Geodesic
Optimal control of inclusions in an elastic body crossing the external boundary
Sibirskij žurnal industrialʹnoj matematiki, Tome 18 (2015) no. 4, pp. 75-87.

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The paper addresses optimal control of the elastic thin inclusions located in an elastic body and crossing the external boundary. The inclusions are assumed to delaminate, thus forming a crack between the inclusions and the matrix. We impose some nonlinear boundary conditions at the crack faces that do not allow the crack faces to penetrate into each other. We prove the solvability of optimal control problems in which the quality functional characterizes the displacement of the points of the elastic inclusions located outside the elastic body, and the length of the inclusions located inside the elastic body is the control function. The case is doscussed of the zero angle between the inclusions and the external boundary.
Keywords: elastic body, elastic inclusion, crack, nonlinear boundary condition, variational inequality, optimal control. 
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A. M. Khludnev. Optimal control of inclusions in an elastic body crossing the external boundary. Sibirskij žurnal industrialʹnoj matematiki, Tome 18 (2015) no. 4, pp. 75-87. http://geodesic.mathdoc.fr/item/SJIM_2015_18_4_a7/

[1] Khludnev A. M., Kovtunenko V. A., Analysis of Cracks in Solids, WIT Press, Southampton–Boston, 2000

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

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

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

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

[6] 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

[7] Kovtunenko V. A., Kunisch K., “Problem of crack perturbation based on level sets and velocities”, Z. Angew. Math. Mech., 87:11–12 (2007), 809–830 | DOI | MR | Zbl

[8] Khludnev A. M., Kovtunenko V. A., Tani A., “On the topological derivative due to kink of a crack with non-penetration. Anti-plane model”, J. Math. Pures Appl., 94 (2010), 571–596 | DOI | MR | Zbl

[9] Lazarev N. P., “Zadacha o ravnovesii plastiny Timoshenko s naklonnoi treschinoi”, Prikl. mekhanika i tekhn. fizika, 54:4 (2013), 171–181 | MR | Zbl

[10] Khludnev A. M., “Thin inclusions in elastic bodies crossing an external boundary”, ZAMM | DOI

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

[12] Gaudiello A., Khludnev A. M., “Crack on the boundary of two overlapping domain”, ZAMP, 61:2 (2010), 341–356 | DOI | MR | Zbl

[13] Khludnev A. M., Negri M., “Crack on the boundary of a thin elastic inclusion inside an elastic body”, Z. Angew. Math. Mech., 92:5 (2012), 341–354 | DOI | MR | Zbl

[14] Neustroeva N. V., “Zhestkoe vklyuchenie v kontaktnoi zadache dlya uprugikh plastin”, Sib. zhurn. industr. matematiki, 12:4 (2009), 92–105 | MR | Zbl

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

[16] Rotanova T. A., “Zadacha ob odnostoronnem kontakte dvukh plastin, odna iz kotorykh soderzhit zhestkoe vklyuchenie”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 11:1 (2011), 87–98 | Zbl

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

[18] 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

[19] Khludnev A. M., “Ob izgibe uprugoi plastiny s otsloivshimsya tonkim zhestkim vklyucheniem”, Sib. zhurn. industr. matematiki, 14:1 (2011), 114–126 | MR | Zbl

[20] Khludnev A. M., “Zadacha o treschine na granitse zhestkogo vklyucheniya v uprugoi plastine”, Izvestiya RAN. Mekhanika tverdogo tela, 2010, no. 5, 98–110

[21] Khludnev A. M., “Contact problems for elastic bodies with rigid inclusions”, Quart. Appl. Math., 70:2 (2012), 269–284 | DOI | MR | Zbl

[22] 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

[23] Khludnev A. M., Leugering G., “On elastic bodies with thin rigid inclusions and cracks”, Math. Meth. Appl. Sci., 33:16 (2010), 1955–1967 | MR | Zbl

[24] Itou H., Khludnev A. M., “On delaminated thin Timoshenko inclusions inside elastic bodies”, Math. Meth. Appl. Sci. | DOI

[25] Khludnev A. M., “Thin rigid inclusions with delaminations in elastic plates”, European J. Mech. A/Solids, 32 (2012), 69–75 | DOI | MR | Zbl

[26] Goldstein R. V., Shifrin E. I., Shushpannikov P. S., “Application of invariant integrals to the problems of defect identification”, Internat. J. Fracture, 147:1–4 (2007), 45–54 | DOI | Zbl

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

[28] Shifrin E. I., “O svyazi mezhdu invariantnymi integralamilineinoi izotropnoi teorii uprugosti i integralami, opredelyaemymi printsipom vzaimnosti”, Prikl. matematika i mekhanika, 73:2 (2009), 325–334 | MR | Zbl

[29] Shifrin E. I., Shushpannikov P. S., “Identification of an ellipsoidal defect in an elastic solid using boundary measurements”, Internat. J. Solids and Structures, 48:7–8 (2011), 1154–1163 | DOI | Zbl

[30] Khludnev A. M., “Shape control of thin rigid inclusions and cracks in elastic bodies”, Arch. Appl. Mech., 83:10 (2013), 1493–1509 | DOI | Zbl

[31] Khludnev A. M., Leugering G., “Optimal control of cracks in elastic bodies with thin rigid inclusions”, Z. Angew. Math. Mech., 91:2 (2011), 125–137 | DOI | MR | Zbl

[32] Scherbakov V. V., “Ob odnoi zadache upravleniya formoi tonkikh vklyuchenii v uprugikh telakh”, Sib. zhurnal industr. matematiki, 16:1 (2013), 138–147 | MR

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

[34] Scherbakov V. V., “Upravlenie zhestkostyu tonkikh vklyuchenii v uprugikh telakh s krivolineinymi treschinami”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 13:1 (2013), 135–149

[35] Scherbakov V. V., Krivorotko O. I., “Optimalnye formy treschin v vyazkouprugomtele”, Trudy In-ta matematiki i mekhaniki UrO RAN, 21, no. 1, 2015, 294–304

[36] Khludnev A. M., Negri M., “Optimal rigid inclusion shapes in elastic bodies with cracks”, Z. Angew. Math. Phys., 64:1 (2013), 179–191 | DOI | MR | Zbl