Large Currents in Random Resistor Networks and Large Tensions in Random Elastic Networks

The statistics of large currents (tensions) in large random resistor (elastic) networks of two finite components are studied. Each bond in the network is occupied either by a large conductor (strong spring) or a small conductor (weak spring). The magnitude of the current (tension) in a bond is determined by the configuration of the composite which surrounds it. The large bond currents (tensions) are most often found in "critical defects", i.e., those configurations which can generate the largest current (tension) in a bond with the smallest number of determined bonds surrounding it. This dissertation addresses the question: Given a random resistor (elastic) network, what is the probability density for large currents (tensions) occurring in a bond? The probability density is obtained by studying the distribution of the critical defects with the aid of numerical simulation and continuum mechanics. The probability density is found to be controlled by the smallest eigenvalue of the Laplace's (biharmonic) equation prescribed by appropriate boundary conditions.