Critical Angles in Bending of Rotationally Inhomogeneous Elastic Wedges

An anisotropic rotationally inhomogeneous wedge bent by either a concentrated couple applied at the tip (Carothers problem) or uniform surface loadings (Levy problem) is considered. The existence criteria for homogeneous solutions describing stresses and strains in both problems are established. In the Levy problem there are two types of critical wedge angles, at which homogeneous solutions break down and become infinite. The first type critical wedge angles of Levy’s problem are shown to be critical also for Carothers’problem whatever the rotational inhomogeneity. Particular solutions to both problems are obtained at the critical wedge angle. The form of these solutions is established to depend on two factors: the multiplicity degree of roots of some eigenvalue equation and the number of independent eigenvectors of some real matrix. It is shown also that the eigenvalue equation does not provide an alternative way to calculate the critical angles and in the first-order perturbation theory results in just the same equations for the critical angles.