The principle of Saint-Venant in linear and non-linear plane elasticity

Joseph J. Roseman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

A two-dimensional elastic isotropic body {Mathematical expression} is considered. The body has self-equilbrated loadings on its short ends and is unstressed on its long sides. If it is assumed that the first four derivatives of the displacement vector are uniformly bounded and the bounds are sufficiently small, it can be shown from standard equations of non-linear elasticity that {Mathematical expression} where (i) e(x, y) is the strain tensor at any point (x, y). (ii) e{open}=sup | e(x,y)| (x, y) ε R (iii) k and a are positive constants which depend only upon the uniform bounds for the derivatives of the displacement vector. If the equations of linear elasticity are used, it need only be assumed that the norm of the strain tensor is uniformly bounded by e{open} (not necessarily small) to obtain {Mathematical expression} where a is an arbitrary positive parameter, k is a positive function of a. The quantity a may be chosen so as to give an optimal value for k. A discussion is also given as to how the above results can be extended to more general domains.

Original languageEnglish
Pages (from-to)142-162
Number of pages21
JournalArchive for Rational Mechanics and Analysis
Volume26
Issue number2
DOIs
StatePublished - Jan 1967
Externally publishedYes

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