Abstract
The article considers the problems of the formation of a neck in a sample of complex shape and Lüders slip bands in a square plate with a central circular hole in a plane-deformed state in finite deformations, taking into account the perfectly plastic flow. The statement of tasks is presented. Due to the symmetry, quarters of the models were considered. Numerical results were obtained in the CAE Fidesys package using the finite element method for the first and second orders, as well as for the spectral element method: for seven orders for the necking problem and the third-order elements for the Lüders slip bands problem. Based on the results obtained, the analysis of the capabilities of the spectral element method relative to the finite element method is carried out. The results of the conducted research can be useful when deciding on the use of the spectral element method in industrial problems.
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References
Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 2nd edn. McGraw-Hill Book Co., New York- Toronto-London (1951). https://doi.org/10.1017/S036839310012471X
Sedov, L.I.: Introduction to the Mechanics of a Continuous Medium, 2nd edn. Addison-Wesley Publishing Co, California (1965)
Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 7th edn. Butterworth-Heinemann, Oxford, United Kingdom (2013). https://doi.org/10.1016/C2009-0-24909-9
Fish, J., Belutschko, T.: A First Course in Finite Elements. John Wiley & Sons Ltd, New York (2007). https://doi.org/10.1002/9780470510858.index
Strang, G., Fix, G.: An Analysis of the Finite Element Method, 2nd edn. Prentice-Hall, Englewood Cliffs, N.J. (1973). https://doi.org/10.1002/zamm.19750551121
Zingerman, K.M., Vershinin, A.V., Levin, V.A.: Comparison of numerically-analytical and finite-element solutions of the Lame problem for nonlinear-elastic cylinder under large strains. J. Phys. Conf. Ser. 1158(4), 042045 (2019). https://doi.org/10.1088/1742-6596/1158/4/042045
Zingerman, K.M., Vershinin, A.V., Levin, V.A.: An approach for verification of finite-element analysis in nonlinear elasticity under large strains. IOP Conf. Ser. Mater. Sci. Eng. 158, 012104 (2016). https://doi.org/10.1088/1757-899x/158/1/012104
Levin, V.A., Vershinin, A.V., Zingerman, K.M.: Numerical analysis of propagation of nonlinear waves in prestressed solids. Modern Appl. Sci. 10(4), 158–167 (2016). https://doi.org/10.5539/mas.v10n4p158
Yakovlev, M.Y., Lukyanchikov, I.S., Levin, V.A., Vershinin, A.V., Zingerman, K.M.: Calculation of the effective properties of the prestressed nonlinear elastic heterogeneous materials under finite strains based on the solutions of the boundary value problems using finite element method. J. Phys. Conf. Ser. 1158(4), 042037 (2019). https://doi.org/10.1088/1742-6596/1158/4/042037
Vdovichenko, I.I., Yakovlev, M.Y., Verchinin, A.V., Levin, V.A.: Calculation of the effective thermal properties of the composites based on the finite element solutions of the boundary value problems. IOP Conf. Ser. Mater. Sci. Eng. 158(1), 012094 (2016). https://doi.org/10.1088/1757-899X/158/1/012094
Rajagopal, K.R., Wineman, A.S.: A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes. Int. J. Plasticity. 8(4), 385–395 (1992). https://doi.org/10.1016/0749-6419(92)90056-I
Misra, A., Placidi, L., dell’Isola, F., et al.: Identification of a geometrically nonlinear micromorphic continuum via granular micromechanics. Z. Angew. Math. Phys. 72(4), 1–21 (2021). https://doi.org/10.1007/s00033-021-01587-7
Javili, A., Steinmann, P., dell’Isola, F.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids. 61(12), 2381–2401 (2013). https://doi.org/10.1016/j.jmps.2013.06.005
Quiligotti, S., Maugin, G., dell’Isola, F.: An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures. Acta Mechanica. 160, 45–60 (2003). https://doi.org/10.1007/s00707-002-0968-z
Levin, V.A., Podladchikov, Y.Y., Zingerman, K.M.: An exact solution to the Lame problem for a hollow sphere for new types of nonlinear elastic materials in the case of large deformations. Eur. J. Mech. A/Solids. 90, 104345 (2021). https://doi.org/10.1016/j.euromechsol.2021.104345
