Planar crack propagation under pure tension loading (mode I) is generally stable. However, it becomes universally unstable with the superposition of a shear stress parallel to the crack front (mode III). Under this mixed-mode (I + III) loading configuration, an initially flat parent crack segments into an array of daughter cracks that rotate towards a direction of maximum tensile stress. This segmentation produces stepped fracture surfaces with characteristic 'lance-shaped' markings observed in a wide range of engineering and geological materials. The origin of this instability remains poorly understood and a theory with which to predict the surface roughness scale is lacking. Here we perform large-scale simulations of mixed-mode I + III brittle fracture using a continuum phase-field method that describes the complete three-dimensional crack-front evolution. The simulations reveal that planar crack propagation is linearly unstable against helical deformations of the crack front, which evolve nonlinearly into a segmented array of finger-shaped daughter cracks. Furthermore, during their evolution, facets gradually coarsen owing to the growth competition of daughter cracks in striking analogy with the coarsening of finger patterns observed in nonequilibrium growth phenomena. We show that the dynamically preferred unstable wavelength is governed by the balance of the destabilizing effect of far-field stresses and the stabilizing effect of cohesive forces on the process zone scale, and we derive a theoretical estimate for this scale using a new propagation law for curved cracks in three dimensions. The rotation angles of coarsened facets are also compared to theoretical predictions and available experimental data.

A planar crack generically segments into an array of "daughter cracks" shaped as tilted facets when loaded with both a tensile stress normal to the crack plane (mode I) and a shear stress parallel to the crack front (mode III). We investigate facet propagation and coarsening using in-situ microscopy observations of fracture surfaces at different stages of quasi-static mixed-mode crack propagation and phase-field simulations. The results demonstrate that the bifurcation from propagating planar to segmented crack front is strongly subcritical, reconciling previous theoretical predictions of linear stability analysis with experimental observations. They further show that facet coarsening is a selfsimilar process driven by a spatial period-doubling instability of facet arrays with a growth rate dependent on mode mixity. Those results have important implications for understanding the failure of a wide range of materials.

We describe theory and experiments concerning a chemical reaction, the alkaline oxidation of glucose with methylene blue as a catalyst, that is hypothesized to drive fluid motion via an overturning instability, as an example of a ''chemoconvective'' process. A theoretical model is developed to explain this phenomenon and linear analyses from steady and pseudosteady states are used to predict the basic length and time scales of the patterns which initially appear. These theoretical predictions, using kinetic parameters from recent independent experiments, are contrasted with results from pattern initiation experiments. Preliminary comparisons indicate good qualitative and quantitative agreement.