Trapping of Vibrational Energy in Crumpled Sheets

We investigate the propagation of transverse elastic waves in crumpled media. We set up the wave equation for transverse waves on a generic curved, strained surface via a Langrangian formalism and use this to study the scaling behaviour of the dispersion curves near the ridges and on the flat facets. This analysis suggests that ridges act as barriers to wave propagation and that modes in a certain frequency regime could be trapped in the facets. A simulation study of the wave propagation qualitatively supported our analysis and showed interesting effects of the ridges on wave propagation.Comment: RevTex 12 pages, 7 figures, Submitted to PR

Anomalous strength of membranes with elastic ridges

We report on a simulational study of the compression and buckling of elastic ridges formed by joining the boundary of a flat sheet to itself. Such ridges store energy anomalously: their resting energy scales as the linear size of the sheet to the 1/3 power. We find that the energy required to buckle such a ridge is a fixed multiple of the resting energy. Thus thin sheets with elastic ridges such as crumpled sheets are qualitatively stronger than smoothly bent sheets.Comment: 4 pages, REVTEX, 3 figure

Phase field model of premelting of grain boundaries

We present a phase field model of solidification which includes the effects of the crystalline orientation in the solid phase. This model describes grain boundaries as well as solid-liquid boundaries within a unified framework. With an appropriate choice of coupling of the phase field variable to the gradient of the crystalline orientation variable in the free energy, we find that high angle boundaries undergo a premelting transition. As the melting temperature is approached from below, low angle grain boundaries remain narrow. The width of the liquid layer at high angle grain boundaries diverges logarithmically. In addition, for some choices of model coupling, there may be a discontinuous jump in the width of the fluid layer as function of temperature.Comment: 6 pages, 9 figures, RevTeX

Properties of Ridges in Elastic Membranes

When a thin elastic sheet is confined to a region much smaller than its size the morphology of the resulting crumpled membrane is a network of straight ridges or folds that meet at sharp vertices. A virial theorem predicts the ratio of the total bending and stretching energies of a ridge. Small strains and curvatures persist far away from the ridge. We discuss several kinds of perturbations that distinguish a ridge in a crumpled sheet from an isolated ridge studied earlier (A. E. Lobkovsky, Phys. Rev. E. 53 3750 (1996)). Linear response as well as buckling properties are investigated. We find that quite generally, the energy of a ridge can change by no more than a finite fraction before it buckles.Comment: 13 pages, RevTeX, acknowledgement adde

The Statistics of Crumpled Paper

A statistical study of crumpled paper is allowed by a minimal 1D model: a self-avoiding line bent at sharp angles -- in which resides the elastic energy -- put in a confining potential. Many independent equilibrium configurations are generated numerically and their properties are investigated. At small confinement, the distribution of segment lengths is log-normal in agreement with previous predictions and experiments. At high confinement, the system approaches a jammed state with a critical behavior, whereas the length distribution follows a Gamma law which parameter is predicted as a function of the number of layers in the system

Crumpling a Thin Sheet

Crumpled sheets have a surprisingly large resistance to further compression. We have studied the crumpling of thin sheets of Mylar under different loading conditions. When placed under a fixed compressive force, the size of a crumpled material decreases logarithmically in time for periods up to three weeks. We also find hysteretic behavior when measuring the compression as a function of applied force. By using a pre-treating protocol, we control this hysteresis and find reproducible scaling behavior for the size of the crumpled material as a function of the applied force.Comment: revtex 4 pages, 6 eps figures submitted to Phys Rev. let

Low temperature dynamics of kinks on Ising interfaces

The anisotropic motion of an interface driven by its intrinsic curvature or by an external field is investigated in the context of the kinetic Ising model in both two and three dimensions. We derive in two dimensions (2d) a continuum evolution equation for the density of kinks by a time-dependent and nonlocal mapping to the asymmetric exclusion process. Whereas kinks execute random walks biased by the external field and pile up vertically on the physical 2d lattice, then execute hard-core biased random walks on a transformed 1d lattice. Their density obeys a nonlinear diffusion equation which can be transformed into the standard expression for the interface velocity v = M[(gamma + gamma'')kappa + H]$, where M, gamma + gamma'', and kappa are the interface mobility, stiffness, and curvature, respectively. In 3d, we obtain the velocity of a curved interface near the orientation from an analysis of the self-similar evolution of 2d shrinking terraces. We show that this velocity is consistent with the one predicted from the 3d tensorial generalization of the law for anisotropic curvature-driven motion. In this generalization, both the interface stiffness tensor and the curvature tensor are singular at the orientation. However, their product, which determines the interface velocity, is smooth. In addition, we illustrate how this kink-based kinetic description provides a useful framework for studying more complex situations by modeling the effect of immobile dilute impurities.Comment: 11 pages, 10 figure

Scaling of the buckling transition of ridges in thin sheets

When a thin elastic sheet crumples, the elastic energy condenses into a network of folding lines and point vertices. These folds and vertices have elastic energy densities much greater than the surrounding areas, and most of the work required to crumple the sheet is consumed in breaking the folding lines or ``ridges''. To understand crumpling it is then necessary to understand the strength of ridges. In this work, we consider the buckling of a single ridge under the action of inward forcing applied at its ends. We demonstrate a simple scaling relation for the response of the ridge to the force prior to buckling. We also show that the buckling instability depends only on the ratio of strain along the ridge to curvature across it. Numerically, we find for a wide range of boundary conditions that ridges buckle when our forcing has increased their elastic energy by 20% over their resting state value. We also observe a correlation between neighbor interactions and the location of initial buckling. Analytic arguments and numerical simulations are employed to prove these results. Implications for the strength of ridges as structural elements are discussed.Comment: 42 pages, latex, doctoral dissertation, to be submitted to Phys Rev

Shapes and energies of icosahedral fullerenes: Onset of ridge sharpening transition

Shapes and energies of icosahedral fullerenes are studied on an atomically detailed level. The numerical results based on the effective binary carbon-carbon potential are related to the theory of elasticity of crystalline membranes with disclinations. Depending on fullerene size, three regimes are clearly identified, each of them characterized by different geometrical properties of the fullerene shape. For extremely large fullerenes (more than about 500000 atoms), transition of fullerene shapes to their asymptotic limit is detected, in agreement with previous predictions based on generic elastic description of icosahedral shells. Quantum effects related to delocalized electrons on the fullerene surface are discussed and a simple model introduced to study such effects suggests that the transition survives even in more general circumstances.Comment: Revised to discuss the effects of electronic delocalization on the ridge-sharpening transition. Submitte