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_iE + χ_i|E|²E, 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)]^t₂ in the conductor/insulator (C/I) limit, and σ_e ∼ [q_c(Δ) − q]^−s and χ_e ∼ [q_c(Δ) − q]^−s₂ in the superconductor/conductor (S/C) limit, where the exponents are found to be t = s = 1 and t₂ = s₂ = 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 = σ₁/σ₂, 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.