. (2004), equations for elastic dielectrics with electric field gradient are specialized to the case of anti-plane deformations of piezoelectric materials with 6 mm symmetry and a general solution was obtained in polar coordinates.

In this paper, we use the solution in Yang *et al*. (2004) to study the effects of electric field gradient in the anti-plane problem of a circular inclusion in an unbounded medium under a uniform electric field and strain field at infinity. The three-dimensional equations for elastic dielectrics with electric field gradient are summarized in §2. Equations for the two-dimensional anti-plane case are given in §3, along with a general solution in polar coordinates. The solution to the inclusion problem is obtained in §4, with numerical results in §5. Finally, some conclusions are drawn in §6.

## 2. Elastic dielectrics with electric field gradient

We summarize the theory of elastic dielectrics with electric field gradient below. Consider the following functional over a volume *V* with *S* with boundary surface and unit exterior normal * n* (Yang

*et al*. 2004):(2.1)where

*u*

_{i}is the mechanical displacement vector,

*ϕ*is the electric potential,

*S*

_{ij}is the strain tensor,

*ϵ*

_{0}is the electric permittivity of free space,

*E*

_{i}is the electric field vector,

*f*

_{i}is the body force,

*q*is the body free charge density and is the surface traction vector. is related to surface free charge. is related to surface electric moments. Mathematically its presence is variationally consistent. Summation convention for repeated tensor indices and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index are used. In equation (2.1),(2.2)where

*H*is the electric enthalpy function of piezoelectric materials. Note that a term has been written separately in the energy density in (2.1). For linear materials

*H*can be written in the following quadratic form (Tiersten 1969):(2.3)where

*c*

_{ijkl}are the elastic constants,

*e*

_{ijk}are the piezoelectric constants and

*Χ*

_{ij}is the dielectric susceptibility tensor.

*γ*

_{ijk}and

*α*

_{ijkl}are new material constants due to the introduction of the electric field gradient in the energy density function.

*γ*

_{ijk}has the dimension of length.

*α*

_{ijkl}has the dimension of (length)

^{2}. Physically they may be related to characteristic lengths of atomic or microstructural interactions of the material. Since

*E*

_{i,j}=

*E*

_{j,i},

*α*

_{ijkl}has the same structure as

*c*

_{ijkl}as required by crystal symmetry and

*γ*

_{ijk}has the same structure as

*e*

_{ijk}. For

*W*to be negative definite in the case of pure electric phenomena without mechanical fields, we require that

*α*

_{ijkl}be positive definite like

*Χ*

_{ij}.

With the following constraints:(2.4)from the variational functional in equation (2.1), for independent variations of *u*_{i} and *ϕ* in *V*, we obtain,(2.5)where we have denoted(2.6)where *ϵ*_{ij}=*ϵ*_{0}(*δ*_{ij}+*Χ*_{ij}). *Π*_{i} and *Π*_{ij} are related to electric dipole and quadrupole densities (Eringen & Maugin 1990). The functional in equation (2.1) also implies the following as possible forms of boundary conditions on *S*:(2.7)where ∇_{s} is the surface gradient operator. One obvious possibility of equation (2.7)_{2} is *δϕ*=0 on *S*. With substitutions from equation (2.6), equation (2.5) can be written as two equations for *u*_{i} and *ϕ*(2.8)

## 3. Anti-plane problems of materials with 6 mm symmetry

For crystals of 6 mm symmetry and ceramics poled in the *x*_{3} direction the material tensors in equation (2.8) are represented by the following matrices under the compact notation (Tiersten 1969; Auld 1973):(3.1)where . We consider anti-plane motions with(3.2)The non-vanishing strain and electric field components are(3.3)where ∇ is the two-dimensional gradient operator. The non-trivial components of *T*_{ij} and *D*_{i} are(3.4)where ∇^{2} is the two-dimensional Laplacian, *c*=*c*_{44}, *e*=*e*_{15}, *ϵ*=*ϵ*_{11} and *α*=*α*_{11}. The non-trivial ones of equation (2.8) take the following form:(3.5)where *f*=*f*_{3}. Equation (3.5) can be decoupled into(3.6)or(3.7)In equations (3.6) and (3.7),(3.8)where *k* is a dimensionless number (the electromechanical coupling factor).

