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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.

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