  Disorder and long-range interactions are two of the key components that make
material failure an interesting playfield for the application of statistical
mechanics. The cornerstone in this respect has been lattice models of the
fracture in which a network of elastic beams, bonds or electrical fuses with
random failure thresholds are subject to an increasing external load. These
models describe on a qualitative level the failure processes of real, brittle
or quasi-brittle materials. This has been particularly important in solving the
classical engineering problems of material strength: the size dependence of
maximum stress and its sample to sample statistical fluctuations. At the same
time, lattice models pose many new fundamental questions in statistical
physics, such as the relation between fracture and phase transitions.
Experimental results point out to the existence of an intriguing crackling
noise in the acoustic emission and of self-affine fractals in the crack surface
morphology. Recent advances in computer power have enabled considerable
progress in the understanding of such models. Among these partly still
controversial issues, are the scaling and size effects in material strength and
accumulated damage, the statistics of avalanches or bursts of microfailures,
and the morphology of the crack surface. Here we present an overview of the
results obtained with lattice models for fracture, highlighting the relations
with statistical physics theories and more conventional fracture mechanics
approaches.
