Variant of the Mathematical Theory of Multilayer Nonlinear Elastic Plates of Non-Symmetric Structures of Arbitrary Constant Thickness

Abstract:

A variant of the mathematical theory of deformation of multilayer nonlinearly elastic (according to Kauderer) plates of arbitrary constant thickness with non-symmetric structure in thickness has been constructed. The transverse load on the horizontal faces can be arbitrary static. The components of the stress-strain state (SSS) and the boundary conditions on the lateral surface are functions of three spatial coordinates. Spatial boundary value problems for multilayer plates are reduced to two-dimensional using three-dimensional equations of the theory of elasticity, the Reissner variational principle, and the expansion of the components of the SSS into infinite mathematical series by combinations of Legendre polynomials within each layer. This approach differs significantly from the approaches of other authors. The main dependencies, boundary conditions and systems of equilibrium differential equations with high-order partial derivatives with respect to the displacement components are derived. All dependencies and equations contain nonlinear terms. The new methodology for constructing a variant of the nonlinear theory makes it possible to accurately satisfy the boundary conditions on the horizontal faces of the plate and on the lateral surface, and to accurately satisfy the conditions of rigid conjugation of adjacent layers. The system of equilibrium equations has a high order. An analytical method for solving these systems is proposed and developed. The method is based on algebraic, differential and operator transformations of the initial systems. They are reduced to two convenient defining systems: one describes the vortex edge effect with a refinement of the SSS, and the other describes a refined internal SSS with a potential edge effect. The order of the systems of differential equations does not depend on the number of layers, but depends only on the number of retained terms in the mathematical seriess. The internal SSS is separated from the potential edge effect. By the method of order reduction, the determining systems are reduced to second-order differential equations. This significantly simplifies the solution of boundary value problems. General solutions for all components of the SSS were found through general solutions of second-order differential equations. For plates with non-symmetric structure, the equations of skew-symmetric and symmetric deformation are interconnected, unlike plates with a symmetric structure. Numerical results are presented for a two-layer linearly elastic plate under cylindrical bending.

• Periodical: Solid State Phenomena (Volume 380)
• Pages: 83-96
• DOI: https://doi.org/10.4028/p-wV2ky6
• Online since: November 2025
• Authors: Anatoly Grigorievich Zelensky*, Sergiy Slobodyanyuk
• Keywords: Differential Equations, Legendre Polynomials, Method, Multilayer Nonlinear-Elastic Plate of Non-Symmetric Structure, Reissner Principle, Solutions, System, Variant of Mathematical Theory

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