Geodesic
Invariant integrals in the plane elasticity problem for bodies with rigid inclusions and cracks
Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 1, pp. 99-109.

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

We consider the plane elasticity problem for a body with a rigid inclusion and a crack along the boundary between the elastic matrix and rigid inclusion. We show that this problem possesses $J$- and $M$-invariant integrals. In particular, we construct an invariant integral of Cherepanov–Rice type for straight cracks.
Keywords: invariant integrals, rigid inclusion, crack, derivative of the energy functional, Cherepanov–Rice integral. 
@article{SJIM_2012_15_1_a9,
     author = {E. M. Rudoǐ},
     title = {Invariant integrals in the plane elasticity problem for bodies with rigid inclusions and cracks},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {99--109},
     publisher = {mathdoc},
     volume = {15},
     number = {1},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2012_15_1_a9/}
}
TY  - JOUR
AU  - E. M. Rudoǐ
TI  - Invariant integrals in the plane elasticity problem for bodies with rigid inclusions and cracks
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2012
SP  - 99
EP  - 109
VL  - 15
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2012_15_1_a9/
LA  - ru
ID  - SJIM_2012_15_1_a9
ER  - 
%0 Journal Article
%A E. M. Rudoǐ
%T Invariant integrals in the plane elasticity problem for bodies with rigid inclusions and cracks
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2012
%P 99-109
%V 15
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2012_15_1_a9/
%G ru
%F SJIM_2012_15_1_a9
E. M. Rudoǐ. Invariant integrals in the plane elasticity problem for bodies with rigid inclusions and cracks. Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 1, pp. 99-109. http://geodesic.mathdoc.fr/item/SJIM_2012_15_1_a9/

[1] Parton V. Z., Morozov E. M., Mekhanika uprugoplasticheskogo razrusheniya, Nauka, M., 1974

[2] Cherepanov G. P., Mekhanika khrupkogo razrusheniya, Nauka, M., 1974 | Zbl

[3] Chen Y.-H., Lu T. J., “Recent developments and applications of invariant integrals”, Appl. Mech. Rev., 56:5 (2003), 515–551 | DOI

[4] Andrieux S., Ben Abda A., Bui Y., “Reciprocity principle and crack identification”, Inverse Problems, 15 (1999), 59–69 | DOI

[5] Goldstein R., Shifrin E., Shushpannikov P., “Application of invariant integrals to the problems of defect identification”, Internat. J. Fracture, 147 (2007), 45–54 | DOI | Zbl

[6] Kaptsov A. V., Shifrin E. I., “Identifikatsiya ploskoi treschiny v uprugom tele s pomoschyu invariantnykh integralov”, Izv. RAN. Mekhanika tverdogo tela, 2008, no. 3, 145–163

[7] Knowels J. K., Stenberg E., “On a class of conservation laws in linearized and finite elastostatics”, Arch. Rat. Mech. Anal., 44:3 (1972), 187–211

[8] Stepanova L. V., Matematicheskie metody mekhaniki razrusheniya, Fizmatlit, M., 2009

[9] Rudoi E. M., “Asimptotika funktsionala energii dlya uprugogo tela s treschinoi i zhestkim vklyucheniem. Ploskaya zadacha”, Prikl. matematika i mekhanika, 75:6 (2011), 1038–1048

[10] Kovtunenko V. A., “Invariantnye integraly energii dlya nelineinoi zadachi o treschine s vozmozhnym kontaktom beregov”, Prikl. matematika i mekhanika, 67:1 (2003), 109–123 | Zbl

[11] Khludnev A. M., Ohtsuka K., Sokolowski J., “On derivative of energy functional for elastic bodies with cracks and unilateral conditions”, Quart. Appl. Math., 60:2 (2002), 99–109 | Zbl

[12] 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

[13] Kovtunenko V. A., “Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration”, IMA J. Appl. Math., 71:5 (2006), 635–657 | DOI | Zbl

[14] Rudoi E. M., “Asimptotika funktsionala energii dlya smeshannoi kraevoi zadachi chetvertogo poryadka v oblasti s razrezom”, Sib. mat. zhurn., 50:2 (2009), 430–445 | MR

[15] Argatov I. I., Nazarov S. A., “Vysvobozhdenie energii pri izlome treschiny v ploskom anizotropnom tele”, Prikl. matematika i mekhanika, 66:3 (2002), 502–514 | Zbl

[16] Nazarov S. A., Shpekovius-Noigebauer M., “Primenenie energeticheskogo kriteriya razrusheniya dlya opredeleniya formy slaboiskrivlennoi treschiny”, Prikl. mekhanika i tekhn. fizika, 47:5 (2006), 119–130 | Zbl

[17] Khludnev A. M., “Zadacha o treschine na granitsezhestkogo vklyucheniya v uprugoi plastine”, Izv. RAN. Mekhanika tverdogo tela, 2010, no. 5, 98–110

[18] Stekina T. A., “Variatsionnaya zadacha ob odnostoronnem kontakte uprugoi plastiny s balkoi”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 9:1 (2009), 45–56

[19] Browder F. E., “On the regularity properties of solutions of elliptic differential equations”, Comm. Pure Appl. Math., 9 (1956), 351–361 | DOI | Zbl

[20] Koshelev A. I., “Apriornye otsenki v $L_p$ i obobschennye resheniya ellipticheskikh uravnenii i sistem”, Uspekhi mat. nauk, 13:4 (1958), 29–88 | MR | Zbl

[21] Khludnev A. M., “Invariantnye integraly v zadache o treschine na granitse razdela dvukh sred”, Prikl. mekhanika i tekhn. fizika, 46:5 (2005), 123–137 | Zbl

[22] Rudoi E. M., “Formula Griffitsa i integral Cherepanova–Raisa dlya plastiny s zhestkim vklyucheniem i treschinoi”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 10:2 (2010), 98–117

[23] Herrmann A. G., Herrmann G., “On energy release rates for a plane crack”, ASME J. Appl. Mech., 48 (1981), 525–528 | DOI | Zbl

[24] Chang J. H., Chien A. J., “Evaluation of $M$-integral for anisotropic elastic media with multiple defects”, Internat. J. Fracture, 114 (2002), 267–289 | DOI