Theory of dynamic crack branching in brittle materials

The problem of dynamic symmetric branching of an initial single brittle crack propagating at a given speed under plane loading conditions is studied within a continuum mechanics approach. Griffith's energy criterion and the principle of local symmetry are used to determine the cracks paths. The bifurcation is predicted at a given critical speed and at a specific branching angle: both correlated very well with experiments. The curvature of the subsequent branches is also studied: the sign of $T$, with $T$ being the non singular stress at the initial crack tip, separates branches paths that diverge from or converge to the initial path, a feature that may be tested in future experiments. The model rests on a scenario of crack branching with some reasonable assumptions based on general considerations and in exact dynamic results for anti-plane branching. It is argued that it is possible to use a static analysis of the crack bifurcation for plane loading as a good approximation to the dynamical case. The results are interesting since they explain within a continuum mechanics approach the main features of the branching instabilities of fast cracks in brittle materials, i.e. critical speeds, branching angle and the geometry of subsequent branches paths.Comment: 41 pages, 15 figures. Accepted to International Journal of Fractur

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

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

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

A statistical approach to close packing of elastic rods and to dna packaging in viral capsids

We propose a statistical approach for studying the close packing of elastic rods. This phenomenon belongs to the class of problems of confinement of low dimensional objects, such as DNA packaging in viral capsids. The method developed is based on Edwards' approach, which was successfully applied to polymer physics and to granular matter. We show that the confinement induces a configurational phase transition from a disordered (isotropic) phase to an ordered (nematic) phase. In each phase, we derive the pressure exerted by the rod (DNA) on the container (capsid) and the force necessary to inject (eject) the rod into (out of) the container. Finally, we discuss the relevance of the present results with respect to physical and biological problems. Regarding DNA packaging in viral capsids, these results establish the existence of ordered configurations, a hypothesis upon which previous calculations were built. They also show that such ordering can result from simple mechanical constraints. C losely packed objects are ubiquitous in nature. Actual examples of such systems are the folding of leaves in buds (1), wing folding of insects in cocoons (2), crumpled paper (3-6), DNA packaging in capsids (7-11), or the confinement of chromatin in the nucleus of a cell (12). In all of these phenomena, the way the object is folded has a role in determining its function or in insuring its integrity during the unfolding process. Although these systems exist at different length scales, they share some common fundamental physical features, such as the symmetries of the folded structure and the dimensionality of the packed objects as well as those of the confining container. Here, we study the packing of a one-dimensional object into a three-dimensional container, the size of which is very small compared to the length of the folded structure. This phenomenon arises in a number of scientific fields such as mechanics We introduce a statistical minimal model based on Edwards' theory to study the conformations of the rod. This approach proved its applicability in other fields such as granular materials (20) and polymer physics This paper is organized as follows: we start by formulating the packing problem by considering an inextensible rod put inside a sphere of a given radius, and we write the reduced free energy of the system as a path integral over all possible configurations of the rod. The only interactions that are allowed are of elastic and self-avoiding nature. The addition of other interactions will be left for future investigations. Then, we proceed to the calculation of the free energy, by evaluating the path integral in a mean field approximation, and we show that a rod packed in a sphere undergoes a continuous transition from an isotropic phase to a nematic phase as the radius of the sphere is decreased below a critical length (put in other words, when the density of the rod on the sphere exceeds a critical value). This transition allows for a reduction in the pressure applied on the sphere as well as in the force needed to inject the rod into it. Let us emphasize here that the present system is different from those studied previously (21-23) because the transition is driven by the macroscopic length scale induced by the container. Finally, we discuss the relevance of our results with respect to physical and biological problems. Especially, the results show that no special intelligence is required on the side of the virus other than being dense enough in the capsid. Mechanical constraints take care of the rest. Formulation of the Packing Problem We consider an inextensible rod of length L put inside a sphere of radius ᐉ and aim at a statistical study of its configurations. A configuration is parameterized by the position vector R(s) as a function of the curvilinear coordinate s along the rod. We assume that ''cut-off'' lengths such as the rod's typical thickness or the ''monomer size'' are very small compared to the geometrical lengths (L and ᐉ), so that the rod can be treated as a material line. We also assume that the forces acting on the rod are only due to elastic stresses and self-repulsion. The energy has two contributions, one that penalizes bending of the rod and the other that penalizes self-intersections. To minimize bending and Author contributions: E.K., M.A.-B., and A.B. designed and performed research

Evolution du Facteur de résistance hygrothermique transitoire en fonction des séquences d'empilement pour des composites stratifiés sous des conditions d'environnement cycliques

Exposé à des conditions d'environnement cycliques, les composites à matrices polymères sont capables d'absorber une quantité d'humidité durant de longues années de service. Suite à ce type d'environnement, il se développe des contraintes hygrothermiques extrêmement critiques aux bords et aux interfaces fibres/résines. Au cours des premiers temps de la diffusion d'humidité, les contraintes hygrothermiques sont assez importantes dans la direction transversale, où elles peuvent engendrées un délaminage par gonflement entre les plis formants le stratifié. Pour avoir une idée sur la probabilité d'endommagement des plaques stratifiées sous des contraintes hygrothermiques transitoires, on a adopté le critère de rupture de Tsai-Wu. Le critère de sécurité sera vérifier en évaluant le facteur de résistance (R) pour chaque séquence d'empilement aux bords de la plaque stratifiée

The spectrum of large powers of the Laplacian in bounded domains

We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. We begin with the one dimensional case and show that the whole spectrum can be obtained in the limit of large N. We also show that it is a useful numerical approach valid for any N. Finally, we discuss implications of this work and present its possible extensions for non integer N and for 3D Laplacian problems.Comment: 13 pages, 2 figure

A prototypical model for tensional wrinkling in thin sheets

The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians and engineers. This has been triggered by the growing interest in developing technologies at ever decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. While the most basic buckling instability of uniaxially compressed plates was understood by Euler more than 200 years ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length - a sheet under axisymmetric tensile loads. This geometry, whose first study is attributed to Lam´e, allows us to construct\ud a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that for thin sheets the far-from-threshold regime is expected to emerge under extremely small compressive loads, emphasizing the relevance of our analysis for nanomechanics applications

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