THE EFFECTIVE CONDUCTIVITY OF DENSELY PACKED HIGH CONTRAST COMPOSITES WITH INCLUSIONS OF OPTIMAL SHAPE

摘要

We consider a mathematical model of a high contrast two phase composite material with inclusions (fibers) close to touching in two space dimensions. The inclusions form a periodic array and have an optimal shape which is curvilinear square with rounded-off angles ("nearly square") described by a flattening parameter m. We present an asymptotic formula for the effective conductivity A_δ of the composite when the interparticle distance δ goes to zero. This formula captures the dependence of A_δ on the parameter m. We also introduce a recurrence sequence of the kth order cell problems (k = 0, 1, 2,...) and present the formula for a full asymptotic expansion of the solution to the problem.