CRM: Centro De Giorgi
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Rate-Independence, Homogenization and Multiscaling

15 November 2007 - 17 November 2007


Hysteresis occurs in many phenomena: ferromagnetism, ferroelectricity, plasticity, undercooling are typical examples. After the pioneering studies of Bouc, Krasnosel'skii and others, hysteresis has been treated as a memory phenomenon. In the last 10 years new approaches for rate-independent processes emerged, like the energetic formulation.

The latter allows to exploit methods from the theory of the calculus of variations and thus has potential to work for general nonconvexnonmonotone systems. Moreover, Gamma convergence and relaxtion techniques are available to treat multiscale problems with hysteresis. Homogenization (i.e., the search for effective models representing the macroscopic behaviour of mesoscopically inhomogeneous materials) has been the object of an intense research, and has extensively been applied to the study of composite materials in the last 30 years. The point of view of two-scale convergence was pioneered by G. Nguetseng, further developed by G. Allaire and others, and nowadays is widely applied in homogenization. This approach leads in a natural way to the formulation of multiscale models.

Multiscaling also arises in a natural way in models representing complex phenomena, in which different phenomena interact. Moreover scaling may be regarded as a unifying feature for hysteresis, homogenization, Gamma-convergence and relaxation.