Some Variational Principles in the Three-Dimensional Micropolar Theories of Solids and Thin Solids

Abstract

In this work, we formulated the variational principles of Lagrange, Castigliano, the generalized Reissner-type variational principles (GRTVP), aswell as the principle of virtual work and the principle of complementary virtual work of threedimensional micropolar mechanics (MM) of solids of some rheologies in the case of potentiality, as well as nonpotentiality of stress and couple stress tensors. Proceeding from them and applying the new parametrization (NP) of the domains of single-layer and multilayer thin bodies, the variational principles corresponding to the theories of single-layer and multilayer thin bodies are formulated. In particular, the generalized Reissner-type operator of three-dimensional MM of solids is constructed, on the basis of which the generalized Reissner-type operators of three-dimensional MM of solid single-layer and multilayer thin bodies with one small size are constructed. From the latter Reissner-type operators, in turn, the GRTVP of three-dimensional MM of solid single-layer and multilayer thin bodies with one small size are derived under the NP of the domains of these bodies. It should be noted that the advantage of the NP is that it is experimentally more accessible than other parameterizations used in the scientific literature. Further, using the method of orthogonal polynomials, from the above-mentioned GRTVP, the GRTVP of MM of solid single-layer and multilayer thin bodies with one small size under the NP of the domains of these bodies in moments with respect to the system of Legendre polynomials are derived. Moreover, in the case of the theory of multilayer thin bodies, the representation of the generalized Reissner-type operator is given and the generalized Reissner-type variational principle is formulated, both in the case of complete contact of adjacent layers of a multilayer structure, and in the presence of zones of weakened adhesion. In addition, the description of obtaining of dual Reissner-type operators and the GRTVP, as well as of Lagrangian and Castiglianian and variational principles of Lagrange and Castigliano is given. The interface (interphase boundary) is described by a surface of zero thickness.