Asymptotic homogenization of a mechanical equilibrium problem of a functionally-graded Euler-Bernoulli beam with non-periodic microstructure

Resumo

To the best of our knowledge, the few classical applications of Keller's two-space method of non-periodic asymptotic homogenization are related to the effective behavior of heterogeneous media in the context of poroelasticity considering fluid flow and saturation. We believe that this is due to the alternative common approach of approximating random or non-periodic microstructures via the periodic replication of a representative volume element, as periodic structures are, generally speaking, much more tractable mathematically and computationally. However, more than 40 years later, a number of preliminary results of applications of the two-space method has arisen on various areas, namely, effective behavior of composite or functionally-graded bars, approximate solution of the electroencephalogram forward problem for neural imaging activity, and modeling of atmospheric pollutant dispersion. These recent applications deal with second-order elliptic or parabolic equations. In this contribution, we present the application of the two-space method to a mechanical equilibrium problem of a functionally-graded Euler-Bernoulli beam with non-periodic microstructure, which relies on a fourth-order elliptic equation. To the best of our knowledge, homogenization of fourth-order equations has been considered only in the periodic case.