Multiplicity results for free and constrained nonlinear elastic rods based on nonsmooth critical point theory

Marco Degiovanni and Friedemann Schuricht

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Submission date: 26. Feb. 1997
Pages: 53
published in: Mathematische Annalen, 311 (1998) 4, p. 675-728 
DOI number (of the published article): 10.1007/s002080050206
Bibtex
with the following different title: Buckling of nonlinearly elastic rods in the presence of obstacles treated by nonsmooth critical point theory

Abstract:
The main goal of this paper is to demonstrate that nonsmooth methods, generalizing the ideas of Ljusternik-Schnirelman theory to merely lower-semicontiuous functionals, can be applied to buckling problems in nonlinear elasticity. Based on some general nonsmooth critical point theory developped by Corvellec, Degiovanni, and Marzocchi we verify nontrivial buckling states of an axially compressed nonlinearly elastic rod. Here we work with the geometrically exact Cosserat theory describing planar deformations of elastic rods that can suffer flexure, extension, and shear and which involves a general nonlinear constitutive relation. In particular we can handle the presence of rigid obstacles and materials with nonsmooth constitutive functions. In that sense we generalize all previous buckling results. In contrast to the usual study of obstacle problems by means of variational inequalities we derive the Euler-Lagrange equations and show that our abstract critical points have physical relevance.