Restrictions on nonlinear constitutive equations for elastic shells
type: Item
instance of: scholarly article
title: Restrictions on nonlinear constitutive equations for elastic shells
zbMATH Open document ID: 0840.73040
DOI: 10.1007/BF00043413
author: P. M. Naghdi
author: M. B. Rubin
published in: Journal of Elasticity
publication date: 1996-07-04T00:00:00Z
review text: It is well known that the problem of the writing the constitutive equations of a shell-like structure is very difficult if the shell theory is based on some Cosserat model, even if the nonlinear three-dimensional constitutive laws are known. The reason comes from the coupling of the three-dimensional laws with the geometrical properties of the shell model. The approach used by the authors, the first one being as a recall one of the pioneers of the model, lies in the search of some restrictions given by the model with respect to the three-dimensional theory. After having described the two-dimensional Cosserat theory (a surface endowed with one director), the restrictions are found in the assumed and simplified case of homogeneous deformations considered as a necessary condition. Besides, the main classical assumption is the existence of a nonlinear strain energy as a function of the deformation unknowns. The definitions are recalled in a rather intrinsic method, some useful change of variables is made, and the associate surface stresses are established through the dynamic equilibrium equations. Some formulae give their values as functions of the Piola-Kirchhoff stresses, the three-dimensional corresponding strains, and the three-dimensional strain energy in the homogeneous deformation case. It is assumed an additional part of the strain energy, so that the restrictions are written for that part owing to the homogeneous strain assumption. A case of simple strain energy function is written for a shell made of a homogeneous material with a general nonlinear anisotropic elastic resonse. First, applications are developed in the case of an infinite plate with a circular hole and compared to the Reissner or the Mindlin plate theory. The second application concerns the torsion of a rectangular plate and is compared with the exact three-dimensional solution. Both comparisons, which are either in statics or in dynamics by matching the first frequencies of free vibrations, show…
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74A20
zbMATH DE Number: 807330
zbMATH Keywords: Piola-Kirchhoff stresses
zbMATH Keywords: Cosserat model
zbMATH Keywords: homogeneous deformations
zbMATH Keywords: nonlinear strain energy
zbMATH Keywords: associate surface stresses
zbMATH Keywords: infinite plate
zbMATH Keywords: circular hole
zbMATH Keywords: rectangular plate
zbMATH Keywords: free vibrations
zbMATH Keywords: spherical shell
zbMATH Keywords: differential geometry
reviewed by: Roger Valid
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00043413
OpenAlex ID: W4256699986
description: scientific article; zbMATH DE number 807330