Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity

A new format of anisotropic gradient elasticity is formulated and implemented to simulate stress concentrations in cortical bone. The higher-order effect of the underlying microstructure in cortical bone is accounted for through the introduction of two length scale parameters and associated strain gradient terms which modify the response of the standard elastic macroscopic continuum: one internal length related to the longitudinal fibres and the other related to the transversal Haversian systems. Thus, anisotropic material behaviour is not only included in the anisotropy of the elastic effective stiffness properties, but also in the anisotropic sources of heterogeneity. The model is validated numerically in tests with bone fractures in the longitudinal and the transversal directions. It was found that the dominant length scale effects are those that coincide with the direction of fracture, as defined by the orientation of a pre-existing crack.