Advertisement

Journal of Materials Science

, Volume 33, Issue 6, pp 1497–1509 | Cite as

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

  • U Akiva
  • H. D Wagner
  • S Weiner
Article

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

Keywords

Fibril Collagen Fibril Hollow Cylinder Lamellar Bone Thin Lamella 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. D. Currey, “The Mechanical Adapatations of Bones” (Princeton University Press, Princeton, NJ, 1984).Google Scholar
  2. 2.
    S. Weiner and W. Traub, FASEB J. 6 (1992) 879.Google Scholar
  3. 3.
    J. D. Currey, J. Bomech 2 (1969) 1.Google Scholar
  4. 4.
    Idem, ibid. 2 (1969) 477.Google Scholar
  5. 5.
    S. Weiner, T. Arad and W. Traub, FEBS Lett. 285 (1991) 49.Google Scholar
  6. 6.
    N. Sasaki, T. Ikawa and A. Fukuda, J. Biomech. 24 (1991) 57.Google Scholar
  7. 7.
    H. D. Wagner and S. Weiner, ibid. 25 (1992) 1311.Google Scholar
  8. 8.
    J. D. Currey, K. Brear and P. Zioupos, ibid. 27 (1994) 885.Google Scholar
  9. 9.
    W. J. Landis, M. J. Song, A. Leith, L. McEwen and B. F. McEwen, J. Struct. Biol. 110 (1993) 39.Google Scholar
  10. 10.
    R. Stuhler, Forscht. Geb. Röntgenstrahlen 57 (1937) 231.Google Scholar
  11. 11.
    R. A. Robinson, J. Bone Surg. 34A (1952) 389.Google Scholar
  12. 12.
    E. Wachtel and S. Weiner, J. Bone. Miner. Res. 9 (1994) 1651.Google Scholar
  13. 13.
    S. Weiner and P. A. Price, Calcif. Tiss. Int. 39 (1976) 365.Google Scholar
  14. 14.
    H. Francillon-ViÉllot, V. De BufferenÍl, J. Castanet, J. Geraudie, F. J. Meunier, J. Y. Sire, L. Zylberberg and A. De RicqlÉs, in “Skeletal Biomineralization Patterns, Processes and Evolutionary Trends”, edited by J. G. Carter, Ch. 20, pp 471-530Google Scholar
  15. 15.
    V. Ziv, H. D. Wagner and S. Weiner, Bone 18 (1996) 417.Google Scholar
  16. 16.
    W. Gebhardt, Arch. Entw. Mech. Org. 20 (1906) 187.Google Scholar
  17. 17.
    M. M. Giraud-Guille, Calcif. Tiss. Int. 42 (1988) 167.Google Scholar
  18. 18.
    S. A. Reid, Anat. Embryol. 174 (1986) 329.Google Scholar
  19. 19.
    J. W. Smith, J. Bone Joint Surg. 42B (1960) 588.Google Scholar
  20. 20.
    G. Marroti, Calicif. Tiss. Int. 53 (1993) 547.Google Scholar
  21. 21.
    S. Weiner, T. Arad, I. Sabanay and W. Traub, Bone, in press.Google Scholar
  22. 22.
    V. Ziv, I. Sabanay, T. Arad, W. Traub and S. Weiner, Microsc. Res. Tech. 33 (1996) 203.Google Scholar
  23. 23.
    D. T. Reilly and A. H. Burstein, J. Biomec-. 8 (1975) 393.Google Scholar
  24. 24.
    W. Bonfield and M. D. Grynpas, Nature 270 (1977) 453.Google Scholar
  25. 25.
    R. M. V. Pidaparti, A. Chandran, Y. Takano and C. H. Turner, J. Biomech. 29 (1996) 909.Google Scholar
  26. 26.
    G. P. Evans, J. C. Behiri, J. D. Currey and W. Bonfield, J. Mater. Sci. Mater. Med. (1990) 38.Google Scholar
  27. 27.
    S. Weiner, T. Arad and W. Traub, Chem. Biol. Mineral. Tiss. (1992) 93.Google Scholar
  28. 28.
    S. Weiner and W. Traub, FEBS Lett. 206 (1986) 262.Google Scholar
  29. 29.
    P. Fratzl, M. Groschner, G. Vogel, H. Plenk, Jr, J. Eschberger, N. Fratzl-Zelman, K. Koller and K. Klaushofer, Bone Miner. Res. 7 (1992) 329.Google Scholar
