STABILITY OF SOLIDS: FROM STRUCTURES TO MATERIALS
Résumé
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. They have also recognized the close association between bifurcation and symmetry in structures. It was for Koiter, beginning with his famous thesis in 1945, to connect the two communities. 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, making it a challenge to present a first course in this topic within the short time available in one semester. The notes that follow are the first attempt to present a comprehensive, modern introduction to the subject of stability of solids. Given the time constraints, only equilibrium configurations of conservative systems will be considered here. These notes start with the introduction of the concepts of stability and bifurcation for conservative elastic systems through finite degree of freedom examples. They continue with the general theory of Lyapunov-Schmidt-Koiter (LSK) asymptotics, followed by examples from continuum mechanics. The presentation subsequently addresses the issue of scale in the stability of solids. In particular the relation between instability at the microstructural level and macroscopic properties of the solid is studied for several types of applications involving different scales: composites (fiber and particle-reinforced), cellular solids and finally SMA's, where temperature- or stress-induced instabilities at the atomic level have macroscopic manifestations visible to the naked eye.