Application of fracture mechanics concepts to hierarchical biomechanics of bone and bone-like materials

  • Huajin Gao
Conference paper


Fracture mechanics concepts are applied to gain some understanding of the hierarchical nanocomposite structures of hard biological tissues such as bone, tooth and shells. At the most elementary level of structural hierarchy, bone and bone-like materials exhibit a generic structure on the nanometer length scale consisting of hard mineral platelets arranged in a parallel staggered pattern in a soft protein matrix. The discussions in this paper are organized around the following questions: (1) The length scale question: why is nanoscale important to biological materials? (2) The stiffness question: how does nature create a stiff composite containing a high volume fraction of a soft material? (3) The toughness question: how does nature build a tough composite containing a high volume fraction of a brittle material? (4) The strength question: how does nature balance the widely different strengths of protein and mineral? (5) The optimization question: Can the generic nanostructure of bone and bone-like materials be understood from a structural optimization point of view? If so, what is being optimized? What is the objective function? (6) The buckling question: how does nature prevent the slender mineral platelets in bone from buckling under compression? (7) The hierarchy question: why does nature always design hierarchical structures? What is the role of structural hierarchy? A complete analysis of these questions taking into account the full biological complexities is far beyond the scope of this paper. The intention here is only to illustrate some of the basic mechanical design principles of bone-like materials using simple analytical and numerical models. With this objective in mind, the length scale question is addressed based on the principle of flaw tolerance which, in analogy with related concepts in fracture mechanics, indicates that the nanometer size makes the normally brittle mineral crystals insensitive to cracks-like flaws. Below a critical size on the nanometer length scale, the mineral crystals fail no longer by propagation of pre-existing cracks, but by uniform rupture near their limiting strength. The robust design of bone-like materials against brittle fracture provides an interesting analogy between Darwinian competition for survivability and engineering design for notch insensitivity. The follow-up analysis with respect to the questions on stiffness, strength, toughness, stability and optimization of the biological nanostructure provides further insights into the basic design principles of bone and bone-like materials. The staggered nanostructure is shown to be an optimized structure with the hard mineral crystals providing structural rigidity and the soft protein matrix dissipating fracture energy. Finally, the question on structural hierarchy is discussed via a model hierarchical material consisting of multiple levels of self-similar composite structures mimicking the nanostructure of bone. We show that the resulting “fractal bone”, a model hierarchical material with different properties at different length scales, can be designed to tolerate crack-like flaws of multiple length scales.

Key words

Biological materials bone buckling flaw tolerance fracture hierarchical materials nacre size effects stiffness strength structural optimization toughness 


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  1. Bao, G. and Suo, Z. (1992). Remarks on crack-bridging concepts. Applied Mechanics Review 45, 355–366.Google Scholar
