Numerical solution of the problem of T-shaped junction of two thin Timoshenko inclusions in a two-dimentional elastic body
https://doi.org/10.25587/2411-9326-2024-3-93-120
Abstract
An algorithm for the numerical solution of the equilibrium problem of a two-dimensional elastic body containing two thin elastic inclusions is developed. The inclusions are modeled within the framework of the theory of Timoshenko beams and intersect at right angle at an internal point of one of them, forming a T-shaped structure in an elastic body. One of the inclusions delaminates from the elastic matrix, forming a crack. On the crack faces, as part of the domain boundary, boundary conditions of the inequality form are specified. The presence of this type of boundary conditions leads to nonlinearity of the problem and formulation in the form of a variational inequality. To develop an algorithm for the numerical solution of the problem, an approximate problem of finding the saddle point of the Lagrangian is formulated. The convergence of solutions of the approximate problem to the solution of the original problem is proven. An iterative Uzawa-type algorithm is constructed and its convergence is shown. Examples of numerical implementation are given.
About the Author
T. S. Popova
Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics
Russian Federation
Russian Federation
Tatiana S. Popova
48 Kulakovsky Street, Yakutsk 677000
References
1. Itou H. and Khludnev A. M., “On delaminated thin Timoshenko inclusions inside elastic bodies,” Math. Meth. Appl. Sci., 39, 4980–4993 (2016).
2. Khludnev A. M. and Leugering G. R., “On Timoshenko thin elastic inclusions inside elastic bodies,” Math. Mech. Solids, 20, No. 5, 495–511 (2015).
3. Khludnev A. M. and Popova T. S., “Timoshenko inclusions in elastic bodies crossing an external boundary at zero angle,” Acta Mech. Solida Sin., 30, No. 3, 327–333 (2017).
4. Khludnev A. M. and Shcherbakov V. V., “Singular invariant integrals for elastic bodies with thin elastic inclusions and cracks,” Dokl. Phys., 61, No. 12, 615–619 (2016).
5. Khludnev A. M., “Thin inclusions in elastic bodies crossing an external boundary,” Z. Angew. Math. Mech., 95, No. 11, 1256–1267 (2015).
6. Khludnev A. M. and Leugering G. R., “Delaminated thin elastic inclusion inside elastic bodies,” Math. Mech. Complex Syst., 2, No. 1, 1–21 (2014).
7. Itou H., Khludnev A. M., Rudoy E. M., et al., “Asymptotic behaviour at a tip of a rigid line inclusion in linearized elasticity,” Z. Angew. Math. Mech., 92, No. 9, 716–730 (2012).
8. Shcherbakov V., “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions,” Z. Angew. Math. Phys., 68, No. 1, article ID 26 (2017).
9. Khludnev A. M. and Popova T. S., “On the hierarchy of thin inclusions in elastic bodies [in Russian],” Mat. Zamet. SVFU, 23, No. 1, 87–107 (2016).
10. Rudoy E. M., “Asymptotic justification of models of plates containing inside hard thin inclusions,” Technologies, 8, 59 (2017).
11. Rudoy E. M., Itou H., and Lazarev N. P., “Asymptotic justification of the models of thin inclusions in an elastic body in the antiplane shear problem,” J. Appl. Ind. Math., 15, No. 1, 129–140 (2021).
12. Ljulj M. and Tambaca J., “3D structure–2D plate–1D rod interaction problem,” Math. Methods Appl. Sci., 46, No. 2 (2023).
13. Popova T. and Rogerson G. A., “On the problem of a thin rigid inclusion embedded in a Maxwell material,” Z. Angew. Math. Phys., 67, No. 4, article ID 105 (2016).
14. Popova T. S., “Problems of thin inclusions in a two-dimensional viscoelastic body,” J. Appl. Ind. Math., 12, 313–324 (2018).
