  We investigate experimentally and theoretically the dynamics of a crack front
during the micro-instabilities taking place in heterogeneous materials between
two successive equilibrium positions. We focus specifically on the
spatio-temporal evolution of the front, as it relaxes to a straight
configuration, after depinning from a single obstacle of controlled strength
and size. We show that this depinning dynamics is not controlled by inertia,
but instead, by the rate dependency of the dissipative mechanisms taking place
within the fracture process zone. This implies that the crack speed
fluctuations around its average value $v_m$ can be predicted from an overdamped
equation of motion $(v-v_m)/v_0 = (G-G_c(v_m))/G_c(v_m)$ involving the
characteristic material speed $v_0 = G_c(v_m)/G_c'(v_m)$ that emerges from the
variation of fracture energy with crack speed. Our findings pave the way to a
quantitative description of the critical depinning dynamics of cracks in
disordered solids and open up new perspectives for the prediction of the
effective failure properties of heterogeneous materials.
