Measure of Unloading Disproportion in the Theory of Small Elastoplastic Deformations

From the standpoint of the theory of small elastoplasticdeformations, the stress-strain state of the continuous mediumalong various unloading trajectories from the state achieved as aresult of a simple active process is analyzed. It is shown that ifthe unloading is disproportionate, then the constitutive relationsconnecting the deviators of stresses and strains are tensoriallynonlinear, i.e., the unit tensors of these deviators do notcoincide. It is demonstrated that in the Il’yushinfive-dimensional deviatoric space there exists only one fullunloading point, and it belongs to the line segment of thepreceding active loading. A measure of the nonproportionality ofthe unloading is introduced, characterizing the degree ofdeviation of the path of the passive deformation process from thepreviously mentioned line segment. This measure is calculated fortwo piece-linear unloadings using the example of a constantannular tube subject to the simultaneous action of \((r\theta)\) -torsion and axial \((rz)\) -shear.