Geodesic
Some Problems in Nonlinear Dynamic Elasticity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 173-199.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper contains a survey of recent studies of small-amplitude quasi-transverse one-dimensional waves in elastic and viscoelastic media. The following issues are addressed: We study small-amplitude nonlinear waves in elastic media under a more accurate consideration of the internal energy compared with the earlier works. We describe new properties of shock waves and Riemann waves of small amplitude in an anisotropic medium whose properties are invariant under the rotation through $120^\circ $ about the wave normal. We formulate similarity conditions for one-dimensional problems of nonlinear elasticity. We discuss reasons for the earlier discovered nonuniqueness of solutions to self-similar problems for waves in elastic media, and formulate a criterion that allows one to predict, based solely on the properties of the shock adiabat, the nonuniqueness or the nonexistence of self-similar solutions to systems of hyperbolic equations that express conservation laws. We consider the structure of shock waves in elastic media in the framework of the Kelvin–Voigt model of a viscous medium. The results of the numerical analysis of the nonlinear stability of the structure of metastable shock waves are also presented. 
@article{TM_2005_251_a7,
     author = {A. G. Kulikovskii and E. I. Sveshnikova and A. P. Chugainova},
     title = {Some {Problems} in {Nonlinear} {Dynamic} {Elasticity}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {173--199},
     publisher = {mathdoc},
     volume = {251},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2005_251_a7/}
}
TY  - JOUR
AU  - A. G. Kulikovskii
AU  - E. I. Sveshnikova
AU  - A. P. Chugainova
TI  - Some Problems in Nonlinear Dynamic Elasticity
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 173
EP  - 199
VL  - 251
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2005_251_a7/
LA  - ru
ID  - TM_2005_251_a7
ER  - 
%0 Journal Article
%A A. G. Kulikovskii
%A E. I. Sveshnikova
%A A. P. Chugainova
%T Some Problems in Nonlinear Dynamic Elasticity
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 173-199
%V 251
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2005_251_a7/
%G ru
%F TM_2005_251_a7
A. G. Kulikovskii; E. I. Sveshnikova; A. P. Chugainova. Some Problems in Nonlinear Dynamic Elasticity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 173-199. http://geodesic.mathdoc.fr/item/TM_2005_251_a7/

[1] Sedov L. I., Mekhanika sploshnoi sredy, 2, Nauka, M., 1994, 560 pp. | MR

[2] Bland D. R., Nonlinear dynamic elasticity, Waltham, Toronto etc., 1969 ; Blend D. R., Nelineinaya dinamicheskaya teoriya uprugosti, Mir, M., 1972, 183 pp. | MR | Zbl | MR

[3] Kulikovskii A. G., Sveshnikova E. I., Nelineinye volny v uprugikh sredakh, Mosk. Litsei, M., 1998, 412 pp.

[4] Kulikovskii A. G., “O nelineinykh volnakh maloi amplitudy v anizotropnom uprugom tele”, Tr. MIAN, 211, 1995, 288–303 | MR | Zbl

[5] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001, 607 pp. | MR | Zbl

[6] Kulikovskii A. G., Lyubimov G. A., Magnitnaya gidrodinamika, Fizmatgiz, M., 1962, 246 pp.

[7] Lax P. D., “Hyperbolic systems of conservation laws”, Commun. Pure and Appl. Math., 10 (1957), 537–566 | DOI | MR | Zbl

[8] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978, 687 pp. | MR | Zbl

[9] Kulikovskii A. G., “Ob uravneniyakh, opisyvayuschikh rasprostranenie nelineinykh kvazipoperechnykh voln v slaboanizotropnom uprugom tele”, PMM, 50:4 (1986), 597–604 | MR | Zbl

[10] Sveshnikova E. I., “Prostye volny v nelineino uprugoi srede”, PMM, 46:4 (1982), 642–646

[11] Kulikovskii A. G., Sveshnikova E. I., “Issledovanie udarnoi adiabaty kvazipoperechnykh udarnykh voln v predvaritelno napryazhennoi uprugoi srede”, PMM, 46:5 (1982), 831–840

[12] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, T. 6: Gidrodinamika, AN SSSR, M., 1953, 788 pp.

