Enrique Cerda

Conical dislocations in crumpling

A crumpled piece of paper is made up of cylindrically curved or nearly planar regions folded along line-like ridges, which themselves pivot about point-like peaks; most of the deformation and energy is focused into these localized objects. Localization of deformation in thin sheets is a diverse phenomenon, and is a consequence of the fact that bending a thin sheet is energetically more favourable than stretching it. Previous studies considered the weakly nonlinear response of peaks and ridges to deformation. Here we report a quantitative description of the shape, response and stability of conical dislocations, the simplest type of topological crumpling deformation. The dislocation consists of a stretched core, in which some of the energy resides, and a peripheral region dominated by bending. We derive scaling laws for the size of the core, characterize the geometry of the dislocation away from the core, and analyse the interaction between two conical dislocations in a simple geometry. Our results show that the initial stages of crumpling (characterized by the large deformation of a few folds) are dominated by bending only. By considering the response of a transversely forced conical dislocation, we show that it is dynamically unstable above a critical load threshold. A similar instability is found for the case of two interacting dislocations, suggesting that a cascade of related instabilities is responsible for the focusing of energy to progressively smaller scales during crumpling.

Comment on noise and bifurcations

We calculate in exact form the first correction in a parameter measuring the strength of the noise to the effective potential for one-variable diffusion processes. The use of this potential to study transitions is discussed.

Asymptotic description of a viscous fluid layer

We prove that the exact non local equation derived by the present authors for the temporal linear evolution of the surface of a viscous incompressible fluid reduces asymptotically for high viscosity to a second order Mathieu type equation proposed recently by Cerda and Tirapegui. The equation describes a strongly damped pendulum and the conditions of validity of the asymptotic regime are given in terms of the relevant physical parameters.

Effective potential and weak noise transitions

We review the notion of effective potential for stochastic processes and discuss its possible applications. We calculate this function up to first order in a parameter measuring the intensity of the noise for a general nonlinear system. The result is applied exhibiting a transition induced by weak noise.

Weak noise expansions through functional integrals for colored noise

We use path integral methods to obtain expansions for the correlation functions of the non-Markovian stochastic processes generated by stochastic differential equations with colored noise.

A Dissipative Model for Parametric Waves in Granular Materials

A simple dissipative model for describing the dynamics of parametric waves in granular materials is proposed. The instability mechanism in the model is due to two elementary processes in competition: a focusing effect that concentrates particles in space, and a diffusion effect that relaxes large thickness gradients in the system. The same mechanism are predicted to occur in a very viscous fluid layer submitted to impulsive forcing.

Effective Potential For Stochastic Processes

The notion and possible uses of the effective potential are considered for Markov processes associated to nonlinear systems in the presence of noise. We calculate this function up to first order in a parameter measuring the intensity of the noise with a method which can be generalized to more complicated cases. Some interpretations of the potential are discussed.