Abstract
The paper considers the contact bending problem for a multilayer composite plate. Each layer of the composite is a material reinforced with thin parallel fibers. The mathematical model is constructed based on the assumptions of the existence of a neutral surface in the plate and the validity of Kirchhoff’s hypotheses. Using the Lagrange variational principle, we obtain a bending equation generalizing the Sophie Germain equation. An elastic energy functional taking into account the different resistance of the material to tension and compression is obtained. The contact problem of plate bending with the aid of a rigid die is considered. To solve the contact problem of plate bending by a rigid die, a Lagrangian with an inequality constraint is constructed. The finite element method using a triangular Bell element is applied for the numerical solution of the problem. The results of calculations for the bending of laminated rectangular plates with various directions of fiber laying and various die shapes are presented.
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Funding
This work was supported by the Russian Foundation for Basic Research, project no. 20-31-90032, and the Krasnoyarsk Mathematical Center under agreement no. 075-02-2022-873 with the Ministry of Education and Science of the Russian Federation.
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Translated by V. Potapchouck
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Petrakov, I.E. Contact Bending Problem for a Multilayer Composite Plate with Allowance for Different Moduli of Elasticity in Tension and Compression. J. Appl. Ind. Math. 16, 751–759 (2022). https://doi.org/10.1134/S1990478922040159
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DOI: https://doi.org/10.1134/S1990478922040159