The material is assumed to have three orthogonal planes of symmetry. Assuming the material coordinate system is perpendicular to these planes of symmetry, the Hill yield criterion reads [Hill, 1983]

(3–59)

where is the equivalent (also called effective) stress

(3–60)

and is the yield stress that can, in general, evolve as a function of some material internal variables. (See Isotropic Hardening below.)

The equivalent stress can likewise be expressed in matrix form as

(3–61)

where the plastic compliance matrix is given by

(3–62)

The coefficients in this yield criterion are defined as

(3–63)

The directional yield ratios are related to the isotropic yield stress parameter by

(3–64)