ИСТИНА

We derive the necessary integral relations for the formulation of the variational principles. Thus, we formulate and prove the variational principles of Lagrange and Castigliano, as well as the generalized variational principles of Reissner type for the three-dimensional micropolar theory. By these three-dimensional principles, we obtain the corresponding variational principles for the theory of thin bodies, from which we get the corresponding variational principles in moments for the theory of thin bodies with respect to systems of orthogonal polynomials. In this case, for the micropolar theory of multilayer thin bodies, both with full contact and in the presence of weak adhesion zones, only generalized variational principles of Reissner type are obtained, since the principles of Lagrange and Castigliano are easily derived from them. We prove the theorems on the minimum of the stationary point of the Lagrangian and on the maximum of the stationary point of Castiglianian, and also the theorem on uniqueness of a generalized solution of boundary value problems. In addition, we consider the variation principles for the second gradient theory of thin bodies.