Micromechanics determinations of thermoelectroelastic fields and effective thermoelectroelastic moduli of piezoelectric composites

By modelling thermal expansion strain and electric field induced by a uniform temperature change as eigenfields, the eigenstrain formulation (T. Mura, Micromechanics of Defects in Solids, 2nd revised edn., Martinus Nijhoff, 1987) for anisotropic inclusion problems is extended to consider the inherently anisotropic coupled behaviour of a piezoelectric composite. Utilizing the extended eigenstrain formulation, a unified explicit expression for the electroelastic tensors analogous to Eshelby tensors for elastic ellipsoidal inclusions is obtained. Particularly, closed forms for the electroelastic Eshelby tensors are presented when both the piezoelectric matrix and piezoelectric inclusions are transversely isotropic, and the shapes of the inclusions are spheroidal, elliptic cylindrical, circular cylindrical, penny-shaped, and ribbon-like. With the resulting tensors, analytical expressions for the thermoelectroelastic fields of the piezoelectric composite can then be obtained. Moreover based on the Mori-Tanaka theory and the equivalent inclusion method to account for interaction at finite concentrations of inhomogeneities, the effective thermal expansion coefficients, pyroelectric coefficients, elastic moduli, piezoelectric coefficients, and dielectric constants of the piezoelectric composite material in terms of phase properties, volume fraction, and the shape of the inhomogeneities are presented in closed forms.