On assymetric measures of stress-strain state and Hooke’s law

Hooke’s law (in a modern tensor form considering different types of material anisotropy, finite or velocity formulation) is widely used in solid mechanics including physical and/or geometrical nonlinear problems. In the recent decades it has also been used in the majority of multilevel models oriented on describing inelastic deformation in mono- and polycrystalline materials. As a rule, in this case Hooke’s law is written using symmetrical measures of stress and strain state that are determined in terms of actual, intermediate (unloaded) or reference configuration. For a material that is elastic according to Green, the elastic potential presence naturally leads to the symmetry of elastic four-valent tensor П in the first and second pare of indices, П ijkl = П klij. However tensor symmetry in the first and second pair of indices is explained only due to the accepted and established agreement in solid mechanics related to symmetry of stresses and strains tensors. It is worth mentioning that the initial Hooke’s law written for uniaxial loading obviously had nothing to do with the symmetry of properties. The specified agreement made it possible to reduce the number of experiments necessary to find tensor elastic properties; and it is especially important for materials studies with an a priory low or unknown symmetry. Stress tensor symmetry results from law of conservation of angular momentum without distributed volume and surface moments. The neglection of the distributed surface moments is based on a hypothesis that two parts of the body interact with distributed forces, which can be put in to the stresses vector on each surface element. This hypothesis again is based on an idea that there is no correlation of distributed surface loadings on any material area element. It is worth stating that already in 1887 V. Voigt suggested to abandon this idea and put the distributed effects of one body part on the other one on any surface element into stresses vector and distributed moments vector. The specified suggestion is in a full compliance with the method related to putting a random system of forces into the principal vector and principal moment (this method is used in theoretical (classical) mechanics). The problem of a simple shear shows that Hooke’s (symmetrical) law leads to the incompliance of the stress state (found with the law in its conventional formulation) and part of boundary conditions. We have considered Hooke’s law which is oriented on application of asymmetrical measures of stresses and strains and elastic properties tensor with symmetry only in a pair of indices. Asymmetrical Cauchy tensor is used as a stress measure, gradient of displacement velocity (displacement velocities with respect to a stiff moving coordinates which is in charge for a rigid displacement of volume element) - as strain velocity measure; all of them do not depend on the reference coordinate. A type of tensor of elastic properties in Hooke’s law oriented on asymmetric measures is proposed.