Rectangular finite elements for modeling the mechanical behavior of auxetic materials

This study explores the application of rectangular finite elements to model the stress-strain behavior of isotropic and orthotropic materials exhibiting negative Poisson's ratio, known as auxetic materials, under static shear conditions within linear elasticity. By employing the classical compatible shape functions for linear interpolation and the incompatible shape functions for quadratic interpolation within a displacement-based finite element framework, the research assesses the effectiveness of these approaches in capturing the mechanical response of auxetic materials. Additionally, the analytical expression for an incompatible rectangular finite element applicable to orthotropic materials is proposed. Hexachiral and re-entrant honeycomb structures, known for their auxetic behavior, are modeled as continuous media with homogenized properties using analytical expressions for their effective material constants. The findings reveal that while the classical shape functions may suffice for displacement modeling, they fall short in accurately predicting stress distributions in auxetic materials. In contrast, the incompatible shape functions demonstrate superior performance in providing appropriate stress and displacement predictions. This work underscores the relevance of using the incompatible rectangular finite elements in the modeling of advanced materials with a negative Poisson's ratio. It provides computationally efficient approaches for calculating auxetic honeycomb structures and their derived multilayer composites.