A methodology for an accurate evaluation of the linearization procedures in nonlinear mean field homogenization

Amna Rekik; Michel Bornert; François Auslender; André Zaoui

doi:10.1016/j.crme.2005.10.002

A methodology for an accurate evaluation of the linearization procedures in nonlinear mean field homogenization
[Une méthodologie pour l'évaluation précise des procédures de linéarisation en homogénéisation non linéaire en champs moyens]

Présenté par : André Zaoui

Amna Rekik ¹ ; Michel Bornert ¹ ; François Auslender ^1,² ; André Zaoui ¹

¹ Laboratoire de mécanique des solides, École polytechnique, CNRS UMR 7649, 91128 Palaiseau cedex, France
² Laboratoire de mécanique et ingénieries, université Blaise Pascal/IFMA, 24, avenue des landais, 63170 Aubière, France

Comptes Rendus. Mécanique, Volume 333 (2005) no. 11, pp. 789-795.

Résumé
Abstract

Une méthodologie systématique pour l'évaluation des procédures de linéarisation sur lesquelles s'appuient les théories d'homogénéisation en champs moyens pour les composites non linéaires est présentée et appliquée à titre d'illustration pour l'élasticité non linéaire aux différentes variantes des formulations << affine >> et << au second ordre >> récemment proposées. Elle repose sur l'analyse de composites pour lesquels on résout numériquement avec la même précision et pour la même microstructure aussi bien le problème d'homogénéisation non linéaire complet que le problème d'homogénéisation linéaire relatif au << milieu linéaire de comparaison >> défini par la procédure de linéarisation. La comparaison des résultats permet ainsi d'évaluer rigoureusement les seuls effets de la méthode de linéarisation.

A systematic methodology for the evaluation of the linearization procedures sustaining mean field homogenization theories for nonlinear composite materials is proposed and applied as an illustration to various recently proposed ‘affine’ and ‘second-order’ formulations for nonlinear elasticity. It relies on the analysis of composites for which both the exact nonlinear homogenization problem and the homogenization problem associated with the ‘linear comparison material’ defined by the linearization procedure can be solved numerically with the same accuracy and for the same microstructure. The comparison of the results then provides a rigorous evaluation of the effects of the sole linearization method.

Reçu le : 2005-06-30
Accepté le : 2005-10-04
Publié le : 2005-11-01

DOI : 10.1016/j.crme.2005.10.002

Keywords: Continuum mechanics, Homogenization, Nonlinear behavior, Linearization, Affine, Second order
Mots-clés : Milieux continus, Homogénéisation, Comportement non linéaire, Linéarisation, Affine, Second ordre

Affiliations des auteurs :

Amna Rekik ¹ ; Michel Bornert ¹ ; François Auslender ^{1, 2} ; André Zaoui ¹

¹ Laboratoire de mécanique des solides, École polytechnique, CNRS UMR 7649, 91128 Palaiseau cedex, France
² Laboratoire de mécanique et ingénieries, université Blaise Pascal/IFMA, 24, avenue des landais, 63170 Aubière, France

@article{CRMECA_2005__333_11_789_0,
     author = {Amna Rekik and Michel Bornert and Fran\c{c}ois Auslender and Andr\'e Zaoui},
     title = {A methodology for an accurate evaluation of the linearization procedures in nonlinear mean field homogenization},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {789--795},
     publisher = {Elsevier},
     volume = {333},
     number = {11},
     year = {2005},
     doi = {10.1016/j.crme.2005.10.002},
     language = {en},
}

TY  - JOUR
AU  - Amna Rekik
AU  - Michel Bornert
AU  - François Auslender
AU  - André Zaoui
TI  - A methodology for an accurate evaluation of the linearization procedures in nonlinear mean field homogenization
JO  - Comptes Rendus. Mécanique
PY  - 2005
SP  - 789
EP  - 795
VL  - 333
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crme.2005.10.002
LA  - en
ID  - CRMECA_2005__333_11_789_0
ER  -

