Geodesic
Special discontinuities in nonlinearly elastic media
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 6, pp. 1023-1032 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems. 
@article{ZVMMF_2017_57_6_a8,
     author = {A. P. Chugainova},
     title = {Special discontinuities in nonlinearly elastic media},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1023--1032},
     year = {2017},
     volume = {57},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_6_a8/}
}
TY  - JOUR
AU  - A. P. Chugainova
TI  - Special discontinuities in nonlinearly elastic media
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2017
SP  - 1023
EP  - 1032
VL  - 57
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_6_a8/
LA  - ru
ID  - ZVMMF_2017_57_6_a8
ER  - 
%0 Journal Article
%A A. P. Chugainova
%T Special discontinuities in nonlinearly elastic media
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2017
%P 1023-1032
%V 57
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_6_a8/
%G ru
%F ZVMMF_2017_57_6_a8
A. P. Chugainova. Special discontinuities in nonlinearly elastic media. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 6, pp. 1023-1032. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_6_a8/

[1] Bakhvalov N. S., Eglit M. E., “Effektivnye uravneniya s dispersiei dlya rasprostraneniya voln v periodicheskikh sredakh”, Dokl. AN, 370:1 (2000), 1–4

[2] Chugainova A. P., “Asimptoticheskoe povedenie nelineinykh voln v uprugikh sredakh s dispersiei i dissipatsiei”, Teor. i matem. fiz., 147:2 (2006), 240–256 | DOI | Zbl

[3] Chugainova A. P., “Avtomodelnye asimptotiki volnovykh zadach i struktury neklassicheskikh razryvov v nelineino-uprugikh sredakh s dispersiei i dissipatsiei”, Prikl. matem. i mekhan., 71:5 (2007), 775–787 | Zbl

[4] Kulikovskii A. G., Chugainova A. P., “Klassicheskie i neklassicheskie razryvy v resheniyakh uravnenii nelineinoi teorii uprugosti”, Uspekhi matem. nauk, 63:2(380) (2008), 85–152 | DOI | Zbl

[5] Kulikovskii A. G., Chugainova A. P., “O statsionarnoi strukture udarnykh voln v uprugikh sredakh i dielektrikakh”, Zh. eksperim. i teor. fiz., 137:4 (2010), 973–985

[6] Kulikovskii A. G., Chugainova A. P., “Avtomodelnye asimptotiki, opisyvayuschie nelineinye volny v uprugikh sredakh s dispersiei i dissipatsiei”, Zh. vychisl. matem. i matem. fiz., 50:12 (2010), 2261–2274 | Zbl

[7] Kulikovskii A. G., “Ob uravneniyakh, opisyvayuschikh rasprostranenie nelineinykh kvazipoperechnykh voln v slaboanizotropnom uprugom tele”, Prikl. matem. i mekhan., 50:4 (1986), 597–604 | Zbl

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

[9] Sveshnikova E. I., “Prostye volny v nelineino uprugoi srede”, Prikl. matem. i mekhan., 46:4 (1982), 642–646

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

[11] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, v. 6, Gidrodinamika, Fizmatlit, M., 1986

[12] Chugainova A. P., “O formirovanii avtomodelnogo resheniya v zadache o nelineinykh volnakh v uprugom poluprostranstve”, Prikl. matem. i mekhan., 52:4 (1988), 692–697