Crack energy density and energy release rate for piezoelectric material

In this work, the concept of crack energy density (CED) was extended so as to be applied to piezoelectric material and its fundamental matters and properties were studied, and taking the knowledge about it into consideration, energy release rate for the material was newly derived. The definitions of CED, its mechanical and electrical contribution are given first and their path independent expressions are derived through the electromechanical energy conservation law. Subsequently, the loading path dependence of mechanical and electrical CEDs is discussed in detail. Some supplementary quantities related to CED and energy release rate are also defined and their path independent expressions are given. Energy release rate is derived through two opposite limit procedures, and the relations between energy release rate and other parameters are elicited through the discussions based on the fundamental properties of energy release rate.