Seminars

Classically, engineers have tried to avoid working with materials and structures under conditions where instabilities are likely to occur. More recently it has become common to take advantage of these instabilities in order to design materials and structures with new and improved properties. When taking advantage of instabilities, one can not longer stop at the onset of bifurcation but needs to go further into the post-bifurcation domain where intricate non-linearities occur. The various proposed applications has in common that they are highly symmetric geometries with periodic structures that present patterns when undergoing instabilities. The very high number of symmetries in these structures creates multiple bifurcation points where several instability modes can coexist for the same buckling load. In this context, traditional perturbation methods appear to be unreliable. Moreover, it often appears that patterning breaks translational symmetries in periodic structures. This behavior requires traditionally to consider various size of periodic unit cell to make sure that the observed bifurcation is the critical bifurcation. This shift in the periodicity of structures is a well-known phenomenon in physics and physicists have efficient tools to cope with these problems.

In this presentation I will briefly review the main principles of bifurcation theory and detail the way symmetries can be taken into account using group theory. With the help of some symmetry considerations, it now becomes possible to pass trough a bifurcation point and explore, exhaustively the various buckling modes that emerges from a multiple bifurcation point. I will also present Bloch-waves and their practical use in bifurcation studies to propose an exhaustive and efficient tool to predict the critical wavelength. Finally I will present various examples where these methods were successfully applied and underline important influencing factors in the models that still need to be explored.