[Polycristaux poreux constitués d'aiguilles orientées de façon uniforme ou axisymétrique : homogénéisation des propriétés élastiques]
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
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.
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Mots-clés : Biomécanique, Polycristal poreux, Distribution des orientations, Micromécanique, Tenseur de Hill, Anisotropie
Andreas Fritsch 1 ; Luc Dormieux 2 ; Christian Hellmich 1
@article{CRMECA_2006__334_3_151_0, author = {Andreas Fritsch and Luc Dormieux and Christian Hellmich}, title = {Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties}, journal = {Comptes Rendus. M\'ecanique}, pages = {151--157}, publisher = {Elsevier}, volume = {334}, number = {3}, year = {2006}, doi = {10.1016/j.crme.2006.01.008}, language = {en}, }
TY - JOUR AU - Andreas Fritsch AU - Luc Dormieux AU - Christian Hellmich TI - Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties JO - Comptes Rendus. Mécanique PY - 2006 SP - 151 EP - 157 VL - 334 IS - 3 PB - Elsevier DO - 10.1016/j.crme.2006.01.008 LA - en ID - CRMECA_2006__334_3_151_0 ER -
%0 Journal Article %A Andreas Fritsch %A Luc Dormieux %A Christian Hellmich %T Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties %J Comptes Rendus. Mécanique %D 2006 %P 151-157 %V 334 %N 3 %I Elsevier %R 10.1016/j.crme.2006.01.008 %G en %F CRMECA_2006__334_3_151_0
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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