Analysis of crack growth in a 3D Voronoi structure: a model for fatigue in low density trabecular bone

Both creep and crack growth contribute to the reduction in modulus associated with fatigue loading in bone. Here we simulate crack growth and subsequent strut failure in fatigue in an open-cell, three-dimensional Voronoi structure which is similar to that of low density, osteoporotic bone. The model indicates that sequential failure of struts leads to a precipitous drop in modulus: the failure of 1% of the struts leads to about a 10% decrease in modulus. A parametric study is performed to assess the influence of normalized stress range, relative density, initial crack size, crack shape and cell geometry on the fatigue life. The fatigue life is most sensitive to the relative density and the initial crack length. The results lead to a quantitative expression for the fatigue life associated with crack growth. Data for the fatigue life of trabecular bone are compared with the crack growth model described in this paper as well as with a previous model for creep of a three-dimensional Voronoi structure. In our models, creep dominates the fatigue behavior in low cycle fatigue while crack growth dominates in high cycle fatigue, consistent with previous observations on cortical bone. The large scatter in the trabecular bone fatigue data make it impossible to identify a transition between creep dominated fatigue and crack growth dominated fatigue. The parametric study of the crack growth model indicates that variations in relative density among specimens, initial crack size within trabeculae and crack shape could easily produce such variability in the test results.