# Metamaterials with Poisson’s ratio sign toggling by means of microstructural duality

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## Abstract

Neither positive nor negative Poisson’s ratio materials are fully advantageous across a wide range of load bearing applications. In some cases, it is advantageous for structures to possess duality in material properties so as to take advantage of auxetic behavior under one loading condition as well as conventional behavior when the loading condition changes. This paper proposes two metamaterials—the hybrid rhombic–re-entrant metamaterial and the hybrid kite-arrowhead metamaterial—in which the prescription of either compressive or tensile load always leads to lateral expansion. Specifically, the Poisson’s ratio of these metamaterials range from negative infinity to zero upon tensile load, and from zero to positive infinity upon compressive load. This duality of auxetic and conventional behavior for each of these metamaterial is achieved by employing a simultaneous lock and slide mechanism that is interchangeable upon load reversal. When loaded in one direction, the locked part takes effect while the sliding part becomes redundant. Upon load direction reversal, the previously locked and sliding parts convert to sliding and locked parts, respectively, thereby activating the redundant part while deactivating the functional part. The change in effective microstructural components, therefore, facilitates Poisson’s ratio sign toggling with load reversal. Due to the indiscriminate lateral expansion regardless of the axial loading direction, such behavior in a fiber is useful to resist both fiber pull-out as well as fiber push-out from the matrix material.

## Keywords

Auxetic Conventional Metamaterials Microstructural duality## 1 Introduction

## 2 Analysis

### 2.1 Preamble

### 2.2 Hybrid rhombic–re-entrant metamaterial

The analysis of compression in x-direction for the hybrid rhombic–re-entrant metamaterial can be established by comparing Fig. 6b against Fig. 6a, in which B is prevented from moving to the left due to a symmetrically opposing vertex; overall contraction in x-direction is therefore made possible with the sliding of vertices A and C in the slots towards O and B, respectively, to their new locations A′ and C′, with the vertex B shifting upward to B′. This causes the inclined rod of length \(l\) to rotate clockwise by an amount \(d\theta\) such that the projections \(y_{0}\) elongates to \(y\) while \(x_{0}\) shortens to \(x\).

### 2.3 Hybrid kite-arrowhead metamaterial

The analysis of compression in x-direction for the hybrid kite-arrowhead metamaterial can be established by contrasting Fig. 7b with reference to Fig. 7a, whereby the length \(l_{3}\) remains constant as A and C displace to A′ and C′, respectively. During this time, OA rotates anticlockwise to OA′ by an angle \(d\theta_{1}\) while AC rotates clockwise to A′C′ by \(d\theta_{3}\). Therefore, the compression analysis considers the movement of linkage OAC while ABC is redundant.

## 3 Results and discussion

### 3.1 Experimental

### 3.2 Hybrid rhombic–re-entrant metamaterial

### 3.3 Hybrid kite-arrowhead metamaterial

## 4 Applications

For the purposes of wrapping a flat sheet onto a curved surface, a positive, a zero or a negative Poisson’s ratio material is advised if the surface takes the form of an anticlastic shape, a cylindrical shape or a synclastic shape in order to reduce bending stress in the sheet material. This line of reasoning implies that the choice of Poisson’s ratio sign is dependent on the application. Likewise, the choice of auxetic fiber is useful to resist fiber pull-out from the matrix material due to the self-locking mechanism in the form of radial expansion during axial pulling; however, auxetic fibers are easily pushed out from matrix material due to radial contraction. In fact, it is the conventional fibers that resist push-out due to radial expansion, although it is also known that conventional fibers are easily pulled out due to the resulting radial contraction. If the fiber behaves as auxetic material during fiber pull-out (\(\sigma_{z} > 0 \Rightarrow v_{zr} < 0\)) but becomes conventional material during fiber push-out (\(\sigma_{z} < 0 \Rightarrow v_{zr} > 0\)), then such a fiber resists both pull-out and push-out as a result of its duality.

## 5 Conclusions

- $$\sigma_{x} < 0 \Rightarrow v_{xy} > 0$$
- $$\sigma_{x} > 0 \Rightarrow v_{xy} < 0$$

## Notes

### Compliance with ethical standards

### Conflict of interest

The corresponding author states that there is no conflict of interest.

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