Figure 2. Snapshots of the atoms in the simulation system with a fracture (white) and damage (dark red) after the crack has propagated through α-quartz. The darker red colored areas and areas with microbranching correspond to damage in the quartz structure by breakage of the atomic bonds. a) With an initial tensile stress of 1060 MPa, the crack propagates along an atomic plane. Area 1 shows a mirror surface, while area 2 shows mist. The yellow lines in the inset figures indicate the atomic layer along which the fracture propagates. b) With an initial tensile stress of 1209 MPa, the crack propagates mostly straight. c) With an initial tensile stress of 1386 MPa, the crack oscillates out-of-plane (hackle pattern). d) With an initial tensile stress of 1604 MPa, crack oscillations induce a more pronounced surface roughening. e) With an initial tensile stress of 1729 MPa, crack branching occurs. f) Amplitude of the oscillations as a function of rupture speed.

We measure the rupture speed in every simulation by sampling the position of the crack tip in the x -direction through time and identify the critical speed for crack oscillations and branching. Figure 3a shows how rupture speed and crack path vary with the initial tensile stress imposed on the system. Based on observations of features (Figure 2a-e), a qualitative difference is observed between Figure 2b and c. In Figure 2b, the crack propagates mostly straight but jump between layers, while in Figure 2c, it never propagates straight. This change in propagation behavior is observed when the initial tensile stress is around 1210 MPa and the rupture speed remains below 460 m/s, equivalent to 0.15C _R (Figure 3a). When the initial tensile stress increases, the rupture speed increases as well, and the crack oscillates (Figure 2c-d). For initial tensile stresses larger than 1600 MPa, crack microbranching occurs, and the rupture speed exceeds 800 m/s, corresponding to 0.26C _R (Figure 3a). During a branching event, two cracks propagate simultaneously until one of the cracks dominates, from which the dynamic rupture continues propagating (Figure 2e). In Figure 3a, we observe that slope of the rupture speed versus initial tensile stress is steepest before oscillations occur. However, when approaching an initial tensile stress where oscillations occur, the rupture speed stagnates before it continues to increase, indicating that there is an energy barrier to overcome before the crack starts to oscillate. Similarly, a stagnation of the cuve is observed around the threshold speed where microbranches start developing.