Nonlinear analysis of a crack in 2D piezoelectric semiconductors with exact electric boundary conditions

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.

A Predictor-Corrector Finite Element Method for Time-Harmonic Maxwell’s Equations in Polygonal Domains

The overall efficiency and accuracy of standard finite element methods may be severely reduced if the solution of the boundary value problem entails singularities. In the particular case of time-harmonic Maxwell’s equations in nonconvex polygonal domains Ω, H1-conforming nodal finite element methods may even fail to converge to the physical solution. In this paper, we present a new nodal finite element adaptation for solving time-harmonic Maxwell’s equations with perfectly conducting electric boundary condition in general polygonal domains. The originality of the present algorithm lies in the use of explicit extraction formulas for the coefficients of the singularities to define an iterative procedure for the improvement of the finite element solutions. A priori error estimates in the energy norm and in the L2 norm show that the new algorithm exhibits the same convergence properties as it is known for problems with regular solutions in the Sobolev space H2Ω2 in convex and nonconvex domains without the use of graded mesh refinements or any other modification of the bilinear form or the finite element spaces. Numerical experiments that validate the theoretical results are presented.

Buckling analysis of nano-scale magneto-electro-elastic plates using the nonlocal elasticity theory

Buckling analysis of nonlocal magneto-electro-elastic nano-plate is investigated based on the higher-order shear deformation theory. The in-plane magnetic and electric fields can be ignored for magneto-electro-elastic nano-plates. According to magneto-electric boundary condition and Maxwell equation, the variation of magnetic and electric potentials along the thickness direction of the magneto-electro-elastic plate is determined. To reformulate the elastic theory of magneto-electro-elastic nano-plate, the nonlocal differential constitutive relations of Eringen is applied. Using the variational principle, the governing equations of the nonlocal theory are derived. The relations between local and nonlocal theories are studied by numerical results. Also, the effects of nonlocal parameters, in-plane load directions, and aspect ratio on buckling response are investigated. Numerical results show the effects of the electric and magnetic potentials. These numerical results can be useful in the design and analysis of advanced structures constructed from magneto-electro-elastic materials.

The fracture behaviour of functionally graded piezoelectric materials containing collinear dielectric cracks

In this paper, the problem of two collinear cracks in functionally graded piezoelectric materials (FGPMs) under in-plane electromechanical loads is examined. The elastic, piezoelectric and dielectric constants of the FGPMs are assumed to vary continuously in space. The theoretical formulations are derived by using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model with deformation-dependent electric boundary condition is adopted in the fracture analysis of FGPMs. Numerical simulations are made to show the effect of the dielectric medium, the material gradient and the geometry of interacting cracks upon the fracture parameters at crack tips. A critical state for applied electromechanical loading is identified, which determines whether the traditionally impermeable (or permeable) crack model serves as the upper or lower bound of the current dielectric crack model.

Plane Electromechanical Fieldes in a Piezoelectric Material with a Crack

A two-dimensional electromechanical analysis is performed on a transversely isotropic piezoelectric material containing a crack based on the impermeable electric boundary condition. By introducing stress function, a general solution is provided in terms of triangle series. It is shown that the stress and electric displacement are all of 1/2 order singularity in front of the crack tip. In addition, the electromechanical fields in the vicinity of the crack when subjected to uniform tensile mechanical load are obtained using boundary collocation method.