A viscoelastic, viscoplastic model of cortical bone valid at low and high strain rates

The stress–strain behavior of cortical bone is well known to be strain-rate dependent, exhibiting both viscoelastic and viscoplastic behavior. Viscoelasticity has been demonstrated in literature data with initial modulus increasing by more than a factor of 2 as applied strain rate is increased from 0.001 to 1500 s−1. A strong dependence of yield on strain rate has also been reported in the literature, with the yield stress at 250 s−1 having been observed to be more than twice that at 0.001 s−1, demonstrating the material viscoplasticity. Constitutive models which capture this rate-dependent behavior from very low to very high strain rates are required in order to model and simulate the full range of loading conditions which may be experienced in vivo; particularly those involving impact, ballistic and blast events. This paper proposes a new viscoelastic, viscoplastic constitutive model which has been developed to meet these requirements. The model is fitted to three sets of stress–strain measurements from the literature and shown to be valid at strain rates ranging over seven orders of magnitude.