On three facts of reticences in the classical mathematical modeling of elastic materials

Abstract

Three facts of reticences (passing over in silence, an absence of comments) in the procedures of mathematical modeling of elastic materials are described and commented. The first fact consists in a reticence of one of the first steps in the mentioned above procedure – an assumption that the kinematics of deformation is described by the linear approximation of motion of material continuum, namely by gradients of deformation. In the paper, a novel nonlinear approach to this procedure is offered. The second and third facts are associated with constitutive relations. The second fact consists in the absence of necessary comments relative to determination of smallness of strains and gradients of displacements (absence of comments relative to a criterion of applicability of the linear model) because the criterion $ |u_{i,k}|\ll 1$ is sufficiently abstract. It is shown that there exists a based on the nonlinear Cauchy relations approximate procedure of determination of threshold values of strains and gradients of deformations starting with which a nonlinearity of process appears. The third fact consists in the absence of comments relative to essential differences between the nonlinear constitutive equations, which are written for the ordered pairs "Lagrange stress tensor – Cauchy-Green strain tensor" and "Kirchhoff stress tensor – gradients of displacements". It is shown on an example of the shear stress and the Murnaghan model of nonlinear elastic deformation that deviation from the corresponding straight lines of linear deformation for different pairs differs in many times in the range of small strains and small gradients of displacements. The general estimate of facts of reticences looks positive, because for one part of scientists-mechanicians the reticences form the comfort feeling of monolithic character of the classical theory of elasticity, whereas for another part the reticences form a space for development of the theory of elasticity.