Rectangular finite elements for modeling the mechanical behavior of auxetic materials

TL;DR

This study evaluates the effectiveness of rectangular finite elements, especially incompatible shape functions, in modeling the stress-strain behavior of auxetic materials like honeycomb structures, highlighting their advantages over classical functions.

Contribution

It introduces and assesses incompatible rectangular finite elements for accurately modeling auxetic materials with negative Poisson's ratio in static conditions.

Findings

Incompatible shape functions accurately predict auxetic behavior.

Classical shape functions are insufficient for stress distribution in auxetics.

The method offers efficient computation for auxetic honeycomb structures.

Abstract

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 and indentation. 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.