Barchiesi, E., Eugster, S.R., dell’Isola, F., Hild, F.: Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenization and experimental validation. Math. Mech. Solids. 25(3), 739–767 (2019). https://doi.org/10.1177/1081286519891228
Vershinin, A.V., Levin, V.A., Zingerman, K.M., Sboychakov, A.M., Yakovlev, M.Y.: Software for estimation of second order effective material properties of porous samples with geometrical and physical nonlinearity accounted for. Adv. Eng. Softw. 86, 80–84 (2015). https://doi.org/10.1016/j.advengsoft.2015.04.007
Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics/Die Nicht-Linearen Feldtheorien der Mechanik. Springer-Verlag, Berlin (1965). https://doi.org/10.1007/978-3-642-46015-9
Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, Chichester (2000)
Vershinin, A.V., Levin, V.A., Kukushkin, A.V., Konovalov, D.A.: Structural analysis of assemblies using non-conformal spectral element method. IOP Conf. Ser. Mater. Sci. Eng. 747, 012033 (2020). https://doi.org/10.1088/1757-899x/747/1/012033
Orel, B., Perne, A.: Chebyshev-Fourier spectral methods for nonperiodic boundary value problems. J. Appl. Math. (2014). https://doi.org/10.1155/2014/572694
Payette, G.: Spectral/hp finite element models for fluids and structures. Doctoral dissertation, Texas A &M University (2012). https://doi.org/10.1088/1969.1/ETD-TAMU-2012-05-10962
Petrovskiy, K.A., Vershinin, A.V., Levin, V.A.: Application of spectral elements method to calculation of stress-strain state of anisotropic laminated shells. IOP Conf. Ser. Mater. Sci. Eng. 158, 012077 (2016). https://doi.org/10.1088/1757-899x/158/1/012077
Karpenko, V.S., Vershinin, A.V., Levin, V.A., Zingerman, K.M.: Some results of mesh convergence estimation for the spectral element method of different orders in FIDESYS industrial package. IOP Conf. Ser. Mater. Sci. Eng. 158, 012049 (2016). https://doi.org/10.1088/1757-899x/158/1/012049
Komatitsch, D., Tromp, J.: Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999). https://doi.org/10.1046/j.1365-246x.1999.00967.x
Patera, A.T.: A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984). https://doi.org/10.1016/0021-9991(84)90128-1
Komatitsch, D., Vilotte, J.P.: The spectral element method: an efficient tool to simulate the seismic response of 3D and 3D geological structures. Bull. Seism. Soc. Am. 88(2), 368–392 (1998). https://doi.org/10.1785/BSSA0880020368
Kachanov, L. M.: Foundations of the Theory of Plasticity. North-Holland, Amsterdam (1971). https://doi.org/10.1007/978-0-387-33599-5_3
Nadai, A.: Plasticity: A Mechanics of the Plastic State of Matter. McGraw-Hill (1931)
Kluth, G., Després, B.: Perfect plasticity and hyperelastic models for isotropic materials. Continuum Mech. Thermodyn. 20, 173 (2008). https://doi.org/10.1007/s00161-008-0078-9
Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998)
Greco, L., Cuomo, M.: An implicit G1-conforming bi-cubic interpolation for the analysis of smooth and folded Kirchhoff-Love shell assemblies. Comput. Methods Appl. Mech. Eng. 373, 113476 (2021). https://doi.org/10.1016/j.cma.2020.113476
Greco, L., Cuomo, M., Contrafatto, L.: Two new triangular G1-conforming finite elements with cubic edge rotation for the analysis of Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 356, 354–386 (2019). https://doi.org/10.1016/j.cma.2019.07.026
Yildizdag, M.E., Ardic, I.T., Ergin, A.: An isogeometric FE-BE method to investigate fluid-structure interaction effects for an elastic cylindrical shell vibrating near a free surface. Ocean Eng. 251, 111065 (2022). https://doi.org/10.1016/j.oceaneng.2022.111065
Zhao, Y., Gu, Y., Guo, Y.: Plasticity and deformation mechanisms of Ultrafine-Grained Ti in Neching region revealed by digital image correlation technique. Nanomaterials 11(3), 574 (2021). https://doi.org/10.3390/nano11030574
Hill, R.: On discontinuous plastic states, with special reference to a localized necking in thin sheets. J. Mech. Phys. Solids. 1, 9–30 (1952). https://doi.org/10.1016/0022-5096(52)90003-3
Lubkova, E.Y., Morozov, E.M., Osintsev, A.V., Plotnikov, A.S.: On the location of a neck formation during the tension of cylindrical specimens. Lett. Mater. (2017). https://doi.org/10.22226/2410-3535-2017-3-260-265
