| Title:
|
A note on bounds for non-linear multivalued homogenized operators (English) |
| Author:
|
Svanstedt, Nils |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
1998 |
| Pages:
|
81-92 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study the behaviour of maximal monotone multivalued highly oscillatory operators. We construct Reuss-Voigt-Wiener and Hashin-Shtrikmann type bounds for the minimal sections of G-limits of multivalued operators by using variational convergence and convex analysis. (English) |
| Keyword:
|
multivalued operators |
| Keyword:
|
highly oscillatory operators |
| Keyword:
|
Reuss-Voigt-Wiener bounds |
| Keyword:
|
Hashin-Shtrikman bounds |
| MSC:
|
35B27 |
| MSC:
|
35J20 |
| MSC:
|
35Q35 |
| MSC:
|
47H04 |
| idZBL:
|
Zbl 0940.47041 |
| idMR:
|
MR1609178 |
| DOI:
|
10.1023/A:1023210332327 |
| . |
| Date available:
|
2009-09-22T17:56:53Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134376 |
| . |
| Reference:
|
[1] G. Allaire, R. V. Kohn: Optimal bounds on the effective behaviour of a mixture of two well-ordered elastic materials.Q. Appl. Math. 51 (1993), 643–674. MR 1247433, 10.1090/qam/1247433 |
| Reference:
|
[2] H. Attouch: Variational Convergence for Functions and Operators.Pitman Publ., 1984. Zbl 0561.49012, MR 0773850 |
| Reference:
|
[3] M. Avellaneda: Optimal bounds and microgeometries for elastic two-phase composites.SIAM J. Appl. Math. 47 (1987), 1216–1228. Zbl 0632.73079, MR 0916238, 10.1137/0147082 |
| Reference:
|
[4] V. Chiado-Piat, G. Dal Maso and A. Defranceschi: G-convergence of monotone operators.Ann. Inst. H. Poincaré, Anal. Non Lineare 7 (1990), 123–160. MR 1065871, 10.1016/S0294-1449(16)30298-0 |
| Reference:
|
[5] V. Chiado-Piat, A. Defranceschi: Homogenization of monotone operators.Nonlinear Anal. 14, 717–732. MR 1049117, 10.1016/0362-546X(90)90102-M |
| Reference:
|
[6] G. Dal Maso: An introduction to $\Gamma $-convergence.Birkhäuser Boston, 1993. Zbl 0816.49001, MR 1201152 |
| Reference:
|
[7] A. Defranceschi: G-convergence of cyclically monotone operators.Asymptotic Anal. 1 (1989), 21–37. Zbl 0681.47023, MR 0991415, 10.3233/ASY-1989-2103 |
| Reference:
|
[8] G. Francfort, F. Murat: Homogenization and optimal bounds in linear elasticity.Arch. Rat. Mech. Anal. 94 (1986), 307–334. MR 0846892, 10.1007/BF00280908 |
| Reference:
|
[9] K. Golden, G. Papanicolaou: Bounds for effective parameters of heterogeneous media by analytic continuation.Comm. Math. Phys. (1983), 473–491. MR 0719428, 10.1007/BF01216179 |
| Reference:
|
[10] Z. Hashin: Analysis of composite materials. A survey.J. Appl. Mech. 50 (1983), 483–505. Zbl 0542.73092, 10.1115/1.3167081 |
| Reference:
|
[11] Z. Hashin, S. Shtrikman: A variational approach to the theory of the elastic behaviour of polycrystals.J. Mech. Phys. Solids. 10 (1962), 343–352. MR 0147032, 10.1016/0022-5096(62)90005-4 |
| Reference:
|
[12] Z. Hashin, S. Shtrikman: A variational approach to the theory of the elastic behaviour of multiphase materials.J. Mech. Phys. Solids. 11 (1963), 127–140. MR 0159459, 10.1016/0022-5096(63)90060-7 |
| Reference:
|
[13] R. Hill: Elastic properties of reinforced solids: Some theoretical principles.J. Mech. Phys. Solids 11 (1963), 357–372. Zbl 0114.15804, 10.1016/0022-5096(63)90036-X |
| Reference:
|
[14] R. V. Kohn, R. Lipton: Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials.Arch. Rat. Mech. Anal. 102 (1988), 331–350. MR 0946964, 10.1007/BF00251534 |
