Onset of propagation of planar cracks in heterogeneous media

The dynamics of planar crack fronts in hetergeneous media near the critical load for onset of crack motion are investigated both analytically and by numerical simulations. Elasticity of the solid leads to long-range stress transfer along the crack front which is nonmonotonic in time due to the elastic waves in the medium. In the quasistatic limit with instantaneous stress transfer, the crack front exhibits dynamic critical phenomenon, with a second-order-like transition from a pinned to a moving phase as the applied load is increased through a critical value. At criticality, the crack front is self-affine, with a roughness exponent zeta=0.34+/-0.02. The dynamic exponent z is found to be equal to 0.74+/-0.03 and the correlation length exponent nu=1.52+/-0.02. These results are in good agreement with those obtained from an epsilon expansion. Sound-travel time delays in the stress transfer do not change the static exponents but the dynamic exponent z becomes exactly one. Real elastic waves, however, lead to overshoots in the stresses above their eventual static value when one part of the crack front moves forward. Simplified models of these stress overshoots are used to show that overshoots are relevant at the depinning transition leading to a decrease in the critical load and an apparent jump in the velocity of the crack front directly to a nonzero value. In finite systems, the velocity also shows hysteretic behavior as a function of the loading. These results suggest a first-order-like transition. Possible implications for real tensile cracks are discussed.