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Journal Articles International Journal of Solids and Structures Year : 2020

The effective conductivity of strongly nonlinear media: The dilute limit

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.

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Materials
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hal-02425307 , version 1 (25-12-2020)

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François Willot. The effective conductivity of strongly nonlinear media: The dilute limit. International Journal of Solids and Structures, 2020, Special Issue on Physics and Mechanics of Random Structures: From Morphology to Material Properties, 184, pp.287-295. ⟨10.1016/j.ijsolstr.2019.06.006⟩. ⟨hal-02425307⟩
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