# A variational problem for an elastic body with periodically located cracks

• Neustroeva Natalia V., nnataliav@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
• Afanas’eva Nadezhda M., afanasieva.nm@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
• Egorova Alena A., alena.egorova@gmail.com Северо-Восточный федеральный университет, Институт математики и информатики, Кулаковского, 42, Якутск 677000
Keywords: elastic body, crack, homogenization, penalty method

### Abstract

We consider a nonlinear problem of equilibrium of an elastic body with periodically located cracks. On the edges of these cracks, non-penetration conditions are given. The nonlinear problem is formulated in the form of variational inequality. The period of distribution of the cracks, as well as their sizes, depends on a small parameter. The behavior of the solution to the problem with periodically located cracks is determined by the first two terms $u^0(x)$ and $u^1(x, y)$ of the asymptotic expansion. In this paper, we study the solution of the variational inequality on a periodicity cell (a local problem). For the first corrector $u^1(x, y)$, we construct a penalty equation and a linear iterative equation in integral form. We prove that the sequence of solutions of the problem with penalty converges to the solution of the problem on the cell when the small regularization parameter tends to zero. We show that the approximate solution of the iteration equation converges strongly to the solution of the penalty equation.

### References

[1]
Sanchez-Palencia E., Non-Homogeneous Media and Vibration Theory [in Russian], Mir, Moscow (1984).

[2]
Pastukhova S. E., “On homogenization of a variational inequality for an elastic body with periodically distributed fissures,” Sb. Math., 191, No. 2, 291–306 (2000).

[3]
Khludnev A. M. and Kovtunenko V. A., Analysis of Cracks in Solids, WIT Press, Southampton; Boston (2000).

[4]
Khludnev A. V., Elasticity Problems in Nonsmooth Domains [in Russian], Fizmatlit, Moscow (2010).

[5]
Kovtunenko V. A., “Numerical simulation of the non-linear crack problem with non-penetration,” Math. Meth. Appl. Sci., 27, 163–179 (2004).

[6]
Rudoy E., “Domain decomposition method for crack problems with nonpenetration condition,” Math. Model. Numer. Anal., 50, No. 4, 995–1009 (2016).

[7]
Kovtunenko V. A., “A method of numerical solution of the problem of contact between an elastic plate and an obstacle [in Russian],” Prikl. Mekh. Tekh. Fiz., 35, No. 5, 142–146 (1994).

[8]
Kovtunenko V. A., “An iterative penalty method for a problem with constraints on the inner boundary,” Sib. Mat. Zh., 37, No. 3, 587–591 (1996).

[9]
Lazarev N. P., “An iterative penalty method for a nonlinear problem of equilibrium of a Timoshenko-type plate with a crack,” Sib. Zh. Vychisl. Mat., 14, No. 4, 397–408 (2011).

[10]
Lions J.-L., Some Methods of Solving Nonlinear Boundary Value Problems [in Russian], Mir, Moscow (1972).

[11]
Cioranescu D., Damlamian A., and Griso G., “The periodic unfolding method in homogenization,” SIAM J. Math. Anal. Soc. Ind. Appl. Math., 40, No. 4, 1585–1620 (2008).

[12]
Cioranescu D., Damlamian A., and Orlik J., “Homogenization via unfolding in periodic elasticity with contact on closed and open cracks,” Asympt. Anal., 82, 201–232 (2013).

[13]
Griso G., Migunova A., and Orlik J., “Homogenization via unfolding in periodic layer with contact,” Asympt. Anal., 99, 23–52 (2016).

[14]
Griso G. and Orlik J., “Homogenization of contact problem with Coulomb’s friction on periodic cracks,” arXiv:1811.06615 [math.AP].
How to Cite
Neustroeva, N., Afanas’eva, N. and Egorova, A. (2019) “A variational problem for an elastic body with periodically located cracks”, Mathematical notes of NEFU, 26(2), pp. 17-30. doi: https://doi.org/10.25587/SVFU.2019.102.31509.
Issue
Section
Mathematics