Mechanics of Hard Tissues

  • Arturo N. Natali
  • Richard T. Hart


The goal of this introduction to the mechanics of bone tissue has been to show the need for interdisciplinary approaches to understanding skeletal mechanics. In the first section, the experimental results were highlighted to demonstrate some of the complexities of bone tissue, including its structural complexity, its sensitivity to moisture, its inhomogeneity, its dependence upon loading rate, its viscoelastic response, its strength dependence upon loading type, and its response to repeated loading. In addition to complexity in mechanical behavior, some of the purely biological aspects of skeletal tissue were introduced with a focus upon the role of the bone cells in changing the material behavior and geometric structure of bone.

These complex mechanical and biological behaviors were employed to motivate the theoretical descriptions that are used to quantify behavior,bone’s behavior, although only the simplest linear elastic behavior was included. A number of ideas about how to simulate the adaptive response of bone tissue was introduced, and the role of numerical simulations in the study of bone and implants and in the study of the adaptive response was highlighted.

Despite the required brevity, this introduction has highlighted the need for a multidisciplinary approach to the study of skeletal mechanics that requires a team with competencies in clinical, mechanical, chemical, experimental, and numerical approaches. Future progress will be increasingly dependent upon collaboration, and hold the promise of prediction of bone responses, study of skeletal disease, and development and manipulation of therapeutic agents to repair, and to perhaps avoid, skeletal disease.


Bone Tissue Cortical Bone Trabecular Bone Cancellous Bone Hard Tissue 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ashman, R.B. 1989. Experimental techniques, in: Bone Mechanics (S.C. Cowin, ed.), pp. 75–96, CRC Press, Inc., Boca Raton.Google Scholar
  2. Ashman, R.B., Rho, J.Y. 1988. Elastic moduli of trabecular bone material, J. Biomech. 21, 177.CrossRefGoogle Scholar
  3. Ashman, R.B., Rho, J.Y., Turner, C.H. 1989. Anatomical variation of orthotropic elastic moduli of the proximal human tibia, J. Biomech, 22, 895–900.CrossRefGoogle Scholar
  4. Beauprè, G.S., Orr, T.E., Carter, D.R. 1990. An approach for time-dependent bone modeling and remodeling — theoretical development, J. Orthop. Res. 8(5), 651–661.Google Scholar
  5. Bowman, S.M., Keaveny, T.M., Gibson, L.J., Hayes, W.C., McMahon, T.A. 1994. Compressive creep behavior of bovine trabecular bone, J. Biomech. 27, 301–310.CrossRefGoogle Scholar
  6. Carter, D.R., Hayes, W.C. 1977. The compressive behavior of bone as a two-phase porous structure, J. Bone Jt. Surg., Am. Vol. 59(7), 954–962.Google Scholar
  7. Carter, D.R., Caler, W.E., Spengler D.M., Frankel, V.H. 1981. Uniaxial fatigue of human cortical bone. The influence of tissue physical characteristics, J. Biomech. 14(7), 461–470.Google Scholar
  8. Carter, D.R., Fyhrie, D.P., Whalen, R.T. 1987. Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy, J. Biomech. 20(8), 785–794.Google Scholar
  9. Cowin, S.C. 1981. Mechanical Properties of Bone, Presented at the Joint ASME-ASCE Applied Mechanics, Fluids Engineering, and Bioengineering Conference, Boulder, Colorado, June 22–24, 1981, New York, N.Y. (345 E. 47th St., New York 10017), American Society of Mechanical Engineers.Google Scholar
  10. Cowin, S.C. 1986. Wolff’s of trabecular architecture at remodeling equilibrium J. Biomech Eng, 108(1), 83–88.CrossRefGoogle Scholar
  11. Cowin, S.C. 1989. The mechanical properties of cortical bone tissue, in: Bone Mechanics (S.C. Cowin, ed.), pp. 97–128, CRC Press, Inc., Boca Raton.Google Scholar
  12. Cowin, S.C. 1997. The false premise of Wolff’s lowForma 12, 247–262.Google Scholar
