Maxwell-Garnett type approximation for nonlinear composites with shape distribution

Maxwell-Gamett type approximation is derived to investigate the effective linear and nonlinear responses of nonlinear composites in which randomly-oriented nonspherical granular inclusions with shape variance distribution are embedded in the host medium. The granular inclusions with the volume fraction f and the host medium with the volume fraction I - f are assumed to obey a current-field J-E relation of the form J = sigma(i)E + chi(i)\E\E-2, where sigma(i) and Xi are the linear conduct'ivity and nonlinear response of component i (i = 1, 2). Within the mean-field approximation, we numerically calculate the effects of the shape variance parameter Delta on the effective linear conductivity sigma(e) and effective nonlinear response chi(e). We find that sigma(e) is a monotonically increasing or decreasing function with Delta, dependent on whether the granular inclusions are a good or poor conductor. For the granular inclusions being nonlinear, chi(e) exhibits monotonic increase; but for the host medium being nonlinear, chi(e) can take on monotonic increase, monotonic decrease and even nonmonotonic behavior. In both nonlinear cases, large enhancement of the effective nonlinear response may be observed when the nonlinear component possesses a poor conductivity. In the dilute limit, an exact formula for the effective nonlinear response is derived. analytically. Exact results are in comparison with those from Maxwell-Gamett type approximation with shape variance distribution, reasonable agreement is found. (C) 2003 Elsevier Science B.V. All rights reserved.