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 0.48cs. 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. Introduction. – The study of crack propagation has gained a lot of interest in the physical community during the recent years [1]. Among the many reasons that have driven this interest is the difficulty to understand the mechanism leading to the branching instability [2, 3] that prevents a crack from reaching its theoretical limiting speed: the Rayleigh wavespeed. During the branching instability, a single straight propagating crack separates into two sub cracks. While, the instability takes place in the process zone (the microscopic region ahead of the crack tip where the interatomic bond breaking leading to the crack propagation occurs), its effects (namely the propagation of two sub cracks) are macroscopic. Hence a full description of the branching instability needs to describe both the macroscopic and the microscopic scales. The classical theory of crack propagation, describes well the macroscopic scale where the linear elasticity theory is valid but describes the crack propagation with laws that postulate the evolution of the crack path is a function of the stress intensity factors (number describing the singularity at the crack tip, that correspond to the three modes of crack propagation (see fig. 1). Such laws that do not describe in any way the small scale of the process zone can not account for the onset of the branching instability unless one explicitly provides a criteria that determines when a crack divides into two sub-cracks. Yet, they provide useful results such as necessary conditions for branching [4, 5]. On the opposite, the use of molecular dynamics simulations can help understanding the way a crack propagates and eventually divides into branches,since it allows to compute what happens at the microscopic scale. [6] Nonetheless, it does not allow to reach neither long enough time scales nor large enough space scales to fully simulate multiscale problems. Therefore, in order to describe crack propagation an alternative approach may be to use phenomenological models of crack propagation. Such models aim at describing the behavior of the elastic material in the process zone with the use of non-linear elasticity. Typically, these models describe the process zone as a region where the material softens gradually from a fully intact state where the law of elasticity are verified to a fully broken state where the crack cannot transmit any stress. A pioneering work was the introduction of the cohesive zone model where the crack line is prolongated by a softening segment [7, 8]. More recently using idea from the phase transition theory, phase-field models of crack propagation were introduced [9–16] to describe crack propagation. They have already proved they can well describe the complex crack path in many cases [10,17]. In addition such models have been able to reproduce the feature of the crack branching in the case of mode III cracks [12,18]. Here, a study of the branching instability using the phasefield model is presented in the case of the inplane crack propagation (mode I and II). It must be emphasized here that while previous works have been devoted to the case of mode III crack propagation (paper tearing mode), this