  13. Cowin, S.C., Hegedus, D.H. 1976. Bone remodeling I: Theory of adaptive elasticity, J. Elast. 6(3), 313–326.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Cowin, S.C., VanBuskirk, W.C. 1979. Surface bone remodeling induced by a medullary pin, J. Biomech, 12(4), 269–276.CrossRefGoogle Scholar
  15. Cowin, S.C., Balser, J.R., Hart, R.T., Kohn, D.H. 1985. Functional adaptation in long bones: establishing in vivo values for surface remodeling rate coefficients, J. Biomech. 18(9), 665–684.CrossRefGoogle Scholar
  16. Cowin, S.C., Sadegh, A.M., Luo, G.M. 1992. An evolutionary Wolff’s law for trabecular architecture, J. Biomech. Eng. 114(1), 129–136.Google Scholar
  17. Currey, J.D. 1995. The validation of algorithms used to explain adaptive remodelling in bone, in: Bone Structure and Remodelling (A. Odgaard, H. Weinans, eds.), pp. 9–13, World Scientific, Singapore.Google Scholar
  18. Currey, J.D. 1997. Was Wolff correct?, Forma 12, 263–266.Google Scholar
  19. Dalstra, M., Huiskes, R., Odgaard, A., van Erning, L. 1993. Mechanical and textural properties of pelvic trabecular bone, J. Biomech. 26, 523–535.CrossRefGoogle Scholar
  20. Davy, D.T., Hart, R.T. 1983. A Theoretical Model for Mechanically Induced Bone Remodeling, American Society of Biomechanics, Rochester, MN.Google Scholar
  21. Eriksen, E.F., Kassem, M. 1992. Editorial, The cellular basis of bone remodeling, Triangle, Sandoz J. Med. Sci. 31, 45–57.Google Scholar
  22. Ford, C.M., Keaveny, T.M. 1996. The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation, J. Biomech. 29, 1309–1317.CrossRefGoogle Scholar
  23. Frost, H.A. 1964. Dynamics of bone remodeling, in: Bone Biodynamics (H.A. Frost, ed.), pp. 315–333, Little, Brown, Boston.Google Scholar
  24. Frost, H.M. 1964. The Laws of Bone Structure, C.C. Thomas, Springfield, IL.Google Scholar
  25. Frost, H.M. 1986. Intermediary Organization of the Skeleton, CRC Press, Boca Raton.Google Scholar
  26. Frost, H.M. 1990. Skeletal structural adaptations to mechanical usage (SATMU): 1. Redefining Wolff’s law, the bone modeling problem, Anat Rec. 226(4), 403–413.Google Scholar
  27. Fyhrie, D.P., Schaffler, M.B. 1994. Failure mechanisms in human vertebral cancellous bone,Bone 15, 105–109.CrossRefGoogle Scholar
  28. Galileo Galilei, 1744, Opere di Galileo Galilei divise in 4 tomi, Dialogo delle scienze nuove, Volume 3, pp. 63 and 87, Stamperia del Seminario, Padova.Google Scholar
  29. Gluer, C.C., Wu, C.Y., Genant, H.K. 1993. Broadband ultrasound attenuation signals depend on trabecular orientation: an in vitro study, Osteoporosis Int. 3, 185–191.Google Scholar
  30. Goldstein, S.A. 1987. The mechanical properties of trabecular bone: dependence on anatomic location and function, J. Biomech. 20, 1055–1061.Google Scholar
  31. Goulet, R.W., Goldstein, S.A., Ciarelli, M.J., Kuhn, J.L., Brown, M.B., Feldkamp, L.A. 1994. The relationship between the structural and orthogonal compressive properties of trabecular bone, J. Biomech. 27, 375–389.Google Scholar
  32. Hall, B.K. (ed.). 1990–1994, Bone, Volumes 1–9,The Telford Press, Inc., Caldwell, N.J., and CRC Press, Inc., Boca Raton.Google Scholar
  33. Hart, R.T. 1995. Review and overview of net bone remodeling, in: Computer Simulations in Biomedicine (H. Power, R.T. Hart, eds.), pp. 267–276, Com putational Mechanics Publications, Southampton, Boston.Google Scholar
  34. Hart, R.T., Rust-Dawicki, A.M. 1995. Computational simulation of idealized long bone realignment, in: Computer Simulations in Biomedicine (H. Power, R.T. Hart, eds.), pp. 341–350, Computational Mechanics Publications, Southampton, Boston.Google Scholar
  35. Hart, R.T., Fritton, S.P. 1997. Introduction to finite element based simulation of functional adaptation of cancellous bone, Forma 12, 277–299.Google Scholar
