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Nonlinear Rayleigh and Love Surface Waves in Elastic Materials

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Nonlinear Elastic Waves in Materials

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

Abstract

This chapter is devoted to analysis of nonlinear elastic Rayleigh and Love surface waves, which corresponds to the Murnaghan model. The analysis is divided into two parts. In the first part, the Rayleigh wave is analyzed. In the first subsection, elastic surface waves are described, and then the basic moments in the theory of elastic linear Rayleigh surface waves are shown. Further, nonlinear elastic Rayleigh surface waves are discussed, which includes general information (new variants of quadratically nonlinear equations describing the two-dimensional motion in dependence on two spatial coordinates x 1, x 3 and time t), basic equations, procedures of solving the nonlinear wave equations, and, finally, the first two approximations are obtained and commented on. The main nonlinear effect is that the second harmonic appears in the description of the wave propagation. An analysis of nonlinear boundary conditions is carried out separately. Here the boundary conditions for cases of small and large deformations are given and then an analysis of boundary conditions is carried out. Finally, a new nonlinear Rayleigh equation is derived and discussed. The main nonlinear effect is that the phase velocity depends nonlinearly on the initial amplitude. In the second part, the problem of elastic Love waves is considered in a classical statement with additional assumption of a presence of nonlinearity in the description of deformation. The nonlinear Murnaghan model is used. A new nonlinear wave equation in displacements is derived that includes a linear part and a part with summands of the third and fifth orders of nonlinearity, only. Where the physical nonlinearity is allowed, the solution of a new nonlinear equation with nonlinear boundary conditions is obtained by the method of successive approximations within the framework of the first two approximations. A new nonlinear equation for determining the wave number is derived, which shows a new factor in an initial wave profile distortion—distortion owing to the wavelength changes with unchanging frequency. This information can be found in scientific publications on the elastic Rayleigh and Love waves, list of which is given in the reference list (48 titles) of the chapter [1–48].

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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

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  1. Jeremiah J. Rushchitsky
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Exercises

Exercises

  1. 1.

    The very successful application of the surface waves is observed in the seismology. Which property of the surface waves is the most important there?

  2. 2.

    Which Rayleigh problem is more popular in applications (not in mechanics)—the problem for the plane boundary or the problem for the cylindrical one?

  3. 3.

    Write the explicit solutions of linear ordinary differential equations with constant coefficients (11.6) and substantiate the necessity of conditions (11.7).

  4. 4.

    Repeat the transition from the first variant of Rayleigh equation [the first line in (11.11)] to the second variant of Rayleigh equation [the second line in (11.11)].

  5. 5.

    The most known ways to prove an existence of the real root of Rayleigh equation are based on finding the interval, where the Rayleigh wave number changes the sign. verify the very original method (described, for example, in [1]), based on the principle of argument.

  6. 6.

    Study more in depth the notion “sagittal plane” and convince that in the general case the Rayleigh wave is formed of the longitudinal and transverse plane waves.

  7. 7.

    See the Stroh formalism (for example, in [13]) and formulate the main advantage of this formalism as compared with the classical approach.

  8. 8.

    Check the transition from (11.18), (11.19), (11.20), and (11.21) to the (11.23).

  9. 9.

    Look for the approach to analysis of Rayleigh waves, which does not use the potentials and compare the procedures.

  10. 10.

    Consider representations (11.44) and (11.45) and the cases 1 and 2. Estimate the utility of analysis of these cases.

  11. 11.

    Suppose the simple case of curvilinear boundary (for example, in the form of parabola) and write the explicit formula for the boundary condition (11.48).

  12. 12.

    Think whether the contradiction exists between the assumption (11.67) and the final formula (11.72).

  13. 13.

    Determine the area of values of wave number k R in the case 1 [see (11.74)].

  14. 14.

    Read more in depth about the anti-plane problem of the theory of elasticity and link this problem with the Love wave statement.

  15. 15.

    Why the boundary conditions (11.83) and (11.84) include only one component of the stress tensor?

  16. 16.

    Substantiate the fact that (11.86) has an infinite number of roots.

  17. 17.

    Check the transition from potential (11.91) to the components of stress tensor (11.92).

  18. 18.

    Compare the steps from the linear standard motion equation in stresses to the nonlinear wave equation (11.95) in analysis of boundary conditions (11.106), (11.107), (11.108), (11.109), (11.110), (11.111), (11.112), (11.113), (11.114), and (11.115) with corresponding steps in analysis of Rayleigh waves. Fix the similarity and difference.

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Rushchitsky, J.J. (2014). Nonlinear Rayleigh and Love Surface Waves in Elastic Materials. In: Nonlinear Elastic Waves in Materials. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-00464-8_11

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  • DOI: https://doi.org/10.1007/978-3-319-00464-8_11

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  • Online ISBN: 978-3-319-00464-8

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