[13] Kulikovskii A. G., Sveshnikova E. I., “Vliyanie izmeneniya entropii na formu udarnoi adiabaty kvazipoperechnykh uprugikh voln”, PMM, 67:1 (2003), 88–98 | MR

[14] Avdeeva A. D., Sveshnikova E. I., “Kvazipoperechnye udarnye volny v uprugoi srede s uslozhnennym uprugim potentsialom”, Izv. RAN. Mekhanika tverdogo tela, 2004, no. 6, 102–113

[15] Sveshnikova E. I., “Volny Rimana v uprugoi srede s maloi kubicheskoi anizotropiei”, PMM, 69:1 (2005), 75–83 | MR | Zbl

[16] Kulikovskii A. G., Sveshnikova E. I., “Avtomodelnaya zadacha o deistvii vnezapnoi nagruzki na granitsu uprugogo poluprostranstva”, PMM, 49:2 (1985), 284–291 | MR | Zbl

[17] Kulikovskii A. G., “Osobennosti povedeniya nelineinykh kvazipoperechnykh voln v uprugoi srede pri maloi anizotropii”, Tr. MIAN, 186, 1989, 132–139 | MR | Zbl

[18] Kulikovskii A. G., Sveshnikova E. I., “Nelineinye volny, voznikayuschie pri izmenenii napryazhenii na granitse uprugogo poluprostranstva”, Voprosy nelineinoi mekhaniki sploshnoi sredy, Valgus, Tallinn, 1985, 133–145

[19] Kulikovskii A. G., Sveshnikova E. I., “O raspade proizvolnogo nachalnogo razryva v uprugoi srede”, PMM, 52:6 (1988), 1007–1012 | MR | Zbl

[20] Kulikovskii A. G., Sveshnikova E. I., “Priznak nesuschestvovaniya i needinstvennosti reshenii avtomodelnykh zadach mekhaniki sploshnoi sredy”, PMM, 65:6 (2001), 971–982 | MR

[21] Galin G. Ya., “K teorii udarnykh voln”, DAN SSSR, 127:1 (1959), 55–58 | Zbl

[22] Kulikovskii A. G., Sveshnikova E. I., “O strukture kvazipoperechnykh uprugikh udarnykh voln”, PMM, 51:6 (1987), 926–932 | MR | Zbl

[23] Chugainova A. P., “Issledovanie struktury kvazipoperechnykh udarnykh voln dlya opredelennogo klassa uprugikh sred”, Problemy mekhaniki, ekologii, tekhnologii, Nauka, M., 1991

[24] Chugainova A. P., “O formirovanii avtomodelnogo resheniya v zadache o nelineinykh volnakh v uprugom poluprostranstve”, PMM, 52:4 (1988), 692–697 | MR

[25] Chugainova A. P., “O vykhode nelineinykh voln na avtomodelnyi rezhim v zadache o deistvii vnezapnogo izmeneniya nagruzki na granitsu uprugogo poluprostranstva”, Izv. AN SSSR. Mekhanika tverdogo tela, 25:3 (1990), 187–189

[26] Chugainova A. P., “O vzaimodeistvii nelineinykh voln v slaboanizotropnoi uprugoi srede”, PMM, 57:2 (1993), 149–156 | MR

[27] Chugainova A. P., “O perestroike nelineinoi uprugoi volny v srede s maloi anizotropiei”, Izv. RAN. Mekhanika tverdogo tela, 28:5 (1993), 75–81

[28] Kulikovskii A. G., Chugainova A. P., “Ob usloviyakh raspada nelineinoi volny v vyazkouprugoi srede”, ZhVM i MF, 38:2 (1998), 315–323 | MR | Zbl

[29] Kulikovskii A. G., Chugainova A. P., “Ob ustoichivosti k dvumernym vozmuscheniyam metastabilnoi udarnoi volny v vyazkouprugoi srede”, PMM, 66:1 (2002), 109–117 | Zbl

[30] Kulikovskii A. G., Chugainova A. P., “Ob ustoichivosti kvazipoperechnykh udarnykh voln v anizotropnykh uprugikh sredakh”, PMM, 64:6 (2000), 1020–1026 | MR | Zbl

[31] Kadomtsev B. B., Petviashvili V. I., “Ob ustoichivosti uedinennykh voln v slabo dispergiruyuschikh sredakh”, DAN SSSR, 192:4 (1970), 753–756 | Zbl

[32] Zabolotskaya E. A., Khokhlov R. V., “Kvaziploskie volny v nelineinoi akustike ogranichennykh puchkov”, Akust. zhurn., 15:1 (1969), 40–47

[33] Samarskii A. A., Popov Yu. P., Raznostnye metody resheniya zadach gazovoi dinamiki, Nauka, M., 1980, 352 pp. | MR