Homogenization of nonlinear problems in the mechanics of composites

A further development of the homogenization method is proposed to solve the physically nonlinear equilibrium problems for the laminated plates or the plates made of functionally graded materials. In the linear case, according to this method, the corresponding solution is a superposition of the solution to the global problem in the entire domain and the solution to the local problem in a representative domain, e.g., in a periodicity cell. In the nonlinear case, such a superposition is not valid, which complicates the application of the homogenization method. In order to eliminate this difficulty, it is possible to combine the homogenization method and the linearization method when solving a boundary value problem or a variational problem. In the mechanics of deformable solids, the constitutive relations can be considered as equations with respect to velocities or the stress and strain differentials in time or in the loading parameter. When these equations are linear with respect to velocities, it is possible to use the homogenization method. In this paper such an approach is illustrated by the example of a symmetric laminated plate bent under a uniformly distributed time-dependent load.