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Wave propagation in anisotropic poroelastic beam with axial–flexural coupling

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Abstract

The exact solution of the governing partial differential equations describing the motion of an anisotropic porous beam with axial-flexural coupling is presented. The motion of the beam is described by the classical Euler–Bernoulli theory. The pore-fluid pressure is governed by the generalized Darcy’s law with relaxation and retardation time parameters to account for the inertia and viscosity of the fluid. Solutions are sought in the frequency domain where the governing equations are converted into a polynomial eigenvalue structure and solved exactly. The wavenumbers and group speeds of propagating waves in the beam are studied in detail. It is found that the presence of fluid-filled porous micro-structure introduces three additional propagating modes, other than the axial and bending modes predicted by the classical beam theory. The effect of diffusion boundary conditions on the transverse motion of a porous beam is investigated in detail. It is also found that the material parameters have considerable influence on the magnitude of the transverse velocity, the group speed of propagation and the behavior of the pressure resultants.

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References

  1. Abousleiman Y, Cui L (1998) Poroelastic solutions in transversely isotropic media for wellbore and cylinder. Int J Solids Struct 35(34-25): 4905–4929

    Article  MATH  Google Scholar 

  2. Abousleiman Y, Cheng A, Cui L, Detournay E, Roegiers J (1996) Mandel’s problem revisited. Geotechnique 46(2): 187–195

    Article  Google Scholar 

  3. Badiey M, Yamamoto T (1985) Propagation of acoustic normal modes in a homogeneous ocean overlying layered anisotropic porous beds. J Acoust Soc Am 77: 954–981

    Article  MATH  Google Scholar 

  4. Bertola V, Cafaro E (2006) Thermal instability in viscoelastic fluids in horizontal porous layers as initial value problem. Int J Heat Mass Transf 49(21-22): 4003–4012

    Article  MATH  Google Scholar 

  5. Biot MA (1941) Consolidation settlement under rectangular load distribution. J Appl Phys 12: 426–430

    Article  Google Scholar 

  6. Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12: 155–165

    Article  Google Scholar 

  7. Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26: 182–185

    Article  MATH  MathSciNet  Google Scholar 

  8. Biot MA (1956) General solutions of the equations of elasticity and consolidation for a porous material. J Appl Mech 23: 91–96

    MATH  MathSciNet  Google Scholar 

  9. Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solids (parts I and II). J Acoust Soc Am 28: 168–191

    MathSciNet  Google Scholar 

  10. Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33: 1482–1498

    Article  MATH  MathSciNet  Google Scholar 

  11. Biot MA, Willis D (1957) The elastic coefficients of the theory of consolidation. J Appl Mech 24: 594–601

    MathSciNet  Google Scholar 

  12. Cederbaum G, Li L, Schulgasser K (2000) Poroelastic structures. Elsevier Ltd, Amsterdam

    Google Scholar 

  13. Chen I, Saha S (1987) Wave propagation characteristics in long bones to diagnose osteoporosis. J Biomech 20(5): 523–527

    Google Scholar 

  14. Cheng A (1997) Material coefficient of anisotropic poroelasticity. Int J Rock Mech Min Sci 34: 199–205

    Article  Google Scholar 

  15. Cheng S, Timonen J, Suominen H (1995) Elastic wave propagation in bone in vivo: methodology. J Biomech 28(4): 471–478

    Article  Google Scholar 

  16. Cowin S (1999) Bone poroelasticity. J Biomech 32: 217–238

    Article  Google Scholar 

  17. Cui L, Abousleiman Y, Cheng A, Roegiers J (1996) Anisotropic effect on one-dimensional consolidation. Eng Mech Vol 1, Proceedings of the 11th Engineering Mechanics Conference, ASCE 1(Y.K. Lin and T.C. Su):471–474

  18. Foldes A, Rimon A, Keinan D, Popovtzer M (1995) Quantitative ultrasound of the tibia: a novel approach for assessment of bone status. Bone 17(4): 363–367

