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

A model for hierarchical patterns under mechanical stresses

We present a model for mechanically-induced pattern formation in growing biological tissues and discuss its application to the development of leaf venation networks. Drawing an analogy with phase transitions in solids, we use a phase field method to describe the transition between two states of the tissue, e.g. the differentiation of leaf veins, and consider a layered system where mechanical stresses are generated by differential growth. We present analytical and numerical results for one-dimensional systems, showing that a combination of growth and irreversibility gives rise to hierarchical patterns. Two-dimensional simulations suggest that such a mechanism could account for the hierarchical, reticulate structure of leaf venation networks, yet point to the need for a more detailed treatment of the coupling between growth and mechanical stresses.Comment: To appear in Philosophical Magazine. 18 pages, 8 figure

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

Seismic Radiation From Simple Models of Earthquakes

We review some basic features of shear wave generation and energy balance for a 2D anti plane rupture. We first study the energy balance for a flat fault, and for a fault that contains a single localized kink. We determine an exact expression for the partition between strain energy flow released from the elastic medium surrounding the fault, radiated energy flow and energy release rate. This balance depends only on the rupture speed and the residual stress intensity factor. When the fault contains a kink, the energy available for fracture is reduced so that the rupture speed is reduced. When rupture speed changes abruptly, the radiated energy flow also changes abruptly. As rupture propagates across the kink, a shear wave is emitted that has a displacement spectral content that decreases like ω^(-2) at high frequencies. We then use spectral elements to model the propagation of an antiplane crack with a slip-weakening friction law. Since the rupture front in this case has a finite length scale, the wave emitted by the kink is smoothed at very high frequencies but its general behavior is similar to that predicted by the simple sharp crack model. A model of a crack that has several kinks and wanders around a mean rupture directions, shows that kinks reduce the rupture speed along the average rupture direction of the fault. Contrary to flat fault models, a fault with kinks produces high frequency waves that are emitted every time the rupture front turns at a kink. Finally, we discuss the applicability of the present results to a 3D rupture model

Cloaking by coating: How effectively does a thin, stiff coating hide a soft substrate?

From human tissue to fruits, many soft materials are coated by a thin layer of a stiffer material. While the primary role of such a coating is often to protect the softer material, the thin, stiff coating also has an important effect on the mechanical behaviour of the composite material, making it appear significantly stiffer than the underlying material. We study this cloaking effect of a coating for the particular case of indentation tests, which measure the `firmness' of the composite solid: we use a combination of theory and experiment to characterize the firmness quantitatively. We find that the indenter size plays a key role in determining the effectiveness of cloaking: small indenters feel a mixture of the material properties of the coating and of the substrate, while large indenters sense largely the unadulterated substrate

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