Second-order estimates for nonlinear isotropic composites with spherical pores and rigid particles

Martín Idiart; Pedro Ponte Castañeda

doi:10.1016/j.crme.2004.12.001

Second-order estimates for nonlinear isotropic composites with spherical pores and rigid particles

Presented by: Pierre Suquet

Martín Idiart ^1,²; Pedro Ponte Castañeda ^1,²

¹ Département de mécanique, École polytechnique, 91128 Palaiseau, France
² Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

Comptes Rendus. Mécanique, Volume 333 (2005) no. 2, pp. 147-154.

Abstract
Résumé

The ‘second-order’ nonlinear homogenization method (Ponte Castañeda, J. Mech. Phys. Solids 50 (2002) 737–757) is used to generate estimates of the Hashin–Shtrikman-type for the effective behavior of viscoplastic materials with isotropically distributed spherical pores or rigid particles. In the limiting case of an ideally plastic matrix with a dilute concentration of pores, the resulting estimates were found to exhibit a linear dependence on the porosity when the material is subjected to axisymmetric shear, but this dependence becomes singular for simple shear. In the process of this work, an alternative prescription for certain reference tensors used in the method is proposed, and shown to lead to more consistent estimates for the effective behavior than the earlier prescription.

On utilise la méthode d'homogénéisation non linéaire proposée par Ponte Castañeda (J. Mech. Phys. Solids 50 (2002) 737–757), dite du second ordre, pour générer des estimations du type Hashin–Shtrikman pour le comportement effectif des matériaux viscoplastiques contenant des pores et des particules rigides sphériques. Dans le cas limite d'une matrice parfaitement plastique à faible concentration de pores, les estimations trouvées présentent une dépendance linéaire de la porosité sous un chargement de cisaillement axisymmétrique ; cependant cette dépendance devient singulière sous cisaillement simple. Lors de ce travail, certaines limites de la formulation de la méthode initialement proposée dans la référence ci-dessus ont été identifiées. En conséquence, des alternatives ont été testées.

Received: 2004-06-22
Accepted: 2004-12-07
Published online: 2005-01-08

DOI: 10.1016/j.crme.2004.12.001

Keywords: Porous media, Homogenization, Nonlinear behavior, Ideal plasticity
Mots-clés : Milieux poreux, Homogénéisation, Comportement non linéaire, Plasticité parfaite

Author's affiliations:

Martín Idiart ^{1, 2}; Pedro Ponte Castañeda ^{1, 2}

¹ Département de mécanique, École polytechnique, 91128 Palaiseau, France
² Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

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     author = {Mart{\'\i}n Idiart and Pedro Ponte Casta\~neda},
     title = {Second-order estimates for nonlinear isotropic composites with spherical pores and rigid particles},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {147--154},
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Martín Idiart; Pedro Ponte Castañeda. Second-order estimates for nonlinear isotropic composites with spherical pores and rigid particles. Comptes Rendus. Mécanique, Volume 333 (2005) no. 2, pp. 147-154. doi : 10.1016/j.crme.2004.12.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.12.001/

References
Cited by

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[2] 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

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

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

[5] J.R. Willis Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. Mech. Phys. Solids, Volume 25 (1977), pp. 185-202

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

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

[8] P. Ponte Castañeda Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: II – Applications, J. Mech. Phys. Solids, Volume 50 (2002), pp. 759-782

[9] D.C. Drucker The continuum theory of plasticity on the macroscale and the microscale, J. Mater., Volume 1 (1966), pp. 873-910

[10] J. Pastor; P. Ponte Castañeda Yield criteria for porous media in plane strain: second-order estimates versus numerical results, C. R. Mecanique, Volume 330 (2002), pp. 741-747

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