Singularity analysis for elastic clamped domains along an edge around which material properties depend on the angular coordinate

The elastic isotropic solution of the displacements for three-dimensional clamped domains in the vicinity of an edge around which the material properties depend on the angular axis is represented by a family of eigen-functions (similar to 2-D domains) complemented by shadow-functions and their associated edge stress intensity functions (ESIFs). The explicit computation of the eigen-pairs and shadow functions for clamped isotropic domains where the elastic modulus, E, change smoothly in the material along the angular axis is presented herein. The computation method is semi-analytical and involves the p-finite element methods. Numerical examples are explicitly provided for cracks and V-notch edges and explore the eigenvalues as a function of the change in material properties in the angular direction. We demonstrate that the singular exponents may change considerably by changing the material properties variation in the angular direction and the eigen-functions are no longer neither symmetric nor asymmetric functions and therefore Mode I and Mode II may no longer be separated. These eigenpairs and their following shadow-functions are necessary to allow the extraction of the edge stress intensity functions.