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STABILITY PROBLEMS IN MECHANICS : MULTIPHYSICS AND MULTISCALE ASPECTS (A MECANICIAN’S PERSPECTIVE…)
Stability is a fascinating topic in solid mechanics that has its roots in the celebrated Euler column buckling problem, which first appeared in 1744. Over the years advances in technology have led to the study of ever more complicated structures first in civil and subsequently in mechanical engineering applications. Aerospace applications, most notably failure of solid propellant rockets, led the way in the 1950s. Problems associated with materials and electronics industries came on stage in the 1970s and 1980s, starting with instabilities associated with thin films and phase transformations in shape memory alloys (SMA’s), just to name some of the most preeminent examples. In a parallel path, starting in the late 19th century, mathematicians studying nonlinear differential equations, developed the concept of a bifurcation (term coined by Poincare) and created powerful techniques to study the associated singularities, followed by advances in group-theoretical methods that exploit the problem’s underlying symmetries. Amazing progress has been made since the early days of structural buckling problems and continues to be made in this field, with applications ranging from atomistic to geological scales. With the advent of new materials, the number of applications in this area continues to progress with an ever- increasing pace.
In this talk we present selected applications of stability problems involving phenomena a) across spatial scales and b) driven by multiphysics coupling. In the first class of applications we visit – by decreasing the size of the underlying scale – the instabilities occurring in fiber reinforced composites, honeycomb and crystal lattices (shape memory effects). In the second class, we present stability problems in magnetoelastic thin films, liquid crystals and step-bunching in epitaxial thin film deposition. In all these applications, we use both continuum description of the problem at hand or appropriate micromechanical models and the mathematical tools of bifurcation theory and symmetry groups.