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R. Lipton
WPI, USA
Bounds and extremal microgeometries for field fluctuations in anisotropic random composites.
Understanding the behavior of local electric fields in random media
is crucial as regions containing high fields are most often the first to
suffer damage during service. Higher moments of the electric field provide
information on the variation of the local electric field that are not
revealed by the effective properties of the composite. Unfortunately
higher moments of the local electric field can not be obtained directly
through simple boundary measurements. On the other hand effective
properties can be measured experimentally. We find bounds on the
covariance tensor of the electric field for anisotropic composites in
terms of the effective dielectric constant. We exhibit microstructures for
which these bounds are optimal. These bounds are used to recover bounds on
the covariance tensor when only the phase volume fractions and the
two-point correlation function are known. For isotropic composites we
obtain a lower bound that is the most restrictive one in terms of the
phase volume fractions. The methodology developed here is new and is based
upon the explicit computation of the convex hull of a curve.
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