  When a crack interacts with material heterogeneities, its front distorts and
adopts complex tortuous configurations that are reminiscent of the energy
barriers encountered during crack propagation. As such, the study of crack
front deformations is key to rationalize the effective failure properties of
micro-structured solids and interfaces. Yet, the impact of a localized
dissipation in a finite region behind the crack front, called the process zone,
has often been overlooked. In this work, we derive the equation ruling 3D
coplanar crack propagation in heterogeneous cohesive materials where the
opening of the crack is resisted by some traction in its wake. We show that the
presence of a process zone results in two competing effects on the deformation
of crack fronts: (i) it makes the front more compliant to small-wavelength
perturbations, and (ii) it smooths out local fluctuations of strength and
process zone size, from which emerge heterogeneities of fracture energy. Their
respective influence on front deformations is shown to strongly impact the
stability of perturbed crack fronts, as well as their stationary shapes when
interacting with arrays of tough obstacles. Overall, our theory provides a
unified framework to predict the variety of front profiles observed in
experiments, even when the small-scale yielding hypothesis of linear elastic
fracture mechanics breaks down.
