Overall electromechanical properties of a binary composite with 622 symmetry constituents.: Antiplane shear piezoelectric state

A binary composite is studied here, where the electroelastic properties of the constituent materials belong to the crystal class 622. A square arrangement of long continuous circular cylinders, the fiber phase, embedded in a homogeneous medium is consider here. The composite is in a state of antiplane shear piezoelectricity, that is, a coupled state of out-of-plane mechanical displacement and in-plane electric field, which is characterized by three electroelastic parameters: longitudinal shear modulus, shear stress piezoelectric coefficient and transverse dielectric constant. Our interest here lies in the determination of its effective properties. They are derived by means of the method of two spatial scales. Closed-form expressions are obtained for them. Only one of the four local (or canonical) problems that arise is needed. Two properties are thus found. The Milgrom–Shtrikman compatibility relation is used to fix the remaining one. The local problem is solved using potential methods of a complex variable. The solution involves doubly periodic Weierstrass elliptic and related functions. The final formulae for the overall properties show explicitly the dependence on (i) the properties of the phases, (ii) the radius of the cylindrical fiber and (iii) the lattice sums associated with the square array. The shear modulus is shown to depend explicitly not only on the rigidity of the phases but also on their piezoelectric and dielectric coefficients. Some natural organic substances have the symmetry 622 like collagen. Recently Silva et al. measured its electroelastic properties. Their data is used to show some numerical results of the derived formulae as a function of the fiber volumetric fraction.