Numerical solution of the equilibrium problem for a two-dimensional elastic body with a thin semirigid inclusion
Аннотация
The equilibrium problem for a two-dimensional elastic body containing a thin semirigid inclusion is considered. The inclusion delaminates from the elastic matrix, forming a crack; therefore, the problem is posed in a nonsmooth domain with a cut. The mathematical model of the delaminated thin semirigid inclusion was developed on the assumption that the rigidity of the material differs in different directions. The problem statement is presented both in the form of a variational inequality and in the form of a boundary value problem. The boundary condition on the crack faces has a form of inequality and, as a result, the problem is non-linear. Consequently, the construction of an algorithm for the numerical solution of the problem requires the use of additional analytical methods. The methods of domain decomposition, the method of Lagrange multipliers, and the finite element method are used. An algorithm for the numerical solution of the problem is constructed and a computational example is provided.
Литература
[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., “Delaminated thin elastic inclusion inside elastic bodies,” Math. Mech. Complex Syst., 2, No. 1, 1–21 (2014).
[3] Khludnev A. M. and Leugering G. R., “On Timoshenko thin elastic inclusions inside elastic bodies,” Math. Mech. Solids, 20, No. 5, 495–511 (2015).
[4] Khludnev A. M. and Popova T. S., “On the hierarchy of thin inclusions in elastic bodies,” Mat. Zametki SVFU, 23, No. 1, 87–107 (2016).
[5] Shield T. W. and Kim K. S., “Beam theory models for thin film segments cohesively bonded to an elastic half space,” Int. J. Solids Struct., 29, No. 9, 1085–1103 (1992).
[6] Khludnev A. M., “Thin inclusions in elastic bodies crossing an external boundary,” Z. Angew. Math. Mech., 95, No. 11, 1256–1267 (2015).
[7] Khludnev A. M., 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).
[8] Shcherbakov V. V., “The Griffith formula and J-integral for elastic bodies with Timoshenko inclusions,” Z. Angew. Math. Mech., 96, No. 11, 1306–1317 (2016).
[9] Rudoy E. M. and Lazarev N. P., “Domain decomposition technique for a model of an elastic body reinforced by a Timoshenkos beam,” J. Comp. Appl. Math., 334, 18–26 (2018).
[10] 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).
[11] Mueller-Hoeppe D., Wriggers P., and Loehnert S., “Crack face contact for a hexahedral-based XFEM formulation,” Comput. Mech., 49 (2012).
[12] Wriggers P. and Zavarise G., “Computational contact mechanics,” in: Encyclopedia of Computational Mechanics, John Wiley and Sons, Ltd. (2004).
[13] Bandeira A., Wriggers P., and Pimenta P., “Numerical derivation of contact mechanics interface laws using a finite approach for large 3D deformation,” Int. J. Numer. Methods Eng., 59, 173–195 (2004).
[14] Andrianov I. V., Danishevskyy V. V., and Topol H., “Local stress distribution in composites for pulled-out fibers with axially varying bonding,” Acta Mech., 231, 2065–2083 (2020).
[15] Itou H., Kovtunenko V. A., and Rajagopal K. R., “Crack problem within the context of implicitly constituted quasi-linear viscoelasticity,” Math. Models Methods Appl. Sci., 29, No. 2, 355–372 (2019).
[16] Jean M., “The non-smooth contact dynamics method,” Comput. Methods Appl. Mech. Eng., 177, No. 3-4, 235–257 (1999).
[17] Rey V., Anciaux G., and Molinari G.-F., “Normal adhesive contact on rough surfaces: efficient algorithm for FFT-based BEM resolution,” Comput. Mech., 60, No. 1, 69–81 (2017).
[18] 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).
[19] Khludnev A. M., Elasticity Problems in Non-Smooth Domains, Fizmatlit, Moscow (2010).
[20] Khludnev A. M. and Kovtunenko V. A., Analysis of Cracks in Solids, WIT Press, Southampton; Boston (2000).
[21] 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).
[22] Khludnev A. M., “Thin rigid inclusions with delaminations in elastic plates,” Eur. J. Mech., A, Solids, 32, 69–75 (2012).
[23] Rudoy E. M., “Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion,” J. Appl. Ind. Math., 10, 264–276 (2016).
[24] Nikolaeva N. A., “On equilibrium of elastic bodies with a cracks crossing thin inclusions,” Sib. J. Ind. Math., 22, No. 4, 68–80 (2019).
[25] Popova T. S., “Problems of thin inclusions in a two-dimensional viscoelastic body,” J. Appl. Ind. Math., 12, 313–324 (2018).
[26] 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).
[27] Khludnev A. M. and Popova T. S., “Junction problem for elastic Timoshenko inclusion and a semirigid inclusion,” Mat. Zametki SVFU, 25, No. 1, 73–86 (2018).
[28] Khludnev A. M and Popova T. S., “On junction problem with damage parameter for Timoshenko and rigid inclusions inside elastic body,” Z. Angew. Math.Mech., 100, No. 8, 202000063 (2020).
[29] Khludnev A. M. and Popova T. S., “Equilibrium problem for elastic body with delaminated T-shape inclusion,” J. Comp. Appl. Math., 376, 112870 (2020).
[30] Popova T. S., “Equilibrium problem for a viscoelastic body with a thin rigid inclusion,” Mat. Zametki SVFU, 21, No. 1, 47–55 (2014).
[31] Popova T. and Rogerson G. A., “On the problem of a thin rigid inclusion embedded in a Maxwell material,” Z. Angew. Math. Phys., 67, 105 (2016).
[32] Shcherbakov V., “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions,” Z. Angew. Math. Phys., 68, No. 1, 26 (2017).
[33] Khludnev A.M. and Popova T. S., “Semirigid inclusions in elastic bodies: mechanical interplay and optimal control,” Comput. Math. Appl., 77, No. 1, 253–262 (2019).
[34] 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, No. 11, 1406–1417 (2017).
[35] Khludnev A. M. and Popova T., “Junction problem for rigid and semi-rigid inclusions in elastic bodies,” Arch. Appl. Mech., 86, No. 9, 1565–1577 (2016).
Поступила в редакцию
14-01-2021
Как цитировать
Popova, T. (2021) Numerical solution of the equilibrium problem for a two-dimensional elastic body with a thin semirigid inclusion, Математические заметки СВФУ, 28(1), сс. 51-66. doi: https://doi.org/10.25587/SVFU.2021.12.25.005.
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