Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties

Andreas Fritsch; Luc Dormieux; Christian Hellmich

doi:10.1016/j.crme.2006.01.008

Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties
[Polycristaux poreux constitués d'aiguilles orientées de façon uniforme ou axisymétrique : homogénéisation des propriétés élastiques]

Présenté par : André Zaoui

Andreas Fritsch ¹ ; Luc Dormieux ² ; Christian Hellmich ¹

¹ Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
² LMSGC, UMR 113, CNRS/ENPC/LCPC, 77455 Marne-la-Vallée, France

Comptes Rendus. Mécanique, Volume 334 (2006) no. 3, pp. 151-157.

Résumé
Abstract

De nombreux matériaux biologiques ou manufacturés présentent une microstructure poreuse à morphologie polycristalline constituée de feuillets ou d'aiguilles. On s'intéresse ici à des cristaux solides élancés, doués d'un comportement linéaire élastique isotrope ou anisotrope, dont les orientations sont distribuées de façon uniforme ou axisymétrique. Dans ce dernier cas, l'approche micromécanique proposée fait appel à la connaissance du tenseur de Hill pour une inclusion ellipsoidale d'élancement infini plongée dans un milieu isotrope transverse. L'expression intégrale de ce dernier donnée par Laws est évaluée numériquement en employant la théorie des fonctions holomorphes. Pour une porosité inférieure à $0, 4$ , les propriétés élastiques du polycristal estimées à partir d'un schéma basé sur des inclusions ellipsoidales sont très proches de celles obtenues avec un schéma basé sur des inclusions sphériques. En revanche, à la différence du schéma basé sur des inclusions sphériques, le schéma basé sur des inclusions ellipsoidales ne prédit pas de seuil de percolation. En ce qui concerne la situation d'une distribution axisymétrique des orientations des cristaux solides, deux effets méritent d'être soulignés. D'une part, l'anisotropie est d'autant plus marquée que l'angle du cône des orientations diminue. D'autre part, à angle de cône donné, l'anisotropie augmente avec la porosité. Les estimations de l'élasticité du polycristal sont très faiblement affectées par l'anisotropie du minéral osseux. Ces résultats confirment le caractère très largement désordonné de l'orientation des cristaux constituant des mousses minérales dans les tissus osseux.

Porous polycrystal-type microstructures built up of needle-like platelets or sheets are characteristic for a number of biological and man-made materials. Herein, we consider (i) uniform, (ii) axisymmetrical orientation distribution of linear elastic, isotropic as well as anisotropic needles. Axisymmetrical needle orientation requires derivation of the Hill tensor for arbitrarily oriented ellipsoidal inclusions with one axis tending towards infinity, embedded in a transversely isotropic matrix; therefore, Laws' integral expression of the Hill tensor is evaluated employing the theory of rational functions. For a porosity lower 0.4, the elastic properties of the polycrystal with uniformly oriented needles are quasi-identical to those of a polycrystal with solid spheres. However, as opposed to the sphere-based model, the needle-based model does not predict a percolation threshold. As regards axisymmetrical orientation distribution of needles, two effects are remarkable: Firstly, the sharper the cone of orientations the higher the anisotropy of the polycrystal. Secondly, for a given cone, the anisotropy increases with the porosity. Estimates for the polycrystal stiffness are hardly influenced by the anisotropy of the bone mineral needles. Our results also confirm the very high degree of orientation randomness of crystals building up mineral foams in bone tissues.

Reçu le : 2006-01-20
Accepté le : 2006-01-31
Publié le : 2006-03-01

DOI : 10.1016/j.crme.2006.01.008

Keywords: Biomechanics, Porous polycrystal, Orientation distribution, Micromechanics, Hill tensor, Anisotropy
Mots-clés : Biomécanique, Polycristal poreux, Distribution des orientations, Micromécanique, Tenseur de Hill, Anisotropie

Affiliations des auteurs :

Andreas Fritsch ¹ ; Luc Dormieux ² ; Christian Hellmich ¹

¹ Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
² LMSGC, UMR 113, CNRS/ENPC/LCPC, 77455 Marne-la-Vallée, France

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     journal = {Comptes Rendus. M\'ecanique},
     pages = {151--157},
     publisher = {Elsevier},
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Andreas Fritsch; Luc Dormieux; Christian Hellmich. Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties. Comptes Rendus. Mécanique, Volume 334 (2006) no. 3, pp. 151-157. doi : 10.1016/j.crme.2006.01.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.01.008/

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