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<p>This paper presents a unified mathematical framework for constructing refined theories and hierarchical models of thin-walled structures in linear case and with essential remarks for nonlinear case. Building upon the foundational work of I.&#xa0;Vekua, we develop a systematic approach using orthogonal polynomials to construct hierarchical models for anisotropic inhomogeneous but not only elastic, for example, magnetoelastic, piezoelastic, and poroelastic bodies. A central contribution is the resolution of the long-standing problem of satisfying boundary conditions when generalized stress vectors are prescribed on the surfaces of elastic plates and shells. This problem has remained open for both refined theories and hierarchical models. In the nonlinear case, where bending and compression–expansion processes are inherently coupled, we present the exact structure of a system of differential equations, constructed without ad hoc assumptions. The proposed theory approximately satisfies the governing system of partial differential equations and boundary conditions on the surfaces of thin-walled structures, providing a mathematically rigorous foundation for both linear and nonlinear analysis of plates and shells.</p>

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TO THE CONSTRUCTION AND FOUNDATION OF THE NONLINEAR REFINED THEORIES AND HIERARCHICAL MODELS FOR THIN-WALLED STRUCTURES

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Tamaz Vashakmadze

摘要

This paper presents a unified mathematical framework for constructing refined theories and hierarchical models of thin-walled structures in linear case and with essential remarks for nonlinear case. Building upon the foundational work of I. Vekua, we develop a systematic approach using orthogonal polynomials to construct hierarchical models for anisotropic inhomogeneous but not only elastic, for example, magnetoelastic, piezoelastic, and poroelastic bodies. A central contribution is the resolution of the long-standing problem of satisfying boundary conditions when generalized stress vectors are prescribed on the surfaces of elastic plates and shells. This problem has remained open for both refined theories and hierarchical models. In the nonlinear case, where bending and compression–expansion processes are inherently coupled, we present the exact structure of a system of differential equations, constructed without ad hoc assumptions. The proposed theory approximately satisfies the governing system of partial differential equations and boundary conditions on the surfaces of thin-walled structures, providing a mathematically rigorous foundation for both linear and nonlinear analysis of plates and shells.