Spectral representation theory of graded composite materials

In graded composite materials, the physical properties can vary continuously in space and it may give different physical phenomena when compared with homogeneous materials. The Bergman-Milton spectral representation is a rigorous mathematical formalism to express the effective dielectric constant of nongraded composite materials [1]. In this study, we consider a material (rather than microsture [2]) graded composites, and generalize the Bergman-Milton spectral representation to extract the spectral density function for the effective dielectric constant of this graded composite material in the frequency domain [3]. Analytic and numerical solution will be presented for graded films and graded spheres. \newline \newline [1]\textbf{ D. J. Bergman,} \textit{Phys. Rev. B }\textbf{\textit{14}}\textit{, 4304 (1976).} \newline [2] \textbf{J. P. Huang, K. W. Yu, G. Q. Gu,} \textbf{M.} \textbf{Karttunen, }\textit{Phys. Rev. E }\textbf{\textit{67}}\textit{, 051405 (2003).} \newline [3] \textbf{L. Gao, J. P. Huang, K.W. Yu,} \textit{Eur. Phys. J. B }\textbf{\textit{36}}\textit{, 475 (2003).}