  2. Barenblatt, G.I. (1985). The formation of equilibrium cracks during brittle fracture: Rectilinear cracks in plane plates. Journal of Applied Mathematics and Mechanics 23, 622–636.CrossRefMathSciNetGoogle Scholar
  3. Bazant, Z.P. (1976). Instability, ductility and size effect in strain-softening concrete. Journal of the Engineering Mechanics Division-ASCE 102, 331–344.Google Scholar
  4. Bazant, Z.P. and Cedolin, L. (1983). Finite element modeling of crack band propagation. Journal of Structural Engineering-ASCE 109, 69–92.Google Scholar
  5. Bazant, Z.P. and Planas, J. (1998). Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton, FL.Google Scholar
  6. Bilby, B.A., Cottrell, A.H. and Swinden, K.H. (1963). The spread of plastic yield from a notch. Proceedings of the Royal Society of London A 272, 304–314.Google Scholar
  7. Bouxsein, M.L. (2003) Bone quality: where do we go from here? Osteoporosis International 14, S118–S127.CrossRefGoogle Scholar
  8. Brett, C. and Waldron, K. (1981). Physiology and Biochemistry of Plant Cell Walls. Chapman & Hall, London.Google Scholar
  9. Camacho, G.T. and Ortiz, M. (1996) Computational modeling of impact damage in brittle materials. International Journal of Solids and Structures 33, 2899–2938.CrossRefGoogle Scholar
  10. Carpinteri, A. (1982). Notch sensitivity in fracture testing of aggregative materials. Engineering Fracture Mechanics 16, 467–481.CrossRefGoogle Scholar
  11. Carpinteri, A. (1997). Structural Mechanics: A Unified Approach. Chapman & Hall, London.MATHGoogle Scholar
  12. Cox, B.N. and Marshall, D.B. (1994). Concepts for bridged cracks in fracture and fatigue. Acta Metallurgica et Materialia 42, 341–363.CrossRefGoogle Scholar
  13. Currey, J.D. (1977). Mechanical properties of mother of pearl in tension. Proceedings of the Royal Society of London B 196, 443–463.Google Scholar
  14. Currey, J.D. (1984). The Mechanical Adaptations of Bones. Princeton University Press, Princeton, NJ, pp. 24–37.Google Scholar
  15. Drugan, W.J. (2001). Dynamic fragmentation of brittle materials: analytical mechanics-based models. Journal of the Mechanics and Physics of Solids 49, 1181–1208.CrossRefGoogle Scholar
  16. Dugdale, D.S. (1960). Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 8, 100–104.CrossRefGoogle Scholar
  17. Evans, A.G. (1990) perspective on the development of high-toughness ceramics. Journal of the American Ceramic Society 73, 187–206.CrossRefGoogle Scholar
  18. Fantner, G.E., Birkedal, H., Kindt, J.H., Hassenkam, T., Weaver, J.C., Cutroni, J.A., Bosma, B.L., Bawazer, L., Finch, M.M., Cidade, G.A.G., Morse, D.E., Stucky, G.D. and Hansma, P.K. (2004). Influence of the degradation of the organic matrix on the microscopic fracture behavior of trabecular bone. Bone 35, 1013–1022CrossRefGoogle Scholar
  19. Fengel, D. and Wegener, G. (1984). Wood Chemistry, Ultrastructure, Reaction. Walter de Gruter, Berlin.Google Scholar
  20. Fratzl, P., Gupta, H.S., Paschalis, E.P. and Roschger, P. (2004a) Structure and mechanical quality of the collagen-mineral nano-composite in bone. Journal of Materials Chemistry 14, 2115–2123.CrossRefGoogle Scholar
  21. Fratzl, P., Burgert, I. and Gupta, H.S. (2004b). On the role of interface polymers for the mechanics of natural polymeric composites. Physical Chemistry Chemical Physics 6, 5575–5579.CrossRefGoogle Scholar
  22. Gao, H. and Chen, S. (2005). Flaw tolerance in a thin strip under tension. Journal of Applied Mechanics 72, 732–737.CrossRefMathSciNetGoogle Scholar
