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

The effects of forcing and dissipation on phase transitions in thin granular layers

Recent experimental and computational studies of vibrated thin layers of identical spheres have shown transitions to ordered phases similar to those seen in equilibrium systems. Motivated by these results, we carry out simulations of hard inelastic spheres forced by homogenous white noise. We find a transition to an ordered state of the same symmetry as that seen in the experiments, but the clear phase separation observed in the vibrated system is absent. Simulations of purely elastic spheres also show no evidence for phase separation. We show that the energy injection in the vibrated system is dramatically different in the different phases, and suggest that this creates an effective surface tension not present in the equilibrium or randomly forced systems. We do find, however, that inelasticity suppresses the onset of the ordered phase with random forcing, as is observed in the vibrating system, and that the amount of the suppression is proportional to the degree of inelasticity. The suppression depends on the details of the energy injection mechanism, but is completely eliminated when inelastic collisions are replaced by uniform system-wide energy dissipation.Comment: 10 pages, 5 figure

Unsteady Crack Motion and Branching in a Phase-Field Model of Brittle Fracture

Crack propagation is studied numerically using a continuum phase-field approach to mode III brittle fracture. The results shed light on the physics that controls the speed of accelerating cracks and the characteristic branching instability at a fraction of the wave speed.Comment: 11 pages, 4 figure

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

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

Geometry of Valley Growth

Although amphitheater-shaped valley heads can be cut by groundwater flows emerging from springs, recent geological evidence suggests that other processes may also produce similar features, thus confounding the interpretations of such valley heads on Earth and Mars. To better understand the origin of this topographic form we combine field observations, laboratory experiments, analysis of a high-resolution topographic map, and mathematical theory to quantitatively characterize a class of physical phenomena that produce amphitheater-shaped heads. The resulting geometric growth equation accurately predicts the shape of decimeter-wide channels in laboratory experiments, 100-meter wide valleys in Florida and Idaho, and kilometer wide valleys on Mars. We find that whenever the processes shaping a landscape favor the growth of sharply protruding features, channels develop amphitheater-shaped heads with an aspect ratio of pi

Estimation of prokaryotic supergenome size and composition from gene frequency distributions

BACKGROUND: Because prokaryotic genomes experience a rapid flux of genes, selection may act at a higher level than an individual genome. We explore a quantitative model of the distributed genome whereby groups of genomes evolve by acquiring genes from a fixed reservoir which we denote as supergenome. Previous attempts to understand the nature of the supergenome treated genomes as random, independent collections of genes and assumed that the supergenome consists of a small number of homogeneous sub-reservoirs. Here we explore the consequences of relaxing both assumptions. RESULTS: We surveyed several methods for estimating the size and composition of the supergenome. The methods assumed that genomes were either random, independent samples of the supergenome or that they evolved from a common ancestor along a known tree via stochastic sampling from the reservoir. The reservoir was assumed to be either a collection of homogeneous sub-reservoirs or alternatively composed of genes with Gamma distributed gain probabilities. Empirical gene frequencies were used to either compute the likelihood of the data directly or first to reconstruct the history of gene gains and then compute the likelihood of the reconstructed numbers of gains. CONCLUSIONS: Supergenome size estimates using the empirical gene frequencies directly are not robust with respect to the choice of the model. By contrast, using the gene frequencies and the phylogenetic tree to reconstruct multiple gene gains produces reliable estimates of the supergenome size and indicates that a homogeneous supergenome is more consistent with the data than a supergenome with Gamma distributed gain probabilities

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

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