Strain rate-dependent non-linear constitutive model of bone: From quasi-static to low-impact loading scenarios

Osteoporotic fractures at the upper and lower extremity are typically caused by falls from standing height involving relatively high strain rates. Finite element models of bone used for fracture risk prediction often underestimate both stiffness and strength in such low-impact fall scenarios due to the absence of strain rate dependency in constitutive models of bone.

In this study, an anisotropic viscoelastoplastic damage model for bone applicable for quasi-static experimental tests, physiological loading and low-impact fall scenarios covering eight orders of magnitude strain rate was developed. Single element tests, as well as homogenised finite element simulations of human distal tibiae ($n$=25) and proximal femora ($n$=14), were performed and validated against literature values and experimental tests performed under quasi-static and high strain rate conditions.

The model reproduces the experimentally observed increase in stiffness and yield stress at higher strain rates both qualitatively and quantitatively. Under quasi-static conditions, high concordance correlation coefficients ($CCC$) confirmed excellent agreement between experimental and simulated apparent stiffness ($CCC$=0.98) and yield force ($CCC$=0.98). For simulations involving high strain rates, both stiffness ($CCC$=0.33) and yield force ($CCC$=0.31) were underestimated when using a rate-insensitive constitutive model. With the viscoelastoplastic model, the apparent stiffness was overestimated ($CCC$=0.53), while the yield force was in fair agreement with the experimental data ($CCC$=0.76).

To conclude, the viscoelastoplastic constitutive model is applicable for finite element analysis involving bone at strain rates ranging from quasi-static experimental tests up to low-impact fall scenarios and substantially improves the prediction of biomechanical outcome parameters relevant for fracture risk prediction.