Homogenization of Elastic Media with Gaseous Inclusions

We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period $\varepsilon > 0$, and a perfect gas placed in each of these perforations, as $\varepsilon$ goes to zero. The model we use is linear, corresponding to deformations around a reference configuration. We apply both two-scale asymptotic expansion and two-scale convergence methods in order to identify the limit behaviors as $\varepsilon$ goes to 0. We state that in the limit, we get a two-scale linear elasticity-like boundary value problem. From this problem, we identify the corresponding homogenized and periodic cell equations which allow us to find the first corrector term. The analysis is performed both in the case of an incompressible and a compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we calculate numerically the homogenized coefficients in the incompressible case for different type...

[1]  Bertrand Maury,et al.  A multiscale/multimodel approach of the respiration tree. New trends in continuum mechanics , 2005 .

[2]  Ivan Hlaváček,et al.  On inequalities of Korn's type , 1970 .

[3]  Salah Naili,et al.  A one-dimensional model for the propagation of transient pressure waves through the lung. , 2002, Journal of biomechanics.

[4]  Jeannine Saint Jean Paulin,et al.  HOMOGENIZATION AND TWO-SCALE CONVERGENCE FOR A STOKES OR NAVIER-STOKES FLOW IN AN ELASTIC THIN POROUS MEDIUM , 1996 .

[5]  S. Mancini,et al.  An image-based computational model of oscillatory flow in the proximal part of tracheobronchial trees , 2005, Computer methods in biomechanics and biomedical engineering.

[6]  Doina Cioranescu,et al.  Homogenization in perforated domains with rapidly pulsing perforations , 2003 .

[7]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[8]  B. Wiggs,et al.  Comparison of the shear modulus of mature and immature rabbit lungs. , 1999, Journal of applied physiology.

[9]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[10]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[11]  Yves Achdou,et al.  Diffusion and propagation problems in some ramified domains with a fractal boundary , 2006 .

[12]  E. Weibel.  Morphometry of the Human Lung , 1965, Springer Berlin Heidelberg.

[13]  Yves Achdou,et al.  A Multiscale Numerical Method for Poisson Problems in Some Ramified Domains with a Fractal Boundary , 2006, Multiscale Model. Simul..

[14]  B. Sapoval,et al.  Interplay between geometry and flow distribution in an airway tree. , 2003, Physical review letters.

[15]  R Takaki,et al.  A three-dimensional model of the human pulmonary acinus. , 2000, Journal of applied physiology.

[16]  Albert Gjedde,et al.  The pathway for oxygen in brain. , 2003, APMIS. Supplementum.

[17]  J. Laidlaw,et al.  ANATOMY OF THE HUMAN BODY , 1967, The Ulster Medical Journal.

[18]  Grégoire Allaire,et al.  Bloch-wave homogenization for a spectral problem in fluid-solid structures , 1996 .

[19]  Nicolas Meunier,et al.  A viscoelastic model with non-local damping application to the human lungs , 2006 .

[20]  Mark A. Lewis,et al.  The Mechanics of Lung Tissue under High-Frequency Ventilation , 2001, SIAM J. Appl. Math..

[21]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[22]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations , 2003, Multiscale Model. Simul..

[23]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[24]  Carlos Conca,et al.  On the application of the homogenization theory to a class of problems arising in fluid mechanics , 1985 .

[25]  R. Bruce Lindsay,et al.  Nonhomogeneous Media and Vibration Theory, by Enrique Sanchez‐Palencia , 1981 .

[26]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[27]  Nicolas Meunier,et al.  OUTLET DISSIPATIVE CONDITIONS FOR AIR FLOW IN THE BRONCHIAL TREE , 2005 .

[28]  Quentin Grimal,et al.  Étude dans le domaine temporel de la propagation d'ondes élastiques en milieux stratifiés : modélisation de la réponse du thorax à un impact , 2003 .

[29]  Carlos Conca.  Numerical results on the homogenization of Stokes and Navier-Stokes equations modeling a class of problems from fluid mechanics , 1985 .

[30]  Yvon Maday,et al.  A Reduced Basis Element Method for Complex Flow Systems , 2006 .

[31]  C. Conca,et al.  Fluids And Periodic Structures , 1995 .

[32]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[33]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[34]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .