By Augusto Visintin

Hysteresis results ensue in technology and engineering: plasticity, ferromagnetism, ferroelectricity are famous examples. Modelling and mathematical research of hysteresis phenomena were addressed through mathematicians only in the near past, yet are actually in complete improvement.

This quantity offers a self-contained and complete advent to the research of hysteresis types, and illustrates numerous new leads to this box. First the classical types of Prandtl, Ishlinskii, Preisach and Duhem are formulated and studied, utilizing the concept that of "hysteresis operator". a brand new version of discontinuous hysteresis is brought. numerous partial differential equations containing hysteresis operators are studied within the framework of Sobolev spaces.

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J ~ ~ . J~~ ..... -:;7 !! - ~ I I (b) Figure 6. Description of the model IIpEP(Ep-B p), representing mechanical damage. l has no atom. Because of symmetry with respect to the origin, we outline just the branch corresponding to E > O. The curve OA is purely elastic. Along the decreasing part CD of the upper branch, fractures prevail over elastic deformation; evolution is controlled by c, and 17 cannot increase. If beyond a threshold ED elastic deformation prevails over fracture, then the curve becomes increasing.

J~~ ..... -:;7 !! - ~ I I (b) Figure 6. Description of the model IIpEP(Ep-B p), representing mechanical damage. l has no atom. Because of symmetry with respect to the origin, we outline just the branch corresponding to E > O. The curve OA is purely elastic. Along the decreasing part CD of the upper branch, fractures prevail over elastic deformation; evolution is controlled by c, and 17 cannot increase. If beyond a threshold ED elastic deformation prevails over fracture, then the curve becomes increasing.

For a finite this is clearly absurd. So it is needed that O'(s) - t -00 as [(B) - t -1+; that is, an infinite pressure is required for the volume to vanish. For instance, Hencky proposed the rheological equation 0'(8) = X log (1 + [(s)). 4) by allowing 0: to be multivalued. More precisely, we assume that the multivalued function 0: : Dom(o:) C R - t peR) corresponds to a maximal monotone (possibly multivalued) function; cf. Sect. S. We shall identify any multi valued function with its graph. , maximal monotone graphs) gives more generality to our developments, and is also convenient, since this class is closed under inversion, in contrast to what occurs for the class of nondecreasing functions.