ИСТИНА

Some integral relations for the statements of variational principles are given. The variational principles of Lagrange and Castigliano are formulated and proved, as well as the generalized variational principles of Reissner type are formulated within the framework of the three-dimensional micropolar theory. From these three-dimensional principles, the corresponding variational principles for the theory of thin bodies are obtained, and from which, in turn, the corresponding variational principles for the theory of thin bodies in moments of systems of orthogonal polynomials are derived. Moreover, for the micropolar theory of multilayer thin bodies, both in full contact and in the presence of zones of weakened adhesion, only generalized variational principles of the Reissner type are obtained, since the principles of Lagrange and Castigliano are easily derived from them. The theorems on the minimum of the stationary point of the Lagrangian and the maximum of the stationary point of the Castiglianian, as well as the uniqueness theorem for the generalized solution of boundary value problems, are proved. In addition, variational principles for the second gradient theory of thin bodies are considered.