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

Fracture surfaces of heterogeneous materials: a 2D solvable model

Using an elastostatic description of crack growth based on the Griffith criterion and the principle of local symmetry, we present a stochastic model describing the propagation of a crack tip in a 2D heterogeneous brittle material. The model ensures the stability of straight cracks and allows for the study of the roughening of fracture surfaces. When neglecting the effect of the non singular stress, the problem becomes exactly solvable and yields analytic predictions for the power spectrum of the paths. This result suggests an alternative to the conventional power law analysis often used in the analysis of experimental data.Comment: 4 pages, 4 figure

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

Roughness of tensile crack fronts in heterogenous materials

The dynamics of planar crack fronts in heterogeneous media is studied using a recently proposed stochastic equation of motion that takes into account nonlinear effects. The analysis is carried for a moving front in the quasi-static regime using the Self Consistent Expansion. A continuous dynamical phase transition between a flat phase and a dynamically rough phase, with a roughness exponent $\zeta=1/2$, is found. The rough phase becomes possible due to the destabilization of the linear modes by the nonlinear terms. Taking into account the irreversibility of the crack propagation, we infer that the roughness exponent found in experiments might become history-dependent, and so our result gives a lower bound for $\zeta$.Comment: 7 page

Study of the branching instability using a phase field model of inplane crack propagation

In this study, the phase field model of crack propagation is used to study the dynamic branching instability in the case of inplane loading in two dimensions. Simulation results are in good agreement with theoretical predictions and experimental findings. Namely, the critical speed at which the instability starts is about $0.48 c_s$. They also show that a full 3D approach is needed to fully understand the branching instability. The finite interface effects are found to be neglectable in the large system size limit even though they are stronger than the one expected from a simple one dimensional calculation

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

Dynamic stability of crack fronts: Out-of-plane corrugations

The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis-Movchan 3D linear perturbation formalism [J. Mech. Phys. Solids {\bf 45}, 591 (1997)] to study the out-of-plane stability of planar crack fronts in the framework of linear elastic fracture mechanics. We discuss a minimal scenario in which linearly unstable crack front corrugations might emerge above a critical front propagation speed. We calculate this speed as a function of Poisson's ratio and show that corrugations propagate along the crack front at nearly the Rayleigh wave-speed. Finally, we hypothesize about a possible relation between such corrugations and the long-standing problem of crack branching.Comment: 5 pages, 2 figures + supplementary informatio