Homogenization Theory and the Assessment of Extreme Field Values in Composites with Random Microstructure

Abstract

Suitable macroscopic quantities are identified and used to assess the field distribution within a composite specimen of finite size with random microstructure. Composites made of N anisotropic dielectric materials are considered. The characteristic length scale of the microstructure relative to the length scale of the specimen is denoted by $\varepsilon$, and realizations of the random composite microstructure are labeled by $\omega$. Consider any cube $C_0$ located inside the composite. The function $P^\varepsilon(t,C_0,\omega)$ gives the proportion of $C_0$ where the square of the electric field intensity exceeds t. The analysis focuses on the case when $0 < \varepsilon \ll 1$. Rigorous upper bounds on ${\lim}_{\varepsilon\rightarrow 0}P^\varepsilon(t,C_0,\omega)$ are found. They are given in terms of the macrofield modulation functions. The macrofield modulation functions capture the excursions of the local electric field fluctuations about the homogenized or macroscopic electric field. Information on the regularity of the macrofield modulations translates into bounds on ${\lim}_{\varepsilon\rightarrow 0}P^\varepsilon(t,C_0,\omega)$. Sufficient conditions are given in terms of the macrofield modulation functions that guarantee polynomial and exponential decay of ${\lim}_{\varepsilon\rightarrow 0}P^\varepsilon(t,C_0,\omega)$ with respect to ``t.' For random microstructure with oscillation on a sufficiently small scale we demonstrate that a pointwise bound on the macrofield modulation function provides a pointwise bound on the actual electric field intensity. These results are applied to assess the distribution of extreme electric field intensity for an L-shaped domain filled with a random laminar microstructure.