On a 3D elastic body with a thin rigid inclusion
https://doi.org/10.25587/2411-9326-2026-2-117-127
Аннотация
We analyze an equilibrium state of a three-dimensional elastic body with a thin two-dimensional inclusion. At the first step we assume the inclusion to be delaminated from the surrounding elastic body, which provides an interfacial crack. To describe a mutual nonpenetration between the crack faces, we impose inequality type boundary conditions at the crack faces. Additional difficulties appear in view of the Neumann type conditions considered at the external boundary of the elastic body. We analyze a passage to the limit provided that the stiffness parameter of the inclusion tends to infinity. The limit model is analyzed, which describes an equilibrium state of the elastic body with a thin rigid inclusion. Various equivalent problem formulations of the limit problem are presented.
Ключевые слова
При поддержке: The study was supported by a grant from the Russian Science Foundation, number 26–11–00009, https://rscf.ru/project/26-11-00009/.
Об авторе
A. M. Khludnev
M. A. Lavrentiev Institute of Hydrodynamics
Россия
Россия
Aleksandr M. Khludnev
15 Lavrentiev Avenue, Novosibirsk 630090
Список литературы
1. Khludnev A. M., “On equilibrium problem for 3D elastic body with thin and volume inclusions in non-coercive case,” Sib. Electron. Math. Rep., 22, No. 2, 1334–1349 (2025).
2. Furtsev A. I., “Equilibrium problem for hyperelastic body with rigid inclusion and crack under nonopenetration condition,” Sib. Electron. Math. Rep., 21, No. 1, 17–40 (2024).
3. Furtsev A., Rudoy E., and Itou H., “Modeling of bonded elastic structures by a variational method: theoretical and numerical simulation,” Int. J. Solids Structures, 182-183, 100–111 (2020).
4. Furtsev A. I., Rudoy E. M., and Sazhenkov S. A., “On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition,” Phil. Trans. R. Soc. A. Math. Phys. Eng. Sci., 382, 20240115 (2024).
5. Lazarev N. P. and Kovtunenko V. A., “Asymptotic analysis of equilibrium of an inhomogeneous body with hinged inclusion of various width,” J. Appl. Mech. Tech. Phys., 64, No. 5, 911–920 (2023).
6. Kovtunenko V. A. and Kunisch K., “Shape derivative for penalty-constrained nonsmooth– nonconvex optimization: cohesive crack problem,” J. Optim. Theory Appl., 194, 597–635 (2022).
7. Rudoy E. M., “Asymptotic modeling of bonded plates by a soft thin adhesive layer,” Sib. Electron. Math. Rep., 17, 615–625 (2020).
8. Shcherbakov V. V., “Shape optimization of rigid inclusions for elastic plates with cracks,” Z. Angew. Math. Phys., 67, article ID 71 (2016).
9. Shcherbakov V. V., “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions,” Z. Angew. Math. Phys., 68, article ID 26 (2017).
10. Shcherbakov V. V., “Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks,” Nonlinear Anal.: Real World Appl., 65, article ID 103505 (2022).
11. Kovtunenko V. A. and Leugering G., “A shape-topological control problem for nonlinear crackdefect interaction: the anti-plane variational model,” SIAM J. Control Optim., 54, 1329–1351 (2016).
12. Popova T. S., “Numerical solution of the equilibrium problem for a two-dimensional elastic body with a delaminated rigid inclusion,” Lobachevskii J. Math., 45, No. 11, 5402–5413 (2024).
13. Popova T. S., “Mathematical modeling of a T-junction of thin anisotropic inclusions in elastic body in the presence of delaminations,” J. Appl. Mech. Tech. Phys., 391, No. 3, P. 192–207 (2025).
14. Lazarev N. and Rudoy E., “Optimal location of a finite set of rigid inclusions in contact problems for inhomogeneous two-dimensional bodies,” J. Comput. Appl. Math., 403, article ID 113710 (2022).
15. Lazarev N., Romanova N., and Semenova G., “Optimal location of a thin rigid inclusion for a problem describing equilibrium of a composite Timoshenko plate with a crack,” J. Inequal. Appl., 2020, article ID 29 (2020).
16. Lazarev N. and Itou H., “Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff–Love plates with a crack,” Math. Mech. Solids, 24, No. 12, 3743– 3752 (2019).
17. Khludnev A. M., “Junction problem for thin elastic and volume rigid inclusions in elastic body,” Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci., 380, article ID 20210360 (2022).
18. Morassi A. and Rosset E., “Detecting rigid inclusions, or cavities, in an elastic body,” J. Elasticity, 73, 101–126 (2003).
19. Alessandrini G., Morassi A., and Rosset E., “Detecting an inclusion in an elastic body by boundary measurements,” SIAM J. Math. Anal., 33, 1247–1268 (2002).
20. Attouch H., Buttazzo G., and Michaille G., Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, SIAM (2014).
21. Khludnev A. M. and Rodionov A. A., “Elasticity tensor identification in elastic body with thin inclusions: non-coercive case,” J. Optim. Theory Appl., 197, No. 3, 993–1010 (2023).
22. Khludnev A. M., “Thin inclusion at the junction of two elastic bodies: non-coercive case,” Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci., 382, article ID 20230296 (2024).
23. Khludnev A. M., Elasticity Problems in Non-Smooth Domains [in Russian], Fizmatlit, Moscow (2010).
Рецензия
Для цитирования:
Khludnev A.M. On a 3D elastic body with a thin rigid inclusion. Математические заметки СВФУ. 2026;33(2):117-127. https://doi.org/10.25587/2411-9326-2026-2-117-127
For citation:
Khludnev A.M. On a 3D elastic body with a thin rigid inclusion. Mathematical notes of NEFU. 2026;33(2):117-127. https://doi.org/10.25587/2411-9326-2026-2-117-127
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