Is it wise to keep laminating ?
ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 4, p. 452-477

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 detP 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.

DOI : https://doi.org/10.1051/cocv:2004015
Classification:  35B27,  74Q15
Keywords: homogenization, bounds, composites, laminates
@article{COCV_2004__10_4_452_0,
     author = {Briane, Marc and Nesi, Vincenzo},
     title = {Is it wise to keep laminating ?},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {4},
     year = {2004},
     pages = {452-477},
     doi = {10.1051/cocv:2004015},
     zbl = {1072.74057},
     mrnumber = {2111075},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2004__10_4_452_0}
}
Briane, Marc; Nesi, Vincenzo. Is it wise to keep laminating ?. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 4, pp. 452-477. doi : 10.1051/cocv:2004015. http://www.numdam.org/item/COCV_2004__10_4_452_0/

[1] G. Alessandrini and V. Nesi, Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal. 158 (2001) 155-171. | MR 1838656 | Zbl 0977.31006

[2] G. Alessandrini and V. Nesi, Univalent σ-harmonic mappings: applications to composites. ESAIM: COCV 7 (2002) 379-406. | Numdam | MR 1925034 | Zbl 1024.30010

[3] P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50 (2001) (Spring). | MR 1871388 | Zbl pre01780879

[4] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland (1978). | MR 503330 | Zbl 0404.35001

[5] M. Briane, Correctors for the homogenization of a laminate. Adv. Math. Sci. Appl. 4 (1994) 357-379. | MR 1294225 | Zbl 0829.35009

[6] M. Briane, G.W. Milton and V. Nesi, Change of sign of the corrector's determinant in three dimensions. Arch. Ration. Mech. Anal. To appear.

[7] A. Cherkaev, Variational methods for structural optimization. Appl. Math. Sci. 140 (2000). | MR 1763123 | Zbl 0956.74001

[8] A. Cherkaev and L.V. Gibiansky, Extremal structures of multiphase heat conducting composites. Internat J. Solids Structures 33 (1996) 2609-2618. | Zbl 0901.73050

[9] L.V. Gibiansky and O. Sigmund, Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48 (2000) 461-498. | MR 1737888 | Zbl 0989.74060

[10] Z. Hashin and S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962) 3125-3131. | Zbl 0111.41401

[11] K.A. Lurie and A.V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportions. Proc. R. Soc. Edinb. A 99 (1984) 71-87. | MR 781086 | Zbl 0564.73079

[12] K.A. Lurie and A.V. Cherkaev, The problem of formation of an optimal isotropic multicomponent composite. J. Opt. Theory Appl. 46 (1985) 571-589. | Zbl 0545.73005

[13] K.A. Lurie and A.V. Cherkaev, Exact estimates of the conductivity of a binary mixture of isotropic materials. Proc. R. Soc. Edinb. A 104 (1986) 21-38. | MR 877890 | Zbl 0623.73011

[14] G.W. Milton, Concerning bounds on transport and mechanical properties of multicomponent composite materials. Appl. Phys A 26 (1981) 125-130.

[15] G.W. Milton and R.V. Kohn, Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36 (1988) 597-629. | MR 969257 | Zbl 0672.73012

[16] F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1981) 69-102. | Numdam | MR 616901 | Zbl 0464.46034

[17] F. Murat, H-convergence. Séminaire d'Analyse Fonctionnelle et Numérique (1977-78), Université d'Alger. English translation: Murat F. and Tartar L., H-convergence. Topics in the Mathematical Modelling of Composite Materials, L. Cherkaev and R.V. Kohn Ed., Birkaüser, Boston, Progr. Nonlinear Differential Equations Appl. (1998) 21-43. | Zbl 0920.35019

[18] F. Murat and L. Tartar, Calcul des variations et homogénéisation, in Les Méthodes de l'homogénéisation : théorie et applications en physique. Eyrolles (1985) 319-369.

[19] V. Nesi, Using quasiconvex functionals to bound the effective conductivity of composite materials. Proc. R. Soc. Edinb. Sect. A 123 (1993) 633-679. | MR 1237607 | Zbl 0791.49042

[20] V. Nesi, Bounds on the effective conductivity of 2d composites made of n3 isotropic phases in prescribed volume fractions: the weighted translation method. Proc. R. Soc. Edinb. A 125 (1995) 1219-1239. | MR 1363001 | Zbl 0852.35016

[21] S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore. Ann. Scuola Norm. Sup. Pisa 3 (1967) 657-699. | Numdam | Zbl 0153.42103

[22] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa 3 (1968) 571-597. | Numdam | MR 240443 | Zbl 0174.42101

[23] L. Tartar, Estimations de coefficients homogénéisés. Lect. Notes Math. 704 (1978) 364-373. English translation: Estimations of homogenized coefficients, in Topics in the mathematical modelling of composite materials. Progr. Nonlinear Differ. Equ. Appl. 31 (1997) 9-20. | MR 540123 | Zbl 0920.35018

[24] L. Tartar, Estimations fines des coefficients homogénéisés, in Ennio De Giorgi's Colloquium, Paris, 1983, P. Kree Ed., Pitman, Boston (1985) 168-187. | Zbl 0586.35004

[25] L. Tartar, Compensated compactness and applications to p.d.e. in nonlinear analysis and mechanics, Heriot-Watt Symposium, Vol. IV, R.J. Knops Ed., Pitman, Boston (1979) 136-212. | MR 584398 | Zbl 0437.35004