Singular asymptotic expansion of the elastic solution along an edge around which material properties depend on the angular coordinate

The eigenvalues of the elastic isotropic solution in the vicinity of an edge are of importance in fracture mechanics and are associated to three modes (I, II & III) according to the symmetry of the corresponding eigen-functions. The explicit computation of the eigen-pairs and shadow functions for isotropic domains where the elastic modulus, , change smoothly in the material along the angular axis is addressed. Although the domain is isotropic, by changing the material properties variation in the angular direction, the singular eigenvalues may be either more or less singular compare to constant material cases. Moreover, the eigen-functions become neither symmetric nor asymmetric functions and therefore mode I & II may no longer be separated. Numerical examples illustrating these phenomena will be presented.