Stress invariants and invariants of the failure envelope as a quadric surface: Their significances in the formulation of a rational failure criterion

When stress invariants up to the second order are employed to construct failure criterion for brittle materials, it involves three independent terms and therefore there are three coefficients to be determined. However, there are only two conditions available associated with the strengths under uniaxial tension and compression. Systematic examinations have given to the invariants of the failure envelope as a quadric surface according to analytic geometry. For the failure envelope to meet the basic assumptions, in particular, infinite strength under and only under hydrostatic compression, one of the coefficients can be eliminated based on rigorous mathematical inferences. As a result, it reproduces the Raghava-Caddell-Yeh criterion, which has never been rationally established before but is now in this paper. The failure envelope takes the form of circular paraboloid for brittle materials in general. The criterion degenerates to the von Mises criterion, giving a circular cylindrical failure envelope for ductile materials as a special case. It is as rational as the von Mises criterion in the sense that the assumptions made and the conditions available are logically sufficient for the complete establishment of the failure criterion without any ambiguity.