The problem of deducing mathematical relationships for predicting the conditions at which plastic yielding begins when a material is subjected to a complex state of stress is an important consideration in the field of plasticity. In uniaxial loading, plastic flow begins at the yield stress, and it is to be expected that yielding under a situation of combined stresses is related to some particular combination of the principal stresses. A yield criterion can be expressed in the general form
F(\sigma_{ij},k_1,k_2,\dots)=0.where k_1, k_2, ... represent variables such as yield stresses, elastic constants, temperature, strain rate, and hardening parameters.
If the material is isotropic, the stress is uniquely determined by the principal stresses. Therefore, the yield criterion can be expressed as
F_1(\sigma_1, \sigma_2, \sigma_3, k_1, \dots) = 0.The above equation represents a surface in the principal stress space. Such a surface is known as the yield surface.
Since the principal stresses can be expressed in terms of stress invariants, the above equation can be expressed as
F_2(I_1, I_2, I_3)=C,where C is some constant, which represents all the variables affecting the yield surface. We can express I2 and I3 in terms of I1, J2 and J3 (see here). Therefore, we
F_3(I_1, J_2, J_3)=C.If the material such as most metals is insensitive to moderate hydrostatic pressure, their yield criteria must be independent of I_1. Therefore, the yield criterion can be written as
F_4(J_2,J_3)=C.If we want the yield surface to remain insensitive to changing the load direction; that is, if \sigma_{ij} is on the yield surface, -\sigma_{ij} also remains on it, then the yield criterion must take the form
f(J_2,J_3^2)=C.- Notice that there is at no theoretical way of calculating the relationship between the stress components to correlate yielding in a three-dimensional state of stress with yielding in the uniaxial tension test. The yielding criteria are therefore essentially empirical relationships.
