  Asymptotic formulae for the mechanical and electric fields in a piezoelectric
body with a small void are derived and justified. Such results are new and
useful for applications in the field of design of smart materials. In this way
the topological derivatives of shape functionals are obtained for
piezoelectricity. The asymptotic formulae are given in terms of the so-called
polarization tensors (matrices) which are determined by the integral
characteristics of voids. The distinguished feature of the piezoelectricity
boundary value problems under considerations is the absence of positive
definiteness of an differential operator which is non self-adjoint. Two
specific Gibbs' functionals of the problem are defined by the energy and the
electric enthalpy. The topological derivatives are defined in different manners
for each of the governing functionals. Actually, the topological derivative of
the enthalpy functional is local i.e., defined by the pointwise values of the
governing fields, in contrary to the energy functional and some other suitable
shape functionals which admit non-local topological derivatives, i.e.,
depending on the whole problem data. An example with the weak interaction
between mechanical and electric fields provides the explicit asymptotic
expansions and can be directly used in numerical procedures of optimal design
for smart materials.
