Analytical formulae for electromechanical effective properties of 3–1 longitudinally porous piezoelectric materials

A unidirectional fiber composite is considered here, the fibers of which are empty cylindrical holes periodically distributed in a transversely isotropic piezoelectric matrix. The empty-fiber cross-section is circular and the periodicity is the same in two directions at an angle π/2 or π/3. Closed-form formulae for all electromechanical effective properties of these 3–1 longitudinally periodic porous piezoelectric materials are presented. The derivation of such expressions is based on the asymptotic homogenization method as a limit of the effective properties of two-phase transversely isotropic parallel fiber-reinforced composites when the fibers properties tend to zero. The plane effective coefficients satisfy the corresponding Schulgasser–Benveniste–Dvorak universal type of relations. A new relation among the antiplane effective constants from the solutions of two antiplane strains and potential local problems is found. This relation is valid for arbitrary shapes of the empty-fiber cross-sections. Based on such a relation, and using recent numerical results for isotropic conductive composites, the antiplane effective properties are computed for different geometrical shapes of the empty-fiber cross-section. Comparisons with other analytical and numerical theories are presented.