Homogeneização matemática de meios unidimensionais microperiódicos lineares e não lineares

Abstract

The Asymptotic Homogenization Method (AHM) has been presented as a good tool for solving problems in media with rapidly oscillating properties, such as micro periodic media, because in it the original material is took as physically equivalent to a certain homogeneous material (before hypotheses about its micro structure), whose problems are easier to solve. In this work, AHM is applied in one-dimensional, linear and non-linear micro periodic media. First of all, was took the continuously differentiable properties media, with the objective of approaching the asymptotic ones to an exact solution of the original problem, which reproduced their local behaviors. For both linearity and non-linearity cases, the asymptotic approximations (first and second order) was presented as good approximations, it was observed that in some cases were determined with the second-order term for a more faithful reproduction of the local details of the solution accurate. Afterwards, was took the constant in parts media, that is a composite material’s representant, considering the cases of perfect and imperfect contact between the phases, in order to evaluate the effective flow, another application of this method. The AHM its proved like effective for this determination, for the different types of compounds (including linear and non-linear ones). At the end, in the context of composites, the non-linear Volokh model was used for the phases of the material to model its global behavior. This model presents more realistic results compared to the one used previously. It was observed that this situation, the composite presented an effective behaviour, in a qualitative way, also followed the Volokh model, both in conditions of perfect contact and imperfect. The final result is that AHM is a useful measure in modeling the behavior of micro periodic media in several situations, which were illustrated in this work, and that the more situations should be used in models to consolidate the results obtained here.