Effective conductivity of loaded granular materials by numerical simulation

Thermal (or electrical) conduction through a stationary, random close packed granular material is studied by numerical simulation. The particles are smooth uniform spheres with either point contacts, or small contact spots under a mechanical load (resulting in practice from the transmitted bed weight or applied pressure), and the particle–to–medium conductivity ratio, γ, is assumed to be high (typically, up to O(103)). To simulate particle arrangements, a recent random close packing algorithm for N ≫ 1 absolutely rigid spheres in a periodic box is used, and small contact deformations are rigorously calculated as a perturbation from the Hertz theory and a mechanical balance of normal contact reactions on the microscale. To solve the conduction boundary–value problem for N ≫ 1 and γ ≫ 1, when all the existing ‘exact’ simulation techniques lead to prohibitive convergence difficulties, a novel, asymptotic algorithm is developed and shown to be highly accurate for all γ ⩾ 100. This algorithm is based on matching the outer solution for perfectly conducting spheres in point contact with the near–contact solutions (that include the effects of finite γ and small deformation), to provide additional equations for unknown particle temperatures (or potentials) in the outer geometry. The outer problem is then ‘regularized’ and solved by a special economical multipole method. Using these techniques for N ⩽ 200 and configurational averaging, the non–dimensional effective conductivity f* is found as a universal function of γ and Π =[〈〉(1−ν2)/E)1/3, where (p) is the average pressure in the material, and ν and E are the particle Poisson ratio and Young's modulus. These results are complemented by the exact values of f*(γ) for a material with point contacts and γ < 100. Conduction through a random close packing of nearly rigid spheres in a vacuum environment is also considered.