By Augusto Visintin

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

This quantity offers a self-contained and finished advent to the research of hysteresis versions, and illustrates a number of new leads to this box. First the classical versions 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. a number of partial differential equations containing hysteresis operators are studied within the framework of Sobolev spaces.

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**Example text**

Fig. 4(b). 5) =X(t)(j(d)(t). 3). Here C(d) = (j(d) E KO. After (j(d) has reached oK, X = I and (j(d)(t) = 0: (C(d)(t)). ° as long as Plasto-Brittleness. Now we combine plastic and brittle elements. Model PIB: Plastic and brittle elements in parallel, cf. Fig. 5(a). Let Kp and K f be the yield and fragility criteria, respectively. 3), this model corresponds to the following rheological law in [0, T]: (j(d)(t) X(t) =(j(d)f(t) + (j(d)p(t) E Kf + K p, =Ho ( sup MK«(j(d)f) [O,t] C(d)(t)[l - X(t)] (j(d)f(t)x(t) =0, =0, [(d/t) E oIKp «(j(d)p(t)).

Fracture is a memory phenomenon, as consequences persist after the event has occurred. Here we deal with a rate independent model of fracture. According to Huber's theory as modified by Reiner and Weissenberg, cf. Reiner [402; pp. 3 Models of Damage 43 critical threshold, named resilience. More precisely, one must distinguish among different thresholds for isotropic compression, isotropic traction and distortion. The first of these thresholds is theoretically unbounded for a homogeneous material, although real materials are not absolutely homogeneous, and so crushing can locally occur.

9) Here the dependence of the functional W f---+ 1/;u(w, T) on the two parameters u, T should be considered globally, and so called elementary catastrophe theory would be quite useful. However, now we confine ourselves to some remarks for constant T and w. First, let us fix T and distinguish the same cases as before: (i) (J(Te) > 0. 10) has a double well; hence hysteresis in the relationship U f---+ W can appear, as we saw in Sect. 3. (ii) (J(T) = < 0, ')'(T) =')' > (S,,), : constants). 11) and we have the following subcases: (a) If T < T e , then 1/;6 is a double well potential, cf.