application of the homogenization method to the elastoplastic bending of a plate

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.