Introducing objective power law rate dependence into a visco-elastic material model of bovine cortical bone

The strain rate dependent behaviour of some visco-elastic materials can be modelled accurately in 1-D by including a stress contribution which depends non-linearly on strain rate. A visco-elastic model by Shim et al. (2005) [3] comprises a Voigt and Maxwell element in parallel and provides an effective representation of cancellous bone from the human cervical spine. The present study demonstrates that the model by Shim et al. is also suitable for modelling the strain rate dependent compression of cortical bone from bovine femurs. Shim et al. found that the model requires a Voigt dash-pot contribution which is proportional to image in order to model specimen response accurately over a large range of strain rates. Shim et al. proposed an expansion of the 1-D formulation to 3-D where the 1-D strain rate is replaced with a function of the strain rate tensor. This paper provides a frame invariant version of the model by Shim et al. which allows general power law rate dependence for the 3-D case. The response of the model is investigated under a load condition which comprises of an axial deformation and a shear twist. The model is implemented in a commercial finite element package and is used to simulate quasi-static and dynamic bovine bone compression experiments.