  23. Gao, H. and Ji, B. (2003). Modeling fracture in nanomaterials via a virtual internal bond method. Engineering Fracture Mechanics 70, 1777–1791.CrossRefGoogle Scholar
  24. Gao, H. and Yao, H. (2004). Shape insensitive optimal adhesion of nanoscale fibrillar structures. Proceedings of the National Academy of Sciences of the United States of America 101, 7851–7856.CrossRefGoogle Scholar
  25. Gao, H., Ji, B., Jäger, I.L., Arzt, E. and Fratzl. P. (2003). Materials become insensitive to flaws at nanoscale: lessons from nature. Proceedings of the National Academy of Sciences of the United States of America 100, 5597–5600.CrossRefGoogle Scholar
  26. Gao, H., Ji, B., Buehler, M.J. and Yao, H. (2004). Flaw tolerant bulk and surface nanostructures of biological systems. Mechanics and Chemistry of Biosystems 1, 37–52.Google Scholar
  27. Gao, H., Wang, X., Yao, H., Gorb, S. and Arzt, E. (2005). Mechanics of hierarchical adhesion structure of gecko, Mechanics of Materials 37, 275–285.CrossRefGoogle Scholar
  28. Goldberg, D. (1989), Genetic Algorithm in Search, Optimization, and Machine Learning. Addison Wesley.Google Scholar
  29. Guo, X. and Gao, H. (2005). Bio-inspired material design and optimization. IUTAM Symposium on topological design optimization of structures, machines and materials-status and perspectives, October 26–29, 2005, Rungstedgaard, Copenhagen, Denmark.Google Scholar
  30. Hassenkam, T., Fantner, G.E., Cutroni, J.A., Weaver, J.C., Morse, D.E. and Hansma, P.K. (2004). Highresolution AFM imaging of intact and fractured trabecular bone. Bone 35, 4–10.CrossRefGoogle Scholar
  31. Hillerborg, A., Modeer, M. and Petersson, P.E. (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6, 773–782.CrossRefGoogle Scholar
  32. Jackson, A.P., Vincent, J.F.V. and Turner, R.M. (1988). The mechanical design of nacre. Proceedings of the Royal Society of London B 234, 415–440.Google Scholar
  33. Jäger, I. and Fratzl, P. (2000). Mineralized collagen Mbrils: a mechanical model with a staggered arrangement of mineral particles. Biophysical Journal 79, 1737–1746.Google Scholar
  34. Ji, B. and Gao, H. (2004a). Mechanical properties of nanostructure of biological materials. Journal of the Mechanics and Physics of Solids 52, 1963–1990.CrossRefGoogle Scholar
  35. Ji, B. and Gao, H. (2004b). A study of fracture mechanisms in biological nano-composites via the virtual internal bond model. Materials Science and Engineering A 366, 96–103.CrossRefGoogle Scholar
  36. Ji, B. and Gao, H. (2006) Elastic properties of nanocomposite structure of bone. Composite Science and Technology, in press.Google Scholar
  37. Ji, B., Gao H. and Hsia, K.J. (2004a). How do slender mineral crystals resist buckling in biological materials? Philosophical Magazine Letters 84, 631–641.CrossRefGoogle Scholar
  38. Ji, B., Gao, H. and Wang, T.C. (2004b). Flow stress of biomorphous metal-matrix composites. Materials Science and Engineering A 386, 435–441.CrossRefGoogle Scholar
  39. Jiang, H.D., Liu, X.Y., Lim, C.T., and Hsu, C.Y. (2005). Ordering of self-assembled nanobiominerals in correlation to mechanical properties of hard tissues. Applied Physics Letters 86, 163901.CrossRefGoogle Scholar
  40. Kamat, S., Su, X., Ballarini, R. and Heuer, A.H. (2000). Structural basis for the fracture toughness of the shell of the conch Strombus gigas. Nature 405, 1036–1040.CrossRefGoogle Scholar
  41. Karihaloo, B.L. (1979). A note on complexities of compression failure. Proceedings of the Royal Society of London A 368, 483–493.Google Scholar
