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Cubic Nonlinearity in Elastic Materials: Theoretical Prediction and Computer Modelling of New Wave Effects

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  • Jeremiah Rushchitsky
  • Carlo Cattani
  • Sergiy Sinchilo

Abstract

Our object of interest is nonlinear interaction of waves in elastic materials. The new model of a material is proposed that takes into account the mechanism of simultaneous quadratic and cubic nonlinear deformations. Introduction of cubic nonlinearity into the model makes the general wave picture more complicated and creates new possibilities for the wave analysis. We present four possibilities for the evolution of profiles of plane harmonic waves. It is noted that quadratic and cubic nonlinearities emerge first of all in the second and third harmonics generation, respectively. Further, we discuss the results of computer modelling of the wave profile evolution. The influence of the progress of second and third harmonics on the wave profile evolution is studied separately. We study separately how second and third harmonics influence the evolution of the wave profile. We also investigate how the progress of harmonics depends on the initial frequency and amplitude. We find two distinct schemes of the evolution progress: the scheme (in) with four stages for the second harmonics and the scheme with three stages for the third harmonics. As a result the influence of both harmonics could be observed simultaneously, and such a case is demonstrated in the paper. Nevertheless this phenomenon is not necessarily present in every material which explains the absence of experimental observations of the third harmonics by this time.

Suggested Citation

  • Jeremiah Rushchitsky & Carlo Cattani & Sergiy Sinchilo, 2004. "Cubic Nonlinearity in Elastic Materials: Theoretical Prediction and Computer Modelling of New Wave Effects," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 10(3-4), pages 331-352, September.
    • Handle: RePEc:taf:nmcmxx:v:10:y:2004:i:3-4:p:331-352
    DOI: 10.1080/13873950412331335298
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