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Exact estimates of the conductivity of a binary mixture of isotropic materials. (English) Zbl 0623.73011
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

MSC:
 74A40 Random materials and composite materials 82C70 Transport processes in time-dependent statistical mechanics 74A60 Micromechanical theories 74M25 Micromechanics of solids
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References:
 [1] Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803 · doi:10.2140/pjm.1952.2.25 [2] DOI: 10.1016/0022-1236(81)90085-9 · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9 [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 · doi:10.1007/BF00934301 [6] Dacorogna, Lecture Notes in Mathematics 922 (1982) [7] DOI: 10.1016/0022-5096(63)90060-7 · Zbl 0108.36902 · doi:10.1016/0022-5096(63)90060-7 [8] DOI: 10.1007/BF00279992 · Zbl 0368.73040 · doi:10.1007/BF00279992 [9] DOI: 10.1007/BF00934300 · Zbl 0504.73060 · doi:10.1007/BF00934300 [10] DOI: 10.1070/RM1979v034n05ABEH003898 · Zbl 0445.35096 · doi:10.1070/RM1979v034n05ABEH003898 [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 · doi:10.1007/BF00934953 [14] Tartar, Non-Linear Analysis and Mechanics (1979) [15] DOI: 10.1002/cpa.3160390107 · Zbl 0609.49008 · doi:10.1002/cpa.3160390107 [16] Lurie, Proc. Roy. Soc. Edinburgh Sect. A 99 pp 71– (1984) · Zbl 0564.73079 · doi:10.1017/S030821050002597X
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