By using nonlinear equations of constraints between macro- and micro-states, the regularities of changes in the limiting values of stress and strain invariants in microinhomogeneous media are studied. It is shown that the extreme relative moduli of stress tensor deviators in polycrystals with a cubic lattice are invariant with respect to external conditions of reversible force and depend only on the crystal anisotropy factor. In the irreversible region of deformation, analytical relations are obtained for bulk and tensile normal stresses. The effect of cyclic change in bulk and tensile stresses in some subelements under external monotonic loading has been established. It is shown that, on the basis of nonlinear equations of constraints, a complex pattern of material failure can be described using the theory of maximum normal stresses at the local level.