Yield criteria representable by elliptic curves and Weierstrass form

Abstract

Elliptic curve terminology comes from the close association with elliptic functions, and not because of any physical resemblance to an ellipse. The curves investigated here represent various yield loci in the plane having cubic algebraic relationships between the second and third invariants of the deviatoric stress tensor. A well-known yield condition attributed to Drucker falls into this classification. In addition, the more commonly used Tresca yield condition represents a limiting case of elliptic curves. All yield criteria based on elliptic curves, including the Tresca, can be parameterized in terms of the Weierstrass elliptic &-function. The properties of elliptic curves as they pertain to the formulation of various plastic yield criteria of materials are the topic of this investigation. Various perfectly plastic solutions of mode I crack problems are discussed.