Spatial scaling in fracture propagation in dilute systems

TL;DR

This study investigates the spatial and temporal scaling behaviors of fracture patterns in dilute elastic networks, revealing a new universality class characterized by specific critical exponents and fractal structures at the transition point.

Contribution

It demonstrates that fracture cluster growth exhibits both spatial and temporal scale invariance, with critical exponents indicating a distinct universality class from similar phenomena.

Findings

Divergence of connectivity length at critical time with exponent ~0.83

Fractal structure of vacancy clusters with dimension ~1.85 at transition

Scaling relation z = ν * d_f confirmed

Abstract

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 $t_c$ at which a dynamical transition occurs with a power 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 cluster growth near $t_c$ also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length $\xi$ of the clusters increses and diverges at $t_c$ as $\xi \sim (t_c-t)^{-\nu}$, with $\nu = 0.83 \pm 0.06$. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at $t_c$ with an effective dimensionality $d_f \sim 1.85 \pm 0.05$. 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 $\nu$ and $d_f$ supports the scaling relation $z=\nu d_f$ with the value of $z$ obtained before.