Nonlinear Conductivity and Dielectric Constant: The Continuum Approach
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In this chapter, we described and discussed general procedures for estimating the effective conductivity and dielectric constant of nonlinear materials. These procedures, which represent the generalization of those described in Chapters 4 and 7 of Volume I for linear materials, provide bounds and estimates for the effective conductivity and dielectric constant. One procedure leads to rigorous bounds and estimates that are exact to first order in the phase property contrast, while the second technique yields estimates that are exact to second order in the contrast.
One important difference between the results obtained by the two procedures must be emphasized. By design, the results presented in Section 10.6 are exact to second-order in the phase contrast, and thus are consistent with the asymptotic results of Blumenfeld and Bergman (1991b), whereas the results presented in Sections 10.4 and 10.5 are nonlinear estimates that are exact only to first order in the phase contrast. On the other hand, while the first-order results provide rigorous bounds for the effective energy function of nonlinear materials (and hence their generalized effective conductivity and dielectric constant), the second-order estimates do not lead to any bound, either the lower or upper bound. Nevertheless, comparison of these estimates with the numerical results and the known bounds suggests that the second-order results provide accurate estimates for the effective conductivity and dielectric constant of general nonlinear materials, and in particular, strongly nonlinear, power-law type composites.
KeywordsDielectric Constant Nonlinear Material Continuum Approach Effective Conductivity Effective Dielectric Constant
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