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June  2008, 1(2): 283-292. doi: 10.3934/dcdss.2008.1.283

Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators

Pavel Krejčí 1, and  Jürgen Sprekels 1,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D–10117 Berlin

Received  October 2006 Revised  August 2007 Published  March 2008

We consider a model for one-dimensional transversal oscillations of an elastic-ideally plastic beam. It is based on the von Mises model of plasticity and leads after a dimensional reduction to a fourth-order partial differential equation with a hysteresis operator of Prandtl-Ishlinskii type whose weight function is given explicitly. In this paper, we study the case of clamped beams involving a kinematic hardening in the stress-strain relation. As main result, we prove the existence and uniqueness of a weak solution. The method of proof, based on spatially semidiscrete approximations, strongly relies on energy dissipation properties of one-dimensional hysteresis operators.
Keywords: von Mises model., Elastoplasticity, beam equation, Prandtl-Ishlinskii model, hysteresis operators.
Mathematics Subject Classification: Primary: 35Q72; Secondary: 74C05; 74K10; 74N30; 34C55; 47J4.
Citation: Pavel Krejčí, Jürgen Sprekels. Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 283-292. doi: 10.3934/dcdss.2008.1.283
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