Asymptotic analysis, polarization matrices and topological derivatives for piezoelectric materials with small voids

Cardone, G.
Nazarov, S. A.
Sokolowski, J.
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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.
Comment: 35 pages
Mathematics - Optimization and Control, Mathematical Physics, 35Q30, 49J20, 76N10 (Primary), 49Q10, 74P15 (Secondary)