Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-12T15:23:35.762Z Has data issue: false hasContentIssue false
  • English
  • Français

Small amplitude homogenization applied to modelsof non-periodic fibrous materials

Published online by Cambridge University Press:  15 December 2007

David Manceau
David Manceau*
Affiliation:
Centre de Mathématiques, I.N.S.A. de Rennes & I.R.M.A.R., 20 avenue des Buttes de Coëmes, CS 14315 - 35043 Rennes Cedex, France. dmanceau@insa-rennes.fr
Get access

Abstract

In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.

Keywords

Non-periodic homogenizationfibrous materialsmall amplitudelow contrastconductionlinear elasticityH-measures.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 6 , November 2007, pp. 1061 - 1087
Copyright
© EDP Sciences, SMAI, 2007

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.)

Article purchase

Temporarily unavailable

References

G. Allaire, Shape Optimization by the Homogenization Method. Springer-Verlag, New York (2002).
G. Allaire and S. Gutiérrez, Optimal design in small amplitude homogenization. R.I. 576, École Polytechnique, C.M.A UMR-CNRS 7641 (2002).
M.J. Arts, A Mathematical Model of the Dynamics of the Left Ventricle. Ph.D. thesis, University of Limburg, The Netherlands (1978).
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for periodic Structures. North-Holland (1978).
M. Briane, Homogénéisation de materiaux fibrés et multi-couches. Ph.D. thesis, Université Paris 6, France (1990).
Briane, M., Three models of non periodic fibrous materials obtained by homogenization. RAIRO Modél. Math. Anal. Numér. 27 (1993) 759–775. CrossRef
Briane, M., Homogenization of a nonperiodic material. J. Math. Pures Appl. 73 (1994) 47–66.
Caillerie, D., Mourad, A. and Raoult, A., Towards a fibre-based constitutive law for the myocardium. ESAIM: Proc. 12 (2002) 25–30. CrossRef
D. Caillerie, A. Mourad and A. Raoult, Cell-to-muscle homogenization. Application to a constitutive law for the myocardium. ESAIM: M2AN 37 (2003) 681–698.
Chadwick, R.S., Mechanics of the left ventricle. Biophys J. 39 (1982) 279–288. CrossRef
Feit, T.S., Diastolic pressure-volume relations and distribution of pressure and fiber extension across the wall of a model of left ventricle. Biophys. J. 28 (1979) 143–166. CrossRef
Francfort, G. and Murat, F., Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94 (1986) 307–334. CrossRef
Gérard, P., Microlocal defect measures. Comm. Partial Diff. Equations 16 (1991) 1761–1794. CrossRef
G.A. Holzapfel, Nonlinear solid mechanics. A continuum approach for engineering. John Wiley and Sons, Ltd., Chichester (2000).
F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, A.V. Cherkaev and R.V. Kohn Eds., Progress in Nonlinear Differential Equations and their Applications, Birkaüser, Boston (1998) 21–43.
R.W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue, in Biomechanics of Soft Tissue in Cardiovascular Systems, G.A. Holzapfel and R.W. Ogden Eds., CISM Courses and Lectures Series 441, Springer, Wien (2003) 65–108.
Peskin, C.S., Fiber architecture of the left ventricular wall: an asymptotic analysis. Commun. Pure Appl. Math. 42 (1989) 79–113. CrossRef
Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellitiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 571–597.
A.J.M. Spencer, Constitutive theory for strongly anisotropic solids, in Continuum Theory of the Mechanics of Fiber-Reinforced Composites, A.J.M. Spencer Ed., CISM Courses and Lectures Notes 282, International Center for Mechanical Sciences, Springer, Wien (1984) 1–32.
D.D. Streeter, Gross morphology and fiber geometry of the heart, in Handbook of physiology. The cardiovascular system, R.M. Berne and N. Sperelakis Eds., Vol. 1, Williams and Wilkins, Baltimore (1979) 61–112.
L. Tartar, H-measures and Small Amplitude Homogenization, in Random Media and Composites, R.V. Kohn and G.W. Milton Eds., SIAM, Philadelphia (1989) 89–99.
L. Tartar, H-measures, a New approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edin. 115-A (1990) 193–230.
Tartar, L., Introduction, An to the Homogenization Method in Optimal Design. Springer Lecture Notes Math. 1740 (2000) 47–156. CrossRef