4. General Concept of Limit Surfaces—Convexity and Normality Rules, Material Stability

General nature of yield or failure criteria terminating elastic range of isotropic or anisotropic materials is summarized. As shown, the hydrostatic pressure sensitivity of anisotropic materials can be captured either by first stress and second common deviatoric invariant direct use (Tsai–Wu), or by the second common stress invariant in an indirect fashion (von Mises). Tension/compression asymmetry in anisotropic materials is accounted for either by presence of first common invariant (only translation, Tsai–Wu) or third common invariant (distortion, Kowalsky). Comparison of two ways to capture anisotropic response, more rigorous explicit common invariants formulations or implicit approaches based on extension of traditional isotropic criteria in terms of transformed invariants (Barlat, Khan) capable of capturing a complete distortion, is shown. Convexity requirement of limit surfaces is discussed and compared for two material behaviors by the use of Drucker’s material stability postulate extended to multi-dissipative response or Sylvester’s stability condition based on positive definiteness of the tangent stiffness or compliance matrices of hyperelastic material. Generalized Drucker’s postulate based on elastic–plastic stiffness matrix is also shown.