| Title:
|
Bounds and estimates on the effective properties for nonlinear composites (English) |
| Author:
|
Wall, Peter |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
45 |
| Issue:
|
6 |
| Year:
|
2000 |
| Pages:
|
419-437 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle together with the bounds presented by Lukkassen, Persson and Wall in 1995. Moreover, we also present some examples where the bounds are so tight that they may be used as a good estimate of the effective behavior. (English) |
| Keyword:
|
homogenization |
| Keyword:
|
effective properties |
| Keyword:
|
variational methods |
| Keyword:
|
nonlinear composites |
| MSC:
|
35B27 |
| MSC:
|
73B27 |
| MSC:
|
74Q20 |
| idZBL:
|
Zbl 0996.74062 |
| idMR:
|
MR1800963 |
| DOI:
|
10.1023/A:1022381416707 |
| . |
| Date available:
|
2009-09-22T18:05:02Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134450 |
| . |
| Reference:
|
[1] A. Braides, D. Lukkassen: Reiterated homogenization of integral functionals.Math. Models Methods Appl. Sci, to appear. MR 1749689 |
| Reference:
|
[2] P. Ponte Castaneda: Bounds and estimates for the properties of nonlinear heterogeneous systems.Philos. Trans. Roy. Soc. London Ser. A. 340 (1992), 531–567. Zbl 0776.73062, MR 1192288, 10.1098/rsta.1992.0079 |
| Reference:
|
[3] P. Ponte Castaneda: A new variational principle and its application to nonlinear heterogeneous systems.SIAM J. Appl. Math. 52 (1992), 1321–1341. Zbl 0759.73064, MR 1182126, 10.1137/0152076 |
| Reference:
|
[4] G. Dal Maso: An Introduction to $\Gamma $-convergence.Birkhäuser, Boston, 1993. Zbl 0816.49001, MR 1201152 |
| Reference:
|
[5] I. Ekeland, R. Temam: Convex analysis and variational problems. Studies in Mathematics and Its Applications, Vol. 1.North-Holland, Amsterdam, 1976. MR 0463994 |
| Reference:
|
[6] V. V. Jikov, S. M. Kozlov and O. A. Oleinik: Homogenization of Differential Operators and Integral Functionals.Springer-Verlag, Berlin-Heidelberg-New York, 1994. MR 1329546 |
| Reference:
|
[7] D. Lukkassen: Formulae and bounds connected to optimal design and homogenization of partial differential operators and integral functionals.(1996), Ph. D. Thesis, Dept. of Math., Tromsö University, Norway. |
| Reference:
|
[8] D. Lukkassen: Some sharp estimates connected to the homogenized $p$-Laplacian equation.Z. Angew. Math. Mech. 76 (S2) (1996), 603–604. Zbl 1126.35303 |
| Reference:
|
[9] D. Lukkassen, L. E. Persson and P. Wall: On some sharp bounds for the $p$-Poisson equation.Appl. Anal. 58 (1995), 123–135. MR 1384593, 10.1080/00036819508840366 |
| Reference:
|
[10] D. R .S. Talbot, J. R. Willis: Variational principles for nonlinear inhomogeneous media.IMA J. Appl. Math. 35 (1985), 39–54. MR 0820896, 10.1093/imamat/35.1.39 |
| Reference:
|
[11] D. R. S. Talbot, J. R. Willis: Bounds and self-consistent estimates for the overall properties of nonlinear composites.IMA J. Appl. Math. 39 (1987), 215–240. MR 0983743, 10.1093/imamat/39.3.215 |
| Reference:
|
[12] J. van Tiel: Convex Analysis.John Wiley and Sons Ltd., New York, 1984. Zbl 0565.49001, MR 0743904 |
| Reference:
|
[13] P. Wall: Optimal bounds on the effective shear moduli for some nonlinear and reiterated problems.Acta Sci. Math. 65 (1999), 553–566. Zbl 0987.35027, MR 1737271 |
| . |
