Geodesic
Shape Sensitivity Analysis of an Equilibrium Problem for a Body with a Thin Rigid Inclusion
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 14 (2014) no. 2, pp. 69-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the paper we consider an equilibrium problem of an elastic body with a thin rigid inclusion. There is a delamination crack between the one side of the inclusion and the elastic body. We investigate the dependence of the solution on the shape perturbation. The main result is the calculation of the material derivative of the solution with respect to the shape perturbation parameter. As an example of the obtained results we calculate the shape derivative of the energy functional. Moreover, we consider the optimization problem of the length of the rigid inclusion. For this problem we get the necessary optimality conditions.
Keywords: thin rigid inclusion, crack, shape variation, sensitivity analysis, optimal control. 
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E. M. Rudoy. Shape Sensitivity Analysis of an Equilibrium Problem for a Body with a Thin Rigid Inclusion. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 14 (2014) no. 2, pp. 69-87. http://geodesic.mathdoc.fr/item/VNGU_2014_14_2_a7/

[1] N. V. Banichuk, Optimization of Shape of Elastic Bodies, Nauka, M., 1980 (in Russian) | MR

[2] Khludnev A. M., Sokolowski J., Modelling and Control in Solids Mechanics, Birkhäuser, Basel–Boston–Berlin, 1997 | MR | Zbl

[3] Khludnev A. M., Kovtunenko V. A., Analysis of Cracks in Solids, WIT-Press, Southampton–N.Y., 2000

[4] E. M. Rudoy, “On a choice of an optimal shape of cracks in 3D elasticity”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 6:2 (2006), 76–87 (in Russian)

[5] Hoemberg D., Khludnev A. M., “On Safe Crack Shapes in Elastic Bodie”, Eur. J. Mech. A/Solids, 21 (2002), 991–998 | DOI | MR | Zbl

[6] Khludnev A., Leontiev A., Herkovits J., “Nonsmooth Domain Optimization for Elliptic Equations with Unilateral Conditions”, J. Math. Pures Appl., 83 (2003), 197–212 | DOI | MR

[7] V. V. Shcherbakov, “On an optimal control problem for the shape of thin inclusions in elastic bodies”, J. Appl. Indust. Math., 7:3 (2013), 435–443 | DOI | MR

[8] V. V. Shcherbakov, “Control of rigidity of thin inclusions in elastic bodies with curvilinear cracks”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 13:1 (2013), 135–149 (in Russian)

[9] Allaire G., Jouve F., Toader A.-M., “Structural Optimization Using Sensitivity Analysis and a Level-Set Method”, J. Comput. Phys., 194 (2004), 363–393 | DOI | MR | Zbl

[10] Delfour M. C., Zolesio J.-P., Shapes and Geometries, SIAM, Philadelphia, PA, 2001 | MR | Zbl

[11] Hintermüller M., Laurain A., “A Shape and Topology Optimization Technique for Solving a Class of Linear Complementarity Problems in Function Space”, J. Comput. Optim. Appl., 46 (2010), 535–569 | DOI | MR | Zbl

[12] Sokolowski J., Zolesio J.-P., Introduction to Shape Optimization. Shape Sensitivity Analisys, Springer, Berlin, 1992 | MR | Zbl

[13] V. A. Kovtunenko, “Regular perturbation methods for a region with a crack”, J. Appl. Mech. Techn. Phys., 43:5 (2002), 748–762 | DOI | MR | Zbl

[14] Kovtunenko V. A., “Sensitivity of Cracks in 2D-Lamé Problem via Material Derivatives”, Z. Angew. Math. Phys., 52 (2001), 1071–1087 | DOI | MR | Zbl

[15] Chen G., Rahman S., Park Y. H., “Shape Sensitivity and Realiability Analyses of Linear-Elastic Cracked Structure”, Int. J. of Fracture, 112 (2001), 223–246 | DOI

[16] Kimura M., Wakano I., “Shape Derivative of Potential Energy and Energy Release Rate in Fracture Mechanics”, J. of Math-for-Industry, 3 (2011), 21–31 | MR | Zbl

[17] Taroco E., “Shape Sensitivity Analysis in Linear Elastic Fracture Mechanics”, Comput. Methods Appl. Mech. Engrg., 188 (2000), 697–712 | DOI | MR | Zbl

[18] Khludnev A. M., “Thin Rigid Inclusions with Delaminations in Elastic Plates”, Europ. J. Mech. A/Solids, 29 (2010), 69–75 | DOI | MR

[19] Khludnev A. M., Leugering G., “On Elastic Bodies with Thin Rigid Inclusions and Cracks”, Math. Meth. Appl. Sci., 33 (2010), 1955–1967 | MR | Zbl

[20] Maz'ya V. G., Poborchi S. V., Differentiable Functions on Bad Domains, World Scientific Pub. Co. Inc., 1997 | MR

[21] E. M. Rudoy, “The Griffith formula and Cherepanov–Rice integral for a plate with a rigid inclusion and a crack”, J. Math. Sci., New York, 186:3 (2012), 511–529 | DOI | MR

[22] E. M. Rudoy, “Asymptotic behavior of the energy functional for a three-dimensional body with a rigid inclusion and a crack”, J. Appl. Mech. Techn. Phys., 52:2 (2011), 252–263 | DOI | MR

[23] Lazarev N. P., Rudoy E. M., “Shape Sensitivity Analysis of Timoshenko's Plate with a Crack under the Nonpenetration Condition”, Z. Angew. Math. Mech., 2013 | DOI

[24] V. A. Kovtunenko, “Invariant energy integrals for the nonlinear crack problem with possible contact of the crack surfaces”, J. Appl. Math. Mech., 67:1 (2003), 99–110 | DOI | MR | Zbl

[25] Itou H., Khludnev A. M., Rudoy E. M., Tani A., “Asymptotic Behaviour at a Tip of a Rigid Line Inclusion in Linearized Elasticity”, Z. Angew. Math. Mech., 96 (2012), 716–730 | DOI | MR