Yield criteria of hexagonal symmetry in the π-plane

The theory of plasticity operates with different yield criteria of incompressible behavior for isotropic materials. Mostly known are the criteria of Tresca, Schmidt-Ishlinsky and von Mises. The first two criteria have a hexagonal symmetry, and the criterion of von Mises has a rotational symmetry in the π-plane. All these criteria do not distinguish between tension and compression (no strength differential effect), but numerous problems are treated in the engineering practice using these criteria. Within this paper, the yield criteria with hexagonal symmetry for isotropic incompressible materials are compared. For this purpose, their geometries in the π-plane will be presented in polar coordinates. The radii at the angles of 15◦ and 30◦ will be related to the radius at 0◦. Based on these two relations, well-known criteria will be shown in one diagram. The extreme shapes of the yield surfaces are restricted by two criteria: the unified yield criterion (UYC) and the multiplicative ansatz criterion (MAC). The examinations of the UYC and MAC depict a linear combination of these extreme yield surfaces. The resulting criterion with two parameters describes all possible convex forms of hexagonal symmetry. On the other hand, this criterion has one disadvantage: It is not possible to solve explicitly the equation for the equivalent stress. Other known criteria (Sokolovsky, Ishlinsky-Ivlev, Dodd-Naruse, Drucker) are depicted in the proposed diagram and compared with the above mentioned criteria. Further criteria are derived from the consideration of solids with orthogonal symmetry planes in the shear stress space. New criteria are introduced for practical applications. The constraints of convexity are established for them. The proposed consideration of the yield criteria simplifies the selection of a proper criterion. The extreme solutions for the analysis of construction parts can be found using these criteria.