×

Domain decomposition technique for a model of an elastic body reinforced by a Timoshenko’s beam. (English) Zbl 1462.74042

Summary: A body made of fiber-reinforced elastic material is considered. It is assumed that a behavior of a fiber is described by the Timoshenko beam. We propose a method of numerical solving to an equilibrium problem of the fiber-reinforced body. The method is based on the Uzawa algorithm and the domain decomposition technique. A numerical examination is carried out to demonstrate the efficiency of the proposed method.

MSC:

74E30 Composite and mixture properties
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S99 Numerical and other methods in solid mechanics

Software:

FreeFem++
Full Text: DOI

References:

[1] Mallick, P. K., Fiber-Reinforced Composites : Materials, Manufacturing, and Design (2007), CRC Press, Taylor & Francis Group: CRC Press, Taylor & Francis Group Boca Raton London New York
[2] Amin, S. S.; Dalir, H.; Farshidianfar, A., Carbon nanotube-reinforced composites: frequency analysis theories based on the matrix stiffness, Comput. Mech., 43, 515-524 (2009)
[3] Rafiee, M.; Yang, J.; Kitipornchai, S., Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams, Comput. Math. Appl., 66, 1147-1160 (2003) · Zbl 1353.82091
[4] Itou, H.; Khludnev, A. M., On delaminated thin Timoshenko inclusions inside elastic bodies, Math. Methods Appl. Sci., 39, 17, 4980-4993 (2016) · Zbl 1356.35241
[5] Eugster, S. R., Geometric Continuum Mechanics and Induced Beam Theories (2015), Springer Cham Heidelberg: Springer Cham Heidelberg New York Dordrecht London · Zbl 1330.74002
[6] Timoshenko, S. P., Vibration Problems in Engineering (1955), Van Nostrand: Van Nostrand Toronto-New York-London · Zbl 0201.27501
[7] Tkacheva, L. A., Unsteady problem of crack propagation in the beam approximation, J. Appl. Mech. Tech. Phys., 49, 5, 853-863 (2008)
[8] Khludnev, A. M.; Negri, M., Crack on the boundary of a thin elastic inclusion inside an elastic body, ZAMM Z. Angew. Math. Mech., 92, 5, 341-354 (2012) · Zbl 1322.74065
[9] Khludnev, A. M.; Leugering, G., Delaminated thin elastic inclusion inside elastic bodies, Math. Mech. Complex Syst., 2, 1, 1-21 (2014) · Zbl 1459.74163
[10] Khludnev, A. M.; Leugering, G., On elastic bodies with thin rigid inclusions and cracks, Math. Methods Appl. Sci., 33, 16, 1955-1967 (2010) · Zbl 1202.35323
[11] Shcherbakov, V., Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions, Z. Angew. Math. Phys., 68 (2017), Article 26 · Zbl 1365.35180
[12] Rudoy, E. M., On numerical solving a rigid inclusions problem in 2D elasticity, Z. Angew. Math. Phys., 68 (2017), Article 19 · Zbl 1440.74094
[13] Gómez, D.; Lobo, M.; Pérez, M. E.; Shaposhnikova, T. A.; Zubova, M. N., On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems, Math. Methods Appl. Sci., 38, 12, 2606-2629 (2015) · Zbl 1332.35025
[14] Chapelle, D.; Ferent, A., Modeling of the inclusion of a reinforcing sheet within a 3d medium, Math. Models Methods Appl. Sci., 13, 4, 573-595 (2003) · Zbl 1057.74021
[15] Rudoy, E. M., Numerical solution of the equilibrium problem for a membrane with embedded rigid inclusions, Comput. Math. Math. Phys., 56, 3, 450-459 (2016) · Zbl 1382.74125
[16] Lazarev, N. P., Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack, ZAMM Z. Angew. Math. Mech., 96, 4, 509-518 (2016) · Zbl 1538.74126
[17] Lazarev, N., Existence of an optimal size of a delaminated rigid inclusion embedded in the Kirchhoff-Love plate, Bound. Value Probl. (2015), Article 180 · Zbl 1336.74043
[18] Lazarev, N. P.; Rudoy, E. M., Optimal size of a rigid thin stiffener reinforcing an elastic plate on the outer edge, ZAMM Z. Angew. Math. Mech. (2017) · Zbl 1538.74127
[19] Geubelle, Ph. H.; Baylor, J. S., Impact-induced delamination of composites: a 2D simulation, Composites Part B., 29B, 589-602 (1998)
