Dinâmica efetiva de barras e vigas de Euler-Bernoulli microperiódicas

Abstract

The asymptotic homogenization method (AHM) allows finding the effective behavior of a periodic micro-heterogeneous medium, characterized by rapid variations in its constitutive properties, by means of a homogeneous medium, which is asymptotically equivalent to the micro-heterogeneous medium. Mathematically, this translates into a powerful tool for finding solutions to differential equations that model physical behavior in a micro-heterogeneous medium. The fundamental steps of AHM include the construction of a formal asymptotic solution of the original problem, finding the problem on the equivalent homogeneous medium and its effective property and finding the problems and solutions of local functions, defined in a portion that is repeats periodically in the middle, called the periodicity cell. To demonstrate the equivalence between these means, we prove the proximity between the solutions on the micro-heterogeneous environment (original problem) and the homogeneous environment (homogenized problem) using some corresponding standard. The present work proposes the application of this method to periodic micro-heterogeneous media for finding higher-order asymptotic solutions to one-dimensional dynamic and static problems of second and fourth order, which model the mechanical behavior of Euler-Bernoulli type bars and beams. In addition, to present numerical examples of the problems addressed.