, Volume 46, Issue 6, pp 1413–1428 | Cite as

Wave propagation modeling in cylindrical human long wet bones with cavity

  • A. M. Abd-Alla
  • S. M. Abo-Dahab
  • S. R. Mahmoud


The wave propagation modeling in cylindrical human long wet bones with cavity is studied. The dynamic behavior of a wet long bone that has been modeled as a piezoelectric hollow cylinder of crystal class 6 is investigated. An analytical solutions for the mechanical wave propagation during a long wet bones have been obtained for the flexural vibrations. The average stresses of solid part and fluid part have been obtained. The frequency equations for poroelastic bones are obtained when the medium is subjected to certain boundary conditions. The dimensionless frequencies are calculated for poroelastic wet bones for various values for non-dimensional wave lengths. The dispersion curves of the dry bone and wet bone for the flexural mode n=2 are plotted. The numerical results obtained have been illustrated graphically.


Elastic Bones Poroelastic Transversely isotropic Mechanical wave Natural frequency Piezoelectric material 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Natali AN, Meroi EA (1989) A review of the biomechanical properties of bone as a material. J Biomed Eng 11(4):266–276 CrossRefGoogle Scholar
  2. 2.
    Thompson GA, Young DR, Orne D (1976) In vivo determination of mechanical properties of human ulna by means of mechanical impedance tests: experimental results and improved mathematical model. Med Biol Eng 14:253–262 CrossRefGoogle Scholar
  3. 3.
    Doherty WP, Boville EG, Wilson EL (1974) Evaluation of the use of resonant frequencies to characterize physical properties of long bones. J Biomech 7:559–561 CrossRefGoogle Scholar
  4. 4.
    Jurist JM (1970) In vivo determination of the elastic response of bone-I. Method of ulnar resonant frequency determination. Phys Med Biol 15:417–426 CrossRefGoogle Scholar
  5. 5.
    Papathanasopoulou VA, Fotiadis DI, Foutsitzi G, Massalas CV (2002) A poroelastic bone model for internal remodeling. Int J Eng Sci 40:511–530 CrossRefGoogle Scholar
  6. 6.
    Fotiadis DI, Fouttsitzi G, Massalas CV (2000) Wave propagation in human long bones of arbitrary cross-section. Int J Eng Sci 38(14):1553–1591 CrossRefGoogle Scholar
  7. 7.
    Fotiadis DI, Foutsitzi G, Massalas CV (1999) Wave propagation modeling in human long bones. Acta Mech 137:65–81 zbMATHCrossRefGoogle Scholar
  8. 8.
    Sebaa N, Fellah Z, Lauriks W, Depollier C (2006) Application of fractional calculus to ultrasonic wave propagation in human cancellous bone. Signal Process 86(10):2668–2677 zbMATHCrossRefGoogle Scholar
  9. 9.
    Padilla F, Bossy E, Haiat G, Jenson F, Laugier P (2006) Numerical simulation of wave propagation in cancellous bone, ultrasonic propagation in cancellous bone. Ultrasonics 44:e239–e243 CrossRefGoogle Scholar
  10. 10.
    Haeat G, Padilla F, Barkmann R, Gluer CC, Laugier P (2006) Numerical simulation of the dependence of quantitative ultrasonic parameters on trabecular bone micro architecture and elastic constants. Ultrasonics 44:e289–e294 CrossRefGoogle Scholar
  11. 11.
    Pithious M, Lasaygues P, Chabrand P (2002) An alternative ultrasonic method for measuring the elastic properties of cortical bone. J Biomech 35:961–968 CrossRefGoogle Scholar
  12. 12.
