Bounds on the self-consistent approximation for nonlinear media and implications for the second-order method

It is shown in this note that the recently proposed ‘second-order’ homogenization method can violate a rigorous bound, when used together with the self-consistent approximation for the relevant ‘linear comparison composite.’ Although the second-order method is known to yield quite accurate results for small to moderate volume fractions of the phases, even for high nonlinearity and high contrast situations, it is shown here to fail near the percolation limit, where it can violate the bound for any level of nonlinearity. This suggests that the second-order method should be amenable to improvement to account for the effect of strong field fluctuations near the percolation threshold.