Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators

The one-dimensional equation for transversal vibrations of an elastoplastic beam is derived from a general three-dimensional system with a single-yield tensorial von Mises plasticity model. It leads after dimensional reduction to a multiyield scalar Prandtl-Ishlinskii hysteresis model whose weight function is explicitly given. The use of Prandtl-Ishlinskii operators in elastoplasticity is thus not just a questionable phenomenological approach, but in fact quite natural. The resulting partial differential equation with hysteresis is transformed into an equivalent system for which the existence and uniqueness of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.