Lattice-Based Continuum Modeling for the Analysis of Geometrically Nonlinear Structures

Resumo

This work proposes a homogenization approach for two-dimensional solids based on periodic cells formed by trusses exhibiting geometric nonlinear behavior. The Finite Element Method with a geometrically nonlinear positional formulation is employed, adopting the Saint Venant–Kirchhoff constitutive law to describe the material behavior. The kinematics of the unit cells are defined by the positions of the nodes, allowing for the capture of large deformations and rotations. The periodic distribution of unit cells enables an equivalent description of the continuous medium's properties, facilitating the efficient modeling of metamaterials with complex mechanical behavior. The influence of the scale factor, corresponding to the size of the unit cells, is analyzed, and model fitting techniques are presented to ensure the accuracy of the homogenization. The nonlinear problem is solved using the Newton-Raphson method with force and displacement control. Numerical examples are presented to demonstrate the applicability and effectiveness of the proposed model, including comparisons with established analytical solutions and literature data. The results highlight the potential of the approach for modeling two-dimensional metamaterials with tunable mechanical properties.