# Mathematical and Numerical Simulation of Equilibrium of an Elastic Body Reinforced by a Thin Elastic Inclusion

• N. A. Kazarinov
• E. M. Rudoy
• V. Yu. Slesarenko
• V. V. Shcherbakov
Article

## Abstract

A boundary value problem describing the equilibrium of a two-dimensional linear elastic body with a thin rectilinear elastic inclusion and possible delamination is considered. The stress and strain state of the inclusion is described using the equations of the Euler–Bernoulli beam theory. Delamination means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions preventing crack face interpenetration are imposed on the crack faces. As a result, problem with an unknown contact domain is obtained. The problem is solved numerically by applying an iterative algorithm based on the domain decomposition method and an Uzawa-type algorithm for solving variational inequalities. Numerical results illustrating the efficiency of the proposed algorithm are presented.

## Keywords

thin elastic inclusion delamination crack nonpenetration condition variational inequality domain decomposition method Uzawa algorithm

## References

1. 1.
A. M. Khludnev and M. Negri, “Crack on the boundary of a thin elastic inclusion inside an elastic body,” Z. Angew. Math. Mech. 92 (5), 341–354 (2012).
2. 2.
A. M. Khludnev and G. Leugering, “Delaminated thin elastic inclusion inside elastic bodies,” Math. Mech. Complex Syst. 2 (1), 1–21 (2014).
3. 3.
H. Itou and A. M. Khludnev, “On delaminated thin Timoshenko inclusions inside elastic bodies,” Math. Methods Appl. Sci. 39 (17), 4980–4993 (2016).
4. 4.
A. M. Khludnev and G. Leugering, “On elastic bodies with thin rigid inclusions and cracks,” Math. Methods Appl. Sci. 33 (16), 1955–1967 (2010).
5. 5.
V. Shcherbakov, “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions,” Z. Angew. Math. Phys. 68, Article 26 (2017).Google Scholar
6. 6.
E. Barbieri and N. M. Pugno, “A computational model for large deformations of composites with a 2D soft matrix and 1D anticracks,” Int. J. Solids Struct. 77, 1–14 (2015).
7. 7.
L. G. S. Leite and W. S. Venturini, “Accurate modeling of rigid and soft inclusions in 2D elastic solids by the boundary element method,” Comput. Struct. 84 (29–30), 1874–1881 (2006).
8. 8.
E. M. Rudoy, “Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion,” J. Appl. Ind. Math. 10 (2), 264–276 (2016).
9. 9.
E. M. Rudoy and V. V. Shcherbakov, “Domain decomposition method for a membrane with a delaminated thin rigid inclusion,” Sib. Electron. Math. Rep. 13, 395–410 (2016).
10. 10.
E. M. Rudoy, “On numerical solving a rigid inclusions problem in 2D elasticity,” Z. Angew. Math. Phys. 68, Article 19 (2017).Google Scholar
11. 11.
V. D. Korneev and V. M. Sveshnikov, “Parallel algorithms and domain decomposition techniques for solving three-dimensional boundary value problems on quasi-structured grids,” Numer. Anal. Appl. 9 (2), 141–149 (2016).
12. 12.
Yu. M. Laevskii and A. M. Matsokin, “Decomposition methods for the solution of elliptic and parabolic boundary value problems,” Sib. Zh. Vychisl. Mat. 2 (4), 361–372 (1999).Google Scholar
13. 13.
A. V. Rukavishnikov, “Domain decomposition method and numerical analysis of a fluid dynamics problem,” Comput. Math. Math. Phys. 54 (9), 1459–1480 (2014).
14. 14.
V. M. Sveshnikov, “Domain decomposition method in problems of high-current electronics,” Sib. Zh. Ind. Mat. 18 (2), 124–130 (2015).
15. 15.
T. K. Dobroserdova and M. A. Olshanskii, “A finite element solver and energy stable coupling for 3D and 1D fluid models,” Comput. Method. Appl. M 259, 166–176 (2013).
16. 16.
J. Koko, “Uzawa conjugate gradient domain decomposition methods for coupled Stokes flows,” J. Sci. Comput. 26 (2), 195–215 (2006).
17. 17.
G. P. Astrakhantsev, “Domain decomposition method for the problem of bending heterogeneous plate,” Comput. Math. Math. Phys. 38 (10), 1686–1694 (1998).
18. 18.
G. Bayada, J. Sabil, and T. Sassi, “Convergence of a Neumann–Dirichlet algorithm for two-body contact problems with nonlocal Coulomb’s friction law,” ESAIM-Math. Model. Numer. 42 (2), 243–262 (2008).
19. 19.
J. Danek, I. Hlavácek, and J. Nedoma, “Domain decomposition for generalized unilateral semi-coercive contact problem with given friction in elasticity,” Math. Comput. Simul. 68, 271–300 (2005).
20. 20.
E. M. Rudoy, “Domain Decomposition method for crack problems with nonpenetration condition,” ESAIMMath. Model. Numer. 50 (4), 995–1009 (2016).
21. 21.
E. M. Rudoy, N. A. Kazarinov, and V. Yu. Slesarenko, “Numerical simulation of equilibrium of an elastic twolayer structure with a through crack,” Numer. Anal. Appl. 10 (1), 63–73 (2017).
22. 22.
M. Hintermüller, V. Kovtunenko, and K. Kunisch, “The primal-dual active set method for a crack problem with non-penetration,” IMA J. Appl. Math. 69 (1), 1–26 (2004).
23. 23.
M. Hintermüller, K. Ito, and K. Kunisch, “The primal-dual active set strategy as a semismooth Newton method,” SIAM J. Optim. 13 (2), 865–888 (2003).
24. 24.
V. A. Kovtunenko, “Numerical simulation of the non-linear crack problem with nonpenetration,” Math. Methods Appl. Sci. 27 (2), 163–179 (2004).
25. 25.
S. R. Eugster, Geometric Continuum Mechanics and Induced Beam Theories (Springer, New York, 2015).
26. 26.
A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT, Southampton, 2000).Google Scholar
27. 27.
J. Céa, Optimisation: Théorie Et Algorithmes (Dunod, Paris, 1971).
28. 28.
E. M. Rudoy, “Domain decomposition method for a model crack problem with a possible contact of crack edges,” Comput. Math. Math. Phys. 55 (2), 305–306 (2015).
29. 29.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems (SIAM, Philadelphia, 1999).
30. 30.
P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1977).Google Scholar
31. 31.
G. Allaire, Numerical Analysis and Optimization: An Introduction to Mathematical Modeling and Numerical Simulation (Oxford University Press, Oxford, 2007).
32. 32.
A. M. Khludnev and V. V. Shcherbakov, “Singular path-independent energy integrals for elastic bodies with Euler–Bernoulli inclusions,” Math. Mech. Solids 22 (11), 2180–2195 (2017).

## Authors and Affiliations

• N. A. Kazarinov
• 1
• E. M. Rudoy
• 1
• 2
• V. Yu. Slesarenko
• 1
• V. V. Shcherbakov
• 1
• 2
1. 1.Lavrentyev Institute of Hydrodynamics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
2. 2.Novosibirsk State UniversityNovosibirskRussia