  30. 30.
    A. Boyde, Cell Tiss. Res. 152 (1974) 543.Google Scholar
  31. 31.
    M. J. Glimcher, Phil. Trans. R. Soc. Lond. B304 (1984) 479.Google Scholar
  32. 32.
    S. Fitton-Jackson, Proc. R. Soc. Lond. Ser. B. 146 (1956) 270.Google Scholar
  33. 33.
    W. Traub, T. Arad and S. Weiner, Conn. Tiss. Res. 28 (1992) 99.Google Scholar
  34. 34.
    A. L. Arsenault, Calcif. Tiss. Res. 6 (1991) 239.Google Scholar
  35. 35.
    E. P. Katz, E. Wachtel, M. Yamauchi and G. L. Mechanic, Conn. Tiss. Res. 21 (1989) 149.Google Scholar
  36. 36.
    M. Yamauchi, E. P. Katz, O. Kazunori, K. Teraoka and G. L. Mechanic, ibid. 18 (1989) 41.Google Scholar
  37. 37.
    M. W. K. Chew and J. M. Squire, Int. J. Biol. Macromol. 8 (1986) 27.Google Scholar
  38. 38.
    H. J. HÖhling, B. A. Ashton and H. D. Koster, Cell. Tiss. Res. 148 (1974) 11.Google Scholar
  39. 39.
    S. Doty, R. A. Robinson and B. Schofield, in “Handbook of Physiology,” (edited by G. D. Aurbach, (American Physiology Society, Washington, DC, 1976) pp. 3-23.Google Scholar
  40. 40.
    R. S. Gilmore and J. L. Katz, J. Mater. Sci. 17 (1982) 1131.Google Scholar
  41. 41.
    A. Tanioka, T. Tazawa, K. Miyasaka and K. Ishikawa, Biopolymers 13 (1974) 735.Google Scholar
  42. 42.
    S. G. Lekhnitskii, “Theory of Elasticity of an Anisotropic Elastic Body” (Holden-Day, San Francisco, 1963) pp. 1-73.Google Scholar
  43. 43.
    U. Akiva, E. Itzhak and H. D. Wagner, Compos. Sci. Tech., in press.Google Scholar
  44. 44.
    J. C. Halpin, “Primer on Composite Materials: Analysis”, Revised Edition (Technomic Publishing, Lancaster, PA, 1984) pp. 125-137.Google Scholar
  45. 45.
    G. E. Padawer and N. Beecher, Polym. Engng Sci. 10 (1970) 185.Google Scholar
  46. 46.
    J. Lusis, R. T. Woodhams and M. Xhantos, ibid. 13 (1973) 139.Google Scholar
  47. 47.
    L. E. Nielsen, J. Appl. Phys. 41 (1970) 4626.Google Scholar
  48. 48.
    L. E. Nielsen, “Mechanical Properties of Polymers and Composites”, Vol. 2 (Marcel-Dekker, 1974) Ch. 8.Google Scholar
  49. 49.
    H. L. Cox, J. Appl. Phys. 3 (1952) 72.Google Scholar
  50. 50.
    H. Goldstein, “Classical Mechanics” (Addison-Wesley, 1980) pp. 143-8 and Appendix B.Google Scholar
  51. 51.
    B. D. Agarwal and L. J. Broutman, “Analysis and Performance of Fiber Composites” (Wiley, New York, 1980).Google Scholar
  52. 52.
    B. Gershon, D. Cohn and G. Marom, Biomaterials 11 (1990) 548.Google Scholar
  53. 53.
    J. D. Currey and K. Brear, J. Mater. Sci. Mater. Med. 1 (1990) 14.Google Scholar
  54. 54.
    G. P. Evans, J. C. Behiri and W. Bonfield, Adv. Biomater. 8 (1988) 311.Google Scholar
  55. 55.
    R. Hodgskinson, J. D. Currey and G. P. Evans, J. Orthop. Res. 7 (1989) 754.Google Scholar
  56. 56.
    K. Hasegawa, C. H. Turner and D. B. Burr, Calcif. Tiss. Int. 55 (1994) 381.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • U Akiva
    • 1
  • H. D Wagner
    • 1
  • S Weiner
    • 2
  1. 1.Department of Materials and InterfacesRehovotIsrael
  2. 2.Department of Structural BiologyThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations