Theoretical Solutions of Dynamic Responses of Cancellous Bone

Human bone may be damaged by impact in the cases of traffic accidents and ship impact. The impact responses of cancellous bone were analyzed based on the two-phase media theory. A direct analytical method is introduced for solving this type of problems. First, flow function and potential function were introduced to decouple the controlling equations. Then direction solving method was used to obtain the solution. The solution is determined by the parameters of a (related with wave speed) and b (related with damping), as well as the boundary conditions. These two parameters a and b determine the propagation speed of the responses along the bone and the attenuation rate. It is shown that the responses: deformation, stress and pressure of the corpus medullae caused by loading, propagate toward the other end when the impact is acted on one end of the bone. The responses are discontinuous during propagate. The discontinuous surface moves with a constant speed. The responses at the cross section increase gradually from the bottom to the top because of the distribution of the loading at the boundary. The solutions can be used as the basis for certification of numerical simulation as well as the design of impact prevention of bone.