Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales

Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart this resides identification two intrinsic, near crack tip length scales -- a nonlinear elastic scale $\ell$ dissipation $\xi$ that do not exist classical theory cracks. In particular, it been shown at high propagation velocity $v$, cracks brittle materials undergo an oscillatory instability whose wavelength varies linearly with $\ell$, yet higher velocities larger loading levels, tip-splitting emerges, both agreements experiments. paper, using phase-field models fracture, we demonstrate following properties instability: (i) It exists also absence near-tip nonlinearity, i.e. limit $\ell\!\to\!0$, determined by $\xi$. This result shows crucially depends on existence intrinsic associated breakdown linear elasticity tips, independently whether latter is related or dissipation. (ii) supercritical Hopf bifurcation, featuring vanishing oscillations amplitude onset. (iii) largely independent fracture energy $\Gamma(v)$ controlled time scale. These results substantiate universal nature ultra-high speed 2D. addition, provide evidence indicating limiting rate transport inside region. Finally, describe detail numerical implementation scheme employed approach.