For problems without body source (*f*=0, *q*=0), a general solution to equation (3.7) in polar coordinates defined by and can be found as (Yang *et al*. (2004))(3.9)where *a*_{n}, *b*_{n}, *g*_{n}, *h*_{n}, *l*_{n} and *p*_{n} are undetermined constants, , and *I*_{n} and *K*_{n} are the first- and second-kind modified Bessel functions of order *n*.

## 4. A circular inclusion

We now consider a circular inclusion under an electric field *E*^{0} and a shear strain *S*^{0} at infinity (see figure 1, where in this section).

### (a) Exterior fields

Far away from the inclusion we have(4.1)We take the following terms from the general solution in equation (3.9):(4.2)which satisfies equation (4.1). Corresponding to equation (4.2) we have(4.3)(4.4)and(4.5)

### (b) Interior fields

We take the following terms from the general solution in equation (3.9):(4.6)which are finite at the origin. Corresponding to equation (4.6) we have(4.7)(4.8)and(4.9)

### (c) Continuity conditions

From equation (2.7) we impose the following continuity conditions at the interface *r*=*R*, which are variationally consistent:(4.10)Substitution of equations (4.1)–(4.9) into equation (4.10) gives the following six equations for *h*_{1}, *d*_{1}, *p*_{1}, *g*_{1}, *c*_{1} and *l*_{1}:(4.11)

## 5. Numerical results

For numerical results PZT-5A and BaTiO_{3} are used for the matrix and the inclusion, respectively. Their linear elastic, piezoelectric and dielectric constants can be found in Auld (1973). *α*_{jikl} could be the atomistic distance, domain size or grain size depending on what microstructural effects we are interested in. For our purposes, what matters is the relative value of *α*_{jikl} with respect to *R*. We artificially choose(5.1)A few different values of *R* are used, ranging from to , where *α* stands for . Only *E*^{0} is applied. *S*^{0}=0.

Equations (4.11) are solved on a computer. Electric field distribution is shown in figure 2. A fundamental difference from the classical inclusion solution is that the electric field in the inclusion is no longer uniform. Note that near the interface the electric field is larger than the near uniform electric field in the central region of the inclusion. Therefore, some field concentration exists near the interface, which is as expected because gradient theories are usually associated with boundary layer effects. Field concentration is important to strength and failure considerations.

To see the effect of on the field distribution, we plot the interior fields for two different values of in figures 3 and 4. It is seen that when is relatively small, the interior field is more uniform. In figures 3 and 4, the electric field in the almost uniform central regions are in fact about the same.

If the piezoelectric constants are set to zero, we have a pure electric inclusion problem of dielectrics. We calculated the ratio of *D*_{1} and *E*_{1} averaged over the square region in figure 1, which represents the effective dielectric constant. The result is shown in figure 5. The figure shows that when is small the effective dielectric constant is not sensitive to . In this case the effective dielectric constant is basically the one predicted by the classical inclusion theory. When is not small, the effective dielectric constant is smaller (size effect) as expected because the electric field is larger near the interface due to the gradient effects. This may be important in nano-scale composites. Note that a smaller dielectric constant may imply a larger electromechanical coupling factor through .

## 6. Conclusion

Electric field gradient or electric quadrupole has important effects in small inclusions. The electric field inside the inclusion is no longer uniform. The effective dielectric constant becomes size dependent, which may imply a higher electromechanical coupling factor for composites of very thin films.

## Acknowledgments

The research was sponsored by the U.S. Army Research Laboratory under the RMAC-RTP Cooperative Agreement no. W911NF-04-2-0011.

## Footnotes

- Received January 13, 2006.
- Accepted March 3, 2006.

- © 2006 The Royal Society