Grabovsky, Yury; Milton, Graeme W. Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation. (English) Zbl 0917.73046 Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 2, 283-299 (1998). Summary: Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the well-known result of R. Hill [J. Mech. Phys. Solids 11, 357-372 (1963; Zbl 0114.15804)] that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of M. Avellaneda et al. [J. Mech. Phys. Solids 44, 1179-1218 (1996)] that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline \(G\)-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic. Cited in 2 ReviewsCited in 3 Documents MSC: 74E30 Composite and mixture properties 74E05 Inhomogeneity in solid mechanics Keywords:elastic percolation problems; polycrystalline \(G\)-closure; effective compliance tensor; pure crystal moduli; shear modulus; isotropic polycrystal; elliptic PDE theory Citations:Zbl 0114.15804 PDFBibTeX XMLCite \textit{Y. Grabovsky} and \textit{G. W. Milton}, Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 2, 283--299 (1998; Zbl 0917.73046) Full Text: DOI References: [1] DOI: 10.1063/1.329385 · doi:10.1063/1.329385 [2] DOI: 10.1016/0022-5096(94)90040-X · Zbl 0803.73047 · doi:10.1016/0022-5096(94)90040-X [3] DOI: 10.1098/rspa.1992.0123 · Zbl 0791.73011 · doi:10.1098/rspa.1992.0123 [4] DOI: 10.1016/0022-5096(93)90006-2 · Zbl 0776.73044 · doi:10.1016/0022-5096(93)90006-2 [5] DOI: 10.1007/s002050050049 · Zbl 0894.73225 · doi:10.1007/s002050050049 [6] DOI: 10.1103/PhysRevLett.44.1285 · doi:10.1103/PhysRevLett.44.1285 [7] Bensoussan, Asymptotic analysis of periodic structures (1978) [8] DOI: 10.1002/cpa.3160390604 · Zbl 0602.35030 · doi:10.1002/cpa.3160390604 [9] DOI: 10.1137/0149048 · Zbl 0702.73078 · doi:10.1137/0149048 [10] DOI: 10.1016/0022-5096(96)00018-X · doi:10.1016/0022-5096(96)00018-X [11] DOI: 10.1063/1.91895 · doi:10.1063/1.91895 [12] DOI: 10.1007/BF00934299 · Zbl 0505.73060 · doi:10.1007/BF00934299 [13] DOI: 10.1007/BF00934301 · Zbl 0525.73102 · doi:10.1007/BF00934301 [14] Lurie, Proc. Roy. Soc. Edinburgh Sect. A 99 pp 71– (1984) · Zbl 0564.73079 · doi:10.1017/S030821050002597X [15] DOI: 10.1016/0022-5096(64)90019-5 · doi:10.1016/0022-5096(64)90019-5 [16] DOI: 10.1016/0022-5096(63)90036-X · Zbl 0114.15804 · doi:10.1016/0022-5096(63)90036-X [17] DOI: 10.1016/0022-5096(95)00017-D · Zbl 0877.73041 · doi:10.1016/0022-5096(95)00017-D [18] DOI: 10.1016/0022-5096(95)00016-C · Zbl 0870.73041 · doi:10.1016/0022-5096(95)00016-C [19] DOI: 10.1098/rspa.1996.0047 · Zbl 0863.53056 · doi:10.1098/rspa.1996.0047 [20] Francfort, C. R. Acad. Sci. Paris Sér. I 312 pp 301– (1991) [21] DOI: 10.1070/RM1991v046n03ABEH002803 · Zbl 0751.15014 · doi:10.1070/RM1991v046n03ABEH002803 [22] Tartar, Ennio de Giorgi’s Colloquium pp 168– (1985) [23] DOI: 10.1093/qjmam/49.4.565 · Zbl 0874.73045 · doi:10.1093/qjmam/49.4.565 [24] DOI: 10.1002/cpa.3160470302 · Zbl 0797.73032 · doi:10.1002/cpa.3160470302 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.