Tangential Stress in Cortical Bone Subjected to Dynamic Axial Loading

Abstract

Cortical bone is a complex hierarchical composite lamallae structure consisting, in general, of a mineral phase (calcium hydroxyapatite), an organic phase (collagen) and fluid [1]. In the analysis of bone, the liquid phase is usually neglected, an assumption that is reasonable for steady state or quasi-static loading. However, when cortical bone is loaded dynamically in the axial direction, the presence of the constrained fluid generates time-dependent stresses in the tangential direction. Since the tangential stress acts perpendicular to the weak transverse direction of the bone, it can create damage in this direction. Cyclic axial compressive loading will result in cyclic tensile loading in the tangential direction which can eventually result in fatigue damage. Such damage has actually been observed in studies conducted on heavily exercised race horses where damage was observed in the form of micro cracks oriented perpendicular to the tangential direction and whose fracture planes lie along the axial direction [2]. In this work, cortical bone is modeled as a biphasic material consisting of a permeable composite material filled with fluid. The geometry considered is that of a hollow cylinder made up of multiple concentric permeable lamellae filled with fluid (Fig. 1). When this structure is loaded axially in compression, a tensile tangential stress is developed which decays with time. The decay rate is a function of permeability and radial position. The greater the permeability, the faster the decay rate. The tangential stress peaks at the inner radius and decreases with radial position (Fig. 2). The tangential stress also peaks earlier at the inner radius. The rate of decay is slower at the outside surface where the bone is subjected to the tangential stress for a much longer time than at the inner surface (Fig. 2). This view of bone as a biphasic structure subjected to dynamic loading may provide a rationale for some of the damage modes observed in vivo in bones subjected to cyclic and impact loading.