Variant of the Mathematical Theory of Non-Thin Multilayer Nonlinearly Elastic Plates of Symmetric Structures

A new version of the mathematical theory of non-thin multilayer nonlinearly elastic Kauderer plates of symmetric structures has been constructed. The components of the stress-strain state (SSS) and boundary conditions on the side surface of the plate are considered functions of three spatial coordinates. The variant is based on the development of components of the SSS of the plate along the transverse coordinate with the help of combinations of Legendre polynomials within each layer. The three-dimensional physically nonlinear boundary value problem for a multilayer non-thin plate is reduced to a two-dimensional one using the three-dimensional equations of the theory of elasticity and Reissner's variational principle. The main dependencies are obtained. Derived systems of high-order differential equations of equilibrium with partial derivatives and boundary conditions containing nonlinear terms from the SSS. The conjugation conditions at the boundary of adjacent layers (rigid connection) are exactly fulfilled for displacements, transverse tangential and normal stresses. The boundary conditions for stresses on the horizontal faces of the plate are also exactly fulfilled. The system of high-order differential equations of equilibrium is transformed into two systems that independently describe skew-symmetric and symmetric deformation relative to the median plane of the plate. In turn, each of these systems is transformed into differential equations that describe vortex boundary effects and equations that describe an internal SSS with a potential boundary layer-type boundary effect. The solution of systems of high-order equations by the developed method is reduced to the solution of second-order differential equations (Poisson and Helmholtz equations). General solutions of the system of differential equilibrium equations are obtained. The validity and accuracy of the variant of the mathematical theory is confirmed by comparisons with exact solutions of bending problems for linearly elastic plates with different mechanical and geometrical parameters. The newly constructed variant of the theory and the developed method provide a real opportunity to solve boundary problems in a spatial formulation with high accuracy for multilayer platens under various loads and boundary conditions on the lateral surface.