### COMPARISON OF FINDINGS OF THE FINITE ELEMENT ANALYSISWITH THE FINDINGS OF THE ASYMPTOTIC HOMOGENIZATIONMETHOD IN RESPECT OF THE PLATE IN ELASTOPLASTIC BENDING

Pages 42-50

The authors present numerical results of the asymptotic homogenization method for elastoplastic bending of the plate. The plate is supposed to be laminated and exposed to the transversal load. Stresses and displacements in the cylindrical bending problem are compared with those calculated using the 2D finite element method. The new trend in the mathematical simulation of structures, made of composite materials, contemplates accurate consideration of their nonlinear properties (for instance, plasticity or damage) on the micro-structural level of materials. The homogenization method provides for the coupling between the microstructural level and the level of the entire structure. The authors have developed a numerical implementation of this coupling. It represents a combination of the homogenization method and linearization with account for the loading parameter. The approach was implemented as a parallel algorithm and applied to the plastic bending simulation of the FGM plate. The parallel algorithm is based on the overlapping subdomain decomposition method and the Euler explicit and implicit integration methods. MPI was used for software development purposes.In this paper, the authors provide a concise description of the proposed method applied to the 3D boundary-value problem. The authors compare numerical solutions obtained through the application of the homogenization approach and the finite element method. Two types of laminated plates are taken as an example. Three-layered plate was exposed to uniformly distributed transversal loading. The second five-layered plate, that was a lot thinner than the first one, was exposed to piecewise constant transversal loading. All layers of both plates are homogenous; they are supposed to be elastic or bilinearly plastic. It was discovered that the asymptotic homogenization technique provides a more accurate solution for the five-layered plate than for the three-layered one. Edge effects near the edges of the plates are smaller for the thin five-layered plate if compared with the thick three-layered plate. The edge effect appears due to the large value of the plate height-to-length ratio. Nevertheless, the first order asymptotic homogenized method provides sufficient accuracy in both cases.

DOI: 10.22227/1997-0935.2013.8.42-50

- Savenkova M.I., Sheshenin S.V., Zakalyukina I.M. Primenenie metoda osredneniya v zadache uprugoplasticheskogo izgiba plastiny [Application of Homogenization Method to Elastoplastic Bending of a Plate]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 9, pp. 156—164.
- Sheshenin S.V., Savenkova M.I. Osrednenie nelineynykh zadach v mekhanike kompozitov [Averaging Method for Nonlinear Problems in Composites Mechanics]. Vestnik Moskovskogo universiteta. Matematika. Mekhanika [Proceedings of Moscow University. Mathematics. Mechanics]. 2012, no. 5, pp. 58—61.
- Barret R. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, SIAM, 1994.
- Sadovnichy V., Tikhonravov A., Voevodin V.l., Opanasenko V. "Lomonosov": Super-computing at Moscow State University. In Contemporary High Performance Computing: from Petascale toward Exascale. Chapman & Hall/CRC Computational Science. 2013, Boca Raton, USA, CRC Press, pp. 283—307.
- Fish J., Shek K., Pandheeradi M., Shephard M.S. Computational Plasticity for Composite Structures Based on Mathematical Homogenization: Theory and Practice. Comput. Methods Appl. Mech. Engrg. 1997, no. 148, pp. 53—73.
- Ghosh S., Lee K., Moorthy S. Two Scale Analysis of Heterogeneous Elastic-plastic Materials with Asymptotic Homogenization and Voronoi Cell Finite Element Model. Comput. Methods Appl. Mech. Enrgr. 1996, no. 132, pp. 63—116.
- Gorbachev V.I., Pobedrya B.E. The Effective Characteristics of Inhomogeneous Media. J. Appl. Math. Mech. 1997, vol. 61, no. 1, pp. 145—151.
- Bakhvalov N.S. Osrednenie differentsial'nykh uravneniy s chastnymi proizvodnymi s bystro ostsilliruyushchimi koeffitsientami [Homogenization of Differential Equations Having Partial Derivatives with Rapidly Ocillating Coefficients]. Doklady AN SSSR [Reports of the Academy of Sciences of the USSR]. 1975, vol. 221, no. 3, pp. 516—519.
- Pobedrya B.E., Gorbachev V.I. Kontsentratsiya napryazheniy i deformatsiy v kompozitakh [Concentration of Stresses and Strains in Composites]. Mekhanika kompozitsionnykh materialov [Mechanics of Composite Materials]. 1984, no. 2, pp. 207—214.
- Kalamkarov A.L., Andrianov I.V., Danishevs'kyy V.V. Asymptotic Homogenization of Composite Materials and Structures. Applied Mechanics Reviews, 2009, v. 63, no. 3, pp. 1—20.
- Sheshenin S.V. Asimptoticheskiy analiz periodicheskikh v plane plastin [Asymptotical Analysis of In-plane Periodical Plates]. Izvestiya RAN. Mekhanika tverdogo tela [RAS News. Mechanics of Solids.], 2006, no. 6, pp. 71—79.
- Sheshenin S.V. Primenenie metoda osredneniya k plastinam, periodicheskim v plane [Application of the Homogenization Method for the In-Plane Periodical Plates]. Vestnik Moskovskogo universiteta. Matematika. Mekhanika [Proceedings of Moscow University. Mathematics. Mechanics]. 2006, no. 1, pp. 47—51.