    Kaczmarek M, Kubik J, Pakula M (2002) Short ultrasonic waves in cancellous bone. Ultrasonics 40:95–100 CrossRefGoogle Scholar
  13. 13.
    Levitsky SP, Bergman RM, Haddad J (2004) Wave propagation in a cylindrical viscous layer between two elastic shells. Int J Eng Sci 42:2079–2086 zbMATHCrossRefGoogle Scholar
  14. 14.
    Tadeu A, Mendes PA, António J (2006) 3D elastic wave propagation modelling in the presence of 2D fluid-filled thin inclusions. Eng Anal Bound Elem 30:176–193 zbMATHCrossRefGoogle Scholar
  15. 15.
    Paul HS, Murali VM (1992) Wave propagation in cylindrical poroelastic bone with cavity. Int J Eng Sci 30:1629–1635 zbMATHCrossRefGoogle Scholar
  16. 16.
    Qina Q, Qua C, Ye J (2005) thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads. Biomaterials 26:6798–6810 CrossRefGoogle Scholar
  17. 17.
    Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33(4):1482–1498 MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Lang SB (1970) Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bovini bones. IEEE Trans Biomed Eng 17:101–105 CrossRefGoogle Scholar
  19. 19.
    Davis CF (1970) On the mechanical properties of bone and a poroelastic theory of stresses in bone. PhD Thesis, Univ of Delaware Google Scholar
  20. 20.
    Ghista DN (1979) Applied Physiological Mechanics. Ellis Horwood, Chichester, pp 31–96 Google Scholar
  21. 21.
    Salzstein RA, Pollack SR, Mak AFT, Petrov N (1987) Electromechanical potentials in cortical bone a continuum approach. J Biomech 20:261–270 CrossRefGoogle Scholar
  22. 22.
    Ding H, Chenbuo L (1996) General solutions for coupled equations for piezoelectric media. Int J Solids Struct 16:2283–2298 Google Scholar
  23. 23.
    Gtizelsu N, Saha S (1981) Electro-mechanical wave propagation in long bones. J Biomech 14:9–33 Google Scholar
  24. 24.
    Mahmoud SR (2010) Wave propagation in cylindrical poroelastic dry bones. Appl Math Inf Sci 4(2):209–226 MathSciNetzbMATHGoogle Scholar
  25. 25.
    Protopappas VC, Vavva MG, Fotiadis DI, Malizos KN (2008) Ultrasonic monitoring of bone fracture healing. IEEE Trans Ultrason Ferroelectr Freq Control 55(6):1243–1255 CrossRefGoogle Scholar
  26. 26.
    Vavva MG, Protopappas VC, Fotiadis DI, Malizos KN (2008) Ultrasound velocity measurements on healing bones using the external fixation pins: a two-dimensional simulation study. J Serb Soc Comput Mech 2(2):1–15 Google Scholar
  27. 27.
    Kauffman JJ (2008) Ultrasonic guided waves in bone. IEEE Trans Ultrason Ferroelectr Freq Control 55(6):1205–1218 CrossRefGoogle Scholar
  28. 28.
    Moilanen P (2008) Ultrasonic guided waves in bone. IEEE Trans Ultrason Ferroelectr Freq Control 55(6):1277–1286 CrossRefGoogle Scholar
  29. 29.
    Vavva MG, Protopappas VC, Gergidis LN, Charalambopoulos A, Fotiadis DI, Polyzos D (2009) Velocity dispersion of guided waves propagating in a free gradient elastic plate: Application to cortical bone. J Acoust Soc Am 125(5):3414–3427 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • A. M. Abd-Alla
    • 1
    • 2
  • S. M. Abo-Dahab
    • 1
    • 4
  • S. R. Mahmoud
    • 2
    • 3
  1. 1.Maths. Dept., Faculty of ScienceTaif UniversityTaifSaudi Arabia
  2. 2.Maths. Dept., Faculty of ScienceSohag UniversitySohagEgypt
  3. 3.Maths. Dept., Faculty of EducationKing Abdul Aziz UniversityJeddahSaudi Arabia
  4. 4.Maths. Dept., Faculty of ScienceSVUQenaEgypt

Personalised recommendations