15. Furtsev A. I., Fankina I. V., Rodionov A. A., and Ponomarev D. A., “Asymptotic modeling of steady vibrations of thin inclusions in a thermoelastic composite,” Z. Angew. Math. Phys., 74, article ID 195 (2023).
16. Rudoy E. and Sazhenkov S., “Variational approach to modeling of curvilinear thin inclusions with rough boundaries in elastic bodies: case of a rod type inclusion,” Mathematics, 11, No. 16, 3447 (2023).
17. Rudoy E. M., “Domain decomposition method for a membrane with a delaminated thin rigid inclusion [in Russian],” Sib. Electron. Math. Rep., 13, 395–410 (2016).
18. Kazarinov N. A., Rudoy E. M., Slesarenko V. Y., et al., “Mathematical and numerical simulation of equilibrium of an elastic body reinforced by a thin elastic inclusion,” Comput. Math. Math. Phys., 58, 761–774 (2018).
19. Rudoy E. M., “Numerical solution of an equilibrium problem for an elastic body with a delaminated thin rigid inclusion,” J. Appl. Ind. Math., 10, No. 2, 264–276 (2016).
20. Rudoy E. M. and Lazarev N. P., “Domain decomposition technique for a model of an elastic body reinforced by a Timoshenko’s beam,” J. Comput. Appl. Math., 334, 18–26 (2018).
21. Khludnev A. M., Faella L., and Popova T. S., “Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies,” Math. Mech. Solids, 22, No. 4, 737–750 (2017).
22. Khludnev A. M. and Popova T. S., “On the mechanical interplay between Timoshenko and semirigid inclusions embedded in elastic bodies,” Z. Angew. Math. Mech., 97, 1406–1417 (2017).
23. Khludnev A. M. and Popova T. S., “On junction problem for elastic Timoshenko inclusion and semi-rigid inclusion [in Russian],” Mat. Zamet. SVFU, 25, No. 1, 73–86 (2018).
24. Popova T. S., “On numerical solving of junction problem for the thin rigid and elastic inclusions in elastic body,” Lobachevskii J. Math., 44, No. 10, 4143–4156 (2023).
25. Khludnev A. M. and Popova T. S., “Equilibrium problem for elastic body with delaminated T-shape inclusion,” J. Comput. Appl. Math., 376, article ID 112870 (2020).
26. Khludnev A. M., “T-shape inclusion in elastic body with a damage parameter,” J. Comput. Appl. Math., 393, article ID 113540 (2021).
27. Popova T. S., “The problem of T-shaped junction of two thin Timoshenko inclusions in a two-dimensional elastic body,” Mat. Zamet. SVFU, 30, No. 2, 40–55 (2023).
28. Dumont S., Rizzoni R., Lebon F., and Sacco E., “Soft and hard interface models for bonded elements,” Composites, Part B: Engineering, 153, 480–490 (2018).
29. Dumont S., Lebon F., and Rizzoni R., “Imperfect interfaces with graded materials and unilateral conditions: theoretical and numerical study,” Math. Mech. Solids, 23, No. 3, 445–460 (2018).
30. Ekeland I. and Temam R., Convex Analysis and Variational Problems, North-Holland, Amsterdam; Elsevier, New York (1976).
31. Popova T. S., Problems of Equilibrium of Inhomogeneous Deformable Bodies with Thin Inclusions in Presence of Delamination [in Russian], Diss. . . . Dokt. Fiz.-Mat. Nauk, Yakutsk (2021).
32. Alexandrov V. M., Smetanin B. I., and Sobol B. V., Thin Stress Concentrators in Elastic Bodies [in Russian], Fizmatlit, Moscow (1993).
Review
For citations:
Popova T.S. Numerical solution of the problem of T-shaped junction of two thin Timoshenko inclusions in a two-dimentional elastic body. Mathematical notes of NEFU. 2024;31(3):93-120. (In Russ.) https://doi.org/10.25587/2411-9326-2024-3-93-120
Views:
8
JATS XML
Email this article 