A Unified Approach for the Finite Element Analysis of Auxetic Materials with Periodic and Multifaceted Symmetric Microstructures

Advanced computational material modeling tools enhanced with artificial intelligence algorithms and manufacturing approaches such as additive manufacturing have resulted in the successful development of many novel metamaterials such as materials with negative Poisson’s ratio (auxetic) or negative coefficient of thermal expansion, which are very promising to be used in extreme environments such as high strain-rate impact and abrupt temperature changes. In the design analysis and optimization of metamaterials, computational homogenization approaches based on the finite element method are commonly employed to predict the macro effective thermoelastic properties using unit cells exhibiting periodic topologies and multifaceted symmetries depending on the microstructures. In the micromechanical analysis of the unit cells, appropriate periodic boundary conditions should be specified as multiple point constraints and symmetric conditions are exploited to reduce the size of the problem and computation time. However, in previous analyses of metamaterials with multifaceted symmetries, mostly a full unit cell or a quarter cell is employed which accounts for the reflection symmetry only. In this study, an approach for the finite element analysis of unit cells with multifaceted symmetries is proposed and illustrated. For a novel auxetic configuration, boundary conditions accounting for the mirror reflection, 0° rotation, 180° rotation, and skew symmetries are determined and the size of the unit cell is reduced up to 1/16th of the full size. Moreover, analysis approaches for both normal and shear loadings are illustrated and the results are validated by comparing with available numerical and test results. Finally, using the proposed approach, an analysis is conducted for a hybrid microstructure manifesting a negative coefficient of thermal expansion. It is demonstrated that a 1/8th model is suffice for the thermal stress analysis under temperature changes.