Quenched disorder and instability control dynamic fracture in three dimensions

Abstract: Materials failure in 3D still poses basic challenges. We study 3D brittle crack dynamics using a phase-field approach, where Gaussian quenched disorder in the fracture energy is incorporated. Disorder is characterized by a correlation length $R$ and strength $\sigma$. We find that the mean crack velocity $v$ is bounded by a limiting velocity, which is smaller than the homogeneous material's prediction and decreases with $\sigma$. It emerges from a dynamic renormalization of the fracture energy with increasing crack driving force $G$, resembling a critical point, due to an interplay between a 2D branching instability and disorder. At small $G$, the probability of localized branching on a scale $R$ is super-exponentially small. With increasing $G$ this probability quickly increases, leading to misty fracture surfaces, yet the associated extra dissipation remains small. As $G$ is further increased, branching-related lengthscales become dynamic and persistently increase, leading to hackle-like structures and to a macroscopic contribution to the fracture surface. The latter dynamically renormalizes the actual fracture energy until eventually any increase in $G$ is balanced by extra fracture surface, with no accompanying increase in $v$. Finally, branching width reaches the system's thickness such that 2D symmetry is statistically restored. Our findings are consistent with a broad range of experimental observations.