Two-time scale fatigue modelling: application to damage

A temporal multiscale modelling applied to fatigue damage evolution in cortical bone is presented. Microdamage accumulation in cortical bone, ensued from daily activities, leads to impaired mechanical properties, in particular by reducing the bone stiffness and inducing fatigue. However, bone damage is also known as a stimulus to bone remodelling, whose aim is to repair and generate new bone, adapted to its environment. This biological process by removing fatigue damage seems essential to the skeleton lifetime. As daily activities induce high frequency cycles (about 10,000 cycles a day), identifying two-time scale is very fruitful: a fast one connected with the high frequency cyclic loading and a slow one related to a quasi-static loading. A scaling parameter is defined between the intrinsic time (bone lifetime of several years) and the high frequency loading (few seconds). An asymptotic approach allows to decouple the two scales and to take into account history effects (Guennouni and Aubry in CR Acad Sci Paris Ser II 20:1765–1767, 1986). The method is here applied to a simple case of fatigue damage and a real cortical bone microstructure. A significant reduction in the amount of computation time in addition to a small computational error between time homogenized and non homogenized models are obtained. This method seems thus to give new perspectives to assess fatigue damage and, with regard to bone, to give a better understanding of bone remodelling.