The yield criterion of a material is mostly determined by the yield (or kind of) strain associated with the actual yield (or kind of) mechanism. Different yield mechanisms might be activated under different loading stress states. Hence, the yield strain of a material might depend on the loading stress state, and the interaction within a material itself (e.g., among grains) also affects the actual yield strain. The traditional Tresca and von Mises criteria are two approximations applicable to many materials with reasonable accuracy.
For instance, we have seen the yield surfaces of shape memory alloys (reorientation transform start or phase transformation start) [1] and various foams [2] do not follow either the von Mises criterion or the Tresca criterion. There is no single explicitly equation that is able to provide a good estimation of the yield surface in three-dimensional stress space for them. Under different loading stress states, the corresponding “yield” mechanism might be different [e.g., metallic or polymeric foam [2] in tension (plastic deformation) or compression (buckling)] or different microstructural change might happen (e.g., different stress-induced martensite variants in shape memory alloys [1]). According to [3], even for materials under pure shearing type of yielding (volumetric change is zero before and after yielding), so that the Tresca criterion is a good approximation if the yield is mostly determined by the yielding of one grain (or kind of), the von Mises criterion, which essentially considers the interaction among multiple-grains, is only accurate for certain yield strains (according to [3]). Thus, instead of based on stress as in [1], it is the yield mechanism and the associated yield strain (or kind of) that determine the yield criterion (yield surface) of a material.
References [1] Huang WM. 'Yield' surfaces of shape memory alloys and their applications. Acta Materialia. 1999;47:2769-76. [2] Huang WM. A simple approach to estimate failure surface of polymer and aluminum foams under multiaxial loads. International Journal of Mechanical Sciences. 2003;45:1531-40. [3] Huang WM, Gao XY. Tresca and von Mises yield criteria: a view from strain space. Philosophical Magazine Letters. 2004;84:625-9.