Modeling and experimental validation of trabecular bone damage, softening and densification under large compressive strains.

Vertebral fractures represent a major health problem and involve the progressive collapse of trabecular bone over large compressive strains. This collapse is driven by local failure and interaction of the trabecular rod and plate elements, which translates into stress softening and densification at the material level. Current constitutive models for trabecular bone are essentially limited to infinitesimal strains. Accordingly, the aim of this work was to extend our current phenomenological model of trabecular bone (Garcia et al., 2009) for the simulation of large compressive strains by including post-yield softening and densification. A constitutive model of trabecular bone based on both volume fraction and trabecular orientation was formulated in a proper theoretical framework, implemented in commercial FE software and validated with human vertebral sections subjected to large compressive strains. As it is for infinitesimal strains, the evolution of plastic strains and damage is described by local internal variables. An isotropic softening rule was controlled by the cumulated plastic strain and a non-linear elastic spring was added to account for densification of the porous material in moderate-to-large compressive strains beyond a given threshold. To avoid convergence problems occurring as a result of softening, a consistent visco-plastic regularization approach was adopted. The experimental results for 37 vertebral sections from previous work (Dall'Ara et al., 2010) were used to validate the constitutive model for compressive loading up to 45% of the average axial deformation. This validation study showed that the model provides both qualitative predictions of damage localization on the cortex and quantitative predictions of dissipated energy (ρ(C)=0.912) of vertebral body behavior under large compressive strains. Since the evolution of the internal variables was considered in local manner, a mesh sensitivity analysis of the finite element model was conducted via two different mesh sizes and revealed that strain localization was dominated by trabecular bone heterogeneity. To our knowledge, this model is the first to simulate collapse of trabecular bone and may help improve the biomechanical understanding of several musculoskeletal conditions such as vertebral fractures or orthopedic implant migration.