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

Pavel Krejčí; Jürgen Sprekels

doi:10.3934/dcdss.2008.1.283

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2008, Volume 1, Issue 2: 283-292. Doi: 10.3934/dcdss.2008.1.283

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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: September 30, 2006

Revised: July 31, 2007

Published: May 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q72; Secondary: 74C05; 74K10; 74N30; 34C55; 47J40.

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