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Ellipsoidal bounds and percolation thresholds of transport properties of composites. (English) Zbl 1401.74246

Summary: Classical Hashin-Shtrikman bounds physically correspond to spherical inclusions distributed in a self-similar pattern. For general ellipsoidal inclusions, new bounds named as ellipsoidal bounds are theoretically derived in this study. As there remains a major theoretical question on rigorous determination of percolation thresholds, this study also fills this major gap between lack of theoretical prediction and the gigantic amount of experimental results produced each year on percolation of composites. Formulae of percolation thresholds are for the first time universally presented for electrical, thermal, magnetic, and hydraulic properties of a variety of composites. New bounds of transport properties and percolation thresholds estimated enable the geometry of fillers or cavities, the most direct and obvious microstructure information, to be explicitly taken into account for both engineering composites and natural media (rocks, soils, and sands) containing spheroidal particles/voids, fibers, cracks, nanotubes, etc.

MSC:

74Q20 Bounds on effective properties in solid mechanics
74E30 Composite and mixture properties
76S05 Flows in porous media; filtration; seepage
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