The evolution of a nonlinear elastic cylindrical displacement wave is analyzed theoretically and numerically using the five-constant Murnaghan model for a noncharacteristic initial profile in the form of the Macdonald function. Unlike most nonlinear waves in materials that have periodical or single humps, this wave has no hump. It monotonically decreases and has a profile concaved downward. The basic novelty of this research is that the evolution of waves is studied using approximate methods and considering the first three approximations. Some essential distinctions of this wave are shown: the noncharacteristic profile evolves in a noncharacteristic way. First, the peculiarities of three types of waves (a harmonic wave (periodically repeated profile) and two single waves with the initial profiles in the form of the Gauss function (symmetric profile) and the Whittaker function (nonsymmetric profile)), which may be considered characteristic properties, are briefly described. Further, a single wave with the initial profile in the form of the Macdonald function is analyzed in detail, both theoretically and numerically. The initial profile distortion caused by the nonlinear interaction of the wave and the decrease in the maximum amplitude during the wave propagation are common to these profiles. Significant features of the Macdonald wave are shown: the noncharacteristic initial profile (without a classical hump) evolves in a noncharacteristic way, as the profile becomes much (soft nonlinearity) or less (hard nonlinearity) steeper, has no hump, and remains convex downward.
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This study was sponsored by the budget program “Support for Priority Areas of Scientific Research” (KPKVK 6541230).
Translated from Prikladnaya Mekhanika, Vol. 57, No. 6, pp. 3–21, November–December 2021
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Rushchytsky, J.J., Yurchuk, V.M., Hryhorchuk, O.M. et al. Noncharacteristic Evolution of a Nonlinear Elastic Single Cylindrical Wave*. Int Appl Mech 57, 619–634 (2021). https://doi.org/10.1007/s10778-022-01112-x
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DOI: https://doi.org/10.1007/s10778-022-01112-x