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Analytical solution for two equal collinear modified strip saturated cracks in 2‐D semipermeable piezoelectric media
Author(s) -
Singh Sandeep,
Sharma Kuldeep,
Bhargava R. R.
Publication year - 2019
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800244
Subject(s) - mathematical analysis , mathematics , saturation (graph theory) , polynomial , piezoelectricity , geometry , materials science , composite material , combinatorics
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.