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Numerical simulation of the bending of a layered beam with prestressed layer under finite strains using the spectral element method

Vladimir A. Levin, Konstantin M. Zingerman, Anatoly V. Vershinin and Dmitriy A. Konovalov

Vol. 10 (2022), No. 1, 85–102
DOI: 10.2140/memocs.2022.10.85
Abstract

The implementation of the spectral element method for the problems of nonlinear elasticity under finite strains is developed. The analysis is performed for structures made by junction of preliminarily strained parts. The bending of a layered beam with a prestressed layer is analyzed as a particular case. It is assumed that the beam consists of weakly compressible nonlinear-elastic material.

The theory of superimposed finite strains is used for the problem statement. The discretization is performed using the Galerkin method. The proposed approach permits one to vary the order of spectral elements without the mesh refinement.

The results of analysis are given. The comparison with a known analytical solution is performed. It is obtained that the results of spectral element analysis are close to the analytical solution. Substantial nonlinear effects are determined. The incomplete junction of layers is analyzed.

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Keywords
variable-order spectral element method, superimposed finite deformations, nonlinear elasticity, layered beam
Mathematical Subject Classification
Primary: 65N35, 74B20
Milestones
Received: 15 June 2021
Revised: 21 April 2022
Accepted: 29 May 2022
Published: 20 October 2022

Communicated by Victor A. Eremeyev
Authors
Vladimir A. Levin
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia
Konstantin M. Zingerman
Faculty of Applied Mathematics and Cybernetics
Tver State University
Tver
Russia		 Institute of Nuclear Physics and Engineering
National Research Nuclear University MEPhI
Moscow
Russia
Anatoly V. Vershinin
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia		 Schmidt Institute of Physics of the Earth
Russian Academy of Sciences
Moscow
Russia
Dmitriy A. Konovalov
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia		 Schmidt Institute of Physics of the Earth
Russian Academy of Sciences
Moscow
Russia