Is it wise to keep laminating?

Abstract

Abstract. We study the corrector matrix P ε to the conductivity equations. We show that if P ε converges weakly to the identity, then for any laminate det P ε ≥ 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 Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]. 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.Mathematics Subject Classification. 35B27, 74Q15.

Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials

“…Laminates have the advantage over most other composites that their effective properties can be readily calculated, which facilitates a mathematically rigorous treatment. Furthermore, despite their simple building principle, they attain many fundamental bounds, i.e., they often realize the most extreme effective properties any composite can fundamentally have [29][30][31].…”

Some open problems in the theory of composites

“…Assuming 〈 E 〉 = I , a perturbation argument [31,32] shows that the sign change of the Hall coefficient is related to the fact that the trace of the cofactor matrix of E ( x ) changes sign, at least in certain regions in the unit cell of periodicity. On the other hand, in any multiple rank laminates (with 〈 E 〉 = I ) Briane and Nesi show that the determinant of E ( x ) remains positive [33], whereas it does take negative values in certain regions in the interlinked tori geometries [34]. While they show that the trace of the cofactor matrix of E ( x ) can change sign in three-phase multiple rank laminates, it is an open question as to whether it can change sign in two-phase multiple rank laminates.…”

Some open problems in the theory of composites

“…Assuming E = I, a perturbation argument [14,16] shows that the sign change of the Hall coefficient is related to the fact that the trace of the cofactor matrix of E(x) changes sign, at least in certain regions in the unit cell of periodicity. On the other hand, in any multiple rank laminates (with E = I) Briane and Nesi show that the determinant of E(x) remains positive [18], whereas it does take negative values in certain regions in the interlinked tori geometries [17]. While they show that the trace of the cofactor matrix of E(x) can change sign in three phase multiple rank laminates, it is an open question as to whether it can change sign in two phase multiple rank laminates.…”