Numerical Studies for Generalized Modified Polarization Saturation (PS) Model in Piezoelectric Media

Abstract

Numerical study is presented for the generalized modified polarization saturation (PS) model in 2-D piezoelectric media using distributed dislocation technique (DDT) and Gauss–Chebychev quadrature numerical scheme. A generalized case of modified PS model for an infinite semipermeable piezoelectric media is proposed here in the center cracked problem by varying the PS condition of the form \(f(|x/c_1 | ) D_s\) in place of constant polarization saturation condition \(D_s\). Here, \(g(x) = f(|x/c_1 | )\) is any arbitrary function of \(x/c_1\) with x as a distance of the arbitrary point on zone length from center of the crack and \(c_1\) is the extended crack length. But obtaining the analytical solution for this generalized model using complex variable and other mathematical techniques is difficult and hence in this paper, the DDT and Gauss–Chebychev quadrature scheme is applied to obtain the numerical solution. Applying DDT, this generalized PS problem is modeled as a continuous distribution of dislocations and by enforcing the crack-face and PS conditions reduced mathematically into simultaneous Cauchy-type singular integral equations in terms of mechanical and electrical dislocation density variables. But before solving them numerically, these developed simultaneous singular integral equations are firstly simplified into separate integral equations according to mechanical and electrical dislocation density parameters. This approach has also helped in evaluating the local stress intensity factor (LIF) obtained at the mechanical crack-tips. Moreover, the saturated zone length which is another important parameter in studying such models is an unknown quantity prior to getting any numerical solution. So, an iterative approach is implemented by varying the zone length and imposing a supplementary condition of finite electric displacement at the outer tips of the zone. Hence, a generalized numerical approach is developed here to obtain the fracture parameters such as saturated zone length and LIF for any varying PS condition of the form \(f(|x/c_1 | ) D_s\). To validate the numerical approach, results are obtained for particular cases, i.e., polynomial varying saturation conditions of degree up to four and compared with the analytical solutions. Excellent agreement of the numerical results has been found with analytical ones and hence showing the efficacy of the approach proposed in this paper. Additionally, the effects of loadings, crack-face conditions and poling direction have been discussed on saturated zone length and LIF.