The Paraboloidal Failure Surface of Initially Anisotropic Elastic Solids

Abstract

The generalization of the paraboloidal failure criterion for isotropic materials in order to cover anisotropic failure is presented in this paper. The resulting limit condition for anisotropic bodies is derived by a straightforward mathematical analysis based on physical grounds. The associated geometric interpretation of the failure condition in the principal stress space is an elliptic paraboloid having its axis of symmetry parallel to the hydrostatic axis and displaced from the origin of the coordinate system. All the geometric features of this failure surface, such as the orientation of the axes of its elliptic cross-sections, or the distance of the axis of symmetry from the hydrostatic axis, as well as the position of the open edge of the elliptic paraboloid with respect to the hydrostatic tension or compres sion half-spaces, depend on the degree of strength-anisotropy and strength-properties of the material.

Then, given the appropriate strength parameters in three mutually orthogonal material directions, coaxial with the resulting principal stress coordinate system, one has a clear and well defined geometric picture of the failure surface for three-dimensional stress systems. This is particularly useful since the effect of a prescribed loading path on the failure behaviour of a composite is determined by the position of its elliptic paraboloid failure surface with respect to material strength-axes.