  A heterogeneous brittle material characterized by a random field of local
toughness Kc(x) can be represented by an equivalent homogeneous medium of
toughness, Keff. Homogenization refers to a process of estimating Keff from the
local field Kc(x). An approach based on a perturbative expansion of the stress
intensity factor along a rough crack front shows the occurrence of different
regimes depending on the correlation length of the local toughness field in the
direction of crack propagation. A `"weak pinning" regime takes place for long
correlation lengths, where the effective toughness is the average of the local
toughness. For shorter correlation lengths, a transition to "strong pinning"
occurs leading to a much higher effective toughness, and characterized by a
propagation regime consisting in jumps between pinning configurations.
