SPATIAL SCALING IN FRACTURE PROPAGATION IN DILUTE SYSTEMS

The geometry of fracture patterns in a dilute elastic network is explo red using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a ch aracteristic time t(c) at which a dynamical transition occurs with a p ower law divergence (with the exponent z) of the average cluster size. Close to t(c), the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the clust er growth near t(c) also exhibits spatial scaling in addition to the t emporal scaling. As fracture develops with time, the connectivity leng th xi of the clusters increases and diverges at t(c) as xi similar to (t(c) - t)(-nu) with nu = 0.83 +/- 0.06. As a result of the scale-inva riant growth, the vacancy clusters attain a fractal structure at t(c) with an effective dimensionality d(f) similar to 1.85 +/- 0.05. These values are independent(within the limit of statistical error) of the c oncentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different fro m the corresponding values in qualitatively similar phenomena suggesti ng a different universality class of the problem. The values of nu and d(f) supports the scaling relation z = nu d(f) with the value of z ob tained before.