Site percolation on central-force elastic networks

Abstract

The elastic properties of model random networks are studied, in which a fraction $p_s$ of the sites are randomly present and are connected to their remaining nearest neighbors by Hooke springs with force constant α. The one-site-defect problem is solved exactly using Green’s-function techniques specialized to the static elastic limit. The location of $p_s^{\mathrm{*}}$, the critical point at which all the elastic moduli vanish, and f($p_s$), the fraction of zero-frequency modes, agree well with the predictions of constraint-counting theory. In contrast to previously studied bond-depletion problems, it is shown both analytically and numerically that Cauchy’s relation ($C_{12}$=$C_{44}$) is strictly disobeyed, even in the one-site-defect limit.