  42. Kauffmann, F., Ji, B., Dehm, G., Gao, H. and Arzt, E. (2005). A quantitative study of the hardness in a superhard nanocrystalline titanium nitride/silicon nitride coating. Scripta Materialia 52, 1269–1274.CrossRefGoogle Scholar
  43. Kendall, K. (1978). Complexities of compression failure. Proceedings of the Royal Society of London A 361, 245–263.Google Scholar
  44. Kessler, H., Ballarini, R., Mullen, R.L., Kuhn, L.T. and Heuer, A.H. (1996). A biomimetic example of brittle toughening: (I) steady state multiple cracking. Computational Materials Science 5, 157–166.CrossRefGoogle Scholar
  45. Kotha, S.P., Kotha, S. and Guzelsu, N. (2000). A shear-lag model to account for interaction effects between inclusions in composites reinforced with rectangular platelets. Composites Science and Technology 60, 2147–2158.CrossRefGoogle Scholar
  46. Landis, W.J. (1995). The strength of a calcified tissue depends in part on the molecular structure and organization of its constituent mineral crystals in their organic matrix. Bone 16, 533–544.CrossRefGoogle Scholar
  47. Landis, W.J., Hodgens, K.J., Song, M.J., Arena, J., Kiyonaga, S., Marko, M., Owen, C., and McEwen, B.F. (1996). Mineralization of collagen may occur on fibril surfaces: evidence from conventional and high voltage electron microscopy and three dimensional imaging. Journal of Structural Biology 117, 24–35.CrossRefGoogle Scholar
  48. Liu, B., Zhang, L. and Gao, H. (2006). Poisson ratio can play a crucial role in mechanical properties of biocomposites. Mechanics of Materials, in press.Google Scholar
  49. Mano, J.F. (2005). Viscoelastic properties of bone: mechanical spectroscopy studies on a chicken model. iMaterials Science and Engineering C 25, 145–152.CrossRefGoogle Scholar
  50. Massabo, R. and Cox, B.N. (1999). Concepts for bridged mode II delamination cracks. Journal of the Mechanics and Physics of Solids 47, 1265–1300.CrossRefGoogle Scholar
  51. Menig, R., Meyers, M.H., Meyers, M.A. and Vecchio, K.S. (2000). Quasi-static and dynamic mechanical response of Haliotis rufescens (abalone) shells. Acta Materialia 48, 2383–2398.CrossRefGoogle Scholar
  52. Menig, R., Meyers, M.H., Meyers, M.A. and Vecchio, K.S. (2001). Quasi-static and dynamic mechanical response of Strombus gigas (conch) shells. Materials Science and Engineering A 297, 203–211.CrossRefGoogle Scholar
  53. Mori, T. and Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusion. Acta Metalurgica 21, 571–574.CrossRefGoogle Scholar
  54. Mulmule, S.V. and Dempsey, J.P. (2000). LEFM size requirement for the fracture testing of sea ice. International Journal of Fracture 102, 85–98.CrossRefGoogle Scholar
  55. Needleman, A. (1987) A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics 54, 525–531.Google Scholar
  56. Neves, N.M. and Mano, J.F. (2005). Structure/mechanical behavior relationships in crossed-lamellar sea shells. Materials Science and Engineering C 25, 113–118.CrossRefGoogle Scholar
  57. Okumura, K. and de Gennes, P.-G. (2001). Why is nacre strong? Elastic theory and fracture mechanics for biocomposites with stratified structures. European Physical Journal E 4, 121–127.CrossRefGoogle Scholar
  58. Pugno, N.M. and Ruoff, R.S. (2004). Quantized fracture mechanics. Philosophical Magazine 84, 2829–2845.CrossRefGoogle Scholar
  59. Rho, J.Y., Kuhn-Spearing, L. and Zioupos, P. (1998). Mechanical properties and the hierarchical structure of bone. Medical Engineering & Physics 20, 92–102.CrossRefGoogle Scholar
  60. Rice, J.R. (1980). The Mechanics of Earthquake Rupture. International School of Physics “E. Fermi”, Course 78, 1979: Italian Physical Society/North Holland Publ. Co.Google Scholar
