Small amplitude homogenization applied to models of non-periodic fibrous materials
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1061-1087.

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.

DOI : 10.1051/m2an:2007050
Classification : 35J25, 74Q15, 74B05
Mots clés : non-periodic homogenization, fibrous material, small amplitude, low contrast, conduction, linear elasticity, $H$-measures
@article{M2AN_2007__41_6_1061_0,
     author = {Manceau, David},
     title = {Small amplitude homogenization applied to models of non-periodic fibrous materials},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1061--1087},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {6},
     year = {2007},
     doi = {10.1051/m2an:2007050},
     mrnumber = {2377107},
     zbl = {1126.92006},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007050/}
}
TY  - JOUR
AU  - Manceau, David
TI  - Small amplitude homogenization applied to models of non-periodic fibrous materials
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2007
SP  - 1061
EP  - 1087
VL  - 41
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007050/
DO  - 10.1051/m2an:2007050
LA  - en
ID  - M2AN_2007__41_6_1061_0
ER  - 
%0 Journal Article
%A Manceau, David
%T Small amplitude homogenization applied to models of non-periodic fibrous materials
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2007
%P 1061-1087
%V 41
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2007050/
%R 10.1051/m2an:2007050
%G en
%F M2AN_2007__41_6_1061_0
Manceau, David. Small amplitude homogenization applied to models of non-periodic fibrous materials. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1061-1087. doi : 10.1051/m2an:2007050. http://www.numdam.org/articles/10.1051/m2an:2007050/

[1] G. Allaire, Shape Optimization by the Homogenization Method. Springer-Verlag, New York (2002). | MR | Zbl

[2] G. Allaire and S. Gutiérrez, Optimal design in small amplitude homogenization. R.I. 576, École Polytechnique, C.M.A UMR-CNRS 7641 (2002).

[3] M.J. Arts, A Mathematical Model of the Dynamics of the Left Ventricle. Ph.D. thesis, University of Limburg, The Netherlands (1978).

[4] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for periodic Structures. North-Holland (1978). | MR | Zbl

[5] M. Briane, Homogénéisation de materiaux fibrés et multi-couches. Ph.D. thesis, Université Paris 6, France (1990).

[6] M. Briane, Three models of non periodic fibrous materials obtained by homogenization. RAIRO Modél. Math. Anal. Numér. 27 (1993) 759-775. | Numdam | Zbl

[7] M. Briane, Homogenization of a nonperiodic material. J. Math. Pures Appl. 73 (1994) 47-66. | Zbl

[8] D. Caillerie, A. Mourad and A. Raoult, Towards a fibre-based constitutive law for the myocardium. ESAIM: Proc. 12 (2002) 25-30. | Zbl

[9] 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. | Numdam | Zbl

[10] R.S. Chadwick, Mechanics of the left ventricle. Biophys J. 39 (1982) 279-288.

[11] T.S. Feit, 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.

[12] G. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94 (1986) 307-334. | Zbl

[13] P. Gérard, Microlocal defect measures. Comm. Partial Diff. Equations 16 (1991) 1761-1794. | Zbl

[14] G.A. Holzapfel, Nonlinear solid mechanics. A continuum approach for engineering. John Wiley and Sons, Ltd., Chichester (2000). | MR | Zbl

[15] 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. | Zbl

[16] 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.

[17] C.S. Peskin, Fiber architecture of the left ventricular wall: an asymptotic analysis. Commun. Pure Appl. Math. 42 (1989) 79-113. | Zbl

[18] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellitiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 571-597. | Numdam | Zbl

[19] 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. | Zbl

[20] 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.

[21] 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. | Zbl

[22] 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. | Zbl

[23] L. Tartar, An Introduction to the Homogenization Method in Optimal Design. Springer Lecture Notes Math. 1740 (2000) 47-156. | Zbl

Cité par Sources :