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Nonlinear Guided Waves and Thermal Stresses

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Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation

Abstract

The first part of the chapter covers theoretical considerations and numerical modeling of higher-harmonic generation in elastic waves propagating in nonlinear prismatic waveguides, including plates, rods, and waveguides of arbitrary cross-sections and/or of inhomogeneous and anisotropic composition. The main purpose of these analyses is to identify suitable combinations of primary and secondary guided modes for the waveguide. The last part of the chapter examines the role of thermal stresses in higher-harmonic wave generation. The latter topic is relevant to the prevention of thermal buckling of slender structural components (e.g., rail tracks).

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Author information

Authors and Affiliations

  1. Department of Structural Engineering, University of California San Diego, La Jolla, CA, USA

    Francesco Lanza di Scalea

  2. Illinois Institute of Technology, Chicago, IL, USA

    Ankit Srivastava

  3. Barclays Investment Bank, London, UK

    Claudio Nucera

Authors
  1. Francesco Lanza di Scalea
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  2. Ankit Srivastava
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  3. Claudio Nucera
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Corresponding author

Correspondence to Francesco Lanza di Scalea .

Editor information

Editors and Affiliations

  1. Department of Civil Engineering and Engineering Mechanics, Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ, USA

    Tribikram Kundu

A.1 Appendix

A.1 Appendix

The second-order and third-order coefficients of the interatomic potential for constrained thermal expansion, for the general case of the Mie potential, are given by the following expressions. These expressions are simply obtained by differentiating the potential VMIE in Eq. (9.145), and calculating the derivatives at the temperature-dependent interatomic position r*, where r* = rABD(T) for the fully constrained case, and r* = rABD-PC(T) for the partially constrained case.

$$ C(T){=}{\left.\frac{\partial^2{V}_{MIE}}{\partial {r}^2}\right|}_{r=r\cdot (T)}{=}\frac{n{\left(\frac{n}{m}\right)}^{\frac{n}{n-m}}\left[n\left(1+n\right){\left(\frac{q}{r\cdot (T)}\right)}^n-m\left(1+m\right){\left(\frac{q}{r\cdot (T)}\right)}^m\right]w}{\left(n-m\right){\left(r\cdot (T)\right)}^2} $$
(9.A1)

and

$$\begin{array}{lll} &&D(T)=\frac{\partial^3{V}_{MIE}}{\partial {r}^3} {\Bigg. \Bigg|}_{r=r\cdot (T)}\\ \nonumber &&{=}\frac{n{\left(\frac{n}{m}\right)}^{\frac{n}{n{-}m}}\left[m\left(1{+}m\right)\left(2{+}m\right){\left(\frac{q}{r\cdot (T)}\right)}^m{-}n\left(1{+}n\right)\left(2{+}n\right){\left(\frac{q}{r\cdot (T)}\right)}^n\right]w}{\left(n{-}m\right){\left(r\cdot (T)\right)}^3} \end{array}$$
(9.A2)

For the specific case of the Lennard-Jones potential (n = 12, m = 6), these expressions simplify to:

$$ {C}_{Lennard- Jones}(T)=\frac{24{wq}^6\left[26{q}^6-7{\left(r\cdot (T)\right)}^6\right]}{{\left(r\cdot (T)\right)}^{14}} $$
(9.A3)

and

$$ {D}_{Lennard- Jones}(T)=\frac{672{wq}^6\left[-13{q}^6+2{\left(r\cdot (T)\right)}^6\right]}{{\left(r\cdot (T)\right)}^{15}} $$
(9.A4)

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Lanza di Scalea, F., Srivastava, A., Nucera, C. (2019). Nonlinear Guided Waves and Thermal Stresses. In: Kundu, T. (eds) Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation. Springer, Cham. https://doi.org/10.1007/978-3-319-94476-0_9

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  • DOI: https://doi.org/10.1007/978-3-319-94476-0_9

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