The identification of anisotropic elastic properties of lamellar bone based on nanoindentation data is an open problem. Therefore, the purpose of this study was to develop a method to estimate the orthotropic elastic constants of human cortical bone secondary osteons using nanoindentation in two orthogonal directions. Since the indentation modulus depends on all elastic constants and, for anisotropic materials, also on the indentation direction, a theoretical model quantifying the indentation modulus from the stiffness tensor of a given material was implemented numerically (Swadener and Pharr, 2001, “Indentation of Elastically Anisotropic Half-Spaces by Cones and Parabolae of Revolution,” Philos. Mag. A, 81(2), pp. 447–466). Nanoindentation was performed on 22 osteons of the distal femoral shaft: A new holding system was designed in order to indent the same osteon in two orthogonal directions. To interpret the experimental results and identify orthotropic elastic constants, an inverse procedure was developed by using a fabric-based elastic model for lamellar bone. The experimental indentation moduli were found to vary with the indentation direction and showed a marked anisotropy. The estimated elastic constants showed different degrees of anisotropy among secondary osteons of the same bone and these degrees of anisotropy were also found to be different than the one of cortical bone at the macroscopic level. Using the log-Euclidean norm, the relative distance between the compliance tensors of the estimated mean osteon and of cortical bone at the macroscopic level was 9.69%: Secondary osteons appeared stiffer in their axial and circumferential material directions, and with a greater bulk modulus than cortical bone, which is attributed to the absence of vascular porosity in osteonal properties. The proposed method is suitable for identification of elastic constants from nanoindentation experiments and could be adapted to other (bio)materials, for which it is possible to describe elastic properties using a fabric-based model.
Elastic anisotropy and off-axis ultrasonic velocity distribution in human cortical bone
