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Shear Waves Dispersion in Cylindrically Structured Cancellous Viscoelastic Bones

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Applied Non-Linear Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 93))

Abstract

In this chapter we study anti-plane shear waves propagating through a cylindrically structured cancellous bone represented by a two-dimensional mesh of elastic trabeculae filled by a viscous marrow. In the long-wave limit, the original heterogeneous medium can be approximately substituted by a homogeneous one characterized by an effective complex shear modulus. The effect of dispersion is caused by the transmission of mechanical energy to heat due to the viscosity of the marrow (viscoelastic damping). We derive an approximate analytical solution using the asymptotic homogenization method; the cell problem is solved by means of a boundary shape perturbation and a lubrication theory approaches. For short waves, when the wavelength is comparable to the trabeculae size, the effect of dispersion is caused by successive reflections and refractions of local waves at the trabecula-marrow interfaces (Bloch dispersion). Decrease in the wavelength reveals a sequence of pass and stop frequency bands, so the heterogeneous bone can act like a discrete wave filter.

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Acknowledgments

This work is supported by the Alexander von Humboldt Foundation (Institutional academic co-operation programme, grant no. 3.4-Fokoop-UKR/1070297), the German Research Foundation (Deutsche Forschungsgemeinschaft, grant no. WE 736/30-1).

Author information

Authors and Affiliations

  1. Department of General Mechanics, RWTH Aachen University, Templergraben 64, 52056, Aachen, Germany

    I. V. Andrianov

  2. Department of Structural Mechanics and Strength of Materials, Prydniprovs’ka State Academy of Civil Engineering and Architecture, Chernyshevs’kogo 24a, Dnipropetrovs’k, 49600, Ukraine

    V. V. Danishevs’kyy

  3. Department of Automation, Lodz University of Technology, Biomechanics and Mechatronics, 1/15 Stefanowski St., 90-924, Lodz, Poland

    J. Awrejcewicz

  4. Department of Vehicles, Warsaw University of Technology, 84 Narbutta Str., 02-524, Warsaw, Poland

    J. Awrejcewicz

Authors
  1. I. V. Andrianov
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  2. V. V. Danishevs’kyy
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  3. J. Awrejcewicz
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Corresponding author

Correspondence to I. V. Andrianov .

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Editors and Affiliations

  1. Department of Automation, Biomechanics and Mechatronics, Łódź University of Technology, Łódź, Poland

    Jan Awrejcewicz

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Andrianov, I.V., Danishevs’kyy, V.V., Awrejcewicz, J. (2014). Shear Waves Dispersion in Cylindrically Structured Cancellous Viscoelastic Bones. In: Awrejcewicz, J. (eds) Applied Non-Linear Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-319-08266-0_7

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