Improved bounds for composites and rigidity of gradient fields

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 463, Issue 2084, 2007, Pages 2031-2048

  Albin, Nathan ^a   Conti, Sergio ^a   Nesi, Vincenzo ^b  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

^b SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

Composites; Gradient Young measures; Homogenization

Indexed keywords

EID: 36348966038     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2007.1863     Document Type: Article

References (38)

1. Albin, N. 2006 Optimality of the translation bounds for linear conducting composites in two and three dimensions. PhD thesis, University of Utah.
2. Albin, N., Cherkaev, A. & Nesi, V. 2007 Multiphase laminates of extremal effective conductivity in two dimensions. J. Mech. Phys. Solids 55, 1513-1553. (doi:10.1016/j.jmps.2006.12.003)
3. Alessandrini, G. & Nesi, V. 2001 Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal. 158, 155-171. (doi:10.1007/PL00004242)
4. Allaire, G. 2002 Shape optimization by the homogenization method, vol. 146. Applied Mathematical Sciences. New York, NY: Springer.
5. Astala, K. & Nesi, V. 2003 Composites and quasiconformal mappings: new optimal bounds in two dimensions. Calc. Var. Partial Diff. Eq. 18, 335-355. (doi:10.1007/s00526-003-0145-9)
6. Bauman, P., Marini, A. & Nesi, V. 2001 Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50, 747-757. (doi:10.1512/iumj.2001.50. 1832)
7. Bendsoe, M. P. & Kikuchi, N. 1988 Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197-224. (doi:10.1016/0045-7825(88)90086-2)
8. Bensoussan, A., Lions, J.-L. & Papanicolaou, G. 1978 Asymptotic analysis for periodic structures, vol. 5. Studies in Mathematics and its Applications. Amsterdam, The Netherlands: North-Holland Publishing Co.
9. Boyarskiǐ, B. V. 1957 Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients. Math. Sb. N.S. 43, 451-503.
10. Braides, A. & Defranceschi, A. 1998 Homogenization of multiple integrals. Oxford, UK: Clarendon Press.
11. Chaudhuri, N. & Müller, S. 2004 Rigidity estimate for two incompatible wells. Calc. Var. Partial Diff. Eq. 19, 379-390. (doi:10.1007/s00526-003-0220-2)
12. Cherkaev, A. 2000 Variational methods for structural optimization, vol. 140. Applied Mathematical Sciences. New York, NY: Springer.
13. Chlebík, M. & Kirchheim, B. 2002 Rigidity for the four gradient problem. J. Reine Angew. Math. 551, 1-9.
14. Cioranescu, D. & Donato, P. 1999 An introduction to homogenization, vol. 17. Oxford Lecture Series in Mathematics and its Applications. New York, NY: Oxford University Press.
15. Dacorogna, B. 1989 Direct methods in the calculus of variations. New York, NY: Springer.
16. Dal Maso, G. 1993 An introduction to Γ-convergence, vol. 8. Progress in Nonlinear Differential Equations and their Applications. Boston, MA: Birkhäuser Boston, Inc.
17. De Lellis, C. & Székelyhidi, L. 2006 Simple proof of two-well rigidity. C.R. Math. Acad. Sci. Paris 343, 367-370.
18. Dolzmann, G. 2003 Variational methods for crystalline microstructure: analysis and computation. Lecture Notes in Mathematics, no. 1803. New York, NY: Springer.
19. 2x2. MPI-MIS Preprint 60.
20. Faraco, D. & Zhong, X. 2005 Geometric rigidity of conformal matrices. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4, 557-585.
21. Friesecke, G., James, R. & Müller, S. 2002 A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math. 55, 1461-1506. (doi:10.1002/cpa.10048)
22. Gibiansky, L. V. & Sigmund, O. 2000 Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48, 461-498. (doi:10.1016/S0022-5096(99)00043-5)
23. Hashin, Z. & Shtrikman, S. 1962 A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125-3131. (doi:10.1063/1.1728579)
24. Jikov, V. V., Kozlov, S. M. & Oleǐnik, O. A. 1994 Homogenization of differential operators and integral functionals. Berlin, Germany: Springer.
25. Kohn, R. V. & Strang, G. 1986 Optimal design and relaxation of variational problems. I, II, III. Comm. Pure Appl. Math. 39, 113-137. (doi:10.1002/cpa.3160390107) See also pp 139-182, 353-377.
26. Lurie, K. A. & Cherkaev, A. V. 1984 Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. Proc. Roy. Soc. Ed. Sect. A 99, 71-87.
27. Lurie, K. A. & Cherkaev, A. V. 1985 Optimization of properties of multicomponent isotropic composites. J. Optim. Theory Appl. 46, 571-580. (doi:10.1007/BF00939160)
28. p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17, 189-206.
29. Milton, G. W. 1981 Concerning bounds on the transport and mechanical properties of multicomponent composite materials. Appl. Phys. A 26, 125-130. (doi:10.1007/BF00616659)
30. Milton, G. W. 2002 The theory of composites, vol. 6. Cambridge Monographs on Applied and Computational Mathematics, Cambridge, UK: Cambridge University Press.
31. Milton, G. W. & Kohn, R. V. 1988 Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36, 597-629. (doi:10.1016/0022-5096(88)90001-4)
32. Müller, S. 1999 Variational models for microstructure and phase transitions. In Calculus of variations and geometric evolution (eds F. Bethuel, G. Huisken, S. Müller & K. Steffen). Springer Lecture Notes in Math, no. 1713, pp. 85-210. Berlin, Germany: Springer.
33. Nesi, V. 1995 Bounds on the effective conductivity of two-dimensional composites made of n>3 isotropic phases in prescribed volume fraction: the weighted translation method. Proc. Roy. Soc. Ed. Sect. A 125, 1219-1239.
34. Pedregal, P. 1997 Parametrized measures and variational principles, vol. 30. Progress in Nonlinear Differential Equations and their Applications. Basel, Germany: Birkhäuser Verlag.
35. Sanchez-Palencia, E. & Zaoui, A. (eds) 1987 Homogenization techniques for composite media. Lecture Notes in Physics, vol. 272. Berlin, Germany: Springer.
36. Tartar, L. 1979 Estimation de coefficients homogéné isés. In Computing methods in applied sciences and engineering (Proc. Third Int. Symp., Versailles, 1977), vol. 704, pp. 364-373. I. Springer Lecture Notes in Math, Berlin, Germany: Springer.
37. Tartar, L. 1985 Estimations fines des coefficients homogéné isés. In Ennio De Giorgi colloquium (Paris, 1983), vol. 125, pp. 168-187. Research Notes in Mathematics, Boston, MA: Pitman.
38. Vigdergauz, S. 1994 Three-dimensional grained composites of extreme thermal properties. J. Mech. Phys. Solids 42, 729-740. (doi:10.1016/0022-5096(94)90040-X)