Critical behaviour in the fracture of disordered media

In this paper we investigate the in ¯ uence of di erent boundary conditions on the ® nal breakdown of a lattice model for the fracture of heterogeneous media. Experimental evidence shows that disordered media subject to stress display some features that are characteristic of critical systems, therefore suggesting an interpretation of the global breakdown of the system as a kind of critical transition. Many of the observed features are well reproduced at least at a qualitative level by lattice models; however, mechanisms at the base of the onset of criticality are not well understood. Besides disorder, there are many parameters that seem to in ¯ uence the critical properties of the system. The system size and the boundary conditions are among these. We ® nd that the statistical properties of the ® nal breakdown are strongly in ¯ uenced by the boundary condition. In particular constant-stress relaxation leads to a ® nal breakdown always involving the breaking of a ® nite number of bonds, which is also large if compared with the number of bonds broken during the formation of each localized crack preceding the ® nal breakdown. When the lattice undergoes constant-strain relaxation instead, the breakdown may involve a vanishingly small number of bond-breaking events