There are many ways of modelling cartilage and other connective tissues. One that, arguably, bears closest resemblance to real tissues is that of a composite material in which collagen fibres, which are strong and stiff, reinforce a mechanically weak proteoglycan gel. The resultant composite is strong, stiff and tough. Similar toughening mechanisms may be found, for instance, in fibreglass. Glass is strong, but brittle, and plastic is weak. Rarely are these used alone in load bearing applications. When combined, however, in the form of fibres of glass embedded in a plastic matrix, the resulting material can be used to make pole-vaulting poles, car bodies and many other structures in which a strong and tough material is required.
Starting with traditional ideas using linear superposition of properties lead to attempts to explain the dramatic changes in the mechanical properties of the uterine cervix that are found at parturition (Aspden 1988a; Aspden 1988b). Studies of spinal ligaments (Aspden et al. 1987; Hukins et al. 1990; Kirby et al. 1989) showed how reorientation of the fibres leads to a nonlinear, J-shaped stress-strain relationship. This provides a control mechanism for constraining motion while minimising the risk of damaging the tissues which might occur if they were to stiffen abruptly. A theory to account for this was developed with reasonable success (Aspden 1986) and was also applied to industrial polymers where a similar reorientation is known to occur. Theories of fibre-composites were developed further to determine the states of stress and strain within a fibre (Aspden 1994b) and clearly showed the confusion that has arisen by mixing the elastic and plastic models of stress transfer. The results were applied to collagen fibres in connective tissue matrices and an estimate made of the molecular interaction forces required to transfer stress effectively to the fibres (Aspden 1994a). Recently these ideas have been developed further in collaboration with David Hukins and Kheng Lim and we have explored the effects of elastic and plastic stress transfer to fibres with different shapes of taper (Goh et al. 1999; Goh et al. 2004; Goh et al. 2000) . Tapered fibres have a number of advantages over cylindrical fibres.
These results have been extending to collagen fibres in connective tissue matrices, which have parabolically tapered ends (Goh et al. 2005) . There are several advantages to having tapered fibres.When loads applied to the tissue are low, stress is transferred elastically from the matrix to the fibre (Fig 1). For a tapered fibril, the axial stress is distributed over more of the length of the fibre, thereby making greater use of the full length of the fibril in reinforcing the ECM. The volume of a paraboloidal fibre is half that of a uniform cylinder of the same length, thus half as much collagen is required to make a paraboloidal fibre as a cylindrical one.
A second advantage of tapered fibres is that they are less likely to fracture than cylindrical fibrils of the same length. When the force applied to a tissue is low, stress will tend to be transferred to its collagen fibrils elastically; increasing the force will lead first to plastic stress transfer (Fig. 2) and, eventually, to failure. During plastic stress transfer, the stress in a fibril is greatest at the centre, and so this is the most likely site for failure. Failure is more likely in a cylindrical fibril, than in a paraboloidal fibril of the same length, because the tensile stress at its centre is 1.5 times as great.
Fig 1. Axial tensile stress produced along the collagen fibre axis by elastic stress transfer for a long fibre embedded in a weak matrix. Only half a fibre is shown, with its centre at the origin and normalized to unit length, because of symmetry about the centre of the fibre. The dashed line is that stress expected in a cylindrical fibre. The effect of a paraboloidal shape is to make more of the fibre carry stress while moving the peak stress close to the fibre end.
Fig 2. Axial tensile stress along the fibre axis due to plastic stress transfer (normalized by the applied shear stress and the length:diameter ratio, q). As before, only half a fibre is shown, with its centre at the origin and normalized to unit length, because of symmetry about the centre of the fibre. The thin line is the stress expected in a cylindrical fibre and the bold line that in the paraboloidal fibre.
References
Aspden, R. M. 1994b, Fibre stress and strain in fibre-reinforced composites, Journal of Materials Science, 29, 1310-1318.
Aspden, R. M. 1986, Relation between structure and mechanical behaviour of fibre reinforced composite materials at large strains, Proceedings of the Royal Society, A406, 287-298.
Aspden, R. M. 1988a, Collagen organisation in the cervix and its relation to mechanical function, Collagen and Related Research, 8, 103-112.
Aspden, R. M. 1988b, The theory of fibre reinforced composite materials applied to changes in the mechanical properties of the cervix during pregnancy, Journal of Theoretical Biology, 130, 213-221.
Aspden, R. M. 1994a, Fibre reinforcing by collagen in cartilage and soft connective tissues, Proceedings of the Royal Society, B-258, 195-200.
Aspden, R. M., Bornstein, N. H., & Hukins, D. W. L. 1987, Collagen organisation in the interspinous ligament and its relationship to tissue function, Journal of Anatomy, 155, 141-151.
Goh, K. L., Aspden, R. M., Mathias, K. J., & Hukins, D. W. L. 1999, Effect of fibre shape on the stresses within fibres in fibre-reinforced composite materials, Proceedings of the Royal Society, A-455, 3351-3361.
Goh, K. L., Aspden, R. M., Mathias, K. J., & Hukins, D. W. L. 2004, Finite-element analysis of the effect of material properties and fibre shape on stresses in an elastic fibre embedded in an elastic matrix in a fibre-composite material, Proceedings of the Royal Society of London Series A- Mathematical Physical and Engineering Sciences, 460, 2339-2352.
Goh, K. L., Mathias, K. J., Aspden, R. M., & Hukins, D. W. L. 2000, Finite element analysis of the effect of fibre shape on stresses in an elastic fibre surrounded by a plastic matrix, Journal of Materials Science, 35, 2493-2497.
Goh, K. L., Meakin, J. R., Aspden, R. M., & Hukins, D. W. L. 2005, Influence of fibril taper on the function of collagen to reinforce extracellular matrix, Proceedings of the Royal Society, B272, 1979-1983.
Hukins, D. W. L., Kirby, M. C., Sikoryn, T. A., Aspden, R. M., & Cox, A. J. 1990, Comparison of structure, mechanical properties and functions of lumbar spinal ligaments, Spine, 15, 787-795.
Kirby, M. C., Sikoryn, T. A., Hukins, D. W. L., & Aspden, R. M. 1989, Structures and mechanical properties of the longitudinal ligaments and ligamenta flava of the spine, Journal of Biomedical Engineering, 11, 192-196.
Department of Orthopaedics, University of
Aberdeen
