A penny-shaped crack in a piezoelectric layer sandwiched between two elastic layers with the electrical saturation and mechanical yielding zones

•We simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by Copson method.•The problem is solved in the real domain by constructing real fundamental solutions.•The numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization.•The influence of the electromechanical loads, the thicknesses of the piezoelectric layer and the elastic layer, and the material parameter ratios on the nonlinear zones is detailedly discussed. This paper deals with an internal penny-shaped crack in a piezoelectric layer sandwiched by two outer elastic layers. It is assumed that the crack faces are electrically impermeable, and the thin annular electrical saturation and mechanical yielding zones appear around the crack front under the uniform distributed normal stress and electric displacement. Three different cases, i.e., the region of electrical saturation is longer, shorter than, or equal to the domain of mechanical yielding are respectively considered in this article. For such an axisymmetric penny-shaped crack problem, we simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by employing Hankel transform technique and the Copson method. The problem is solved in the real domain by constructing real fundamental solutions and the numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization. When the outer elastic layers vanish and the piezoelectric layer-thickness approaches infinity, the relation between the nonlinear zone-lengths and the loadings obtained in this paper reduces to the analytical expression in literature. The influence of the electromechanical loads, the thicknesses of the piezoelectric layer and the elastic layer, and the material parameter ratios on the nonlinear zones is discussed in details. The results of paper are believed to have some guiding references for the design of the piezoelectric sandwich plate.