  61. Roschger, P., Grabner, B.M., Rinnerthaler, S., Tesch, W., Kneissel, M., Berzlanovich, A., Klaushofer, K. and Fratzl, P. (2001). Structural development of the mineralized tissue in the human L4 vertebral body. Journal of Structural Biology 136, 126–136.CrossRefGoogle Scholar
  62. Roschger, P., Matsuo, K., Misof, B.M., Tesch, W., Jochum, W., Wagner, E.F., Fratzl, P. and Klaushofer, K. (2004) Normal mineralization and nanostructure of sclerotic bone in mice overexpressing Fra-1. Bone 34, 776–782.CrossRefGoogle Scholar
  63. Smith, B.L., Schaeffer, T.E., Viani, M., Thompson, J.B., Frederick, N.A., Kindt, J., Belcher, A., Stucky, G.D., Morse, D.E. and Hansma, P.K. (1999). Molecular mechanistic origin of the toughness of natural adhesive, fibres and composites. Nature 399, 761–763.CrossRefGoogle Scholar
  64. Song, F., Soh, A.K. and Bai, Y.L. (2003). Structural and mechanical properties of the organic matrix layers of nacre. Biomaterials 24, 3623–3631.CrossRefGoogle Scholar
  65. Suo, Z., Ho, S. and Gong, X. (1993). Notch ductile-to-brittle transition due to localized inelastic band. Journal of Engineering Materials and Technology 115, 319–326.Google Scholar
  66. Tada, J., Paris, P.C. and Irwin, G.R. (1973). The Stress Analysis of Cracks Handbook. Del Research Corporation, St. Louis (2nd edition, 1985).Google Scholar
  67. Tang, R.K., Wang, L.J., Orme, C.A., Bonstein, T., Bush, P.J. and Nancollas, G.H. (2004). Dissolution at the nanoscale: Self-preservation of biominerals. Angewandte Chemie-International Edition 43, 2697–2701.CrossRefGoogle Scholar
  68. Tang, T., Hui, C.-Y. and Glassmaker, N.J. (2005) Can a fibrillar interface be stronger and tougher than a non-fibrillar one? Journal of the Royal Society Interface 2, 505–516.CrossRefGoogle Scholar
  69. Tesch, W., Eidelman, N., Roschger, P., Goldenberg, F., Klaushofer, K. and Fratzl, P. (2001). Graded microstructure and mechanical properties of human crown dentin. Calcified Tissue International 69, 147–157.CrossRefGoogle Scholar
  70. Thompson, J.B., Kindt, J.H., Drake, B., Hansma, H.G., Morse, D.E. and Hansma, P.K. (2001) Bone indentation recovery time correlates with bond reforming time. Nature 414, 773–776.CrossRefGoogle Scholar
  71. Tvergaard, V. and Hutchinson, J.W. (1992) The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. Journal of the Mechanics and Physics of Solids 40, 1377–1397.CrossRefGoogle Scholar
  72. Wang, R.Z., Suo, Z., Evans, A.G., Yao, N. and Aksay, I.A. (2001). Deformation mechanisms in nacre. Journal of Materials Research 16, 2485–2493.Google Scholar
  73. Wang, L.J., Tang, R.K., Bonstein, T., Orme, C.A., Bush, P.J. and Nancollas, G.H. (2005). A new model for nanoscale enamel dissolution. Journal of Physical Chemistry B 109, 999–1005.CrossRefGoogle Scholar
  74. Warshawsky, H. (1989). Organization of crystals in enamel. Anatomical Record 224, 242–262.CrossRefGoogle Scholar
  75. Weiner, S. and Wagner, H.D. (1998). The material bone: structure-mechanical function relations. Annual Review of Materials Science 28, 271–298.CrossRefGoogle Scholar
  76. Xu, X.P. and Needleman, A. (1994). Numerical simulations of fast crack-growth in brittle solids. Journal of the Mechanics and Physics of Solids 42, 1397–1434.CrossRefGoogle Scholar
  77. Yao, H. and Gao, H. (2006). Mechanics of robust and releasable adhesion in biology: bottom-up designed hierarchical structures of gecko. Journal of the Mechanics and Physics of Solids, in press.Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Huajin Gao
    • 1
  1. 1.Max Planck Institute for Metals ResearchStuttgartGermany

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