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' homogeni zation method can violate a rigorous bound, when used together with the sel f-consistent approximation for the relevant 'linear comparison composite.' Although the second-order method is known to yield quite accurate results f or small to moderate volume fractions of the phases, even for high nonlinea rity and high contrast situations, it is shown here to fail near the percol ation limit, where it can violate the bound for any level of nonlinearity. This suggests that the second-order method should be amenable to improvemen t to account for the effect of strong field fluctuations near the percolati on threshold. (C) 2001 Academie des sciences/Editions scientifiques et medi cales Elsevier SAS.