Analytical solution for two equal collinear modified strip saturated cracks in 2‐D semipermeable piezoelectric media

Abstract

An analytical solution and study is presented for two equal collinear cracks in 2‐D semipermeable piezoelectric media based on modified strip saturation models. The constant saturated condition defined in strip saturation model is modified here by considering the symmetric and polynomial varying saturation conditions. These proposed saturated conditions are multiplicative of saturated electric displacement value and polynomial (constant to cubic order) of a variable defined as ratio of distance of a point on saturated zone length to the extended half‐crack length. Moreover, these are interpolated on the basis of possible saturated values near the crack‐tips and zone tips. To present the analytical study, a problem of two cracks of equal length situated in series is considered in 2‐D arbitrary polarized infinite piezoelectric domain subjected to semipermeable crack‐face and electromechanical loading conditions. Applying the extended Stroh formalism and complex variable approach, these problems are mathematically modeled into non‐homogeneous Riemann Hilbert problems in terms of unknown complex functions representing stress and electric displacement components at any point within the domain. Using Muskhelishvili [1] and Collin's [2] mathematical techniques, these complex functions are obtained from the developed non‐homogeneous Riemann Hilbert problems. Hence, the explicit expressions for saturated zone lengths (inner and outer), crack opening potentials (COPs), crack opening displacements (CODs) and local stress intensity factors (LSIFs) are obtained. Further, the impact of polynomial varying saturated conditions is highlighted on these fracture parameters with respect to inter crack space distance, crack‐face conditions, polarization angle by presenting the numerical applications in infinite 2‐D PZT‐4 media.

Simplified numerical algorithms for generalized strip saturated two equal collinear cracks in piezoelectric media using distributed dislocation technique

“…13: Let's say (𝐾 𝐼𝑉 ) (𝐿 2 ) is evaluated and then check the sign of (𝐾 𝐼𝑉 ) (𝐿 2 ) whether (𝐾 𝐼𝑉 ) (𝐿 2 ) < 0 or not. loading, crack-face conditions, poling directions and the variable saturated conditions on the behavior of fracture parameters have been observed same as established/concluded in [17].…”

“…In particular for numerical validation and studies, the particular case of GSS model that is with polynomial varying saturation condition is considered having degree equals to 0 to 3. It is in general defined as: $$$$\begin{equation} g(x)= {\begin{cases} {\left|\frac{\mu -x}{\mu -a}\right|}^n,\nobreakspace \nobreakspace a\le {\left|x\right|} \le c \\[9pt] {\left|\frac{x-\mu }{b-\mu }\right|}^n,\nobreakspace \nobreakspace d\le {\left|x\right|} \le b \end{cases}} \end{equation}$$$$The geometric parameter, loading conditions and electric displacement saturated value are considered the same as in [17] and tabulated in Table 1. Here, the study is presented on PZT‐4 material with material constants given in Table 2.…”

“…The geometric parameter, loading conditions and electric displacement saturated value are considered the same as in [17] and tabulated in Table 1. Here, the study is presented on PZT-4 material with material constants given in Table 2.…”

“…Singh et al [16] applied the complex variable technique to study the polynomial varying modified SS model in 2-D semipermeable center cracked piezoelectric media. Also, Singh et al [17][18][19] extended these modified models to two equal/unequal collinear cracks problems in 2-D infinite piezoelectric media and employing Riemann-Hilbert approach obtained the explicit form of solutions for saturated zone lengths and other fracture parameters.…”

“…Also, Singh et al. [17–19] extended these modified models to two equal/unequal collinear cracks problems in 2‐D infinite piezoelectric media and employing Riemann‐Hilbert approach obtained the explicit form of solutions for saturated zone lengths and other fracture parameters.…”