The derivative of the energy functional in an equilibrium problem for a~Timoshenko plate with a~crack on the boundary of an elastic inclusion

N. V. Neustroeva ; N. P. Lazarev

Sibirskij žurnal industrialʹnoj matematiki, Tome 20 (2017) no. 2, pp. 59-70.

Résumé

We consider an equilibrium of a composite plate containing a through vertical crack of variable length on the separation boundary between the matrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion is described by the Kirchhoff–Love model. We obtain a formula for the derivative of the energy functional with respect to the crack length.

Détail

Keywords: plate, crack, nopenetration condition, elastic inclusion, derivative of the energy functional.

Bibliographie

[1] Cherepanov G. P., Mekhanika khrupkogo razrusheniya, Nauka, M., 1974

[2] Parton V. Z., Morozov E. M., Mekhanika uprugoplasticheskogo razrusheniya, Nauka, M., 1985

[3] Khludnev A. M., Zadachi teorii uprugosti v negladkikh oblastyakh, Fizmatlit, M., 2010

[4] Lazarev N. P., “Zadacha o ravnovesii plastiny Timoshenko, soderzhaschei treschinu na granitse uprugogo vklyucheniya s beskonechnoi zhestkostyu poperechnogo sdviga”, Prikl. matematika i tekhn. fizika, 54:2 (2013), 179–189 | MR | Zbl

[5] Khludnev A. M., Kovtunenko V. A., Analysis of cracks in solids, WIT-Press, Southampton–Boston, 2000

[6] Rotanova T. A., “O postanovkakh i razreshimosti zadachi o kontakte dvukh plastin, soderzhaschikh zhestkie vklyucheniya”, Sib. zhurn. industr. matematiki, 15:2 (2012), 107–118 | MR | Zbl

[7] Scherbakov V. V., “Suschestvovanie optimalnoi formy tonkikh zhestkikh vklyuchenii v plastine Kirkhgofa–Lyava”, Sib. zhurn. industr. matematiki, 16:4 (2013), 142–151 | MR

[8] Neustroeva N. V., “Odnostoronnii kontakt uprugikh plastin s zhestkim vklyucheniem”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 9:4 (2009), 51–64 | Zbl

[9] Lazarev N. P., Itou H., Neustroeva N. V., “Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle”, Japan J. Industr. Appl. Math., 33:1 (2016), 63–80 | DOI | MR | Zbl

[10] Khludnev A. M., Sokolowski J., “The Griffith formula and the Rice–Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains”, Europ. J. Appl. Math., 10:4 (1999), 379–394 | DOI | MR | 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 (2002), 99–109 | DOI | MR | Zbl

[12] Rudoi E. M., “Formula Griffitsa dlya plastiny s treschinoi”, Sib. zhurn. industr. matematiki, 5:3 (2002), 155–161 | MR | Zbl

[13] Lazarev N. P., “Differentsirovanie funktsionala energii v zadache o ravnovesii plastiny Timoshenko, soderzhaschei treschinu”, Prikl. matematika i tekhn. fizika, 53:2 (2012), 175–185 | MR | Zbl

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

[15] Rudoi E. M., “Differentsirovanie funktsionalov energii v dvumernoi teorii uprugosti dlya tel, soderzhaschikh krivolineinye treschiny”, Prikl. matematika i teor. fizika, 45:6 (2004), 83–94 | MR | Zbl

[16] 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

[17] Rudoi E. M., “Asimptotika funktsionala energii dlya trekhmernogo tela s zhestkim vklyucheniem i treschinoi”, Prikl. matematika i tekhn. fizika, 52:2 (2011), 114–127 | MR

[18] Lazarev N. P., “Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion”, Z. Angew. Math. Phys., 66:4 (2015), 2025–2040 | DOI | MR | Zbl

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

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

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

[22] 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

[23] Mazya V. G., Nazarov S. A., Plamenevskii V. A., Asimptotika reshenii ellipticheskikh kraevykh zadach pri singulyarnykh vozmuscheniyakh oblasti, Izd-vo Tbilis. un-ta, Tbilisi, 1981 | MR

[24] Mazya V. G., Slutskii A. C., Fomin B., “Ob asimptotike funktsii napryazhenii vblizi vershiny treschiny v zadache krucheniya pri ustanovivsheisya polzuchesti”, Izv. RAN. Mekhanika tverdogo tela, 1986, no. 4, 170–176

[25] Mazya V. G., Nazarov S. A., “Asimptotika integralov energii pri malykh vozmuscheniyakh vblizi uglovykh i konicheskikh tochek”, Tr. Mosk. mat. o-va, 50, 1987, 79–129 | MR | Zbl

[26] Ohtsuka K., “Mathematics of brittle fracture”, Theoretical Studies on Fracture Mechanics in Japan, Hiroshima-Denki Institute of Technology, Hiroshima, 1997, 99–172

[27] Kozlov V. A., Khludnev A. M., “Asimptotika resheniya uravneniyaPuassona vblizi vershiny treschiny s nelineinymi kraevymi usloviyami na beregakh”, Dokl. AN, 411:5 (2006), 583–586 | MR | Zbl

[28] Khludnev A. M., Kozlov V. A., “Asymptotics of solutions near crack tips for Poisson equation with inequlity type boundary conditions”, Z. Angew. Math. Phys., 59:2 (2008), 264–280 | DOI | MR | Zbl

[29] Pelekh B. L., Teoriya obolochek s konechnoi sdvigovoi zhestkostyu, Nauk. dumka, Kiev, 1973