On the Theory and Practice of Thin Walled Structures

TL;DR

This paper develops a comprehensive mathematical theory for refined and hierarchical models of elastic plates and shells, addressing boundary condition satisfaction in both linear and nonlinear cases, with practical parameter selection and experimental validation.

Contribution

It introduces a unified approach to refined and hierarchical theories for elastic plates and shells, including nonlinear cases, with a method for optimal model selection based on experimental data.

Findings

Developed a mathematical framework for refined theories in linear and nonlinear elasticity.

Created a procedure for selecting optimal models via experimental measurements.

Extended the theory to hierarchical models with boundary condition satisfaction.

Abstract

We consider the problem of satisfaction of boundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. This problem was open also both for refined theories in the wide sense and hierarchical type models. This one for hierarchical models was formulated by Vekua. In nonlinear cases the bending and compression-extension processes did not split and for this aim we cited von Karman type system without variety of ad hoc assumptions since, in the classical form of this system of DEs one of them represents the condition of compatibility but it is not an equilibrium equation. Thus, we created the mathematical theory of refined theories both in linear and nonlinear cases for anisotropic nonhomogeneous elastic plates and shells, approximately satisfying the corresponding system of partial differential equations and boundary conditions on the surfaces. The optimal and convenient refined theory might be chosen easily by selection of arbitrary parameters; preliminarily a few necessary experimental measurements have been made without using any simplifying hypotheses. The same problem is solved for hierarchical models too.