Second-order estimates for the large-deformation response of particle-reinforced rubbers

Oscar Lopez-Pamies; Pedro Ponte Castañeda

doi:10.1016/S1631-0721(03)00021-4

Second-order estimates for the large-deformation response of particle-reinforced rubbers
[Estimations homogénéisées pour le comportement mécanique des caoutchoucs renforcés en grandes déformations]

Présenté par : Évariste Sanchez-Palencia

Oscar Lopez-Pamies ¹ ; Pedro Ponte Castañeda ¹

¹ Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

Comptes Rendus. Mécanique, Volume 331 (2003) no. 1, pp. 1-8.

Abstract (VO)
Résumé

This paper presents the application of a recently proposed ‘second-order’ homogenization method (J. Mech. Phys. Solids 50 (2002) 737–757) to the estimation of the effective behavior of hyperelastic composites subjected to finite deformations. The main feature of the method is the use of ‘generalized’ secant moduli that depend not only on the phases averages of the fields, but also on the phase covariance tensors. The use of the method is illustrated in the context of particle-, or fiber-reinforced elastomers and estimates analogous to the well-known Hashin–Shtrikman estimates for linear-elastic composites are generated. The new estimates improve on earlier estimates (J. Mech. Phys. Solids 48 (2000) 1389–1411) neglecting the use of fluctuations. In particular, the new estimates, unlike the earlier ones, are capable of recovering the exact incompressibility constraint when the matrix is also taken to be incompressible.

Cet article présente l'application d'une méthode d'homogénéisation récente, dite de deuxième ordre (J. Mech. Phys. Solids 50 (2002) 737–757), afin d'estimer le comportement effectif de matériaux composites hyperélastiques sujets à des grandes déformations. La principale caractéristique de cette méthode est l'utilisation d'un module sécant généralisé par phase qui dépend non seulement des moyennes des champs par phase mais aussi des tenseurs de covariance des fluctuations des phases. L'utilisation de cette procédure est illustrée dans le cas particulier d'élastomères renforcés à particules ou à fibres. Ces nouvelles estimations généralisent celles bien connues de Hashin–Shtrikman obtenues pour des composites élastiques linéaires et améliorent certaines estimations décrites antérieurement (J. Mech. Phys. Solids 48 (2000) 1389–1411) qui, elles, négligeaient l'existence de fluctuations des champs. En particulier, ces nouvelles estimations, contrairement a celles étudiées précédemment, sont capables d'établir la contrainte exacte d'incompressibilité lorsque la matrice est elle aussi supposée incompressible.

Reçu le : 2003-01-05
Accepté le : 2003-01-07
Publié le : 2003-01-01

DOI : 10.1016/S1631-0721(03)00021-4

Keywords: Computational solid mechanics, Homogenization, Finite deformations, Elastomers
Mots-clés : Mécanique des solides numérique, Homogénéisation, Déformations, Matériaux élastomères

Affiliations des auteurs :

Oscar Lopez-Pamies ¹ ; Pedro Ponte Castañeda ¹

¹ Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

@article{CRMECA_2003__331_1_1_0,
     author = {Oscar Lopez-Pamies and Pedro Ponte Casta\~neda},
     title = {Second-order estimates for the large-deformation response of particle-reinforced rubbers},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {1--8},
     publisher = {Elsevier},
     volume = {331},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-0721(03)00021-4},
     language = {en},
}

TY  - JOUR
AU  - Oscar Lopez-Pamies
AU  - Pedro Ponte Castañeda
TI  - Second-order estimates for the large-deformation response of particle-reinforced rubbers
JO  - Comptes Rendus. Mécanique
PY  - 2003
SP  - 1
EP  - 8
VL  - 331
IS  - 1
PB  - Elsevier
DO  - 10.1016/S1631-0721(03)00021-4
LA  - en
ID  - CRMECA_2003__331_1_1_0
ER  -

%0 Journal Article
%A Oscar Lopez-Pamies
%A Pedro Ponte Castañeda
%T Second-order estimates for the large-deformation response of particle-reinforced rubbers
%J Comptes Rendus. Mécanique
%D 2003
%P 1-8
%V 331
%N 1
%I Elsevier
%R 10.1016/S1631-0721(03)00021-4
%G en
%F CRMECA_2003__331_1_1_0

Oscar Lopez-Pamies; Pedro Ponte Castañeda. Second-order estimates for the large-deformation response of particle-reinforced rubbers. Comptes Rendus. Mécanique, Volume 331 (2003) no. 1, pp. 1-8. doi : 10.1016/S1631-0721(03)00021-4. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00021-4/

Bibliographie
Cité par

[1] R. Hill On constitutive macro-variables for heterogeneous solids at finite strain, Proc. Roy. Soc. London Ser. A, Volume 326 (1972), pp. 131-147

[2] S. Müller Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., Volume 99 (1987), pp. 189-212

[3] G. Geymonat; S. Müller; N. Triantafyllidis Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity, Arch. Rational Mech. Anal., Volume 122 (1993), pp. 231-290

[4] R. Ogden Extremun principles in non-linear elasticity and their application to composites – I, Theory, Int. J. Solids Structures, Volume 14 (1978), pp. 265-282

[5] P. Ponte Castañeda The overall constitutive behaviour of nonlinearly elastic composites, Proc. Roy. Soc. London Ser. A, Volume 422 (1989), pp. 147-171

[6] L. Mullins; N.R. Tobin Stress softening in rubber vulcanizates. Part I. Use of strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber, J. Appl. Polymer Sci., Volume 99 (1965), pp. 189-212

