Generalised strip-saturation zone models for piezoelectric strip weakened by non-centric semi-permeable crack

Abstract

In this paper, we have proposed generalised strip-saturation zone models for non-centric semi-permeable straight hairline crack weakening a piezoelectric strip. Strip-saturation zone model is generalised by considering three different situations: when the distributed in-plane normal yield point of electric-displacement over developed saturation zone rims are linear, quadratic, and cubic interpolating polynomials times. The piezoelectric strip is considered under out-of-plane mechanical and in-plane electrical loadings and crack-faces are assumed to be parallel to the strip boundaries. Fourier integral transform technique is adopted to express the solutions in Fredholm integral equation of second kind. Under small-scale electrical yielding, the analytical expression for size of developed saturation zone is determined for constant, linear, quadratic, and cubic varying yield point conditions. Closed-form analytical expressions are also formed for numerous fracture parameters viz. crack-sliding displacement, crack opening potential drop, field intensity factors, and energy release rates. A numerical case study is demonstrated for the PZT-5H, PZT-4, PZT-6B, and PZT-7A piezoelectric ceramics strip to investigate the impact of material properties, strip width, prescribed electro-mechanical loadings, crack-face boundary conditions on fracture parameters. Apart from this, influence of prescribed loadings on electric crack condition parameter is also presented graphically for all generalised strip-saturation zone models. All the obtained results are analyzed and compared graphically.

Appendix

$$\begin{aligned} L_{1} &= \displaystyle \int _{t = 0}^{1}K_{1}(1, t)\Omega _{1}(t)dt \\ L_{2} &= \displaystyle \int _{t = 0}^{1}K_{2}(1, t)\Omega _{2}^{I}(t)dt \\ L_{3} &= \displaystyle \int _{t = 0}^{1}K_{2}(1, t)\Omega _{2}^{II}(t)dt \\ L_{4} &= \displaystyle \int _{t = 0}^{1}K_{2}(1, t)\Omega _{2}^{III}(t)dt \\ L_{5} &= \displaystyle \int _{t = 0}^{1}K_{2}(1, t)\Omega _{2}^{IV}(t)dt \\ K_{1}(1, t) &= \sqrt{t}\displaystyle \int _{y = 0}^{\infty }y\{\Upsilon (y/a) - 1\}J_{0}(y)J_{0}(ty)dy \\ K_{2}(1, t) &= \sqrt{t}\displaystyle \int _{y = 0}^{\infty }y\{\Upsilon (y/c) - 1\}J_{0}(y)J_{0}(ty)dy \\ \Upsilon (y) &= 2\tanh (h_{1}y)/\{1 + M_{12}(y)\},\;\;M_{12}(y) = \tanh (h_{1}y)\coth (h_{2}y) \end{aligned}$$