Solving metamaterials-based structures via second order homogenization

Abstract

This bachelor’s thesis presents the development and implementation of a Second-Order Homogenization (SOH) model for predicting the mechanical behavior of metamaterials, which are engineered materials with exotic properties not found in nature, such as a negative Poisson’s ratio, due to their microstructural architecture. The main objective of the work was to implement this model within an existing finite element program developed in MATLAB. Originally designed for first-order homogenization, the environment was upgraded to incorporate second-order theory, allowing for the reproduction of higher-order effects that could not previously be captured, especially in the presence of complex geometries. To achieve this, an extensive theoretical review was conducted, covering metamaterials and their applications, as well as various numerical methods used to analyze structures composed of such materials. The theoretical foundations of both first-order and second-order homogenization were then studied in depth. After that, the second-order model was implemented in an existing MATLAB code, originally developed for first-order analysis. Results for three different geometries were computed and visualized using ParaView, producing fluctuation fields, macroscopic, and total displacement fields. Some challenges were encountered in visualizing macroscopic and total displacement fields, which could be due to limitations in post-processing. Nevertheless, a strong agreement between theoretical predictions and simulation results was observed, confirming the validity of the approach. In conclusion, this project contributes to the growing field of multiscale modeling of metamaterials and lays the foundation for future development of more advanced structural analysis tools based on SOH, which offers significant advantages over traditional modeling techniques such as Direct Numerical Simulation (DNS).