In this paper, taking the exact electric boundary conditions into account, we propose a double iteration method to analyze a crack problem in a two-dimensional piezoelectric semiconductor. The method consists of a nested loop process with internal and outside circulations. In the former, the electric field and electron density in governing equations are constantly modified with the fixed boundary conditions on crack face and the crack opening displacement; while in the latter, the boundary conditions on crack face and the crack opening displacement are modified. Such a method is verified by numerically analyzing a crack with an impermeable electric boundary condition. It is shown that the electric boundary condition on crack face largely affects the electric displacement intensity factor near a crack tip in piezoelectric semiconductors. Under exact crack boundary conditions, the variation tendency of the electric displacement intensity factor versus crack size is quite different from that under an impermeable boundary condition. Thus, exact crack boundary conditions should be adopted in analysis of crack problems in a piezoelectric semiconductor.
An iteration approach in combination with the boundary element method is proposed to analyze a crack with exact crack face boundary conditions (BCs) in a finite magnetoelectroelastic solid. The crack opens under an applied load and the opened cavity is considered as a single domain filled with air or vacuum. The electric and magnetic fields inside a crack cavity affect the crack opening displacement (COD), which is a geometrically nonlinear problem. When establishing a boundary integral equation for inner and outer domains bounded by opening crack faces, nearly singular integrals occur due to the very thin domain of a crack cavity. However, the nearly singular integrals require no special treatment by employing intelligent adaptive algorithms in software Mathematica. The proposed approach is based on iteration of boundary elements for a crack-cavity domain and sub-region boundary elements for an outer magnetoelectroelastic solid with the crack faces changing during the iterative process. In this approach, exact crack face BCs are used in iteration, and the exact electric displacement and magnetic induction across the crack face as well as the COD can be determined. Furthermore, extended stress intensity factors are calculated and finally, the effects of BCs and the crack size are discussed.
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