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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.

The authors present a method of homogenization used to solve nonlinear equilibrium problems
of laminated plates exposed to transversal loads.
The homogenization technique is a general and mathematically rigorous solution to elasticity
problems. It describes the processes of deformation of composite structural elements. It was originally
developed for linear problems. This method encompasses the calculation of all characteristics
related to deflection by combining solutions to local and global homogenization problems. Thus, it
implements the general idea of the domain decomposition into subdomains.
The homogenization method has been most widely used in cases of periodical heterogeneity
because of significant simplification that happens due to periodicity. This simplification implies that
any cell of periodicity appears to be the material representative volume element (RVE). Therefore,
it is sufficient to solve local problems within a single periodicity cell. Hence, with reference to local
problems, conditions of periodicity are a mere consequence of the periodicity of the material
structure. Decomposition of the domain causes decomposition of the solution. The latter means that
displacements, stresses and strains are represented by functions that depend on both global and local
coordinates. Global coordinates are associated with the whole body scale and local coordinates
vary in the periodicity cell, i.e. in RVE only.
If the material structure is not periodic, but its properties do not depend on global coordinates,
material effective properties can be determined by solving local problems in any RVE. That is not the ase of nonlinear materials. Now local problems have to be solved in every RVE because of the homogenized
properties dependence on global coordinates. Another complication arises due to nonlinearity.
Indeed, the homogenization method employs the superposition principle to represent the solution to the
elasticity problem as summarized solutions to global and local problems. This principle doesn't work in
the case of nonlinearity. We suggest combining the standard homogenization technique with linearization
by using the loading history to solve the nonlinear problem. On the contrary, local linear problems
have to be solved in every RVE. Certainly, this method involves numerous calculations.
As for the problem considered in the paper, its nonlinearity is caused by material plastic properties.
Most plasticity-related principles are formulated as tensorial linear relationships between the
stress and strain rates. Hence, here we identify a perfect opportunity to employ the homogenization
method combined with linearization with regard to the load parameter. This combined technique is
implemented to resolve the heterogeneous plate bending problem. Heterogeneous materials are of
the two types: laminates and functionally graded materials (FGM).
The computer code is developed for the purpose of numerical plate bending simulation. It employs
the parallel programming MPI technique and the Euler type explicit and implicit methods. For
example, laminated plate bending due to the distributed transversal load was the subject of research.
Each layer of the plate was composed of FGM or a homogeneous material. The authors have discovered
that FGM plates have a higher yield stress then the plates composed of homogeneous layers.