Development of the “Separated Anisotropy” Concept in the Theory of Gradient Anisotropic Elasticity

Abstract

A variation model of the gradient anisotropic elasticity theory is constructed. Its distinctive feature is the fact that the density of potential energy, along with symmetric fourth-and sixth-rank stiffness tensors, also contains a nonsymmetric fifth-rank stiffness tensor. Accordingly, the stresses in it also depend on the curvatures and the couple stresses depend on distortions. The Euler equations are three fourth-order equilibrium equations. The spectrum of boundary-value problems is determined by six pairs of alternative boundary conditions at each nonsingular surface point. At each special point of the surface (belonging to surface edges), in the general case, additional conditions arise for continuity of the displacement vector and the meniscus force vector when crossing the surface through the edge. In order to reduce the number of the physical parameters requiring experimental determination, particular types of the fifth-and sixth-rank stiffness tensors are postulated. Along with the classical tensor of anisotropic moduli, it is proposed to introduce a first-rank stiffness tensor (a vector with a length dimension), with the help of which the fifth-rank tensor is constructed from the classical fourthrank tensor by means of tensor multiplication. The sixth-rank tensor is constructed as the tensor product of the classical fourth-rank tensor and two first-rank tensors.

Foundations of Continuum Mechanics and Mathematical Physics—Editorial 2021–2023

“…In [3][4][5][6][7], it is shown that for the extensive class of elasticity models, including those regarding gradient elasticity, additional symmetry conditions from classical elasticity models must be met. Otherwise, models intended to simulate a deformable, defect-free medium become erroneous because the condition of the continuity of distortions, i.e., the condition of the absence of a field of defects, is not satisfied.…”

“…Therefore, it also extends to non-gradient models of anisotropic media, for which the general structure of elastic modulus tensors is very complex. These issues were partly considered in [4,6]. The same papers discuss the issues of symmetry in the structure of elastic modulus tensors related to the potentiality of the strain energy density and the issues of the inheritance of the symmetrical anisotropy properties from the classical theory in a gradient anisotropic medium.…”