Non-linear viscoelastic constitutive model for bovine cortical bone tissue

In the paper a constitutive law formulation for bovine cortical bone tissue is presented. The formulation is based on experimental studies performed on bovine cortical bone samples. Bone tissue is regarded as a non-linear viscoelastic material. The constitutive law is derived from the postulated strain energy function. The model captures typical viscoelastic effects, i.e. hysteresis, stress relaxation and rate-dependence. The elastic and rheological constants were identified on the basis of experimental tests, i.e. relaxation tests and monotonic uniaxial tests at three different strain rates, i.e. λ˙=0.1 min−1, λ˙=0.5 min−1 and λ˙=1.0 min−1. In order to numerically validate the constitutive model the fourth-order stiffness tensor was analytically derived and introduced to Abaqus® finite element (FE) software by means of UMAT subroutine. The model was experimentally validated. The validation results show that the derived constitutive law is adequate to model stress–strain behaviour of the considered bone tissue. The constitutive model, although formulated in the strain rate range λ˙=0.1−1.0 min−1, is also valid for the strain rate values slightly higher than λ˙=1.0 min−1.The work presented in the paper proves that the formulated constitutive model is very useful in modelling compressive behaviour of bone under various ranges of load.