The influence of elasticity on analysis: Modern developments
Stuart S. Antman · Bulletin of the American Mathematical Society · 1983
Contents 1. Introduction 2. Connectivity questions of global bifurcation theory 3. The peculiarities of nonlinear elasticity 4. Calculus of variations.Weak convergence 5. Variational inequalities 6.Other contributions 7. Conclusion Introduction.This article is one of a pair devoted to an historical study of the influence of the theory of elasticity upon the development of mathematical analysis.In the accompanying article (which appears in this issue of the Bulletin) C. Truesdell describes the contribution made by elasticity to analysis through the middle of the nineteenth century.By that time, the three-dimensional theories of linear and nonlinear elasticity had been well established.From then on, linear elasticity has enjoyed an extensive development, characterized by the use of increasingly sophisticated methods of linear analysis.Nonlinear elasticity, on the other hand, was not to receive sustained scrutiny until after the Second World War.In this article I refrain from discussing developments in the century beginning in 1855 in order to take up the more fascinating tale of the modern interaction of nonlinear elasticity with nonlinear analysis.(St.Venant's memoir on torsion, discussed by Truesdell, appeared in 1855.Cauchy died in 1857.)Despite this gap, the subject of my account is a fitting complement to that of Truesdell, because the modern problems of nonlinear elasticity are much closer in spirit to those studied by the Bernoullis and Euler than to the linear problems studied by the successors of Cauchy.Below we examine several areas of modern analysis to which elasticity has made crucial contributions. Connectivity questions of global bifurcation theory.An elastica is a mathematical model for a thin, flexible beam.Its configuration is described by a curve.Bifurcation theory began with Euler's (1744) analysis of the planar equilibrium configurations of the elastica subjected solely to end forces.