WIAS Preprint No. 1143, (2006)

The von Mises model for one-dimensional elastoplastic beams and Prandtl--Ishlinskii hysteresis operators


  • Krejčí, Pavel
    ORCID: 0000-0002-7579-6002
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 74C05 35Q72 74N30 34C55 47J40


  • beam equation, elastoplasticity, hysteresis operators, Prandtl-Ishlinskii model, von Mises model




In this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general three-dimensional system. The plastic behavior is modeled using the classical three-dimensional von Mises plasticity model. It turns out that this single-yield model without hardening leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model with kinematic hardening, given by a hysteresis operator of Prandtl-Ishlinskii type whose density function can be determined explicitly. This result indicates that the use of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be 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.

Appeared in

  • Math. Methods Appl. Sci., 30 (2007) pp. 2371--2393.

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