Effects of Buckling on Stress and Strain in Thin Randomly Disordered   Tension-Loaded Sheets
  We study how crack buckling affects stress and strain in a thin sheet with random disorder. The sheet is modeled as an elastic lattice of beams where each of the beams have individual thresholds for breaking. A statistical distribution with an exponential tail towards either weak or strong beams is used to generate the thresholds and the magnitude of the disorder can be varied arbitrarily between zero and infinity. Applying a uniaxial force couple along the top and bottom rows of the lattice, fracture proceeds according to where the ratio of the stress field to the local strength is most intense. Since breakdown is initiated from an intact sheet where the first crack appears at random, the onset and mode of buckling varies according to where and how the cracks grow. For a wide range of disorders the stress-strain relationships for buckling sheets are compared with those for non-buckling sheets. The ratio of the buckling to the non-buckling value of the maximum external force the system can tolerate before breaking is found to decrease with increasing disorder, as is the ratio for the corresponding displacement. 
Bjorn Skjetne, Torbjorn Helle and Alex Hansen