Closed form solution for generalized modified polarization saturation model in piezoelectric media

The polarization saturation (PS) model is modified here by considering generalized polynomial-varying saturation condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:msub> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> in place of constant saturated electric displacement ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ) condition on the developed saturation zone length. The analytical study is presented on an infinite arbitrary polarized piezoelectric domain weakened by a Mode-III semipermeable center crack subjected to out-of-plane mechanical and in-plane electric displacement loadings. After mathematical formulation using complex variables and extended Stroh’s formalism approach, the problem is reduced into simultaneous non-homogeneous Riemann-Hilbert (R-H) problems in terms of complex variable functions, and the solution of these R-H problems is obtained in closed-form expressions. Explicit expressions for fracture parameters such as saturated zone lengths, crack sliding displacement (CSD), local stress intensity factor (LSIF), and electric displacement intensity factor (EDIF) are also evaluated for the generalized modified PS model. To validate the generalized solution against existing solutions in the literature, explicit analytical solutions for fracture parameters were derived from the generalized solution for specific cases of the modified PS model, including up to degree-five polynomial varying saturated conditions. The validation confirms the accuracy of the derived analytical solution for the generalized modified PS model in piezoelectric media. In addition, numerical studies were conducted on piezoelectric materials to investigate the effects of prescribed electromechanical loadings, crack-face conditions, and polarization angle on the defined fracture parameters. The results are presented graphically, analyzed, and discussed.