  The interaction of crack fronts with asperities is central to the criteria of
fracture in heterogeneous materials and for the prediction of fracture surface
formation. It is known how dynamic crack fronts respond to small, 1st-order,
perturbations. Large and localized disturbances to crack motion, however,
induce dynamic and geometric nonlinear effects that are beyond the existing
linear theories. Because the determination of the 3D elastic fields surrounding
perturbed crack fronts is a necessary step towards any theoretical study of
crack front dynamics, we develop a 2nd-order perturbation theory for the
asymptotic fields of planar crack fronts. Based on previous work, we consider
two models of fracture. In the so-called scalar elastic model, which is
analogous to anti-plane (Mode III) fracture, the stress and displacement fields
are obtained through matched asymptotic expansions. A self-consistent expansion
is used to resolve the fields near tensile (Mode I) crack fronts. Both methods
can be extended to higher perturbation orders. The main results of this work
are the explicit 2nd-order expressions of the local dynamic
energy-release-rates for perturbations of straight fronts. These general
formulae recover the known energy-release-rates of curved quasi-static fronts
and of straight dynamic fronts.
