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Nonlinear Plane Longitudinal Waves in Elastic Materials (John Model, Two-Constant Model and Signorini Model, Three-Constant Model)

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Nonlinear Elastic Waves in Materials

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Abstract

This chapter is devoted to an analysis of nonlinear elastic plane longitudinal harmonic waves, which corresponds to the John and Signorini models. The statement is divided into two parts.

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Authors and Affiliations

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

    Jeremiah J. Rushchitsky

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  1. Jeremiah J. Rushchitsky
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Exercises

Exercises

  1. 1.

    Compare M L from (6.6) with M from (5.22) and note the difference between the Murnaghan and John approaches.

  2. 2.

    Derive formulas (6.18), (6.19), and (6.20) for components of the stress tensor from the representation of Murnaghan potential (6.17).

  3. 3.

    Try to realize in steps one of six shown on page 170 possibilities (e.g., consider the 4th standard problem).

  4. 4.

    Compare solution (6.27), obtained for the cubically nonlinear approach, with solution (5.22), obtained for the quadratically nonlinear approach, and analyze the differences and similarities.

  5. 5.

    Define simple shear and pure shear. Formulate distinctions between the two.

  6. 6.

    Substantiate the statement from page 174 that the potential as the work of internal forces is always positive.

  7. 7.

    Define omni-dimensional (uniform) tension.

  8. 8.

    Describe the Pointing and Kelvin effects. Formulate the distinctions between the two.

  9. 9.

    Try to find the dependence among algebraic invariants of Almansi and Cauchy–Green strain tensors (6.41), (6.42), and (6.43) in books not mentioned on page 178.

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Rushchitsky, J.J. (2014). Nonlinear Plane Longitudinal Waves in Elastic Materials (John Model, Two-Constant Model and Signorini Model, Three-Constant Model). In: Nonlinear Elastic Waves in Materials. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-00464-8_6

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  • DOI: https://doi.org/10.1007/978-3-319-00464-8_6

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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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