Symmetry and applied variational models for strain gradient anisotropic elasticity

Variational gradient models of the theory of anisotropic elasticity are constructed. It is shown that the symmetry conditions for the elastic moduli tensors follow from the integrability condition of the variational form corresponding to the principle of virtual work for reversible deformation processes in the general case of physically nonlinear processes. More general conditions of symmetry, characteristic of gradient defect-free media, are discussed. The generalized Hooke’s law for linear physical processes is established based on the assumption of integrability of virtual work and linearity in the absence of initial stresses. It is shown that the variational approach allows one to formulate a complete system of governing equilibrium equations and boundary conditions in terms of generalized stresses. Clapeyron and Betty variational theorems are presented. Applied gradient models of anisotropic continuum are formulated by introducing gradient tensors of elastic moduli of the sixth rank with a special symmetric structure which are constructed based on the procedure of convolution of the fourth-rank tensors of elastic moduli of an anisotropic body. In the proposed models, cohesive/scale effects are modeled by an anisotropic tensor of the second rank with the squared length dimension. We show that there is a wide class of boundary-value problems for which boundary-value problems for displacement fields of classical elasticity and fields of cohesive displacements are completely separated. It leads to a significant simplification for the solution of applied problems. An example of the problem of thermoelasticity for a rectangular region, which can be considered as a fragment of an inhomogeneous structure, is studied. © 2021 Begell House, Inc.