Seminar: About non-regular dynamics

By Alain Léger
Director of Research at the CNRS

Wednesday 28 March 2018 at 14:00
IEMN Ampli LCI - Villeneuve d'Ascq


This talk will present some aspects, first introductory, then more recent, of non-regular mechanics. A number of situations, edge conditions or behaviour laws provide examples of non-regularity in mechanics. We will concentrate mainly on the case of contact and friction, but several fundamental aspects would be identical in the cases of plasticity, damage, etc. In all cases, the introduction of non-regular conditions in continuum mechanics leads to open and difficult mathematical problems. To this end, an attempt will be made to give a detailed account of the current state of affairs, in the form of a list of problems that have been solved or are still open, so as to clarify the situations in which it is legitimate or not to use results from different areas of physics, and it will be observed that it is in these cases simple models which, provided they are well chosen, provide qualitative information where models closer to physics would be inaccessible.

It will be recalled that non-regularity removes the possibility of referring to the classical framework of the theory of differential equations or partial derivatives. After a few results, stated in the case of a very simple mechanical system but generalisable to all discrete problems, a large part of the presentation will be devoted to the study of the response to a periodic load, as is traditional in the qualitative study of dynamic systems.

Initially, the mechanical system will be linear, which will make the results usable qualitatively in many areas of physics, acoustics or vibrations. Particular attention will be paid to the transition between zones of different behaviour, and it should be noted that no transition to chaos is observed when the only non-linearity is due to contact and friction. We will then add a regular non-linearity of the large deformation type. We will then see that the response can include zones of non-periodic behaviour, which will lead us to question the coupling between different types of non-linearity.