Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T10:27:51.564Z Has data issue: false hasContentIssue false
  • English
  • Français

Bloch wave homogenization of linear elasticity system

Published online by Cambridge University Press:  15 September 2005

Sista Sivaji Ganesh  and
Muthusamy Vanninathan
Sista Sivaji Ganesh
Affiliation:
IISc-TIFR Mathematics Programme, TIFR Centre, P.O. Box 1234, Bangalore 560012, India; [email protected]; [email protected]
Muthusamy Vanninathan
Affiliation:
IISc-TIFR Mathematics Programme, TIFR Centre, P.O. Box 1234, Bangalore 560012, India; [email protected]; [email protected]
Get access

Abstract

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 thelower Bloch spectrum. For the three dimensional linear elasticity system,the first eigenvalue is degenerate of multiplicity three and henceexistence of such a regular Bloch spectrum is not guaranteed. Theaim here is to develop all necessary spectral tools to overcome thesedifficulties. The existence of a directionally regular Bloch spectrum isproved and isused in the homogenization. As a consequence an interesting relation betweenhomogenization process and wave propagation in the homogenized medium isobtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequenceit is proved that higher modes apart from the first three do not contribute to the homogenization process.

Keywords

Bloch waveshomogenizationlinear elasticity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 4 , October 2005 , pp. 542 - 573
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., Homogenization and two scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. CrossRef
Allaire, G. and Conca, C., Bloch wave homogenization for a spectral problem in fluid-solid structures. Arch. Rational Mech. Anal. 135 (1996) 197–257. CrossRef
Allaire, G. and Conca, C., Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal. 29 (1997) 343–379. CrossRef
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North Holland, Amsterdam (1978).
C. Conca, S. Natesan and M. Vanninathan, 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.
Conca, C., Orive, R. and Vanninathan, M., Bloch approximation in homogenization and applications. SIAM J. Math. Anal. 33 (2002) 1166–1198. CrossRef
C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures. John Wiley & Sons, New York, and Masson, Paris (1995).
Conca, C. and Vanninathan, M., Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 (1997) 1639–1659. CrossRef
Conca, C. and Vanninathan, M., Fourier approach to homogenization. ESAIM: COCV 8 (2002) 489–511. CrossRef
A.P. Cracknell and K.C. Wong, The Fermi surface. Clarendon press, Oxford (1973).
G. Dal maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993).
Gérard, P., Microlocal defect measures. Commun. PDE 16 (1991) 1761–1794. CrossRef
Gérard, P., Markowich, P.A., Mauser, N.J. and Poupaud, F., Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323–379. 3.0.CO;2-C>CrossRef
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential operators and Integral functionals. Berlin, Springer-Verlag (1994).
T. Kato, Perturbation theory for linear operators. 2nd edition, Springer-Verlag, Berlin (1980).
F. Murat and L. Tartar, H-Convergence, Topics in the Mathematical Modeling of Composite Materials, A. Charkaev and R. Kohn Eds. PNLDE 31, Birkhäuser, Boston (1997).
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. CrossRef
O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization. North Holland, Amsterdam (1992).
F. Rellich, Perturbation theory of eigenvalue problems. Gordon and Breach science publishers, New York (1969).
M. Roseau, Vibrations in Mechanical systems: Analytical methods and applications. Springer-Verlag, Berlin (1987).
W. Rudin, Functional analysis. 2nd edition, Mc-Graw Hill, New York (1991).
J. Sínchez-Hubert and E. Sínchez-Palencia, Vibration and coupling of continuous systems: asymptotic methods. Springer-Verlag, Berlin (1989).
E. Sínchez-Palencia, Non-homogeneous media and vibration theory. Lect. Notes Phys. 127 (1980).
Santosa, F. and Symes, W.W., A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51 (1991) 984–1005. CrossRef
Sivaji Ganesh, S. and Vanninathan, M., Bloch wave homogenization of scalar elliptic operators. Asymptotic Analysis 39 (2004) 15–44.
Tartar, L., 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. CrossRef
Turbé, N., Applications of Bloch decomposition to periodic elastic and viscoelastic media. Math. Meth. Appl. Sci. 4 (1982) 433–449. CrossRef