The limits of change in stress/strain invariants in the phases of polycrystalline materials with cubic lattices are investigated. The relationship between the local and macroscopic parameters is established on the basis of the following principles: averaged connections, orthogonality of fluctuations of the stress and strain tensors, extremum of discrepancy between the macroscopic measures and suitable average values of microscopic analogues. General expressions for extreme values of stress/strain deviator invariants for the polycrystal phases are obtained. The non-monotonic nature of changes in the extreme values of the invariants of stress/strain deviators and volumetric stresses/strains depending on the phase concentration is revealed. In case of a two-phase polycrystal, as the harder phase increases, the invariants first increase, reaching their maximum value at a concentration of less than 5%, and then, monotonically decrease. Volumetric macrostress has a nonlinear effect on the patterns of changes in volumetric stresses in the grains of a polycrystalline material.