Effect of the Third Approximation in the Analysis of the Evolution of a Nonlinear Elastic P-wave. Part 1*
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International Applied Mechanics, Vol. 56, No. 5, September, 2020
EFFECT OF THE THIRD APPROXIMATION IN THE ANALYSIS OF THE EVOLUTION OF A NONLINEAR ELASTIC P-WAVE. PART 1* J. J. Rushchitsky* and V. N. Yurchuk
A nonlinear plane longitudinal elastic displacement wave is studied theoretically and numerically using the Murnaghan model for two forms of initial profile: harmonic and bell-shaped. A major novelty is that the evolution of waves is analyzed by approximate methods taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolution of waves are shown. The initially symmetric profiles transform differently due to distortion: symmetrically (for the harmonic profile) and asymmetrically (for the bell-shaped profile). The third approximation introduces the fourth harmonic for the harmonic wave when this wave is analyzed by the method of successive approximations, while the bell-shaped wave is characterized in the third approximation differently when using the method of constraints on the displacement gradient. At relatively long distances from the beginning of the propagation, the one-hump bell-shaped wave transforms into a two-hump one. These humps adjoin each other halving their lengths. The third approximation allows us to observe new wave effects: the asymmetry of the left and right humps about their peaks and the asymmetry of the humps about each other; the lowering of the left hump and the rise of the right one. The results obtained are analyzed. Keywords: nonlinear elastic P-wave; Murnaghan potential; approximate method; harmonic and bell-shaped initial wave profiles; evolution; distortion 1. Introduction. In the presented study, the five-constant Murnaghan model of nonlinear elastic deformation of a material is used [1–3, 5, 7–11]. The Murnaghan elastic potential is known to be quadratically and cubically nonlinear in the components of the Cauchy–Green strain tensor e nm = (1/ 2)( u n ,m + u m ,n + u k ,n u k ,m ) (u k are the components of the displacement vector) W ( e ik ) = (1/ 2)l( e mm ) 2 + m ( e ik ) 2 + (1/ 3) Ae ik e im e km + B ( e ik ) 2 e mm + (1/ 3)C ( e mm ) 3
(1)
(l, m , A , B , C are the Murnaghan elastic constants). We consider the case where the Murnaghan potential is expressed in terms of the displacement gradients taking into account only the quadratically and cubically nonlinear components: W = (1/ 2)l( u m ,m ) 2 + (1/ 4 )m ( u i ,k + u k ,i ) 2 + (m + (1/ 4 ) A)u i ,k u m ,i u m ,k
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, Ukraine 03057; *e-mail: [email protected]. Translated from Prikladnaya Mekhanika, Vol. 56, No. 5, pp. 65–77, September–October 2020. Original article submitted October 30, 2019.
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This study was sponsored by the budget program “Support for Priority Areas of Scientific Research” (KPKVK 6541230).
1063-7095/20/5605-0581 ©2020 Springer Science+Business Media, LLC
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