A general homogenization result of spectral problem for linearized elasticity in perforated domains

Mohamed Mourad Lhannafi Ait Yahia; Hamid Haddadou

Applications of Mathematics (2021)

  • Volume: 66, Issue: 5, page 701-724
  • ISSN: 0862-7940

Abstract

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The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the -convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor , the -limit of , is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method of test functions, and some general arguments of spectral analysis used in the literature on the homogenization of eigenvalue problems. Moreover, we give a result on a particular case of a simple eigenvalue of the homogenized problem. We conclude our work by some comments and perspectives.

How to cite

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Ait Yahia, Mohamed Mourad Lhannafi, and Haddadou, Hamid. "A general homogenization result of spectral problem for linearized elasticity in perforated domains." Applications of Mathematics 66.5 (2021): 701-724. <http://eudml.org/doc/297514>.

@article{AitYahia2021,
abstract = {The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^0$-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^0$, the $H^0$-limit of $A^\{\varepsilon \}$, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method of test functions, and some general arguments of spectral analysis used in the literature on the homogenization of eigenvalue problems. Moreover, we give a result on a particular case of a simple eigenvalue of the homogenized problem. We conclude our work by some comments and perspectives.},
author = {Ait Yahia, Mohamed Mourad Lhannafi, Haddadou, Hamid},
journal = {Applications of Mathematics},
keywords = {homogenization; $H$-convergence; perforated domain; linear elasticity; eigenvalue problem},
language = {eng},
number = {5},
pages = {701-724},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A general homogenization result of spectral problem for linearized elasticity in perforated domains},
url = {http://eudml.org/doc/297514},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Ait Yahia, Mohamed Mourad Lhannafi
AU - Haddadou, Hamid
TI - A general homogenization result of spectral problem for linearized elasticity in perforated domains
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 5
SP - 701
EP - 724
AB - The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^0$-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^0$, the $H^0$-limit of $A^{\varepsilon }$, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method of test functions, and some general arguments of spectral analysis used in the literature on the homogenization of eigenvalue problems. Moreover, we give a result on a particular case of a simple eigenvalue of the homogenized problem. We conclude our work by some comments and perspectives.
LA - eng
KW - homogenization; $H$-convergence; perforated domain; linear elasticity; eigenvalue problem
UR - http://eudml.org/doc/297514
ER -

References

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