Indentation of ellipsoidal and cylindrical elastic shells

Thin shells are found in nature at scales ranging from viruses to hens’ eggs; the stiffness of such shells is essential for their function. We present the results of numerical simulations and theoretical analyses for the indentation of ellipsoidal and cylindrical elastic shells, considering both pressurized and unpressurized shells. We provide a theoretical foundation for the experimental findings of Lazarus et al. [Phys. Rev. Lett. (submitted)] and for previous work inferring the turgor pressure of bacteria from measurements of their indentation stiffness; we also identify a new regime at large indentation. We show that the indentation stiffness of convex shells is dominated by either the mean or Gaussian curvature of the shell depending on the pressurization and indentation depth. Our results reveal how geometry rules the rigidity of shells

Measuring order in the isotropic packing of elastic rods

The packing of elastic bodies has emerged as a paradigm for the study of macroscopic disordered systems. However, progress is hampered by the lack of controlled experiments. Here we consider a model experiment for the isotropic two-dimensional confinement of a rod by a central force. We seek to measure how ordered is a folded configuration and we identify two key quantities. A geometrical characterization is given by the number of superposed layers in the configuration. Using temporal modulations of the confining force, we probe the mechanical properties of the configuration and we define and measure its effective compressibility. These two quantities may be used to build a statistical framework for packed elastic systems.Comment: 4 pages, 5 figure

Rescaling the dynamics of evaporating drops

The dynamics of evaporation of wetting droplets has been investigated experimentally in an extended range of drop sizes, in order to provide trends relevant for a theoretical analysis. A model is proposed, which generalises Tanner's law, allowing us to smooth out the singularities both in dissipation and in evaporative flux at the moving contact line. A qualitative agreement is obtained, which represents a first step towards the solution of a very old, complex problem

Buckling of swelling gels

The patterns arising from the differential swelling of gels are investigated experimentally and theoretically as a model for the differential growth of living tissues. Two geometries are considered: a thin strip of soft gel clamped to a stiff gel, and a thin corona of soft gel clamped to a disk of stiff gel. When the structure is immersed in water, the soft gel swells and bends out of plane leading to a wavy periodic pattern which wavelength is measured. The linear stability of the flat state is studied in the framework of linear elasticity using the equations for thin plates. The flat state is shown to become unstable to oscillations above a critical swelling rate and the computed wavelengths are in quantitative agreement with the experiment

Finite-distance singularities in the tearing of thin sheets

We investigate the interaction between two cracks propagating in a thin sheet. Two different experimental geometries allow us to tear sheets by imposing an out-of-plane shear loading. We find that two tears converge along self-similar paths and annihilate each other. These finite-distance singularities display geometry-dependent similarity exponents, which we retrieve using scaling arguments based on a balance between the stretching and the bending of the sheet close to the tips of the cracks.Comment: 4 pages, 4 figure

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

A comparative study of crumpling and folding of thin sheets

Crumpling and folding of paper are at rst sight very di erent ways of con ning thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly ine cient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than 6 or 7 times. Here we show that the sti ness that builds up in the two processes is of the same nature, and therefore simple folding models allow to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We nd that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.Comment: 5 page

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

Fourier analysis of wave turbulence in a thin elastic plate

The spatio-temporal dynamics of the deformation of a vibrated plate is measured by a high speed Fourier transform profilometry technique. The space-time Fourier spectrum is analyzed. It displays a behavior consistent with the premises of the Weak Turbulence theory. A isotropic continuous spectrum of waves is excited with a non linear dispersion relation slightly shifted from the linear dispersion relation. The spectral width of the dispersion relation is also measured. The non linearity of this system is weak as expected from the theory. Finite size effects are discussed. Despite a qualitative agreement with the theory, a quantitative mismatch is observed which origin may be due to the dissipation that ultimately absorbs the energy flux of the Kolmogorov-Zakharov casade.Comment: accepted for publication in European Physical Journal B see http://www.epj.or

Statistical distributions in the folding of elastic structures

The behaviour of elastic structures undergoing large deformations is the result of the competition between confining conditions, self-avoidance and elasticity. This combination of multiple phenomena creates a geometrical frustration that leads to complex fold patterns. By studying the case of a rod confined isotropically into a disk, we show that the emergence of the complexity is associated with a well defined underlying statistical measure that determines the energy distribution of sub-elements,``branches'', of the rod. This result suggests that branches act as the ``microscopic'' degrees of freedom laying the foundations for a statistical mechanical theory of this athermal and amorphous system