Saint-Venant's principle in linear isotropic elasticity for incompressible or nearly incompressible materials
type: Item
instance of: scholarly article
title: Saint-Venant's principle in linear isotropic elasticity for incompressible or nearly incompressible materials
zbMATH Open document ID: 0907.73010
DOI: 10.1023/A:1007361306189
published in: Journal of Elasticity
publication date: 1999-02-14T00:00:00Z
review text: (From authors' abstract.) One of the unresolved issues on Saint-Venant's principle concerns the energy decay estimates established in the literature for the traction boundary value problem of three-dimensional linear isotropic elastostatics for a cylinder. For the semi-infinite cylinder with traction-free lateral surface and self-equilibrated loads at the near end, it has been shown that the stresses decay exponentially from the end and results are available for estimated decay rate, which is a lower bound for the exact decay rate. These results are, however, generally conservative in that they underestimate the exact decay rate. Another shortcoming, which motivated the present investigation, is that the estimated decay rates tend to zero as the Poisson's ratio \(\nu\) tends to \(\frac 12\). Thus for the limiting case of an incompressible material, these methods fail to establish exponential decay. The purpose of this paper is to remedy this defect. In particular, an exponential decay estimate is established with estimated decay rate independent of Poisson's ratio. Thus, in particular, the results here hold in the incompressible limit as \(\nu\rightarrow\frac 12\). An alternative treatment directly for the incompressible case has been given recently. It should be noted that the stresses in the three-dimensional traction boundary value problem do depend on Poisson's ratio \(\nu\) and that stress decay estimates for the cylinder problem with estimated decay rates dependent on \(\nu\) are, in fact, to be expected. However, in the absence of such results that do not deteriorate as \(\nu\rightarrow\frac 12\), we obtain here an estimated decay rate that is independent of \(\nu\).
Mathematics Subject Classification ID: 74G50
Mathematics Subject Classification ID: 74B05
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 1077053
zbMATH Keywords: energy decay estimate
zbMATH Keywords: boundary value problem
zbMATH Keywords: Poisson's ratio
author: Cornelius O. Horgan
author: L. E. Payne
reviewed by: R. P. Srivastav
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1077053