A comprehensive study of nonlinear perturbations in the dynamics of planar crack fronts

The interaction of crack fronts with asperities is central to fracture criteria in heterogeneous materials and for predicting fracture surface formation. It is known how dynamic crack fronts respond to small, 1st-order, perturbations. However, large and localized disturbances to crack motion induce dynamic and geometric nonlinear effects beyond the existing linear theories. Because the determination of the 3D elastic fields surrounding perturbed crack fronts is a necessary step toward 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: (1) Fracture in a scalar elastic solid which is an analog of antiplane shear fracture (Mode III). In this model, the near-crack fields are obtained via matched asymptotic expansions. (2) Tensile Mode I fracture, in which a self-consistent expansion is used to resolve the fields near the crack front. These methods can be readily 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 arbitrary perturbations of straight fronts. The formulae recover the known energy-release-rates of curved quasi-static fronts and of simple 2D cracks. We show that the expressions are separable as a product of a dynamical prefactor that only depends on the instantaneous local normal front velocity, and a history functional that integrates past front configurations. To gain insight, the energy-release-rates in the two models are computed for a traveling wave perturbation. While similar at low wave velocities, the two theories behave differently for fast waves. In scalar elasticity, the 2nd-order contributions are always sub-dominant. However, in the Mode I theory, the 2nd-order correction becomes the dominant term at the crack front wave velocity, where the 1st-order term is zero. We discuss employing the energy-release-rate expressions to predict crack front dynamics via energy balance with dissipation.