    Article  Google Scholar 

  19. Gopalakrishnan S, Chakraborty A, RoyMahapatra D (2008) Spectral finite element method. Springer, New York

    MATH  Google Scholar 

  20. Gu W, Lai W, Mow V (1993) Transport of fluid and ions through a porous-permeable charged-hydrated tissue, and streaming potential data on normal bovine articular cartilage. J Biomech 26: 709–723

    Article  Google Scholar 

  21. Kazi-Aoual M, Bonnet G, Jouanna P (1988) Green’s function in an infinite transversely isotropic saturated poroelastic medium. J Acoust Soc Am 84: 1883–1889

    Article  Google Scholar 

  22. Khan M, Saleem M, Fetecau C, Hayat T (2007) Transient oscillatory and constantly accelerated non-newtonian flow in a porous medium. Int J Nonlinear Mech 42(10): 1224–1239

    Article  Google Scholar 

  23. Kohles S, Vanderby R, Ashman R, Manley P, Markel M, Heiner J (1994) Ultrasonically determined elasticity and cortical density in canine femora after hip arthroplasty. J Biomech 27(2): 137–144

    Article  Google Scholar 

  24. Krone R, Schuster P (2006) An investigation on the importance of material anisotropy in fe modeling of the human femur. SAE SP-1995(2006-01-0064):1–9

  25. Menahem AB, Gibson R (1993) Directional attenuation of sh waves in anisotropic poroelastic media. J Acoust Soc Am 93: 3057–3065

    Article  Google Scholar 

  26. Mourtada FA, Beck TJ, Hauser D, Ruff C, Bao G (1996) Curved beam model of the proximal femur for estimating stress using dual-energy X-ray absorptiometry derived structural geometry. J Orth Res 14: 483–492

    Article  Google Scholar 

  27. Nowinski J (1971) Bone articulations as systems of poroelastic bodies in contact. AIAA J 9: 62–69

    Article  Google Scholar 

  28. Nowinski J (1972) Stress concentrations around a cylindrical cavity in a bone treated as a poroelastic body. Acta Mech 13: 281–292

    Article  MATH  Google Scholar 

  29. Nowinski J, Davis C (1970) A model of the human skull as a poroelastic spherical shell subjected to a quasistatic load. Math Biosci 8: 397–416

    Article  MATH  Google Scholar 

  30. Protopappas V, Fotiadis D, Malizos K (2006) Guided ultrasound wave propagation in intact and healing long bones. Ultrasound Med Biol 32(5): 693–708

    Article  Google Scholar 

  31. Rice JR, Cleary M (1976) Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev Geophys Space Phys 14(4): 227–241

    Article  Google Scholar 

  32. Schmitt D (1989) Acoustic multipole logging in transversely isotropic poroelastic formations. J Acoust Soc Am 86: 2397–2421

    Google Scholar 

  33. Sharma MD, Gogna M (1991) Wave propagation in anisotropic liquid-saturated porous solids. J Acoust Soc Am 90(2): 1068–1073

    Article  Google Scholar 

  34. Thompson M, Willis J (1991) A reformulation of the equations of anisotropic poroelasticity. J Appl Mech 58: 612–616

    Article  MATH  Google Scholar 

  35. Vashishth AK, Khurana P (2004) Waves in stratified anisotropic poroelastic media: a transfer matrix approach. J Sound Vib pp 239–275

  36. Wang Y, Zhang Z (1997) Propagation of love waves in a transversely isotropic fluid-saturated porous layered half-space. J Acoust Soc Am 103(2): 695–701

    Google Scholar 

  37. Zhang D, Cowin S (1994) Oscillatory bending of a poroelastic beam. J Mech Phys Solids 42(10): 1575–1599

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. India Science Lab, General Motors R&D, Bangalore, India

    A. Chakraborty

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  1. A. Chakraborty
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Correspondence to A. Chakraborty.

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Chakraborty, A. Wave propagation in anisotropic poroelastic beam with axial–flexural coupling. Comput Mech 43, 755–767 (2009). https://doi.org/10.1007/s00466-008-0343-6

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  • DOI: https://doi.org/10.1007/s00466-008-0343-6

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