Lurie, K. A.; Cherkaev, A. V. Exact estimates of the conductivity of a binary mixture of isotropic materials. (English) Zbl 0623.73011 Proc. R. Soc. Edinb., Sect. A 104, 21-38 (1986). Consider a mixture of two isotropic conducting mixtures of concentration \(m_ 1\) and \(m_ 2\), taking \(m_ 1+m_ 2=1\) and \(0\leq m_ i\leq 1\); and the conductivities as \(u_ 1\) and \(u_ 2\). The materials have an arbitrary microstructure including alignments. The main result proved is that the effective conductivity tensor forms a closed set. The Voigt- Reuss bounds are generalized for various possible microstructures in the form of upper and lower bounds. Special cases of layering and networking are considered as limiting cases and possible extensions when the materials are anisotropic are indicated. Reviewer: E.S.R.Gopal Cited in 2 ReviewsCited in 45 Documents MSC: 74A40 Random materials and composite materials 82C70 Transport processes in time-dependent statistical mechanics 74A60 Micromechanical theories 74M25 Micromechanics of solids Keywords:effective medium theory; microstructure effect; conduction problems; inhomogeneity; isotropic conducting mixtures; arbitrary microstructure; effective conductivity tensor; closed set; Voigt-Reuss bounds; layering; networking Citations:Zbl 0564.73079 PDFBibTeX XMLCite \textit{K. A. Lurie} and \textit{A. V. Cherkaev}, Proc. R. Soc. Edinb., Sect. A, Math. 104, 21--38 (1986; Zbl 0623.73011) Full Text: DOI References: [1] Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803 [2] DOI: 10.1016/0022-1236(81)90085-9 · Zbl 0459.35020 [3] Lurie, Dokl. Akad. Nauk SSSR 264 pp 1128– (1982) [4] Lurie, Exact estimates of conductivity of a binary mixture of isotropic compounds (1984) [5] DOI: 10.1007/BF00934301 · Zbl 0525.73102 [6] Dacorogna, Lecture Notes in Mathematics 922 (1982) [7] DOI: 10.1016/0022-5096(63)90060-7 · Zbl 0108.36902 [8] DOI: 10.1007/BF00279992 · Zbl 0368.73040 [9] DOI: 10.1007/BF00934300 · Zbl 0504.73060 [10] DOI: 10.1070/RM1979v034n05ABEH003898 · Zbl 0445.35096 [11] Murat, Encyclopedia of Systems and Control (1983) [12] Tartar, Ennio DeGiorgi Colloquium 125 pp 168– (1985) [13] DOI: 10.1007/BF00934953 · Zbl 0464.73109 [14] Tartar, Non-Linear Analysis and Mechanics (1979) [15] DOI: 10.1002/cpa.3160390107 · Zbl 0609.49008 [16] Lurie, Proc. Roy. Soc. Edinburgh Sect. A 99 pp 71– (1984) · Zbl 0564.73079 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.