On the solution technique for the interior electric displacement in a piezoelectric crack with the semi-permeable electric condition

In the study of crack problems in piezoelectric materials, the choice of electric conditions on the crack surface is crucial. The semi-permeable condition is believed to be more consistent with the real physical situation than the impermeable and permeable. In the semi-permeable crack condition, one task is to solve the crack interior electric displacement. However, there is no discussion in literature on the possible general method for its solution. A feasible method is proposed in this paper, addressed by taking as an example a penny-shaped crack in a piezoelectric layer with strip-like electrical saturation and mechanical yielding zones. The mixed boundary value problem is transformed into a set of coupling Fredholm integral equations of the second kind by Hankel transform and Copson method. With appropriate numerical discretization, the bisection method is adopted to solve the crack interior electric displacement. The algorithmic flow of the problem is given and some degenerated examples are presented to illustrate the effectiveness of the proposed technique. Finally, numerical results are presented for the relation of external load as well as the crack face electric displacement versus zone-lengths. Comparison and discussion are presented for the results of semi-permeable and impermeable crack condition. Results show that the crack interior electric displacement increases with electrical load but decreases with mechanical load. The electrical saturation zone-length is affected by the mechanical load as well as electrical load, regardless of the thickness of piezoelectric layer as well as the size relation of the mechanical and electrical zone-length. The present solution technique has a direct reference value for solving the problem of semi-permeable crack with distinct strip-like yielding zones in the finite piezoelectric material.