%0 Journal Article
%A Amna Rekik
%A Michel Bornert
%A François Auslender
%A André Zaoui
%T A methodology for an accurate evaluation of the linearization procedures in nonlinear mean field homogenization
%J Comptes Rendus. Mécanique
%D 2005
%P 789-795
%V 333
%N 11
%I Elsevier
%R 10.1016/j.crme.2005.10.002
%G en
%F CRMECA_2005__333_11_789_0

Amna Rekik; Michel Bornert; François Auslender; André Zaoui. A methodology for an accurate evaluation of the linearization procedures in nonlinear mean field homogenization. Comptes Rendus. Mécanique, Volume 333 (2005) no. 11, pp. 789-795. doi : 10.1016/j.crme.2005.10.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.10.002/

Bibliographie
Cité par

[1] P. Ponte Castañeda; P. Suquet Nonlinear composites, Adv. Appl. Mech., Volume 34 (1998), pp. 171-302

[2] P. Ponte Castañeda The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids, Volume 39 (1991), pp. 45-71

[3] R. Masson; M. Bornert; P. Suquet; A. Zaoui An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals, J. Mech. Phys. Solids, Volume 48 (2000), pp. 1203-1227

[4] P. Ponte Castañeda Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. Mech. Phys. Solids, Volume 44 (1996), pp. 827-862

[5] M. Bornert; R. Masson; P. Ponte Castañeda; A. Zaoui Second-order estimates for the effective behaviour of viscoplastic polycrystalline materials, J. Mech. Phys. Solids, Volume 49 (2001), pp. 2737-2764

[6] P. Ponte Castañeda Second-order homogenisation estimates for nonlinear composites incorporating field fluctuations: I – Theory, J. Mech. Phys. Solids, Volume 50 (2002), pp. 737-757

[7] J.L. Chaboche; P. Kanouté Sur les approximations « isotrope » et « anisotrope » de l'opérateur tangent pour les méthodes tangentes incrémentale et affine, C. R. Mécanique, Volume 331 (2003), pp. 857-864

[8] N. Lahellec; P. Suquet Nonlinear composites: a linearization procedure, exact to second-order in contrast and for which the strain-energy and affine formulations coincide, C. R. Mécanique, Volume 332 (2004), pp. 693-700

[9] M. Bornert; P. Ponte Castañeda Second-order estimates of the self-consistent type for viscoplastic polycrystals, Proc. Roy. Soc. London Ser. A, Volume 454 (1998), pp. 3035-3045

[10] P. Ponte Castañeda; P. Suquet On the effective mechanical behavior of weakly inhomogeneous nonlinear composites, Eur. J. Mech. A Solids, Volume 14 (1995), pp. 205-236

[11] H. Moulinec; P. Suquet Intraphase strain heterogeneity in nonlinear composites: a computational approach, Eur. J. Mech. A Solids, Volume 22 (2003), pp. 751-770

[12] T. Kanit; S. Forest; I. Galliet; V. Mounoury; D. Jeulin Determination of the size of the representative volume element for random composites: statistical and numerical approach, Int. J. Solids Struct., Volume 40 (2003), pp. 3647-3679

[13] P. Suquet Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castañeda's nonlinear variational procedure, C. R. Mécanique, Volume 320 (1995), pp. 563-571

[14] J.C. Michel; H. Moulinec; P. Suquet Composites périodiques (M. Bornert; T. Bretheau; P. Gilormini, eds.), Homogénéisation en mécanique des matériaux, tome 1, Hermes Science Publications, 2001, pp. 57-94

[15] L.J. Walpole Elastic behavior of composite materials: theoretical foundations, Adv. Appl. Mech., vol. 21, Academic Press, New York, 1981, pp. 160-242

[16] P. Suquet; M. Bornert Calcul tensoriel et élasticité (M. Bornert; T. Bretheau; P. Gilormini, eds.), Homogénéisation en mécanique des matériaux, tome 2, Hermes Science Publications, 2001, pp. 171-202

