Electric and elastic singularity eigenanalysis of the V-notch in a piezoelectric Reissner plate

The singular eigenstate with respect to the stress and electric displacement at the vertex of a V-notch in a piezoelectric plate is investigated. The displacement and electric potential near the vertex of a V-notch are expressed as series asymptotic expansions, and their typical terms are substituted into the governing equations to establish the singularity eigenequations, from which the singularity orders and angular eigenfunctions can be yielded by the interpolating matrix method. The coupling effect of elastic and electric singularities can be found by distinguishing the singularity orders from their angular eigenfunctions. There are most four singularity orders for the V-notch in a piezoelectric Reissner plate, which is one more than the one in a homogeneous plate. Two orders are corresponding to the singularity of bending moment and electric displacement in the thickness direction, and another two orders are with respect to the singularity of shear force and electric displacement in the mid-plane. The free and electrically opened, and clamped and electrically closed boundary conditions are respectively introduced to investigate the influence of boundary conditions on the singularity. It is found that the singularity orders with respect to bending moment and electric displacement in the thickness direction change with boundary conditions, while the singularity orders with respect to shear force and electric displacement in the mid-plane are identical under these two kinds of boundary conditions.