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Nonlinear Cylindrical and Torsional Waves in Hyperelastic Materials

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Nonlinear Elastic Waves in Materials

Part of the book series: Foundations of Engineering Mechanics ((FOUNDATIONS))

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Abstract

This chapter is devoted to cylindrical and torsional waves and is divided into three parts. Both waves have the common feature that they are traditionally described by cylindrical coordinates.

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Author information

Authors and Affiliations

  1. Department of Rheology, S.P.Timoshenko Institute of Mechanics The National Academy of Sciences, Kyiv, Ukraine

    Jeremiah J. Rushchitsky

Authors
  1. Jeremiah J. Rushchitsky
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Exercises

Exercises

  1. 1.

    Point out the configurations (states) distinguishing from the four ones analyzed in the chapter.

  2. 2.

    Repeat more in detail the evaluation of invariants (10.15) and (10.16).

  3. 3.

    Repeat more in detail the evaluation of components of stress tensor (10.18).

  4. 4.

    Verify whether the simplest types of nonlinearity in (10.19), (10.20), and (10.21) number just 19.

  5. 5.

    Equations (10.32) and (10.33) are usually called the equations in stresses. Try calling this term in question basing on the fact that they include the displacements, too.

  6. 6.

    Write the fourth summand in representation (10.51) and refine the formula (10.54). Sketch the refined plot from Fig. 10.1.

  7. 7.

    The characteristic size of internal structure is introduced for the quadratic arrangement of fibers. Look for other types of arrangement.

  8. 8.

    Write the formula for evaluation of the limit frequency.

  9. 9.

    Write a definition of the true stress tensors.

  10. 10.

    Repeat the procedure of evaluation of the components of stress tensors (10.83) and (10.84).

  11. 11.

    Write in detail the procedure of transition from the motion equation in the concomitant coordinates to (10.85).

  12. 12.

    Show a transition from relationships (10.83) and (10.84) to the nonlinear wave equation in displacements (10.86).

  13. 13.

    Consider the nonlinear wave equations based on Murnaghan (five elastic constants) and Signorini (three elastic constants) models, represented at the end of Sect. 10.2. Propose the way to identify the Signorini elastic constant through the Murnaghan elastic constants.

  14. 14.

    Repeat a transition from (10.88) and (10.89) to the nonlinear wave equations in displacement (10.90).

  15. 15.

    Sketch the plots of Bessel functions J 0(x), J 1(x) from (10.93).

  16. 16.

    Comment solution (10.99) from the point of view that the first harmonic amplitude includes only the odd degrees of mr, whereas the second harmonic amplitude includes only the even degrees.

  17. 17.

    The potential (10.100) is written for the general case of anisotropy. Write it for the partial case of transversally isotropic materials.

  18. 18.

    Repeat the procedure of derivation of formulas for the components of stress tensor (10.102) starting with representation (10.101) of the potential.

  19. 19.

    Compare the nonlinear wave equations for the isotropic (10.90) and the transversely isotropic (10.103) cases and discuss a difference.

  20. 20.

    Verify the evaluation of expressions A k in a solution of the second approximation equation (10.106).

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Rushchitsky, J.J. (2014). Nonlinear Cylindrical and Torsional Waves in Hyperelastic Materials. In: Nonlinear Elastic Waves in Materials. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-00464-8_10

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  • DOI: https://doi.org/10.1007/978-3-319-00464-8_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-00463-1

  • Online ISBN: 978-3-319-00464-8

  • eBook Packages: EngineeringEngineering (R0)

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