Levin, V.A., Zingerman, K.M., Vershinin, A.V.: Geomechanical modelling of fracture propagation under finite strain. Prefracture Zones. Seism. Technol. 11(4), 1–11 (2014). https://doi.org/10.3997/2405-7495.2015102
Battista, A., Della Corte, A., dell’Isola, F., et al.: Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams. Z. Angew. Math. Phys. 69(3), 52 (2018). https://doi.org/10.1007/s00033-018-0946-5
Han, J., Lu, C., Wu, B., Li, J., Li, H., Lu, Y., Gao, Q.: Innovative analysis of Luders band behavior in X80 pipeline steel. Mater. Sci. Eng. A. 683, 123–128 (2017). https://doi.org/10.1016/j.msea.2016.12.008
Mazière, M., Forest, S.: Strain gradient plasticity modeling and finite element simulation of Lüders band formation and propagation. Continuum Mech. Thermodyn. 27, 83–104 (2015). https://doi.org/10.1007/s00161-013-0331-8
Zhang, Y., Ding, H.: Ultrafine also can be ductile: On the essence of Luders band elongation in ultrafine-grained medium manganese steel. Mater. Sci. Eng. A. 733, 220–223 (2018). https://doi.org/10.1016/j.msea.2018.07.052
Contrafatto, L., Cuomo, M.: A globally convergent numerical algorithm for damaging elasto-plasticity based on the Multiplier method. Int. J. Numer. Methods Eng. 63(8), 1089–1125 (2005). https://doi.org/10.1002/nme.1235
Kukushkin, A.V., Konovalov, D.A., Vershinin, A.V., Levin, V.A.: Numerical simulation in CAE Fidesys of bonded contact problems on non-conformal meshes. J. Phys. Conf. Ser. 1158(2), 032022 (2019). https://doi.org/10.1088/1742-6596/1158/3/032022
CAE Fidesys: User Guide version 4.1. https://www.cae-fidesys.com/documentation/ (2021). Accessed 25 January 2022
Konovalov, D., Vershinin, A., Zingerman, K., Levin, V.: The implementation of spectral element method in a CAE system for the solution of elasticity problems on hybrid curvilinear meshes. Modell. Simul. Eng. (2017). https://doi.org/10.1155/2017/1797561
Bernardi, C., Debit, N., Maday, Y.: Coupling finite element and spectral methods. Math. Comput. 54(189), 21–39 (1990). https://doi.org/10.1090/S0025-5718-1990-0995205-7
Giorgio, I., De Angelo, M., Turco, E., Misra, A.: A Biot-Cosserat two-dimensional elastic nonlinear model for a micromorphic medium. Continuum Mech. Thermodyn. 32(5), 1357–1369 (2020). https://doi.org/10.1007/s00161-019-00848-1
Giorgio, I.: Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial. Continuum Mech. Thermodyn. 33(4), 1063–1082 (2021). https://doi.org/10.1007/s00161-020-00955-4
Andreaus, U., dell’Isola, F., Giorgio, I., Placidi, L., Lekszycki, T., Rizzi, N. L.: Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int. J. Eng. Sci. 108, 34–50 (2016). https://hal.archives-ouvertes.fr/hal-01378498
Levin, V.A., Zubov, L.M., Zingerman, K.M.: Large bending strains in an orthotropic beam with a preliminarily stretched or compressed layer: Exact solution. Doklady Phys. 61(8), 407–411 (2016). https://doi.org/10.1134/S1028335816080127
Levin, V.A., Zingerman, K.M., Krapivin, K.Y., Ryabova, O.A., Kukushkin, A.V.: A Model of material microstructure formation on selective laser sintering with allowance for large elastoplastic strains. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki 161(2), 191–204 (2019). https://doi.org/10.26907/2541-7746.2019.2.191-204
Babuška, I., Szabó, B.: Introduction to Finite Element Analysis: Formulation, Verification and Validation. John Wiley & Sons Ltd, New York (2011). https://doi.org/10.1002/9781119993834
Acknowledgements
The authors are grateful to Professor of Lomonosov Moscow State University Vladimir Levin for the problem statement and constant attention to this work.
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The research for this article was performed in Lomonosov Moscow State University and was financially supported by the Ministry of Education and Science of the Russian Federation under the agreement No 075-15-2019-1890.
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Kozlov, V.V., Komolova, E.D., Kartsev, M.A. et al. Analysis of the capabilities of the spectral element method in solving physically and geometrically nonlinear problems of mechanics using the CAE Fidesys package. Continuum Mech. Thermodyn. 35, 1263–1273 (2023). https://doi.org/10.1007/s00161-022-01121-8
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DOI: https://doi.org/10.1007/s00161-022-01121-8