A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor

Abstract

The fabric tensor is employed as a quantitative stereological measure of the structural anisotropy in the pore architecture of a porous medium. Earlier work showed that the fabric tensor can be used additionally to the porosity to describe the anisotropy in the elastic constants of the porous medium. This contribution presents a reformulation of the relationship between fabric tensor and anisotropic elastic constants that is approximation free and symmetry-invariant. From specific data on the elastic constants and the fabric, the parameters in the reformulated relationship can be evaluated individually and efficiently using a simplified method that works independent of the material symmetry. The well-behavedness of the parameters and the accuracy of the method was analyzed using the Mori-Tanaka model for aligned ellipsoidal inclusions and using Buckminster Fuller's octet-truss lattice. Application of the method to a cancellous bone data set revealed that employing the fabric tensor allowed explaining 75-90% of the total variance. An implementation of the proposed methods was made publicly available.

Keywords: Anisotropy; Cancellous bone; Elastic constants; Fabric tensor; Orthogonal basis; Parameters.

Fig. 1

Parameters p₁ to p₉ determined for the Mori–Tanaka model.

Fig. 2

The squared norms ‖q⁽ⁱ⁾‖² with i = 1 … 9 determined for the Mori–Tanaka model.

Fig. 3

The magnitudes of the ratios $p_i^2{\Vert {\mathbf{q}}^{(i)}\Vert }^2/(𝒞\colon\colon 𝒞)$ for i = 1 … 9 for the Mori–Tanaka model.

Fig. 4

Unit cell of the octet-truss lattice.

Fig. 5

Parameters p₁ to p₉ determined for the octet-truss model.

Fig. 6

Relative approximation error for the octet-truss model.

Fig. 7

The magnitudes of the ratios $p_i^2{\Vert {\mathbf{q}}^{(i)}\Vert }^2/(𝒞\colon\colon 𝒞)$ for i = 1 … 9 for the octet-truss model.

Fig. 8

Correlations of the predicted versus calculated entries, C_iiii, C_iijj and C_ijij, obtained when using the analytical model developed in Kabel et al. (1999) based on the original set of parameters k_i obtained by Cowin (1985).

Fig. 9

Orthogonal parameters p_i (i = 1 … 9) as function of the volume fraction derived from the analytical model developed in this study and data set in Kabel et al. (1999). Dashed lines represent 95%t confidence intervals on the predictions by the mentioned power-law equations.

Fig. 10

Correlations of the predicted versus calculated entries, C_iiii, C_iijj and C_ijij, obtained when using the analytical model developed in this study. All parameters p₁ to p₉ were taken into account.

Fig. 11

Correlations of the predicted versus calculated entries, C_iiii, C_iijj and C_ijij, obtained when using the analytical model developed in this study. Third and fourth order terms were neglected.

Fig. 12

Correlations of the predicted versus calculated entries, C_iiii, C_iijj and C_ijij, obtained when using the analytical model developed in this study. Only the first order terms (p₁ to p₄) were considered.