Creasing, point-bifurcations, and the spontaneous breakdown of scale-invariance

About this book

Symmetry and symmetry breaking are important in condensed matter theory; they explain how continuum phases of matter emerge from molecular-scale chaos and lead directly to the classes of Partial Differential Equations (PDEs) that describe them. Here I suggest a new kind of symmetry breaking, the spontaneous breaking of scale symmetry, which is an asymptotic symmetry of all continuum models at small scales, and explains why things like fracture and first order phase transitions have macroscopic robustness. I formalize this idea with the notions of a point-instability and a point-bifurcation, which are sudden changes in the state of a system that develop from a point manifesting as a kind of topological change. As a non-trivial example of an equilibrium point-bifurcation, I discuss creasing in the surfaces of soft rubber-like materials. A crease is a singular, self-contacting fold, such as in the cup of the hand. Numerical methods (which rely on a novel kind of pseudo arc-length continuation for variational inequalities, called rough continuation) agree with experiments that show that creases in bent PDMS blocks form suddenly, but vanish continuously to a point, showing an effect called perfect hysteresis. Like a phase transition, creasing is characterized by intensive criterion--critical stresses--but the "phases" are a smooth surface and an intrinsically localized crease. Indeed I prove that for a wide class of PDEs of any differential and in any dimension, the only kinds of instabilities are ordinary linear instabilities and point-instabilities. I show that while linear stability is almost never sufficient to prove actual stability, linear stability (linear hyperbolicity) and a condition called a point-Lipshitz condition (or metastability) are necessary and sufficient conditions. I show this constructively: generically, when an equilibrium system loses metastability but retains linear stability, a bifurcation occurs where the bifurcating branch develops as a function of a control parameter by the self-similar growth of a "defect" from a point. Whether or not this happens is directly related to specific auxiliary problem, which I also formulate. This auxiliary problem, in turn, predicts the critical conditions and the shape of the growing defect.