| Reference:
|
[15] D. Lukkassen, L. E. Persson and P. Wall: On some sharp bounds for the homogenized p-Poisson equation.Applicable Analysis 58 (1995), 123–135. MR 1384593, 10.1080/00036819508840366 |
| Reference:
|
[16] K. A. Lurie, A. V. Cherkaev: Exact estimates of the conductivity of composites formed by two isotropically conducting media taken in prescribed proportion.Proc. Roy. Soc. Edinburgh 99A (1984), 71–87. MR 0781086 |
| Reference:
|
[17] G. Milton: On characterizing the set of possible effective tensors of composites: The variational method and the translation method.Comm. Pure Appl. Math. 43 (1990), 63–125. Zbl 0751.73041, MR 1024190, 10.1002/cpa.3160430104 |
| Reference:
|
[18] P. Ponte Castaneda: The effective mechanical properties of nonlinear isotropic composites.J. Mech. Phys. Solids 39 (1991), 45–71. Zbl 0734.73052, MR 1085622, 10.1016/0022-5096(91)90030-R |
| Reference:
|
[19] P. Ponte Castaneda: New variational principles in plasticity and their application to composite materials.J. Mech. Phys. Solids 40 (1992), 1757–1788. Zbl 0764.73103, MR 1190849, 10.1016/0022-5096(92)90050-C |
| Reference:
|
[20] P. Suquet: Plasticite et homogeneisation.These, Paris VI (1982). |
| Reference:
|
[21] P. Suquet: Overall potentials and extremal surfaces of power law or ideally plastic composites.J. Mech. Phys Solids 41 (1993), 981–1002. Zbl 0773.73063, MR 1220788, 10.1016/0022-5096(93)90051-G |
| Reference:
|
[22] N. Svanstedt: Bounds for homogenized constitutive laws in non-linear elasticity and plasticity (in preparation).. |
| Reference:
|
[23] D. R. S. Talbot, J. R. Willis: Some simple explicit bounds for the overall behaviour of nonlinear composites.Int. J. Solids Struct. 29 (1992), 1981–1987. MR 1173106 |
| Reference:
|
[24] D. R. S. Talbot, J. R. Willis: Upper and lower bound for the overall properties of a nonlinear composite dielectric I. Random microgeometry.Proc. Royal Soc. London A 447 (1994) (to appear), 365–384. |
| Reference:
|
[25] L. Tartar: Estimations fines des coefficients homogenises.Ennio De Giorgi Colloquium, P. Kree (ed.), Pitman Publ., London, 1985, pp. 168–187. MR 0909716 |
| Reference:
|
[26] L. Tartar: H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations.Proc. Roy. Soc. Edinburgh 115A (1990), 193–230. MR 1069518 |
| Reference:
|
[27] S. Torquato: Random Heterogeneous media: Microstructures and improved bounds on effective properties.Appl. Mech. Rev. 44 (1991), 37–76. MR 1092035, 10.1115/1.3119494 |
| Reference:
|
[28] L. J. Walpole: On bounds for the overall elastic moduli of anisotropic composites.J. Mech. Phys. Solids 14 (1966), 151–162. 10.1016/0022-5096(66)90035-4 |
| Reference:
|
[29] J. R. Willis: Variational and related methods for the overall properties of composites.Advances in Applied Mechanics 21 (1981), 1–78. Zbl 0476.73053, MR 0706965, 10.1016/S0065-2156(08)70330-2 |
| Reference:
|
[30] J. R. Willis: The overall elastic response of composite materials.J. Appl. Mech. 50 (1983), 1202–1209. Zbl 0539.73003, 10.1115/1.3167202 |
| Reference:
|
[31] J. R. Willis: The structure of overall constitutive relations for a class of nonlinear composites.IMA J. appl. Math. 43 (1989), 231–242. Zbl 0695.73005, MR 1042634, 10.1093/imamat/43.3.231 |
| Reference:
|
[32] E. Zeidler: Nonlinear Functional Analysis and its Applications, Volume III.Springer-Verlag, 1990. |
| . |
