Critical behaviour of electric failure thresholds in percolation

In this comment we show that the scaling behaviour of the failure threshold of a random lattice whose individual éléments have a rupture threshold will be différent for the mechanical and electrical problems in percolation conditions. This is due to the fact that only torques and angular distortions are effective in bringing about rupture in the mechanical case. J. Physique 48 (1987) 903-904 J U 1 N 1987, 11 Classification Physics Abstracts . 05.50 62.20M 64.60A The difference in the critical behaviours of the conductivity and the elastic moduli of random lattices above the percolation threshold has already been emphasized in several theoretical [1-3], numerical [4-6], and experimental [7-8] works. The differences basically stem from the fact that, whereas the conductivity properties come from a scalar (potential) problem, the elasticity comes from a vector (displacement) problem. In this comment, we stress that this difference also introduces a difference in critical behaviour between the electrical and mechanical rupture properties of these percolation structures. Preliminary studies on this, or related, problem have recently been reported [9]. Using the well-known node, link and blob picture of the percolation backbone [10-131, one can relate the current i flowing into a macro-bond of length (the correlation length), to the current density j imposed on the boundary by : where d is the space-dimension. In a random fuse percolation problem [14] it has been assumed that rupture will occur in any bond of a random lattice if the current flowing through it reaches a threshold value ic. Therefore as there exist singly connected bonds in the macro-bonds, rupture will develop macroscopically when the current in an individual bond reaches its critical value ; i = tc! or Within the node-link-blob model, we cannot distinguish between the onset of failure (first broken bond) and the macroscopic failure. The density of current for failure, jf, will scale as as suggested in the work of reference [14]. A similar scaling relation was proposed for the critical current in a superconductor random lattice [15]. Similarly, in the elastic problem if a stress cr is applied at the boundary the force f that will be carried in a macro-link is In addition, the torques m that will be transmitted in the macro-link will be such that : Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004806090300