# Hierarchical Fibers with a Negative Poisson’s Ratio for Tougher Composites

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Hierarchical Structures with Negative Poisson’s Ratio

#### 2.1. Design of Hierarchical NPR Tubes

^{(i)}< 0°. Similar to the fabrication of hierarchically branched nanotubes [50] and STs [13], that of hierarchical NPR fibers could be realized in the future.

**Figure 2.**(

**a**) Schematics of the ith ($1\le i\le N$) level NPR sheet and the corresponding ith ($1\le i\le N$) level NPR tube made by rolling the NPR sheet; (

**b**) the force diagram of a representative junction of the ith ($1\le i\le N$) level NPR sheet or tube subject to the y-axis tension.

#### 2.2. Elasticity of the Hierarchical NPR Tubes

#### 2.2.1. The Level 1 NPR Tube

#### 2.2.2. The Level N NPR Tube

#### 2.2.3. Effects of the Parameters ${\theta}^{\left(N\right)}$, ${\alpha}^{\left(N-1\right)}$ and N

**Figure 3.**Schematics of (

**a**) ${E}^{\left(1\right)}{\beta}^{\left(1\right)}$ vs. ${\alpha}^{\left(0\right)}$ and (

**b**) ${\nu}^{\left(1\right)}$ vs. ${\alpha}^{\left(0\right)}$.

**Figure 4.**Schematics of (

**a**) ${E}^{\left(1\right)}{\beta}^{\left(1\right)}$ vs. ${\theta}^{\left(1\right)}$ and (

**b**) ${\nu}^{\left(1\right)}$ vs. ${\theta}^{\left(1\right)}$.

**Figure 5.**Schematics of the axial rigidity ${E}^{\left(1\right)}{A}^{\left(1\right)}$ vs. the hierarchical level N.

## 3. Conclusions

## Acknowledgements

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Sun, Y.; Pugno, N.
Hierarchical Fibers with a Negative Poisson’s Ratio for Tougher Composites. *Materials* **2013**, *6*, 699-712.
https://doi.org/10.3390/ma6020699

**AMA Style**

Sun Y, Pugno N.
Hierarchical Fibers with a Negative Poisson’s Ratio for Tougher Composites. *Materials*. 2013; 6(2):699-712.
https://doi.org/10.3390/ma6020699

**Chicago/Turabian Style**

Sun, Yongtao, and Nicola Pugno.
2013. "Hierarchical Fibers with a Negative Poisson’s Ratio for Tougher Composites" *Materials* 6, no. 2: 699-712.
https://doi.org/10.3390/ma6020699