Types of Physical Nonlinearity in the Theory of Constitutive Relations and the Generalized Poynting Effect

The certain class of constitutive relations are considered that connect the symmetric stress tensor and the symmetric strain tensor by means of isotropic potential tensor nonlinear functions in three-dimensional space. The various definitions of tensor nonlinearity are given as well as their equivalence is shown. From the perspective of mathematical theory about the tensor nonlinear functions, an interpretation of the Poynting effect is given, which is well known in experimental mechanics. It is demonstrated that such an effect is not necessarily the consequence of tensor nonlinearity in constitutive relations; instead, it is effected by the quadratic dependence on invariants in certain material functions. Therefore, in the physically linear case for a small strain, this dependence is absent. Concerning this “order of smallness,” the Poynting effect is investigated and a possibility is discussed for simulating such an effect by means of the tensor linear constitutive relations.