[20] Lapczyk, I.; Hurtado, J. A., Progressive damage modeling in fiber-reinforced materials, Composites A, 38, 2333-2341 (2007)
[21] Larsson, R., A discontinuous shell-interface element for delamination analysis of laminated composite structures, Comput. Methods Appl. Mech. Engrg., 193, 3173-3194 (2004) · Zbl 1085.74043
[22] Pietropaoli, E., (Tesinova, Dr. Pavla, Virtual Crack Closure Technique and Finite Element Method for Predicting the Delamination Growth Initiation in Composite Structures. Virtual Crack Closure Technique and Finite Element Method for Predicting the Delamination Growth Initiation in Composite Structures, Advances in Composite Materials - Analysis of Natural and Man-Made Materials (2011), InTech)
[23] Steigmann, D. J., Theory of elastic solids reinforced with fibers resistant to extension. Flexure and twist, Int. J. Non-Linear Mech., 47, 734-742 (2012)
[24] Lanzoni, L.; Radi, E., A loaded Timoshenko beam bonded to an elastic half plane, Int. J. Solids Struct., 92-93, 76-90 (2016)
[25] Vorel, J.; Boshoff, W. P., Numerical simulation of ductile fiber-reinforced cement-based composite, J. Comput. Appl. Math., 270, 433-442 (2017) · Zbl 1321.74075
[26] Sapountzakis, E. J.; Mokos, V. G., Shear deformation effect in plates stiffened by parallel beams, Arch. Appl. Mech., 79, 893-915 (2009) · Zbl 1184.74045
[27] Dobroserdova, T. K.; Olshanskii, M. A., A finite element solver and energy stable coupling for 3D and 1D fuid models, Comput. Methods Appl. Math., 259, 166-176 (2013) · Zbl 1286.76082
[28] Bayada, G.; Sabil, J.; Sassi, T., Convergence of a Neumann-Dirichlet algorithm for two-body contact problems with non local Coulomb’s friction law, ESAIM: Math. Model. Numer., 42, 2, 243-262 (2008) · Zbl 1133.74042
[29] Korneev, V. D.; Sveshnikov, V. M., 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) · Zbl 1349.65678
[30] Koko, J., Uzawa conjugate gradient domain decomposition methods for coupled Stokes flows, J. Sci. Comput., 26, 2, 195-215 (2006) · Zbl 1203.76116
[31] Goessens, T.; Malengier, B.; Constales, D.; De Staelen, R. H., A volume averaging and overlapping domain decomposition technique to model mass transfer in textiles, J. Comput. Appl. Math., 275, 456-464 (2015) · Zbl 1302.65204
[32] Le Tallec, P.; De Roeck, Y. H.; Vidrascu, M., Domain decomposition methods for large linearly elliptic three-dimensional problems, J. Comput. Appl. Math., 34, 1, 93-117 (1991) · Zbl 0719.65083
[33] Rudoy, E. M., 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)
[34] Rudoy, E. M., Domain decomposition method for crack problems with nonpenetration condition, ESAIM-Math. Model. Numer., 50, 4, 995-1009 (2016) · Zbl 1457.65242
[35] Rudoy, E. M., Domain decomposition method for a model crack problem with a possible contact of crack edges, Comput. Math. Math. Phys., 55, 2, 305-316 (2015) · Zbl 1426.74321
[36] Rudoy, E. M.; Kazarinov, N. A.; V. Yu, Slesarenko, Numerical simulation of equilibrium of an elastic two-layer structure with a through crack, Numer. Analys. Appl., 10, 1, 63-73 (2017)
[37] Hintermüller, M.; Kovtunenko, V.; Kunisch, K., The primal-dual active set method for a crack problem with non-penetration, IMA J. Appl. Math., 69, 1-26 (2004) · Zbl 1084.49029
[38] Kovtunenko, V. A., Numerical simulation of the non-linear crack problem with nonpenetration, Math. Methods Appl. Sci., 27, 2, 163-179 (2004) · Zbl 1099.74056
[39] Ciarlet, Ph., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0383.65058
[40] Allaire, G., Numerical Analysis and Optimization: An Introduction To Mathematical Modelling and Numerical Simulation (2007), Univ. Press: Univ. Press Oxford · Zbl 1120.65001
[41] Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 3-4, 251-265 (2012) · Zbl 1266.68090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.