On two accurate methods for computing 3D Green׳s function and its first and second derivatives in piezoelectricity

Abstract

In this paper, we present two accurate methods for the calculation of the Green׳s function and its derivatives for three-dimensional anisotropic piezoelectric solids. In the first method, the Stroh formalism is used. The Green׳s function is expressed explicitly in terms of the Stroh eigenvectors, which are eigenvectors of the fundamental piezoelectricity matrix. The explicit derivatives of the 3D Green׳s function in terms of the derivatives of the Stroh eigenvalues and Stroh eigenvectors are derived for generally anisotropic piezoelectric materials. In the second method, we first express the Green׳s function and its derivatives in terms of novel infinite line integrals. Then the explicit expressions are obtained by the application of the Cauchy׳s residue theorem. The accuracies of both methods are verified by the numerical results compared with analytical solutions. Both explicit expressions are only applicable when the Stroh eigenvalues are distinct, which can be ensured by a small perturbation on some material constants in the case of degenerated eigenvalues.