On some features of the effective behaviour of porous solids with J2- and J3-dependent yielding matrix behaviour

Ahmed Benallal

doi:10.1016/j.crme.2017.11.002

On some features of the effective behaviour of porous solids with J₂- and J₃-dependent yielding matrix behaviour

Ahmed Benallal ¹

¹ LMT, ENS Paris-Saclay/CNRS/Université Paris-Saclay, France

Comptes Rendus. Mécanique, Volume 346 (2018) no. 2, pp. 77-88.

Abstract
Résumé

Some features od the constitutive behaviour of voided materials taking into account possible effects of the Lode angle in the yielding behaviour of the matrix are discussed. The Gurson approach is used to this end. After providing a parametric representation of the effective behaviour of such materials, some closed-form results are given for pure shear stress states and also at very high stress triaxialities. In the former case corresponding to a zero macroscopic mean stress, the contour of the yield domain in the π-plane has exactly the shape of the yield surface of the matrix in the deviatoric plane, but a size reduced by a factor $1 - f$ , with f the porosity of the voided material. In the latter, effective yield stresses for the voided material are slightly different from the Gurson result and found to be set by the yield stress at a microscopic stress Lode angle $\frac{π}{3}$ for very high positive triaxiality and by the yield stress at a microscopic stress Lode angle 0 for very high negative triaxiality. This last result is extended for porous materials with yielding depending further on the hydrostatic stress, fully exhibiting the interaction between volumetric and shear interactions on the yielding behaviour of isotropic porous materials. Applications to many usual yielding criteria for the matrix are also provided.

Les effets du troisième invariant des contraintes sur la surface de charge macroscopique d'un matériau ductile poreux sont analysés dans le cadre de l'approche de Gurson. Ces effets proviennent du processus d'homogénéisation lui-même à cause de l'hétérogénéité des contraintes dans la cellule de Gurson ou encore lorsque le comportement plastique de la matrice dépend du troisième invariant des contraintes. On fournit une représentation paramétrique de la surface de charge valable pour un comportement de la matrice assez général qui permet d'exhiber quelques résultats analytiques, en particulier pour les états de contrainte hydrostatiques et pour des états de cisaillement. Les résultats obtenus pour les chargements hydrostatiques sont étendus au cas où la matrice a un comportement dépendant aussi de la contrainte hydrostatique.

Received: 2017-02-18
Accepted: 2017-11-08
Published online: 2017-12-19

DOI: 10.1016/j.crme.2017.11.002

Keywords: Porous material, Lode angle, Effective yielding
Mot clés : Matériau poreux, Angle de Lode, Comportement effectif

Author's affiliations:

Ahmed Benallal ¹

¹ LMT, ENS Paris-Saclay/CNRS/Université Paris-Saclay, France

@article{CRMECA_2018__346_2_77_0,
     author = {Ahmed Benallal},
     title = {On some features of the effective behaviour of porous solids with {\protect\emph{J}\protect\textsubscript{2}-} and {\protect\emph{J}\protect\textsubscript{3}-dependent} yielding matrix behaviour},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {77--88},
     publisher = {Elsevier},
     volume = {346},
     number = {2},
     year = {2018},
     doi = {10.1016/j.crme.2017.11.002},
     language = {en},
}

TY  - JOUR
AU  - Ahmed Benallal
TI  - On some features of the effective behaviour of porous solids with J2- and J3-dependent yielding matrix behaviour
JO  - Comptes Rendus. Mécanique
PY  - 2018
SP  - 77
EP  - 88
VL  - 346
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crme.2017.11.002
LA  - en
ID  - CRMECA_2018__346_2_77_0
ER  -

%0 Journal Article
%A Ahmed Benallal
%T On some features of the effective behaviour of porous solids with J2- and J3-dependent yielding matrix behaviour
%J Comptes Rendus. Mécanique
%D 2018
%P 77-88
%V 346
%N 2
%I Elsevier
%R 10.1016/j.crme.2017.11.002
%G en
%F CRMECA_2018__346_2_77_0

Ahmed Benallal. On some features of the effective behaviour of porous solids with J₂- and J₃-dependent yielding matrix behaviour. Comptes Rendus. Mécanique, Volume 346 (2018) no. 2, pp. 77-88. doi : 10.1016/j.crme.2017.11.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.11.002/

