Degeneracy, Derivative Rule, and Green’s Functions of Anisotropic Elasticity

Abstract

Degenerate and extra-degenerate anisotropic elastic materials have repeated material eigenvalues whose multiplicity is greater than the number of independent eigensolutions. Using basic elasticity relations, a simple, direct proof is given to show that higher-order eigensolutions may be obtained from the analytical expressions of the zeroth-order eigensolutions according to the derivative rule. These higher-order eigensolutions contribute to the complexity of the general solutions of degenerate and extra-degenerate materials, and to the analytical difficulties inherent in such cases including isotropic elasticity. For all types of anisotropic materials, the general solution is given specific forms to obtain Green’s functions of several domains with straight or elliptical boundaries. These results, presented in fully explicit expressions, extend Green’s functions of nondegenerate materials to degenerate and extra-degenerate cases that have not been explored previously.