FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
The evolution equations of intensive deformation problems of elastic inhomogeneous medium |
Ragozina V. E., Ivanova Yu. E. |
2015, issue 1, P. 76-90 |
Abstract |
| The motion problems of the plane longitudinal or transverse shock wave for the nonlinear elastic medium model with inhomogeneous properties, which are represented by a continuous changes of the elastic moduli and density are considered. Changing of the medium properties is assumed in the direction of the wave fronts motion. The method of matched asymptotic expansions allows to determine the problems evolution equations, reflecting nonlinear wave processes and the inhomogeneity of the medium. The most interesting variant of the evolution equation occurs whenthe intensity of the impact process and the small inhomogeneity havethe same order. The transition to the limiting inner problem of the smallparameter method is dictated by the chain of inner problems for which it is necessary to change all the independent variables and their scales. |
Keywords: nonlinear elastic medium, inhomogeneity, shock waves, volume and shear deformation, evolution equation |
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References |
| [1] D. Blend, Nelinejnaja dinamicheskaja teorija uprugosti, Mir, Moskva, 1972, 183 s. [2] A. G. Kulikovskij, E. I. Sveshnikova, Nelinejnye volny v uprugih sredah, Moskovskij licej, Moskva, 1998, 412 s. [3] A. A. Burenin, A. D. Chernyshov, “Udarnye volny v izotropnom uprugom prostranstve”, Prikl. matematika i mehanika, 42:4 (1978), 711–717. [4] M. Van-Dajk, Metody vozmushhenij v mehanike zhidkosti, Mir, Moskva, 1967, 239 s. [5] Dzh. Koul, Metody vozmushhenij v prikladnoj matematike, Mir, Moskva, 1972, 275 s. [6] A. H. Najfe, Vvedenie v metody vozmushhenij, Mir, Moskva, 1984, 535 s. [7] J. D. Achenbach, D. P. Reddy, ""Note of wave propogation in lineary viscoelastic media"", ZAMP, 18:1 (1967), 141-144. [8] L. A. Babicheva, G. I. Bykovcev, N. D. Vervejko, “Luchevoj metod reshenija dinamicheskih zadach v uprugovjazkoplasticheskih sredah”, Prikl. matematika i mehanika,37:1 (1973), 145–155. [9] A. A. Burenin, Ju.A. Rossihin, “Luchevoj metod reshenija odnomernyh zadach nelinejnoj dinamicheskoj teorii uprugosti s ploskimi poverhnostjami sil'nyh razryvov”, Prikladnye zadachi mehaniki deformiruemyh sred, 1991, 129–137. [10] A. A. Burenin, “Ob odnoj vozmozhnosti postroenija priblizhennyh reshenij nestacionarnyh zadach dinamiki uprugih sred pri udarnyh vozdejstvijah”, Dal'nevostochnyj mat. sb., 1999, № 8, 49–72. [11] Dzh. Uizem, Linejnye i nelinejnye volny, Mir, Moskva, 1977, 622 s. [12] V. E. Ragozina, Ju.E. Ivanova, “Ob jevoljucionnyh uravnenijah zadach udarnogo deformirovanija s ploskimi poverhnostjami razryvov”, Vychislitel'naja mehanika sploshnyh sred, 2:3 (2009), 82–95. [13] A. A. Burenin, Ju.A. Rossihin, “O vlijanii vjazkosti na harakter rasprostranenija ploskoj prodol'noj volny”, Prikl. mehanika i tehn. fizika, 1990, №6, 13–17. [14] Ju.E. Ivanova, “O strukture udarnoj volny deformacij izmenenija formy”, Materialy Vserossijskoj konferencii ""Fundamental'nye i prikladnye voprosy mehaniki"", posvjashhennoj 70-letiju so dnja rozhdenija akademika V.P. Mjasnikova, 2006,52–54. [15] Ju.N. Pelinovskij, V. E. Fridman, Ju.K. Jengel'breht, Nelinejnye jevoljucionnye uravnenija, Valgus, Tallinn, 1984, 164 s. [16] A. I. Lur'e, Nelinejnaja teorija uprugosti, Nauka, Moskva, 1980, 512 s. [17] T. Tomas, Plasticheskoe techenie i razrushenie v tverdyh telah, Mir, Moskva, 1964, 308 s. [18] G. I. Bykovcev, D. D. Ivlev, Teorija plastichnosti, Dal'nauka, Vladivostok, 1998,528 s. |