Far Eastern Mathematical Journal

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The evolution equation of transverse shock waves in solids


Ragozina V. E., Ivanova Yu. E.

2013, issue 1, P. 116-126


Abstract
Solution of a number of boundary value problems by the method of matched asymptotic expansions for single-wave processes in incompressible nonlinear elastic media is carried out. The frontal area of the wave is defined by the nonlinear evolution equation, which is different from the Cole – Hopf equation. This demonstrates the fundamental differences in the mechanisms of formation and subsequent movement of volume and shear shock waves. The authors propose the inclusion of particular solutions of the evolution equation in the additional parametric method for the determination of the displacement field and medium strains.

Keywords:
nonlinear elasticity, incompressibility, shock wave, perturbation method, evolution equation

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References

[1] A. G. Kulikovskii, E. I. Sveshnikova, Nelineinye volny v uprugikh sredakh, Moskovskii litsei, Moskva, 1998, 412 s.
[2] A. A. Burenin, A. D. Chernyshov, “Udarnye volny v izotropnom uprugom prostranstve”, Prikl. matematika i mekhanika, 42:4 (1978), 711–717.
[3] J. D. Achenbach, D.P. Reddy, “Note of wave propogation in lineary viscoelastic media”, ZAMP, 18:1 (1967), 141-144.
[4] L. A. Babicheva, G. I. Bykovtsev, N. D. Verveiko, “Luchevoi metod resheniia dinamicheskikh zadach v uprugoviazkoplasticheskikh sredakh”, Prikl. matematika i mekhanika, 37:1 (1973), 145–155.
[5] M. Van-Daik, Metody vozmushchenii v mekhanike zhidkosti, Mir, Moskva, 1967, 239 s.[6] A. A. Burenin, Iu.A. Rossikhin, “K resheniiu odnomernoi zadachi nelineinoi dinamicheskoi teorii uprugosti so strukturnoi udarnoi volnoi”, Prikl. mekhanika, 26:1 (1990), 103–108.
[7] Dzh. Uizem, Lineinye i nelineinye volny, Mir, Moskva, 1977, 622 s.
[8] L. I. Sedov, Mekhanika sploshnoi sredy. T. 1,2. Izd-ie 2-oe ispr. i dopoln., Nauka, Moskva, 1973, T.1. 536 T.2. 584 s.
[9] T. Tomas, Plasticheskoe techenie i razrushenie v tverdykh telakh, Mir, Moskva, 1964, 308 s.
[10] Dzh. Koul, Metody vozmushchenii v prikladnoi matematike, Mir, Moskva, 1972, 275 s.
[11] V. E. Ragozina, Iu.E. Ivanova, “Ob udarnykh osesimmetricheskikh dvizheniiakh neszhimaemoi uprugoi sredy pri udarnykh vozdeistviiakh”, PMTF, 47:6 (2006), 144–151.
[12] V. E. Ragozina, I. I. Voronin, E. L. Vekovshinin, “Ob ispol'zovanii prifrontovoi asimptotiki v chislennykh resheniiakh dinamicheskikh zadach teorii uprugosti s udarnymi volnami”, Problemy estestvoznaniia i proizvodstva, 1995, №115, 25–27.
[13] A. A. Burenin, P. V. Zinov'ev, “K probleme vydeleniia poverkhnostei razryvov v chislennykh metodakh dinamiki deformiruemykh sred”, Problemy mekhaniki. Sbornik statei k 90-letiiu A.Iu. Ishlinskogo, 2003, 146–155.

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