Basic Solutions of Elastic and Electric Fields of Piezoelectric Materials with Inclusions and Defects

Abstract

The solution of elastic and electric fields in solids with heterostructures is the foundation of micromechanics for predicting the overall behavior of materials. For practical considerations, no purely homogeneous material exists. The effect of defects such as dislocations, cracks, and inhomogeneities has always been one of the most important issues in materials science. Even for an assumed homogeneous body, different distributions of phases or preloads will have a significant impact on the fields. For piezoelectric solids, the following three factors provide new challenges for deriving the solutions of the elastic and electric field in the materials: (1) the materials are intrinsically anisotropic; (2) the elastic field and electric field are coupled; (3) sometimes it is extremely difficult to determine the exact boundary conditions for the elastic and electric fields. In the early 1980’s, researchers started to establish continuum models for the coupled elastic and electric field in piezoelectric materials. In the early 1990’s, the research in this area was significantly enhanced, and some efficient solution methods for the coupled fields were proposed and some milestone analytical solutions were derived. In this chapter, we intend to summarize some basic solution approaches and list some basic important solutions. The solution methods and the basic solutions are classified into two groups: two-dimensional problems and three-dimensional problems. Solutions for some one-dimensional problems have already been well documented in Tiersten (1969).