Effective medium approximation for two-component nonlinear composites with shape distribution

The effective medium approximation (EMA) is derived to investigate the effective linear and nonlinear responses of two-component composites in which one component is nonspherical and distributed in shape. Both components with the volume fractions p and q are assumed to obey a current–field J – E relation of the form J = σ i E + χ i | E | 2 E , where σ i and χ i are the linear conductivity and nonlinear response of the component i ( i = 1, 2) respectively. As the percolation threshold p c (or q c ) is approached from above (or below), the effective linear conductivity σ e and effective nonlinear response χ e behave as σ e ∼ [ p − p c (Δ)] t and χ e ∼ [ p − p c (Δ)] t 2 in the conductor/insulator (C/I) limit, and σ e ∼ [ q c (Δ) − q ] − s and χ e ∼ [ q c (Δ) − q ] − s 2 in the superconductor/conductor (S/C) limit, where the exponents are found to be t = s = 1 and t 2 = s 2 = 2, independent of the shape variance parameter Δ, and p c (Δ) (or q c (Δ)) is a monotonically decreasing (or increasing) function with Δ. For a finite-conductivity ratio h = σ 1 /σ 2 , numerical results show that σ e may be increased or decreased with increasing Δ, dependent on whether the first component is a good or a poor conductor, while χ e can exhibit a monotonic increase, monotonic decrease and nonmonotonic behaviour. Therefore, χ e can be greatly enhanced by the adjustment of the shape variance parameter, and thereby provides an alternative way to achieve large enhancement of effective nonlinear response. The results of EMA with shape distribution are also compared with exact solutions in the dilute limit and reasonable agreement is found.