Analytical relationships for re-entrant honeycombs

Abstract

Mechanical metamaterials have emerged in the last few years as a new type of artificial material which show properties not usually found in nature. Such unprecedented properties include negative stiffness, negative Poisson’s ratio, negative compressibility and fluid-like behaviors. Unlike normal materials, materials with negative Poisson’s ratio (NPR), also known as auxetics, shrink laterally when a compressive load is applied to them. The 2D re-entrant honeycombs are the most prevalent auxetic structures and many studies have been dedicated to study their stiffness, large deformation behavior, and shear properties. Analytical solutions provide inexpensive and quick means to predict the behavior of 2D re-entrant structures. There have been several studies in the literature dedicated to deriving analytical relationships for hexagonal honeycomb structures where the internal angle θ is positive (i.e. when the structure has positive Poisson’s ratio). It is usually assumed that such solutions also work for corresponding re-entrant unit cells. The goal of this study was to find out whether or not the analytical relationships obtained in the literature for θ>0 are also applicable to 2D-reentrant structures (i.e. when θ<0). Therefore, this study focused on unit cells with a wide range of internal angles from very negative to very positive values. For this aim, new analytical relationships were obtained for hexagonal honeycombs with possible negativity in the internal angle θ in mind. Numerical analyses based on finite element (FE) method were also implemented to validate and evaluate the analytical solutions. The results showed that, as compared to analytical formulas presented in the literature, the analytical solutions derived in this work give the most accurate results for elastic modulus, Poisson’s ratio, and yield stress. Moreover, some of the formulas for yield stress available in the literature fail to be valid for negative ranges of internal angle (i.e. for auxetics). However, the yield stress results of the current study demonstrated good overlapping with numerical results in both the negative and positive domains of θ.