A structural theory for a 3D isotropic linear-elastic finite body

TL;DR

This paper develops a nonlinear structural theory for 3D isotropic linear-elastic finite bodies using Taylor expansion and weak formulation, resulting in a set of ODEs for deformation analysis.

Contribution

It introduces a new nonlinear theory for 3D elastic bodies with a specific second order case and derives a system of ODEs for deformation analysis.

Findings

Derivation of 30 ODEs for deformation fields

Theory consistent with classical elasticity

Potential for analytical and numerical solutions

Abstract

The development of a nonlinear structural theory (model) for isotropic linear-elastic finite continua is the main objective of the study. To derive the theory, we used Taylor's multivariable expansion and Bubnov-Galerkin's weak formulation. The resulting formulation is consistent with elasticity theory. As a special case, we consider second order theory, resulting in 30 internal degrees of freedom (IDF). A set of 30 ordinary differential equations (ODE) is derived and must be solved to determine the deformation field. ODEs depend on the initial and actual geometry of the structure, loads and material properties. Our theory can be used to derive analytical or numerical solutions. Aspects of generalization of the method to solids with nonlinear constitutive relations are presented.