The effective conductivity of strongly nonlinear media: The dilute limit. (February 2020) - Willot, François
Abstract: This work is a combined numerical and analytical investigation of the effective conductivity of strongly nonlinear media in two dimensions. The nonlinear behavior is characterized by a threshold value for the maximal absolute current. Our main focus is on random media containing an infinitesimal proportion f ≪ 1 of insulating phase. We first consider a random conducting network on a square grid and establish a relationship between the length of minimal paths spanning the network and the network's effective response. In the dilute limit f ≪ 1, the network's effective conductivity scales, to leading-order correction in f, as ∼ f ν with ν = 1 or ν = 1 / 2, depending on the direction of the applied field with respect to the grid. Second, we introduce coupling between local bonds, and observe an exponent ν ≈ 2/3. To interpret this result, we derive an upper-bound for the length of geodesics spanning random media in the continuum, relevant to media with a dilute concentration of heterogeneities. We argue that ν = 2 / 3 for random composites in the continuum with homogeneously-distributed, monodisperse particles, in two dimensions.