Twelve-dimensional Stroh-like formalism for Kirchhoff anisotropic piezoelectric thin plates

A twelve-dimensional Stroh-like formalism is developed for the coupled stretching, bending and polarization of a Kirchhoff anisotropic piezoelectric thin plate which is inhomogeneous and laminated along the thickness direction. The structure and explicit expressions of the fundamental piezoelectric plate matrix and its inverse form are established. The formalism in a rotated coordinate system is presented. Then the fundamental piezoelectric plate matrix in dual coordinate systems can be addressed. Some identities associated with the new formalism are derived. It is rigorously proved that each 3 × 3 partitioned matrix of the introduced three 6 × 6 real matrices S, H and L and the 6 × 6 Hermitian matrix M is a second-order tensor. Sixty-four permuted forms of the twelve-dimensional formalism are presented. Finally, to demonstrate the applications of the new formalism, we investigate the effective electroelastic properties of a microcracked anisotropic piezoelectric thin plate within the framework of non-interaction approximation (NIA) and study the asymptotic problem associated with a semi-infinite interface crack between two dissimilar piezoelectric thin plates.