THE BEHAVIOR OF ELASTIC FIELDS AND BOUNDARY INTEGRAL MELLIN TECHNIQUES NEAR CONICAL POINTS

For the computation of the local singular behaviour of an homogeneous anisotropic elastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use:a boundary integral e quation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the int ersection curve gamma of the cone with the unit sphere. The Mellin tra nsformed operator defines the singular exponents and Jordan chains, wh ich provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yield s a holomorphic operator pencil of elliptic boundary value problems on the cross-sectional interior and exterior intersection of the unit sp here with the conical interior and exterior original cones, respective ly, and using results by MAZ'YA and KOZLOV, it can be shown how the Jo rdan chains of the one-dimensional boundary integral equation are rela ted to the corresponding Jordan chains of the operator pencil and thei r jumps across gamma. This allows a new and detailed analysis of the a symptotic behaviour of the boundary integral equation solutions near t he vertex of the Cone. In particular, the integral equation method dev eloped by SCHMITZ. VOLK and WENDLAND for the special case of the elast ic Dirichlet problem in isotropic homogeneous materials could be compl eted and generalized to the anisotropic case.