  Traditional computational approaches in simulating crack propagation in
perfectly brittle materials rely on the estimate of stress intensity factors
along the rupture front. This proves highly challenging in 3D when the crack
geometry departs from very specific cases for which analytical solutions are
available, like e.g. the penny-shaped crack geometry. Here, we extend the
first-order theory of Gao and Rice (1987), and predict the distribution of the
mode I stress intensity factor $K_\mathrm{I}$ along the front of a tensile
coplanar crack that is slightly perturbed from a reference penny-shaped
configuration, up to second order in the perturbation amplitude. Our theory is
validated against analytical solutions available for embedded elliptical
cracks, and its range of validity is further assessed using numerical
simulations performed on cosine front perturbations of varying mode and
amplitude. It is then used to develop a homogenization framework for the
toughness of weakly disordered media. The effective toughness and its
fluctuations are bridged quantitatively to the intensity of the toughness
fluctuations and their spatial structure. Our theoretical predictions are
compared to the results of ~1 million simulations of crack propagation building
on our second-order theory and Fast Fourier Transforms. We show that the impact
of toughness heterogeneities is size-dependent, as they generally weaken the
material when the crack size is lower or comparable to the typical
heterogeneity size, but reinforces it otherwise. It results in an apparent
R-curve behavior of the brittle composite at the macroscale.
