Family of invariant stress surfaces

A family of invariant stress surfaces with a cubic dependence on the deviatoric stress components is expressed as a linear combination of the second and third deviatoric stress invariants. The general format is described in terms of two functions of the mean stress. A simple geometric derivation demonstrates the convexity of the contours in the deviatoric plane. An explicit representation of the deviatoric contours in terms of a size and a shape parameter is given. The shape parameter effects a continuous transition from a triangle to a circle in the deviatoric plane. An explicit format in terms of the triaxial compression and tension generators is derived, and the plane stress contour is given in explicit form. Many previously proposed failure and yield criteria are obtained as special cases.