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

Porous polycrystal-type microstructures built up of needle-like platelets or sheets are characteristic for a number of biological materials, such as bone [

2

] or eggs [

5

]. Herein, we consider (i) uniform, (ii) axisymmetrical orientation distribution of linear elastic, isotropic as well as anisotropic needles. The latter requires derivation of the Hill tensor for arbitrarily oriented ellipsoidal inclusions with one axis tending towards infinity, embedded in a transversely isotropic matrix; this is accomplished by a new semi-analytical technique based on the work of Laws [

3

]. 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. These results confirm the very high degree of orientation randomness of crystals [

4

] building up mineral foams [

2

,

1

] in bone tissues.