On a discrete-to-continuum convergence result for a two dimensional   brittle material in the small displacement regime
  We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Gamma-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets. 
Manuel Friedrich, Bernd Schmidt
math.AP cond-mat.mtrl-sci math-ph math.MP