Homogenized strain gradient remodeling model for trabecular bone microstructures

Constitutive models for bone remodeling are established from micromechanical analyses at the scale of individual trabeculae defining the representative unit cell (RUC), accounting for both first- and second-order deformation gradients. On the microscale, trabeculae undergo apposition of new bone modeled by a surface growth velocity field driven by a mechanical stimulus identified to the surface divergence of an Eshelby-like tensor. The static and evolutive first and second gradient effective properties of a periodic network of bone trabeculae are evaluated by numerical simulations with controlled imposed first and second displacement gradient rates over the RUC. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae relates the (average) first and second growth strain rates to the homogenized stress and hyperstress tensors. The constitutive model is identified relying on the framework of TIP (thermodynamics of irreversible processes), adopting a split of the kinematic and static tensors into their deviator and hydrostatic contributions. The obtained results quantify the strength and importance of strain gradient effects on the overall remodeling process.