In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process.
Mots-clés : Bloch waves, homogenization, linear elasticity
@article{COCV_2005__11_4_542_0,
author = {Ganesh, Sista Sivaji and Vanninathan, Muthusamy},
title = {Bloch wave homogenization of linear elasticity system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {542--573},
publisher = {EDP-Sciences},
volume = {11},
number = {4},
year = {2005},
doi = {10.1051/cocv:2005018},
mrnumber = {2167874},
zbl = {1103.35017},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2005018/}
}
TY - JOUR AU - Ganesh, Sista Sivaji AU - Vanninathan, Muthusamy TI - Bloch wave homogenization of linear elasticity system JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 542 EP - 573 VL - 11 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005018/ DO - 10.1051/cocv:2005018 LA - en ID - COCV_2005__11_4_542_0 ER -
%0 Journal Article %A Ganesh, Sista Sivaji %A Vanninathan, Muthusamy %T Bloch wave homogenization of linear elasticity system %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 542-573 %V 11 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2005018/ %R 10.1051/cocv:2005018 %G en %F COCV_2005__11_4_542_0
Ganesh, Sista Sivaji; Vanninathan, Muthusamy. Bloch wave homogenization of linear elasticity system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 4, pp. 542-573. doi : 10.1051/cocv:2005018. http://archive.numdam.org/articles/10.1051/cocv:2005018/
[1] , Homogenization and two scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | Zbl
[2] and, Bloch wave homogenization for a spectral problem in fluid-solid structures. Arch. Rational Mech. Anal. 135 (1996) 197-257. | Zbl
[3] and, Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal. 29 (1997) 343-379. | Zbl
[4] , and, Asymptotic analysis for periodic structures. North Holland, Amsterdam (1978). | MR | Zbl
[5] , and, Numerical solution of elliptic partial differential equations by Bloch waves method, XVII CEDYA: Congress on differential equations and applications/VII CMA: Congress on applied mathematics, Dep. Mat. Appl., Univ. Salamanca, Salamanca (2001) 63-83. | Zbl
[6] , and, Bloch approximation in homogenization and applications. SIAM J. Math. Anal. 33 (2002) 1166-1198. | Zbl
[7] , and, Fluids and periodic structures. John Wiley & Sons, New York, and Masson, Paris (1995). | MR | Zbl
[8] and, Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 (1997) 1639-1659. | Zbl
[9] and, Fourier approach to homogenization. ESAIM: COCV 8 (2002) 489-511. | Numdam | Zbl
[10] and, The Fermi surface. Clarendon press, Oxford (1973).
[11] maso, An introduction to -convergence. Birkhäuser, Boston (1993). | MR | Zbl
[12] , Microlocal defect measures. Commun. PDE 16 (1991) 1761-1794. | Zbl
[13] ,, and, Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323-379. | Zbl
[14] , and, Homogenization of Differential operators and Integral functionals. Berlin, Springer-Verlag (1994). | MR | Zbl
[15] , Perturbation theory for linear operators. 2nd edition, Springer-Verlag, Berlin (1980). | Zbl
[16] and, H-Convergence, Topics in the Mathematical Modeling of Composite Materials, A. Charkaev and R. Kohn Eds. PNLDE 31, Birkhäuser, Boston (1997). | MR | Zbl
[17] , A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | Zbl
[18] , and, Mathematical problems in elasticity and homogenization. North Holland, Amsterdam (1992). | MR | Zbl
[19] , Perturbation theory of eigenvalue problems. Gordon and Breach science publishers, New York (1969). | MR | Zbl
[20] , Vibrations in Mechanical systems: Analytical methods and applications. Springer-Verlag, Berlin (1987). | MR | Zbl
[21] , Functional analysis. 2nd edition, Mc-Graw Hill, New York (1991). | MR | Zbl
[22] and, Vibration and coupling of continuous systems: asymptotic methods. Springer-Verlag, Berlin (1989). | MR | Zbl
[23] , Non-homogeneous media and vibration theory. Lect. Notes Phys. 127 (1980). | Zbl
[24] and, A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51 (1991) 984-1005. | Zbl
[25] and, Bloch wave homogenization of scalar elliptic operators. Asymptotic Analysis 39 (2004) 15-44. | Zbl
[26] , H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edin. Sect. A 115 (1990) 193-230. | Zbl
[27] , Applications of Bloch decomposition to periodic elastic and viscoelastic media. Math. Meth. Appl. Sci. 4 (1982) 433-449. | Zbl
Cité par Sources :
