Some aspects related to the indentation-based viscoelastic modelling of trabecular bone tissue

Purpose The research aims to verify to what extent the shape of an indenter tip influences the final form of the constitutive equation for the trabecular bone. Methods Trabecular bone was formulated as a non-linear viscoelastic material with Mooney-Rivlin hyperelastic model to describe the purely elastic response of the bone tissue. Tests of the mechanical properties of the trabecular bone, resected from the femoral head of a 56-year-old patient, were carried out with two types of indenter: the spherical tip of a diameter of 200 µm and pyramid Vickers tip with 136º between plane faces. Tests with both indenters included loading and unloading phases with no hold at peak force and with hold time t=20 s and were conducted with a maximum load P_max = 500 mN and loading/unloading rate V = 500 mN/min. Results The formulated constitutive model describes the trabecula behaviour very well. The model curves match the experimental results in the loading phase, holding period and most of the unloading ramp. The purely viscoelastic material constants are very close in value for both considered tips, but purely elastic constants differ. Conclusions The results indicate that the constitutive model based on the indentation with the Vickers tip does not cover the plastic residual deformation. When a viscoelastic response of bone is expected, a model with constants calibrated for the spherical tip should be used, and the other set of parameters values (Vickers tip) when trabeculae may undergo plastic deformation.