  36. Hart, R.T., Davy, D.T., Heiple, K.G. 1984. A computational method for stress analysis of adaptive elastic materials with a view toward applications in strain-induced bone remodeling, J. Biomech. Eng. 106, 342–350.CrossRefGoogle Scholar
  37. Hart, R.T., Hennebel, V.V., Thongpreda, N., Dulitz, D.A. 1990. Computer simulation of cortical bone remodeling, in: Science and Engineering on Supercomputers (E.J. Pitcher, ed.), pp. 57–66, 565–566, Computational Mechanics Publications, Southampton, Boston.Google Scholar
  38. Huiskes, R. 1995. The law of adaptive bone remodelling, A case for crying Newton?, in: Bone Structure and Remodelling (A. Odgaard, H. Weinans, eds.), pp. 15–24, World Scientific, Singapore.Google Scholar
  39. Huiskes, R., Weinans, H., Grootenboer, H.J., Dalstra, M., Fudala, B., Slooff, T.J. 1987. Adaptive bone-remodeling theory applied to prosthetic-design analysis, J. Biomech. 20(11–12), 1135–1150.Google Scholar
  40. Katz, J.L. 1995. Mechanics of hard tissue, in: The Biomedical Engineering Handbook (J.D. Bonzino, ed.), CRC Press, Inc., Boca Raton.Google Scholar
  41. Kuhn, J.L., nee Ku, J.L., Goldstein, S.A., Choi, K.W., Landon, M., Herzig, M.A., Matthews, L.S. 1987. The mechanical properties of single trabeculae, pp. 12–48, Trans. 33rd Annu. Meet. Orthop. Res. Soc.Google Scholar
  42. Langton, C.M., Njeh, C.F., Hodgskinson, R., Currey, J.D. 1996. Prediction of mechanical properties of the human calcaneus by broadband attenuation, Bone 18, 495–503.CrossRefGoogle Scholar
  43. Lanyon, L.E., Goodship, A.E., Pye, C.J., MacFie, J.H. 1982. Mechanically adaptive bone remodelling, J. Biomech. 15(3), 141–154.CrossRefGoogle Scholar
  44. Martin, R.B. 1972. The effects of geometric feedback in the development of osteoporosis, J. Biomech. 5(5), 447–455.CrossRefGoogle Scholar
  45. Mattheck, C., Huber-Betzer, H. 1991. CAO: Computer simulation of adaptive growth in bones and trees, in: Computers in Biomedicine (K.D. Held, C.A. Brebbia, R.D. Ciskowski, eds.), pp. 243–252, Computational Mechanics Publications, Southampton, Boston.Google Scholar
  46. McNamara, B.P., Prendergast, P.J., Taylor, D. 1992. Prediction of bone adaptation in the ulnar-osteotomized sheep’s forelimb using an anatomical finite element model, J. Biomed. Eng. 14(3), 209–216.Google Scholar
  47. Mente, P.L., Lewis, J.L. 1987. Young’smodulus of trabecular bone tissue, pp. 112–149, Trans. 33rd Annu. Meet. Orthop. Res. Soc.Google Scholar
  48. Meroi, E.A., Natali, A.N., Schrefler, B.A. 1998. A porous media approach to finite deformation behaviour in soft tissues, Comp. Meth. Biomech. Biomed. Eng. 2(2), 157–170.Google Scholar
  49. Natali, A.N. 1999. The simulation of load bearing capacity of dental implants, in: Computer Technology in Biomaterials Science and Engineering, John Wiley & Sons, New York.Google Scholar
  50. Natali, A.N., Meroi, E.A. 1990. Nonlinear analysis of intervertebral disk under dynamic load, ASME J. Biomech Eng. 112, 358–363.Google Scholar
  51. Natali, A.N., Meroi, E.A. 1993. The mechanical behaviour of bony endplate and annulus in prolapsed disc configuration, J. Biomed. Eng. 15, 235–239.Google Scholar
  52. Natali, A.N., Meroi, E.A. 1996. Biomechanical analysis of dental implant in its interaction with bone tissue, in: Ceramics, Cells and Tissues-Bioceramic Coatings for Guided Bone Growth, pp. 223–240, Irtec CNR, Faenza.Google Scholar
  53. Natali, A.N., Meroi, E.A. 1997. Numerical formulation of intervertebral joint with regard to ageing problems of soft and hard tissues, in: Ceramics, Cells and Tissues, pp. 101–108, Irtec CNR, Faenza.Google Scholar
  54. Natali, A.N., Meroi, E.A. 1998. Numerical formulation for biomechanical analysis of spinal motion segment, Proc. Mathematical Theory of Networks and Systems MTNS98-13th Int. Symp. on Math Theory of Networks and Systems, pp. 1051–1054.Google Scholar
  55. Natali, A., Trebacz, H. 1999. The ultrasound velocity and attenuation in cancellous bone samples from lumbar vertebra and calcaneus, Osteoporosis Int. 9(2), 99–105.Google Scholar
