This paper is devoted to the exploration of rectangular finite elements' ability to model the stress-strain state of isotropic and orthotropic materials with a negative Poisson's ratio, known as auxetic materials. By employing linear elasticity in the plane stress formulation, the research evaluates the linear compatible and the quadratic incompatible shape functions in describing the mechanical behavior of auxetic materials within a displacement-based finite element method under static shear conditions. Additionally, the analytical expression of an incompatible rectangular finite element is adapted to accommodate an orthotropic case. Hexachiral and re-entrant honeycomb structures, characterized by 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 be sufficient for displacement modeling, they are ineffective in accurately predicting the characteristic auxetic behavior and stress distributions in auxetic materials. In contrast, the incompatible shape functions prove to be effective in providing appropriate stress modeling in both cases. This work underscores the relevance of the incompatible rectangular finite elements in the analysis of advanced materials with a negative Poisson's ratio. It provides computationally efficient approaches for the calculation of auxetic honeycomb structures and multilayer composites based on them.

翻译：本文致力于探索矩形有限元在模拟具有负泊松比的各向同性及正交各向异性材料（即拉胀材料）应力-应变状态的能力。研究基于平面应力公式的线弹性理论，在静态剪切条件下通过位移基有限元法，评估了线性协调形函数与二次非协调形函数在描述拉胀材料力学行为中的表现。此外，研究还调整了非协调矩形有限元的解析表达式以适应正交各向异性情形。通过有效材料常数的解析表达式，将具有拉胀特性的六手性结构和凹角蜂窝结构建模为具有均匀化特性的连续介质。研究结果表明：经典形函数虽可用于位移建模，但无法准确预测拉胀材料的特征性拉胀行为与应力分布；相比之下，非协调形函数在两种情况下均能实现有效的应力建模。本工作凸显了非协调矩形有限元在分析具有负泊松比的先进材料中的适用性，为基于拉胀蜂窝结构和多层复合材料的计算提供了高效的计算方法。