Rigid line inclusions under anti-plane deformation and in-plane electric field in piezoelectric materials

The problems of the rigid line inclusions under anti-plane deformation and in-plane electric field in piezoelectric materials are studied by using the complex variable method of Muskhelishvili. The rigid line inclusions are considered, respectively, as a dielectric and conductor. A square-root singularity of the field variables is identified. The singularity exists only if the shear loading is acting perpendicular to the rigid lines and the electric loading is parallel to them. For the rigid dielectric line inclusion, the singularity coefficients of both stress and electric field depend on the material properties, and the mechanical and electrical loads applied at infinity, whereas the electric displacement is a constant on the entire plane. The values of the J-integral are always negative. For the rigid conductor line inclusion, the field singularity coefficients corresponding to the field variables applied at infinity become independent of the material properties and other respective loading parameters. The values of the J-integral can be either positive or negative.