THE ANALYSIS OF DISLOCATION, CRACK, AND INCLUSION PROBLEMS IN PIEZOELECTRIC SOLIDS.

摘要

This thesis is to serve as a framework for performing stress analyses of defects in piezoelectric materials. After a short literature review, the basic equations of piezoelectric elasticity are presented. These equations are a generalized form of Hooke's Law, the equilibrium equations, and the definition of the free energy potential for a piezoelectric solid. A short-hand notation is introduced by combining stress and electric displacement in a 3 x 4 matrix, elastic displacement and electric potential in a four component column vector, and by defining an electro-elastic constant matrix. When using this short-hand notation, the basic equations of piezoelectric elasticity take on a form very similar to that of the analogous equations encountered in anisotropic elasticity theory. This similarity is used to analyze various problems of interest in any basic elasticity theory.; First the Green's function for an infinite, homogeneous, piezoelectric solid is determined directly from the equations of equilibrium using the Radon integral transform. This Green's function is the elastic displacement and electric potential at a point P due to a unit point force and a unit point charge located at the point S.; Next this Green's function is used to analyze a D-CD defect L composed of a collinear dislocation and charge dipole line. Stress, electric displacement, elastic displacement, and electric potential are determined for the cases in which the defect L is either an infinite straight line or an infinitesimal line segment. The results for the infinitesimal D-CD defect are then used to analyze both planar and three dimensional D-CD defect arrangements in piezoelectric solids. This yields the piezoelectric analogues of the Brown-Lothe and Indenbom-Orlov theorems.; In addition, the stress-electric displacement matrix and the elastic displacement-electric potential column vector for a general defect composed of a collinear combination of a dislocation, a charge dipole line, a line of force, and a line of charge, as first presented by Barnett and Lothe in 1975, are reviewed and some minor corrections included. The results for this general defect are used to analyze a slit-like crack in an infinite, piezoelectric solid subject to a remote constant stress-electric displacement state. The stress and electric displacement intensity factors for this crack are determined. They are found to be independent of the electro-elastic constants for a slit-like crack. It is found that by a suitable choice of the externally applied electric displacement, crack growth can be arrested in a piezoelectric solid.; To complete this overview of piezoelectric elasticity theory, the Eshelby inclusion problem is generalized to the piezoelectric case using the Green's function mentioned earlier. An ellipsoidal inclusion, undergoing a change in shape, size, and electric potential, referred to as the transformation, in the presence of an infinite matrix, is analyzed. First matrix and inclusions are assumed to have the same electro-elastic constants and the transformation considered is a polynomial of degree P in the position coordinates. This result is simplified for the case P = 1 and used to analyze the case in which the inclusion and matrix have different electro-elastic coefficients. Finally, the electro-elastic fields due to the presence of an ellipsoidal peizoelectric inhomogeneity in an infinite, piezoelectric solid subject to a constant elastic strain-electric field state throughout, are determined. For each of the inclusion problems, free energy calculations are also carried out.