  56. Natali, A.N., Meroi, E.A., Donà, S.,1997a. Tissue-implant interaction phenomena for dental implants: a numerical approach, in: Ceramics, Cells and Tissues, pp. 93–100, Irtec CNR, Faenza.Google Scholar
  57. Natali, A.N., Meroi, E.A., Trebacz, H. 1997b. The influence of ageing on mechanical behaviour of intervertebral segment, Proc. 3rd Int. Symp. on Computer Methods in Biomechanics & Biomedical Engineering, pp. 323–330.Google Scholar
  58. Natali, A.N., Meroi, E.A., Williams, K.R., Calabrese, L. 1998. Investigation of the integration process of dental implants by means of a numerical analysis of dynamic response, Dent. Mater. 13(5), 325–337.Google Scholar
  59. Nicholson, P.H.F., Haddaway, M.J., Davie, M.W.J. 1994. The dependence of ultrasonic properties on orientation in human vertebral bone, Phys. Med. Biol. 39, 1013–1024.CrossRefGoogle Scholar
  60. Oden, Z.M., Hart, R.T., Forwood, M.R., Burr, D.B. 1995. A priori prediction of functional adaptation in canine radii using a cell based mechanistic approach, Trans. 41st Orthop. Res. Soc.Google Scholar
  61. Odgaard, A., Kabel, J., van Rietbergen, B., Dalstra, M., Huiskes, R. 1997. Fabric and elastic principal directions of cancellous bone are closely related, J. Biomech. 30(5), 487–495.CrossRefGoogle Scholar
  62. Pugh, J.W., Rose, R.M., Radin, E.L. 1973. Elastic and viscoelastic properties of trabecular bone: dependence of structure, J. Biomech. 6, 475.Google Scholar
  63. Reilly, D.T., Burstein, A.H., Fankel, V.H. 1974. The elastic modulus for bone, J. Biomech. 7, 271–275.CrossRefGoogle Scholar
  64. Rice, J.C., Cowin, S.C., Bowman, J.A. 1988. On the dependence of the elasticity and strength of cancellous bone on apparent density, J. Biomech. 21, 155–161.CrossRefGoogle Scholar
  65. Roesler, H. 1981. Some historical remarks on the theory of cancellous bone structure (Wolff’s law), in: Mechanical Properties of Bone (S.C. Cowin, ed.), pp. 27–42, American Society of Mechanical Engineers, New York.Google Scholar
  66. Rubin, C.T., Lanyon, L.E. 1984. Dynamic strain similarity in vertebrates: an alternative to allometric limb bone scaling, J. Theor Biol. 107(2), 321–327.Google Scholar
  67. Runkle, J.C., Pugh, J.W. 1975. The micromechanics of cancellous bone. II. Determination of the elastic modulus of individual trabeculae by buckling analysis, Bull. Hosp. Jt. Dis. 36, 2.Google Scholar
  68. Ryan, S.D., Williams, J.L. 1986. Tensile testing of individual bovine trabeculae, Proc. 12th NE Bio-Engineering Conference, 35.Google Scholar
  69. Sadegh, A.M., Luo, G.M., Cowin, S.C. 1993. Bone ingrowth, an application of the boundary element method to bone remodeling at the implant interface, J. Biomech. 26(2), 167–182.CrossRefGoogle Scholar
  70. Townsend, P.R., Rose, R.M., Radin, E.L. 1975. Buckling studies of a single human trabecula, J. Biomech. 8, 199.Google Scholar
  71. Treharne, R.W. 1981. Review of Wolff’s proposed means of operation, Orthop. Rev. 10(1), 35–47.Google Scholar
  72. Turner, C.H. 1997. The relationship between cancellous bone architecture and mechanical properties at the continuum level, Forma 12, 225–233.Google Scholar
  73. Whitehouse, W.J. 1974. The quantitative morphology of anisotropic trabecular bone, J. Microsc. 101, 153–168.Google Scholar
  74. Williams, J.L., Lewis, J.L. 1982. Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis, J Biomech. Eng. 104, 50.CrossRefGoogle Scholar
  75. Wolff, J. 1892. Das Gesetz der Transformation der Knochen, Hirschwald, Berlin.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Arturo N. Natali
    • 1
  • Richard T. Hart
    • 2
  1. 1.Centre of Mechanics of Biological Materials, Dipartimento di Costruzioni e TrasportiUniversitàPadovaPadovaItaly
  2. 2.Department of Biomedical Engineering, Boggs Center, Suite 500Tulane UniversityNew OrleansUSA

Personalised recommendations