By Pasquale Porcelli

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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)).

32) Mil +,u + Eu + 8h(u) 3 f(t), More general models of elasto-plasticity (for instance, Prandtl-Ishlinskil models, which we shall introduce in the next chapter) correspond to laws of the form F = F(u), where F is a hysteresis operator (this concept will be defined in Sect. 1). This yields a first order differential equation with a hysteresis term: Mil + ,u + Eu + F(u) = f(t). 33) The latter equation can be compared with that of the ferroelectric oscillator, we shall describe later on, cf. 35).

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. ferential of a proper, convex, lower semicontinuous function E : R -+ R U {+oo }; cf.