ИСТИНА

The variational principles of Lagrange and Castigliano, as well as generalized variational principles of the Reissner type in the framework of the three-dimensional micropolar theory are formulated and proved. The original forms of writing are given for the compatibility of deformation with respect to strain and bending-shearing tensors, as well as with respect to stress and couple stress tensors (analogs to the Beltrami-Mitchell equations). Moreover, the equations for stress and couple stress tensors are represented by both asymmetric differential operators and symmetric ones. From them, as a special case, the Beltrami-Michell equations by both asymmetric and symmetric differential operators are obtained. New formulation of the boundary value problem with respect to stress and couple stress tensors are given, from which a new formulation of B.E. Pobedria of the classical stress problem is obtained as a special case. From the above three-dimensional principles, the corresponding variational principles for the theory of thin bodies are derived. Of the last, in turn, the corresponding variational principles for the theory of thin bodies in moments with respect to 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, 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 some gradient theories of thin bodies are considered.