Electro-elastic analysis of cracks in piezoelectric solids
Athanasius, Louis Commillus
Date of Issue2011
School of Mechanical and Aerospace Engineering
In this thesis, several important classes of electro-elastic crack problems involving an arbitrary number of arbitrarily orientated planar cracks are solved. In all the problems considered here, the relevant boundary conditions on the cracks are formulated in terms of a system of hypersingular integral equations which are solved by using an accurate collocation technique. The problems under consideration are as follows. Firstly, the electro-elastostatic interaction of multiple planar cracks in an infinitely long piezoelectric strip is considered. The cracks are acted upon by suitably prescribed internal stresses and are electrically either permeable or impermeable. To solve the problem, a Green's function which satisfies the conditions of vanishing traction and normal electric displacement on the edges of the strip is first derived using a Fourier transform technique. It is then used to derive an explicit semi-analytic solution for the electro-elastostatic fields around the cracks in the strip. The solution is expressed in terms of integrals over lines representing the cracks. The jumps in the displacement and electric potential across opposite crack faces are the unknown functions in the integrals. They are to be determined by solving a system of hypersingular integral equations which are derived from the boundary conditions on the cracks. Secondly, special electro-elastostatic Green's functions satisfying stress-free and electrically either permeable or impermeable conditions on multiple planar cracks in a piezoelectric domain of infinite extent are constructed numerically. From the boundary conditions on the cracks, the task of constructing the Green's functions requires solving a system of hypersingular integral equations. Once the hypersingular integral equations are solved, explicit formulae can be obtained for computing numerically the Green's functions. The numerical Green's functions are used to develop a simple but accurate boundary element method for analyzing numerically multiple planar cracks in a piezoelectric solid of finite extent. As the singular behaviors of the stress and electric displacement are analytically built into the Green's functions, the boundary element procedure does not require the crack faces to be discretized into boundary elements. Furthermore, the relevant crack tip stress and electric displacement intensity factors can be extracted very accurately. Thirdly, the numerical Green's function boundary element approach for electrically impermeable cracks is adapted to deal with electro-elastostatic plane problem involving electrically semi-permeable cracks in a piezoelectric solid of finite extent. An iterative procedure for treating the nonlinear boundary conditions on the semi-permeable cracks is proposed. Lastly, a semi-analytic method of solution is successfully developed for analyzing the multiple cracks in an infinite piezoelectric domain under dynamic loading. The Laplace transform technique is employed to suppress the second order time derivatives of the displacement in the governing partial differential equations. The displacement and electric potential in the Laplace transform domain are then expressed using suitably constructed exponential Fourier transform representations. The unknown functions in the Fourier transform representations are directly related to the jumps in the Laplace transforms of the displacement and electric potential across opposite crack faces and are determined by solving a system of hypersingular integral equations. Once the displacement and electric potential are determined in the Laplace transform domain, they can be recovered in the physical time domain by using a numerical formula for inverting Laplace transforms. For some specific cases of the problems under consideration, the computed crack tip intensity factors are compared with those published in the literature. Some new results are also obtained for certain configurations of the planar cracks.
DRNTU::Engineering::Mechanical engineering::Mechanics and dynamics