Geodesic
Numerical investigation of a model problem for deforming an elastoplastic body with a crack under non-penetration condition
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 4, pp. 335-344 
E. V. Vtorushin. Numerical investigation of a model problem for deforming an elastoplastic body with a crack under non-penetration condition. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 4, pp. 335-344. http://geodesic.mathdoc.fr/item/SJVM_2006_9_4_a1/
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     title = {Numerical investigation of a~model problem for deforming an elastoplastic body with a~crack under non-penetration condition},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Lamé system is considered in a two-dimensional domain with a crack. The Dirichlet and the Neuman conditions are held on the exterior boundary, and non-penetration condition is assumed to be on a crack. The convolution product of the deviator of the stress tensor is restricted by a certain constant within the domain. Thus, we have a model problem for deforming an ideal elastoplastic body with a crack (the Henky model) subject to the Mises yield criterion. Simultaneously, the non-penetration condition is held on a crack. The problem is formulated as a variational one. We find a displacement vector as solution to minimization problem for the energy functional over a convex set. Discretization of the problem is provided by a finite element method. Examples of calculation are obtained using the Udzava algorithm.

[1] Morozov N. F., Matematicheskie voprosy teorii treschin, Nauka, M., 1984 | MR

[2] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980 | MR

[3] Temam R., Matematicheskie zadachi teorii plastichnosti, Nauka, M., 1991 | MR | Zbl

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

[5] Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979 | MR

[6] Glovinski R., Lions Zh.-L., Tremoler R., Chislennoe issledovanie variatsionnykh neravenstv, Mir, M., 1979 | MR

[7] Syarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[8] Leontiev A., Herskovits J., Eboli C., “Optimization theory application to slitted plate bending problems”, Int. J. Solids Structures, 35:20 (1998), 2679–2694 | DOI | MR | Zbl

[9] Hintermuller M., Kovtunenko V., Kunisch K., “The primal-dual active set method for a crack problem with non-penetration”, IMA J. of Appl. Math., 69 (2004), 1–26 | DOI | MR

[10] Rabotnov Yu. N., Mekhanika deformiruemogo tverdogo tela, Nauka, M., 1988 | Zbl

[11] Ortega Dzh., Vvedenie v parallelnye i vektornye metody resheniya lineinykh sistem, Mir, M., 1991 | MR