[17] R.M. Christensen; K.H. Lo Solution for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, Volume 27 (1979), pp. 315-330

[18] A. Rekik, M. Bornert, F. Auslender, A. Zaoui, Evaluation précise des procédures de linéarisation en homogénéisation nonlinéaire, in: 17ème Congres Français de Mécanique, Troyes, France, 2005

Martín I. Idiart; Loïc Chaix; Pedro Ponte Castañeda A homogenization scheme for viscoplastic composites based on the plane-wave decomposition of constitutive potentials, Mechanics Research Communications, Volume 142 (2024), p. 104349 | DOI:10.1016/j.mechrescom.2024.104349
Yi Cheng; Kaifu Zhang; Biao Liang; Hui Cheng; Ping Liu Incremental sequential homogenization on effective elastoplastic properties of composites with functionally graded interphase, Mechanics of Materials, Volume 167 (2022), p. 104211 | DOI:10.1016/j.mechmat.2022.104211
Manuel Araújo; Markus Merkel; Andreas Öchsner Laser Beam Drilling of Cellular Metals: Numerical Simulation, Computational Modeling, Optimization and Manufacturing Simulation of Advanced Engineering Materials, Volume 49 (2016), p. 271 | DOI:10.1007/978-3-319-04265-7_10
A. Tchalla; W.L. Azoti; Y. Koutsawa; A. Makradi; S. Belouettar; H. Zahrouni Incremental mean-fields micromechanics scheme for non-linear response of ductile damaged composite materials, Composites Part B: Engineering, Volume 69 (2015), p. 169 | DOI:10.1016/j.compositesb.2014.08.055
Amna Rekik; François Auslender; Michel Bornert A set of enhanced formulations for existing nonlinear homogenization schemes and their evaluation, European Journal of Mechanics. A. Solids, Volume 50 (2015), pp. 1-16 | DOI:10.1016/j.euromechsol.2014.10.001 | Zbl:1406.74565
W.L. Azoti; Y. Koutsawa; A. Tchalla; A. Makradi; S. Belouettar Micromechanics-based multi-site modeling of elastoplastic behavior of composite materials, International Journal of Solids and Structures, Volume 59 (2015), p. 198 | DOI:10.1016/j.ijsolstr.2015.02.002
Manuel António Ferreira Araújo; Markus Merkel; Andreas Öchsner Heat Conductivity of Metallic Hollow Sphere Structures: Numerical Simulation and Experimental Validation, Defect and Diffusion Forum, Volume 336 (2013), p. 209 | DOI:10.4028/www.scientific.net/ddf.336.209
Amna Rekik; Michel Bornert; François Auslender A critical evaluation of local field statistics predicted by various linearization schemes in nonlinear mean-field homogenization, Mechanics of Materials, Volume 54 (2012), p. 1 | DOI:10.1016/j.mechmat.2012.05.011
Serge Dumont; Frédéric Lebon Effective properties of linear random materials: Application to Al/SiC and resin/glass composites, Computational Mechanics, Volume 42 (2008) no. 5, pp. 775-786 | DOI:10.1007/s00466-008-0280-4 | Zbl:1163.74541
E. Héripré; M. Dexet; J. Crépin; L. Gélébart; A. Roos; M. Bornert; D. Caldemaison Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials, International Journal of Plasticity, Volume 23 (2007) no. 9, p. 1512 | DOI:10.1016/j.ijplas.2007.01.009
Noël Lahellec; Pierre Suquet On the effective behavior of nonlinear inelastic composite. II: A second-order procedure, Journal of the Mechanics and Physics of Solids, Volume 55 (2007) no. 9, pp. 1964-1992 | DOI:10.1016/j.jmps.2007.02.004 | Zbl:1170.74045

Cité par 11 documents. Sources : Crossref, zbMATH

Commentaires - Politique