Under certain conditions, metals and other solids deform plastically. There are two conditions which promote plastic deformation: the condition of stress and the mechanism which determines how that deformation takes place. The critical condition of stress which produces yielding is due to the net effect of the combination of stresses acting at a point. A mathematical theory which predicts the critical condition of stress for which plastic flow occurs is a yield criterion.A new yield criterion is introduced. It is hyperbolic in form so it is significantly different from the criteria developed by Rankine, Tresca, Saint-Venant, or von Mises. But like these criteria, the new yield condition is also limited to a biaxial stress condition.The new yield criterion, because of its form, serves as an instructive model for the development of an analytical equation of equivalent form. As a result, the new yield criterion can be interpreted as an energy equation which includes a mechanism related to how metals flow plastically. This mechanism can be related to small angle (gonio) distortions which develop in materials according to a pattern or form (morphous) that is either fixed or controlled by the basic structure of the material and the condition of stress. As a result, this new criterion is identified as a goniomorphous yield criterion.The analytical equation is used as a pattern for two additional yield criteria. These new criteria predict yield for a three-dimensional condition of stress for two different deformation patterns.