Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored 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 characteristic time tc at which a dynamical transition occurs with a power law divergence (with the exponent z) of the average cluster size. Close to tc, the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the cluster growth near tc also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length xi of the clusters increases and diverges at tc as xi ∼ (tc − t)−ν, with ν = 0.83 ± 0.06. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at tc with an effective dimensionality df ∼ 1.85 ± 0.05. These values are independent (within the limit of statistical error) of the concentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem. The values of ν and df supports the scaling relation z = νdf with the value of z obtained before.