References
Cited by

[1] A. Benallal; R. Desmorat; M. Fournage An assessment of the roles of the third invariant of stress in the Gurson approach for ductile fracture, Eur. J. Mech. A, Solids, Volume 47 (2014), pp. 400-414

[2] A.L. Gurson Continuum theory of ductile rupture by void nucleation and growth – Part I. Yield criteria and flow rules for porous ductilemedia, J. Eng. Mater. Technol., Volume 99 (1977), pp. 2-15

[3] F.A. McClintock Plasticity aspects of fracture (H. Leibowitz, ed.), Fracture, vol. 3, Academic Press, 1971, pp. 47-225

[4] G.R. Johnson; W.H. Cook Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Eng. Fract. Mech., Volume 21 (1985) no. 1, pp. 31-48

[5] Y. Bao; T. Wierzbicki On fracture locus in the equivalent strain and stress triaxiality space, Int. J. Mech. Sci., Volume 46 (2004), pp. 81-98

[6] I. Barsoum; J. Faleskog Rupture in combined tension and shear: experiments, Int. J. Solids Struct., Volume 44 (2007), pp. 1768-1786

[7] T. Borvik et al. A computational model of viscoplasticity and ductile damage for impact and penetration, Eur. J. Mech. A, Solids, Volume 20 (2001), pp. 685-712

[8] K. Nahshon; J.W. Hutchinson Modification of the Gurson model for shear failure, Eur. J. Mech. A, Solids, Volume 27 (2008), pp. 1-17

[9] K. Danas; M.I. Idiart; P. Ponte Castaneda A homogenization-based constitutive model for isotropic viscoplastic porous media, Int. J. Solids Struct., Volume 45 (2008), pp. 3392-3409

[10] K. Danas; P. Ponte Castañeda A finite-strain model for anisotropic viscoplastic porous media: I – Theory, Eur. J. Mech. A, Solids, Volume 28 (2009), pp. 387-401

[11] E. Lemarchand; L. Dormieux; D. Kondo Lode's angle effect on the definition of the strength criterion of porous media, Int. J. Numer. Anal. Methods Geomech., Volume 39 (2015) no. 14, pp. 1506-1526

[12] M. Kroon; J. Faleskog Numerical implementation of a $J_{2}$ and $J_{3}$ -dependent plasticity model based on spectral decomposition of the stress deviator, Comput. Mech., Volume 2013 (2013) no. 52, pp. 1059-1070

[13] K.J. Willam; E.P. Warnke Constitutive models for the triaxial behavior of concrete, Proceedings of the International Assoc. for Bridge and Structural Engineering, 1975

[14] A.V. Hershey; V.A. Dahlgren The plasticity of an isotropic aggregate of anisotropic face-centered cubic crystals, J. Appl. Mech., Volume 76 (1954), p. 241

[15] W.F. Hosford A generalized isotropic yield criterion, J. Appl. Mech., Volume 39 (1972), p. 607

[16] N.S. Ottosen A failure criterion for concrete, J. Eng. Mech. Div., ASCE, Volume 103 (1977), pp. 527-535 (EM4, Proc. Paper 13111)

[17] J.R. Rice; D.M. Tracey On the ductile enlargement of voids in triaxial stress fields, J. Mech. Phys. Solids, Volume 17 (1969), pp. 201-217

[18] T.F. Guo; J. Faleskog; C.F. Shih Continuum modeling of a porous solid with pressure sensitive dilatant matrix, J. Mech. Phys. Solids, Volume 56 (2008), pp. 2188-2212

[19] P. Thoré; F. Pastor; J. Pastor; D. Kondo Closed-form solutions for the hollow sphere model with Coulomb and Drucker–Prager materials under isotropic loadings, C. R. Mecanique, Volume 337 (2009), pp. 260-267

[20] H.Y. Jeong; J. Pan A macroscopic constitutive law for porous solids with pressure-sensitive matrices and its implications to plastic flow localization, Int. J. Solids Struct., Volume 32 (1995), pp. 3669-3691

[21] S. Brach; L. Dormieux; D. Kondo; G. Vairo Nanoporous materials with a general isotropic plastic matrix: exact limit state under isotropic loadings, Int. J. Plast., Volume 89 (2017), pp. 1-28

[22] B. Anoukou; F. Pastor; P. Dufrenoy; D. Kondo Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation, J. Mech. Phys. Solids, Volume 91 (2016), pp. 145-171

Cited by Sources:

Comments - Policy