Measure of unloading disproportion in the theory of small elastoplastic deformations

From the standpoint of the theory of small elastoplastic deformations, the stress-strain state of the continuous medium along various unloading trajectories from the state achieved as a result of a simple active process is analyzed. It is shown that if the unloading is disproportionate, then the constitutive relations connecting the deviators of stresses and strains are tensorially nonlinear, i.e., the unit tensors of these deviators do not coincide. It is shown that in the Ilyushin five-dimensional deviatoric space there exists only one full unloading point, and it belongs to the line segment of the preceding active loading. A measure of the non-proportionality of the unloading is introduced, characterizing the degree of deviation of the path of the passive deformation process from the previously mentioned line segment. This measure is calculated for two piece-linear unloadings using the example of a constant annular tube subject to the simultaneous action of $ (r\theta) $-torsion and axial $(rz)$-shear.