Evaluation of intensities of singularity at three-dimensional piezoelectric bonded joints using a conservative integral

A conservative integral based on the Betti reciprocal principle is extended for calculating intensities of singularity at vertices of interfaces in three-dimensional piezoelectric bonded joints. Regarding this method, eigenanalysis, formulated using a three-dimensional finite element method (FEM), is used for calculating the orders of stress singularity, angular functions of mechanical displacements, stresses, electric displacements and electric potentials. This method was initially developed for bonded joints between isotropic materials. For piezoelectric bonded joints, in addition to the stress singularities, the singularities associated with the electric field are also important. Hence, both the singularities associated with the mechanical stresses and the electric displacements are investigated and discussed in this study. The interface between piezoelectric and isotropic materials with one-term of stress singularity, and the interface between piezoelectric and piezoelectric materials with three-terms of stress singularity, are analyzed. Several models with various element sizes and integral areas are examined to investigate the influence of mesh refinement and integral area on the accuracy of the results. In order to summarize the results of one-term and three-terms of stress singularity in a general form, a unified singular equation is proposed.