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Is it wise to keep laminating? (English) Zbl 1072.74057

Summary: We study the corrector matrix \(P^c\) to the conductivity equations. We show that if \(P^c\) converges weakly to the identity, then for any laminate \(\det P^c\geq 0\) at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work M. Briane et al. [Arch. Ration. Mech. Anal. 173, No. 1, 133–150 (2004; Zbl 1118.78009)]. In two dimensions it holds true for any microgeometry as a corollary of the work in G. Alessandrini and V. Nesi [Arch. Ration. Mech. Anal. 158, No. 2, 155–171 (2001; Zbl 0977.31006)]. We use this property of laminates to prove that, in any dimension, the classical Hashin-Shtrikman bounds are not attained by laminates, in certain regimes, when the number of phases is greater than two. In addition we establish new bounds for the effective conductivity, which are asymptotically optimal for mixtures of three isotropic phases among a certain class of microgeometries, including orthogonal laminates, which we then call quasiorthogonal.

MSC:

74Q20 Bounds on effective properties in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
74E30 Composite and mixture properties
74A40 Random materials and composite materials
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