EIGENFUNCTIONS AT A SINGULAR POINT IN TRANSVERSELY ISOTROPIC MATERIALS AND COMPOSITES UNDER AXISYMMETRIC DEFORMATIONS (STRESS, NOTCH, CRACK, SINGULARITY)

In finding stress distribution in an elastic solid numerically by a finite element scheme, special elements are used at a singular point such as a notch or a crack in which the singular nature of the stress is given by an analytical expression. When a two-dimensional elastic body that contains a notch or a crack is under a plane stress or plane strain deformation, the asymptotic solution of the stress near the apex of notch or crack is simply a series of eigenfunctions of the form (rho)('(delta))((psi),(delta))in which ((rho),(psi)) is the polar coordinate with origin at the apex and (delta) is the eigenvalue. If the body is a three-dimensional elastic solid that contains axisymmetric notches or cracks and subjected to an axisymmetric deformation, the eigenfunctions associated with an eigenvalue contain not only the (rho)('(delta)) term, but also (rho)('(delta)+1), (rho)('(delta)+2) ... terms. Therefore, the second and higher-order terms of the asymptotic solution are not simply the second and subsequent eigenfunctions. We present here the eigenfunctions for transversely isotropic materials under an axisymmetric deformation. The degenerate materials in which the eigenvalues, p(,1) and p(,2) of the elasticity constants are identical are also considered. The latter include the isotropic material as a special case. The solution is applied to eigenfunctions at a singular point in composites. To avoid the unrealistic inter-penetration of displacement at an interface crack, the assumption of a contact zone is introduced. Alternative formulas for the singularities at an interface crack with contact are derived by using the Stroh formalism.