Due to its catastrophic character, brittle fracture has been an important object of investigation in fracture mechanics. The idea of a fracture energy proportional to the measure of the fracture surface is due to Griffith. The revisitation of the subject by Francfort and Marigo offered a genuinely variational formulation of the problem.
For the mathematical community, the interest in fracture problems originated from its reformulation as a free discontinuity problem (FDP), following the approach by De Giorgi and Ambrosio to analyze the Mumford and Shah image segmentation functional.
After introducing the mathematical setting for FDPs, I will survey on several basic problems to allow the use of variational methods to establish existence of weak/strong minimizers and their phase-field approximation, both for Griffith type energies and for cohesive fracture models. The latter have been the subject of much attention recently and are based on FDPs with surface densities that are concave and bounded functions of the jump amplitude.