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.

翻译：本研究探讨了在线弹性范围内，应用矩形有限元对具有负泊松比（即拉胀材料）的各向同性及正交各向异性材料在静态剪切条件下的应力-应变行为进行建模。研究在基于位移的有限元框架内，采用经典的协调形函数进行线性插值，以及非协调形函数进行二次插值，评估了这些方法在捕捉拉胀材料力学响应方面的有效性。此外，本文提出了适用于正交各向异性材料的非协调矩形有限元的解析表达式。研究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内凹蜂窝结构建模为具有均匀化属性的连续介质。结果表明，经典形函数虽足以进行位移建模，但在准确预测拉胀材料的应力分布方面存在不足。相比之下，非协调形函数在提供合理的应力和位移预测方面表现出更优的性能。本工作强调了采用非协调矩形有限元对具有负泊松比的先进材料进行建模的重要性，并为计算拉胀蜂窝结构及其衍生的多层复合材料提供了计算高效的方法。