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Numerical simulation of superimposed finite strains using spectral element method

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Abstract

A novel approach to the numerical modeling of superimposed finite strains is proposed in the article. The spectral element method is used for analysis. This method is an effective modification of the finite element method that provides exponential decrease of computational error as the order of element. The discretization of static problems of nonlinear elasticity under superimposed finite strains is carried out for spectral elements using the Galerkin method. The developed approach is further used for solution of some model problems of superimposed finite strains. The presented examples include the insertion of hyperelastic cylinder into the prestressed cylinder, the formation of a hole in a preloaded nonlinear-elastic sample, and the nonlinear bending of the layered beam with a prestressed layer. The examples demonstrate the robustness of the proposed algorithms. The results obtained are in good agreement with the analytical solutions.

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Acknowledgements

The research for this article was performed in Lomonosov Moscow State University and was financially supported by the Ministry of Science and Higher Education of the Russian Federation as part of the program of the Mathematical Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2019-1621 (Sects. 1, 2), by the Ministry of Science and Higher Education of the Russian Federation under the agreement No. 075-15-2019-1890 (Sect. 3), by the grant of the President of the Russian Federation for young scientists—doctors of sciences MD-208.2021.1.1. (Sect. 5) and by Russian Scientific Foundation (Project No. 19-71-10008, Sects. 4, 6).

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russian Federation

    V. A. Levin, A. V. Vershinin & D. A. Konovalov

  2. Faculty of Applied Mathematics and Cybernetics, Tver State University, Tver, Russian Federation

    K. M. Zingerman

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  1. V. A. Levin
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  2. K. M. Zingerman
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  3. A. V. Vershinin
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  4. D. A. Konovalov
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Correspondence to V. A. Levin.

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Communicated by Andreas Öchsner.

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Levin, V.A., Zingerman, K.M., Vershinin, A.V. et al. Numerical simulation of superimposed finite strains using spectral element method. Continuum Mech. Thermodyn. 34, 1205–1217 (2022). https://doi.org/10.1007/s00161-022-01115-6

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