Dynamic Crack Tip Equation of Motion: High-speed Oscillatory Instability

A dynamic crack tip equation of motion is proposed based on the autonomy of the near-tip nonlinear zone of scale $\ell_{nl}$, symmetry principles, causality and scaling arguments. Causality implies that the asymptotic linear-elastic fields at time $t$ are determined by the crack path at a {\bf retarded time} $t-\tau_d$, where the delay time $\tau_d$ scales with the ratio of $\ell_{nl}$ and the typical wave speed $c_{nl}$ within the nonlinear zone. The resulting equation is shown to agree with known results in the quasi-static regime. As a first application in the fully dynamic regime, an approximate analysis predicts a high-speed oscillatory instability whose characteristic scale is determined by $\ell_{nl}$. This prediction is corroborated by experimental results, demonstrating the emergence of crack tip inertia-like effects.