# Indentation of bone tissue

These tests have been conducted on cortical bone samples at the micro scale using Nano Indenter II (Nano Instruments Inc., USA) equipped with a Berkovich diamond tip following Oliver and Pharr (1992) guidelines. Indentation tests were performed under displacement control according the following protocol: A 0.05 s−1 constant strain rate was applied to a peak of displacement of 5000 nm, followed by a 10 s dwell at peak displacement to limit the viscous behavior of bone tissue, a 45 s withdrawal to 10% of maximum displacement, a 50-s hold period for thermal drift calculation and final withdrawal to zero displacement.

The measurements were done at relatively high load (∼500 mN) to overcome the effects related to the heterogeneity of bone tissue at lamellar level and in Continuous Stiffness Mode (CSM), to measure the mechanical properties continuously along the depth of the sample and assure the homogeneity of bone tissue. This mode consist on applying a sinusoidal solicitation during loading to measure mechanical properties for each micro-unloading.

The system was calibrated with fused silica according Oliver and Pharr (1992) protocol. The 150 bone structural units selected according to their mineral density measurements were indented. The indent location was chosen at sites distant from visible lacunae or other discontinuities.

Indentation curves were analyzed using a homemade program developed with Matlab R2010a (The MathWorks Inc., Natik MA, USA). For each indent, elastic modulus, E and contact hardness, Hc were calculated according the method described by Oliver and Pharr (1992). Assuming bone as an isotropic material, the elastic modulus was calculated from the following equation:

where Ei and νi are the elastic modulus and Poisson’s ratio of the diamond indenter (1140 GPa and 0.07 respectively). For isotropic models, the Poisson modulus of bone ν is usually assumed to be 0.3 (Zysset et al., 1999). Er is the reduced modulus, calculated using the formula:

where A is the projected contact area, β is an empirical indenter shape factor Berkovich tip (1.034) and S the stiffness of the sample derived from the initial tangent of the unloading segment of the load/depth curve. The stiffness was calculated from a power fit to the unloading curve. The regression was performed with a Levenberg–Marquardt algorithm. Contact hardness was calculated from the equation:

where Pmax is the maximum load. The values of E and Hc given by Oliver and Pharr method were highly correlated. Based on the Sakai model, Oyen and Cook dissociate elastic and plastic deformations in mineralized tissue (Oyen, 2006; Oyen and Cook, 2003; Sakai, 1999). In this model, bone is considered to be an elasto–plastic material and its mechanical behavior during loading of the indentation test is modeled as a purely elastic element connected to a purely plastic element in series in a sense of the Maxwell combination.

The elastic element is characterized by the elastic modulus E, and the plastic one by the true hardness H. Developing the constitutive relationships for pyramidal indentation for both components allow to derive Hc as a function of E and H (Sakai, 1999):

where α1 = 24.5 and α2 = 4.4 are adimensional constants associated with Berkovich indentation tip and E∗ was the plain strain modulus . The true hardness has been calculated based on this equation. The work of indentation has also been obtained from the loading and unloading curves. The total work of indentation (Wtot) is defined as the area under the loading curve, the reversible work (Wu) as the area under the unloading curve and the irreversible work (Wp) as the area enclosed by the loading and unloading curve.

Oliver, W.C., Pharr, G.M., 1992. An improved technique for determining hardness and elastic modulus. J. Mater. Res. 7, 1564–1583.

Oyen, M.L., 2006. Nanoindentation hardness of mineralized tissues. J. Biomech. 39, 2699–2702.

Oyen, M.L., Cook, R.F., 2003. Load-displacement behavior during sharp indentation of viscous-elastic–plastic materials. J. Mater. Res. 18, 139–150.

Zysset, P.K., Guo, X.E., Hoffler, C.E., Moore, K.E., Goldstein, S.A., 1999. Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J. Biomech. 32, 1005–1012.