Nonlinear Tensor Functions of Two Arguments and Some “Orthogonal Effects” of the Stress–Strain State

Abstract

The general representation of a symmetric isotropic tensor-function of the second rank in three-dimensional space, depending on two tensor arguments of the second rank, is investigated. This representation includes eight scalar material functions of ten invariants of dependent tensors, including four joint invariants. The tensor function is interpreted as the constitutive relation of an isotropic nonlinear material, which expresses the relationship between deformations and stresses and stress rates. A certain type of stress state is selected, corresponding to a combination of axial tension of the rod by a constant force and (θz) -torsion by a time-periodic moment. It is shown that with a proper selection of the above-mentioned eight material functions as functions of the stress state invariants, it is possible to describe a much stronger change in axial deformations under joint tension and cyclic torsion than under separate tension without torsion. Such an “orthogonal effect” of the stress-strain state is close in form to the ratchetting and vibration creep known in experimental mechanics.