Aplicações do método de homogeinização assintótica e a técnicas de camadas fictícias em formulações variacionais zig-zag de alta ordem para vigas compósitas laminadas

Abstract

In the field of materials mechanics, many structural problems exhibit periodic characteristics or can be approximated as such, related to the variation of elastic modules due to the heterogeneity of the internal structure, external forces on the body and/or geometry, and which may also occur over time in a manner that is not necessarily periodic (combined effects of seasonality, temporal variation of external forces and/or, for example, fatigue or stiffening/softening due to the evolution of the internal structure). The more repetitive and complex the structure, the more challenging it becomes to solve the problem, as classical numerical techniques require modeling in two or three dimensions and very fine discretization of the domain, resulting in high computational costs. Additionally, it is observed in nature that most materials are heterogeneous at some scale, and their properties may vary randomly or in a standardized manner. Examples include concrete, whose behavior varies with its components and proportions; wood, with its anisotropic characteristics; and composite materials reinforced with fibers or particles, which are essential for ensuring the mechanical efficiency of structures. To avoid direct modeling of problems with rapidly oscillatory coefficients, mathematical homogenization methods are essential, as they aim to transform these coefficients into homogeneous equivalents. This work proposes a twodimensional modeling of heterogeneous materials beams by concurrently utilizing high-order zig-zag multilayer theories and the one-dimensional asymptotic homogenization method, allowing the description of periodic or random materials through fictitious layer techniques. Based on problems resolved using experimental results from the literature, the results demonstrate that the methodology is highly efficient, as it maintains good accuracy in twodimensional models even when using one-dimensional functions for composite, laminated and periodic materials problems.