Mechanics of disordered solids. I. Percolation on elastic networks with central forces

Abstract

Bond and site percolation on two- and three-dimensional elastic and superelastic percolation networks with central forces are studied using large-scale Monte Carlo simulations and finite-size scaling analysis. A highly accurate method of estimating the elastic percolation threshold ${\mathit{p}}_{\mathit{c}\mathit{e}}$ is proposed. For bond percolation (BP) on a triangular network we find ${\mathit{p}}_{\mathit{c}\mathit{e}}$\ensuremath{\simeq}0.641\ifmmode\pm\else\textpm\fi{}0.001, for site percolation (SP), ${\mathit{p}}_{\mathit{c}\mathit{e}}$ \ensuremath{\simeq}0.713\ifmmode\pm\else\textpm\fi{}0.002, and for BP on a bcc network we obtain, ${\mathit{p}}_{\mathit{c}\mathit{e}}$\ensuremath{\simeq}0.737\ifmmode\pm\else\textpm\fi{}0.002. We calculate the force distribution (FD) in these networks, i.e., the distribution of forces that the intact bonds of the networks suffer, near and away from ${\mathit{p}}_{\mathit{c}\mathit{e}}$. Far from ${\mathit{p}}_{\mathit{c}\mathit{e}}$ the FD is unimodal, but as ${\mathit{p}}_{\mathit{c}\mathit{e}}$ is approached, it becomes bimodal. We find that for BP on the triangular network near ${\mathit{p}}_{\mathit{c}\mathit{e}}$, the zeroth and second moments of the FD belong to the universality class of bond-bending models discussed in paper II. However, this is not the case for SP on the triangular network and BP on the bcc network. In particular, for the bcc network we find f/${\ensuremath{\nu}}_{\mathit{e}}$\ensuremath{\simeq}2.1, where f and ${\ensuremath{\nu}}_{\mathit{e}}$ are the critical exponents of the elastic moduli and the correlation length of the system, respectively. This value of f/${\ensuremath{\nu}}_{\mathit{e}}$ is distinctly different from that of bond-bending models, which is about 4.3.