[7] L.R. Treolar The Physics of Rubber Elasticity, Oxford University Press, Oxford, 1975

[8] E.A. Meinecke; M.I. Taftaf Effect of carbon-black on the mechanical properties of elastomers, Rubber Chem. Technol., Volume 61 (1988), pp. 534-547

[9] S. Govindjee; J. Simo A micromechanically based continuum damage model for carbon black-filled rubbers incoporating Mullins' effect, J. Mech. Phys. Solids, Volume 39 (1991), pp. 87-112

[10] P. Ponte Castañeda; E. Tiberio A second-order homogenization procedure in finite elasticity and applications to black-filled elastomers, J. Mech. Phys. Solids, Volume 48 (2000), pp. 1389-1411

[11] N. Lahellec Estimates of the homogenized hyperelastic behavior of periodic fiber-reinforced composites using the second-order procedure, C. R. Acad. Sci. Paris IIb, Volume 329 (2001), pp. 67-73

[12] 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

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

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

[15] N. Laws On the thermostatics of composite materials, J. Mech. Phys. Solids, Volume 21 (1973), pp. 9-17

[16] P. Suquet; P. Ponte Castañeda Small-contrast perturbation expansions for the effective properties of nonlinear composites, C. R. Acad. Sci. Paris II, Volume 317 (1993), pp. 1515-1522

[17] Z. Hashin; S. Shtrikman A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, Volume 11 (1963), pp. 127-140

[18] V.M. Levin Thermal expansion coefficients of heterogeneous materials, Mekh. Tverd. Tela, Volume 2 (1967), pp. 83-94

S. Firooz; P. Steinmann; A. Javili Homogenization of Composites With Extended General Interfaces: Comprehensive Review and Unified Modeling, Applied Mechanics Reviews, Volume 73 (2021) no. 4 | DOI:10.1115/1.4051481
Donghai Li; Yin Yao An approximate method to predict the mechanical properties of small volume fraction particle-reinforced composites with large deformation matrix, Acta Mechanica, Volume 230 (2019) no. 9, p. 3307 | DOI:10.1007/s00707-019-02444-5
Saba Saeb; Paul Steinmann; Ali Javili Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound, Applied Mechanics Reviews, Volume 68 (2016) no. 5 | DOI:10.1115/1.4034024
Jaehyung Ju; Joshua D. Summers Hyperelastic Constitutive Modeling of Hexagonal Honeycombs Subjected to In-Plane Shear Loading, Journal of Engineering Materials and Technology, Volume 133 (2011) no. 1 | DOI:10.1115/1.4002640
I. Doghri; L. Adam; N. Bilger Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method, International Journal of Plasticity, Volume 26 (2010) no. 2, pp. 219-238 | DOI:10.1016/j.ijplas.2009.06.003 | Zbl:1426.74111
Qingsheng Yang; Fang Xu Numerical modeling of nonlinear deformation of polymer composites based on hyperelastic constitutive law, Frontiers of Mechanical Engineering in China (2009) | DOI:10.1007/s11465-009-0067-0
J. C. Michel; O. Lopez-Pamies; P. Ponte Castañeda; N. Triantafyllidis Microscopic and macroscopic instabilities in finitely strained porous elastomers, Journal of the Mechanics and Physics of Solids, Volume 55 (2007) no. 5, pp. 900-938 | DOI:10.1016/j.jmps.2006.11.006 | Zbl:1170.74018
Qing Sheng Yang; Fang Xu Effective Hyperelastic Behaviour of Fiber Reinforced Polymer Composite Materials, Key Engineering Materials, Volume 334-335 (2007), p. 473 | DOI:10.4028/www.scientific.net/kem.334-335.473
Boris Guiot; Carole Nadot-Martin; André Dragon Towards a non-linear micromechanics-based analysis for particulate composites, Composites Science and Technology, Volume 66 (2006) no. 15, p. 2726 | DOI:10.1016/j.compscitech.2006.03.006
N. Triantafyllidis; M. D. Nestorović; M. W. Schraad Failure Surfaces for Finitely Strained Two-Phase Periodic Solids Under General In-Plane Loading, Journal of Applied Mechanics, Volume 73 (2006) no. 3, p. 505 | DOI:10.1115/1.2126695
Q.-C. He; H. Le Quang; Z.-Q. Feng Exact results for the homogenization of elastic fiber-reinforced solids at finite strain, Journal of Elasticity, Volume 83 (2006) no. 2, pp. 153-177 | DOI:10.1007/s10659-006-9049-1 | Zbl:1103.74049
Boris Guiot; Carole Nadot-Martin; André Dragon A multiscale (visco)-hyperelastic modelling for particulate composites, Mechanics Research Communications, Volume 33 (2006) no. 4, p. 441 | DOI:10.1016/j.mechrescom.2006.01.004
P. Ponte Castañeda Linear Comparison Methods for Nonlinear Composites, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, Volume 170 (2005), p. 247 | DOI:10.1007/1-4020-2623-4_11
Oscar Lopez-Pamies; Pedro Ponte Castañeda Second-order estimates for the macroscopic response and loss of ellipticity in porous rubbers at large deformations, Journal of Elasticity, Volume 76 (2004) no. 3, pp. 247-287 | DOI:10.1007/s10659-005-1405-z | Zbl:1086.74032

Cité par 14 documents. Sources : Crossref, zbMATH

Commentaires - Politique