Two-dimensional piezoelectricity. Part II: general solution,Green’s function and interface cracks

The complete solution space of a piezoelectric material is the direct sum of several orthogonal eigenspaces, one for each distinct eigenvalue. Each one of the 14 different classes of piezoelectric materials has a distinct form of the general solution, expressed in terms of the eigenvectors of the zeroth and higher orders and a kernel matrix containing analytic functions. When these functions are chosen to be logarithmic, one obtains, in a unified way, Green’s function of the infinite space as a single 8 × 8 matrix function G_∞ for the various load cases of concentrated line forces, dislocations, and a line charge. This expression of Green’s function is valid for all classes of nondegenerate and degenerate materials. With an appropriate choice of the parameters, it reduces to the solution of a half space with concentrated (line) forces at a boundary point, and with dislocations in the displacements. As another application, eigenvalues and eigensolutions are obtained for the bimaterial interface crack problem.