Geodesic
A choice of an optimal form of surface cracks in 3D solids
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 2, pp. 76-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider 3D-elastic body with a surface crack satisfying a nonpenetration conditions (Signorini-type conditions). In the paper, we investigate two optimization problems. The first problem is to find the crack shape which gives the minimum deviation of the energy functional derivative from a given critical value. The second one is to investigate a solvability of an optimization problem of a crack path with a cost functional depending on the derivative of the energy functional and a crack surface area. 
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E. M. Rudoy. A choice of an optimal form of surface cracks in 3D solids. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 2, pp. 76-87. http://geodesic.mathdoc.fr/item/VNGU_2006_6_2_a5/

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

[2] A. M. Khludnev, J. Sokolowski, Modelling and control in solid mechanics, Birkhauser, Basel, 1997 | MR | Zbl

[3] N. V. Banichuk, Optimizatsiya form uprugikh tel, Nauka, M., 1980 | MR

[4] A. M. Khludnev, “Ob ekstremalnykh formakh razrezov v plastine”, Izv. RAN., MTT, 1992, no. 1, 170–176 | Zbl

[5] E. M. Rudoi, “Differentsirovanie funktsionalov energii v trekhmernoi teorii uprugosti dlya tel, soderzhaschikh poverkhnostnye treschiny”, Sib. zhurn. industr. matematiki, 8:1 (2005), 106–116

[6] V. Z. Parton, E. M. Morozov, Mekhanika uprugo-plasticheskogo razrusheniya, Nauka, M., 1974

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

[8] V. A. Kovtunenko, “Shape sensitivity of a plane crack front”, Math. Meth. Appl. Sci., 26 (2003), 359–374 | DOI | MR | Zbl

[9] K. Ohtsuka, “Mathematics of brittle fracture”, Theoretical Studies on Fracture Mechanics in Japan, 1997, 99–172

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

[11] S. Garreau, P. Guillaume, M. Masmoudi, “The topological asymptotic for PDE system: the elasticity case”, SIAM J. Control Optim., 39:6 (2001), 1756–1778 | DOI | MR | Zbl

[12] J. Sokolowski, A. Zochowski, “On the topological derivative in shape optimization”, SIAM J. Control Optim., 37:4 (1999), 1251–1272 | DOI | MR | Zbl

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

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

[15] Ya. Sokolovskii, A. M. Khludnev, “O differentsirovanii funktsionalov energii v teorii treschin s vozmozhnym kontaktom beregov”, Dokl. RAN, 374:6 (2000), 776–779 | MR

[16] E. M. Rudoi, “Formula Griffitsa dlya plastiny s treschinoi”, Sib. zhurn. industr. mat., 5:3 (2002), 155–161

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

[18] V. A. Kovtunenko, “Sensitivity of interfacial cracks to non-linear crack front perturbations”, Z. angew. Math. Mech., 82 (2002), 387–398 | 3.0.CO;2-I class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

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

[20] E. M. Rudoi, “Invariantnye integraly dlya zadachi ravnovesiya plastiny s treschinoi”, Sib. mat. zhurn., 45:2 (2004), 466–477 | MR | Zbl

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

[22] A. Khludnev, A. Leontiev, J. Herskovits, “Nonsmooth domain optimization for elliptic equations with unilateral conditions”, J. Math. Pures Appl., 83 (2003), 197–212 | DOI | MR

[23] D. Hoemberg, A. M. Khludnev, “On safe crack shapes in elastic bodies”, Eur. J. Mech. A/Solids, 21 (2002), 991–998 | DOI | MR | Zbl