New analytical solutions for modified polarization saturation models in piezoelectric materials

Abstract

In this paper, we present new analytical solutions for modified polarization saturation (PS) models for arbitrary polarized and semipermeable 2-D piezoelectric media. The PS model is modified to various other non-linear fracture models by varying the normal electric displacement saturated conditions in place of a constant saturated value. These variations are hereby defined as interpolating linear, quadratic and cubic type’s polynomials into saturated value interpolated on the basis of possible electric displacement values at the centre of the crack, actual crack tip and saturated zone tip. To present the analytical study, a centre crack problem in 2-D semipermeable piezoelectric media under arbitrary poling direction and combined in-plane mechanical and electric displacement loadings is considered. Using extended Stroh formalism and complex variable approach, these models are mathematically reduced into generalized non-homogeneous Riemann–Hilbert problems. Applying Riemann–Hilbert technique, the solution of these problems is obtained in terms of generalized complex potential functions representing traction forces and electric displacement components. These lead to obtain the explicit expressions for saturated zone length, crack opening displacement, crack opening potential and local intensity factor in each case. Also, the results of saturation zones and local intensity factor are obtained in each case and compared with the established numerical results. The studies show that saturated zone length increases with increasing electric loading for each case and also depends on the polynomial varying saturation condition. For a particular electric loading applied within the range of critical value, it increases with increasing degree of polynomial type saturated condition i.e. from constant to cubic type’s normal electric displacement saturated condition.