Fracture is a fundamental mechanism of materials failure. Propagating cracks can exhibit a rich dynamical behavior controlled by a subtle interplay between microscopic failure processes in the crack tip region and macroscopic elasticity. We review recent approaches to understand crack dynamics using the phase field method. This method, developed originally for phase transformations, has the well-known advantage of avoiding explicit front tracking by making material interfaces spatially diffuse. In a fracture context, this method is able to capture both the short-scale physics of failure and macroscopic linear elasticity within a self-consistent set of equations that can be simulated on experimentally relevant length and time scales. We discuss the relevance of different models, which stem from continuum field descriptions of brittle materials and crystals, to address questions concerning crack path selection and branching instabilities, as well as models that are based on mesoscale concepts for crack tip scale selection. Open questions which may be addressed using phase field models of fracture are summarized.

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase-field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

The strength and stability of frictional interfaces, ranging from tribological systems to earthquake faults, are intimately related to the underlying spatially extended dynamics. Here we provide a comprehensive theoretical account, both analytic and numeric, of spatiotemporal interfacial dynamics in a realistic rate-and-state friction model, featuring both velocity-weakening and velocity-strengthening behaviors. Slowly extending, loading-rate-dependent creep patches undergo a linear instability at a critical nucleation size, which is nearly independent of interfacial history, initial stress conditions, and velocity-strengthening friction. Nonlinear propagating rupture fronts\char22{}the outcome of instability\char22{}depend sensitively on the stress state and velocity-strengthening friction. Rupture fronts span a wide range of propagation velocities and are related to steady-state-front solutions.

The prediction of the degradation of lithium-ion batteries is essential for various applications and optimized recycling schemes. In order to address this issue, this study aims to predict the cycle lives of lithium-ion batteries using only data from early cycles. To reach such an objective, experimental raw data for 121 commercial lithium iron phosphate/graphite cells are gathered from the literature. The data are analyzed, and suitable input features are generated for the use of different machine learning algorithms. A final accuracy of 99.81% for the cycle life is obtained with an extremely randomized trees model. This work shows that data-driven models are able to successfully predict the lifetimes of batteries using only early-cycle data. That aside, a considerable reduction in errors is seen by incorporating data management and physical and chemical understanding into the analysis.