Geodesic
Optimal shapes of cracks in a viscoelastic body
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 294-304 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider an optimal control problem for equations describing the quasistatic deformation of a linear viscoelastic body. There is a crack in the body, and displacements of opposite faces of the crack are constrained by the nonpenetration condition. The continuous dependence of the solution to the equilibrium problem on the shape of the crack is established. In particular, we prove the existence of a shape for which the crack opening is minimal
Keywords: viscoelasticity; crack; nonpenetration condition; optimal control; fictitious domain method. 
@article{TIMM_2015_21_1_a28,
     author = {V. V. Shcherbakov and O. I. Krivorot'ko},
     title = {Optimal shapes of cracks in a viscoelastic body},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {294--304},
     year = {2015},
     volume = {21},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a28/}
}
TY  - JOUR
AU  - V. V. Shcherbakov
AU  - O. I. Krivorot'ko
TI  - Optimal shapes of cracks in a viscoelastic body
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2015
SP  - 294
EP  - 304
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a28/
LA  - ru
ID  - TIMM_2015_21_1_a28
ER  - 
%0 Journal Article
%A V. V. Shcherbakov
%A O. I. Krivorot'ko
%T Optimal shapes of cracks in a viscoelastic body
%J Trudy Instituta matematiki i mehaniki
%D 2015
%P 294-304
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a28/
%G ru
%F TIMM_2015_21_1_a28
V. V. Shcherbakov; O. I. Krivorot'ko. Optimal shapes of cracks in a viscoelastic body. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 294-304. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a28/

[1] Khludnev A.M., “On equilibrium problem for a plate having a crack under the creep condition”, Control and Cybern., 25:5 (1996), 1015–1030 | MR

[2] Khludnev A.M., Zadachi teorii uprugosti v negladkikh oblastyakh, Fizmatlit, M., 2010, 252 pp.

[3] Khludnev A.M., Kovtunenko V.A., Analysis of cracks in solids, WIT Press, Southampton; Boston, 2000, 408 pp.

[4] Kovtunenko V.A., “Shape sensitivity of curvilinear cracks on interface to non-linear perturbations”, Z. Angew. Math. Phys., 54:3 (2003), 410–423 | DOI | MR | Zbl

[5] Rudoi E.M., “Vybor optimalnoi formy poverkhnostnykh treschin v trekhmernykh telakh”, Vestn. NGU. Matematika, mekhanika, informatika, 6:2 (2006), 76–87

[6] Rudoi E.M., “Differentsirovanie funktsionalov energii v zadache o krivolineinoi treschine s vozmozhnym kontaktom beregov”, Izv. RAN. Mekhanika tverdogo tela, 2007, no. 6, 113–127

[7] Vtorushin E.V., “Upravlenie formoi treschiny v uprugom tele pri uslovii vozmozhnogo kontakta beregov”, Sib. zhurn. industr. matematiki, 9:2 (2006), 20–30 | MR | Zbl

[8] Lazarev N.P., “Suschestvovanie ekstremalnoi formy treschiny v zadache o ravnovesii plastiny Timoshenko”, Vestn. NGU. Matematika, mekhanika, informatika, 11:4 (2011), 49–62 | MR | Zbl

[9] Lazarev N.P., “Formula Griffitsa dlya plastiny Timoshenko s krivolineinoi treschinoi”, Sib. zhurn. industr. matematiki, 16:2 (2013), 98–108 | MR

[10] Lazarev N.P., Rudoy E.M., “Shape sensitivity analysis of Timoshenko's plate with a crack under the nonpenetration condition”, Z. Angew. Math. Mech., 94:9 (2014), 730–739 | DOI | MR | Zbl

[11] Arutyunyan N.Kh., Shoikhet B.A., “Asimptoticheskoe povedenie resheniya kraevoi zadachi teorii polzuchesti neodnorodnykh stareyuschikh tel s odnostoronnimi svyazyami”, Izv. AN SSSR. Mekhanika tverdogo tela, 1981, no. 3, 31–48 | MR

[12] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980, 384 pp. | MR

[13] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972, 587 pp. | MR

[14] Goldshtein R.V., Entov V.M., Kachestvennye metody v mekhanike sploshnykh sred, Nauka, M., 1989, 224 pp. | MR

[15] Hoffmann K.-H., Khludnev A.M., “Fictitious domain method for the Signorini problem in a linear elasticity”, Adv. Math. Sci. Appl., 14:2 (2004), 465–481 | MR | Zbl

[16] Alekseev G.V., Khludnev A.M., “Treschina v uprugom tele, vykhodyaschaya na granitsu pod nulevym uglom”, Vestn. NGU. Matematika, mekhanika, informatika, 9:2 (2009), 15–29 | Zbl

[17] Popova T.S., “Metod fiktivnykh oblastei v zadache Sinorini dlya vyazkouprugikh tel”, Mat. zametki YaGU, 13:1 (2006), 87–106 | MR

[18] Khludnev A.M., “Ob ekstremalnykh formakh razrezov v plastine”, Izv. RAN. Mekhanika tverdogo tela, 1992, no. 1, 170–176 | MR