A 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 deviatori stress invariants. 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 compresson and tension generators is derived, and the plane stress contour is given in explicit form. Several special cases are considered: a generalized Drucker-Prager criterion with straight generators and a smooth triangular deviatoric contour, surfaces with parabolic compression and tension generators, and the Lade failure surface for cohesionless soils. The use of an asymptotic tension cut-off condition in triaxial tension is discussed.