INVERSION THEOREM FOR CONICAL REGIONS IN ELASTICITY THEORY

Two-dimensional stress singularities in wedges have already drawn attention since a long time. An inverse square-root stress singularity (in a 360° wedge) plays an important role in fracture mechanics. Recently some similar three-dimensional singularities in conical regions have been investigated, from which one may be also important in fracture mechanics. Spherical coordinates are r, θ, φ{symbol}. The conical region occupied by the elastic homogeneous body (and possible anisotropic) has its vertex at r=0. The mantle of the cone is described by an arbitrary function f(θ, φ{symbol})=0. The displacement components be uξ. For special values of λ (eigenvalues) there exist states of displacements (eigenstates) {Mathematical expression},which may satisfy rather arbitrary homogeneous boundary conditions along the generators. The paper brings a theorem which expresses that if λ is an eigenvalue, then also-1-λ is an eigenvalue. Though the theorem is related to a known theorem in Potential Theory (Kelvin's theorem), the proof has to be given along quite another line. © 1979 Sijthoff & Noordhoff International Publishers.