Modelling the three-dimensional elastic constants of parallel-fibred and lamellar bone

Abstract

The complex hierarchical structure of lamellar bone makes understanding structure–mechanical function relations, very difficult. We approach the problem by first using the relatively simple structure of parallel-fibred bone to construct a mathematical model for calculating Young's moduli in three-dimensions. Parallel-fibred bone is composed essentially of arrays of mineralized collagen fibrils, which are also the basic structural motif of the individual lamellae of lamellar bone. Parallel-fibred bone structure has orthotropic symmetry. As the sizes and shapes of crystals in bone are not well known, the model is also used to compare the cases of platelet-, ribbon- and sheet-reinforced composites. The far more complicated rotated plywood structure of lamellar bone results in the loss of the orthotropic symmetry of individual lamellae. The mathematical model used circumvents this problem by sub-dividing the lamellar unit into a thin lamella, thick lamella, transition zone between them, and the recently observed “back-flip” lamella. Each of these is regarded as having orthotropic symmetry. After the calculation of their Young's moduli they are rotated in space in accordance with the rotated plywood model, and then the segments are combined to present the overall modulus values in three-dimensions. The calculated trends compare well with the trends in microhardness values measured for circumferential lamellar bone. Microhardness values are, as yet, the only measurements available for direct comparison. Although the model is not directly applicable to osteonal bone, which is composed of many hollow cylinders of lamellar bone, the range of calculated modulus values and the trends observed for off-axis calculations, compare well with measured values. © 1998 Chapman & Hall