AN EXACT CORRESPONDENCE BETWEEN PLANE PIEZOELECTRICITY AND GENERALIZED PLANE-STRAIN IN ELASTICITY

We consider an anisotropic body bounded by a cylindrical surface, whic h is infinitely long in the axial direction. Suppose the body is loade d in such a way that the field variables do not vary along the generat ors. An exact correspondence is established between the plane piezoele ctric equations and those of generalized plane strain in elasticity. I n particular, we show that by setting a linkage between the two sets o f material constants, any problem of a plane deformation in piezoelect ricity may be solved as a generalized plane strain in elasticity and v ice versa. The assertation is true for rectilinearly, as well as for c ylindrically, anisotropic solids. The equivalence is found for the mos t general anisotropic case, which links the fields between a monoclini c piezoelectric body of class m and a fully anisotropic (triclinic) el astic solid. A few degenerate systems are also identified. In addition , these correspondences can be extended to inhomogeneous media. Applie d to composite materials or polycrystalline aggregates, they imply tha t results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors (and vice versa). We also dem onstrate that the connection can be used to extend the scope of the in variant stress theorem proved by Cherkaev, Lurie & Milton. In illustra tion, we present solutions for the plane problem of an elliptical inho mogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity. Lekhnitskii's complex potential approach, togethe r with the conformal mapping technique, are employed. General solution s of the fields inside the inhomogeneity and the matrix are obtained. The results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem.