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

In this paper, a new kind of hierarchical tube with a negative Poisson’s ratio (NPR) is proposed. The first level tube is constructed by rolling up an auxetic hexagonal honeycomb. Then, the second level tube is produced by substituting the arm of the auxetic sheet with the first level tube and rolling it up. The Nth (N≥1) level tube can be built recursively. Based on the Euler beam theory, the equivalent elastic parameters of the NPR hierarchical tubes under small deformations are derived. Under longitudinal axial tension, instead of shrinking, all levels of the NPR hierarchical tubes expand in the transverse direction. Using these kinds of auxetic tubes as reinforced fibers in composite materials would result in a higher resistance to fiber pullout. Thus, this paper provides a new strategy for the design of fiber reinforced hierarchical bio-inspired composites with a superior pull-out mechanism, strength and toughness. An application with super carbon nanotubes concludes the paper.


Introduction
In the last few years, due to their special mechanical and electronic properties, hierarchical covalent two-dimensional (2D) and three-dimensional (3D) networks based on one-dimensional (1D) nanostructures have attracted much research attention. One relevant example is carbon nanotube (CNT) networks, in which carbon nanotubes are covalently connected through different nanojunctions, such as X-, Y-, T-shape [1][2][3][4][5][6], even hierarchically [7][8][9][10][11]. Coluci et al. [12] proposed self-similar hierarchical super carbon nanotubes (STs) and showed that they are stable and could present metallic or semiconducting behavior. Then, through fractal and fracture mechanics, Pugno [13] evaluated the strength, toughness and stiffness of the STs-reinforced composites and revealed that the optimized hierarchical tubes under small deformations are calculated. Such auxetic hierarchical fibers are ideal to increase the pull-out resistance and, thus, the toughness of bio-inspired composites. Figure 1 shows the scheme of a N-level ( 1 N  ) hierarchical tube with a negative Poisson's ratio. The N-level ( 1 N  ) hierarchical tube is fabricated through iterating N times the process of rolling a NPR sheet to a tube along the x-axis, see Figures 1 and 2a.

Design of Hierarchical NPR Tubes
At first, based on the considered 1D nanostructure (e.g., a solid nanorod or a thin hollow cylinder, such as CNT), the first level NPR sheet that mimics the NPR hexagonal honeycomb is constructed. Then, rolling up the first level NPR sheet gives the first level NPR tube. The second level NPR tube is constructed by substituting the arm tube of the first level NPR sheet with the first level NPR tube and then rolling it up. Iteratively, repeating the above process N times, we can build the Nth level tube. A representative junction of the ith (1 i N   ) level NPR sheet or tube is shown in Figure 2b, in which  is the length of the arms and −30° < θ (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.  (1 i N   ) level NPR tube made by rolling the NPR sheet; (b) the force diagram of a representative junction of the ith ( 1 i N   ) level NPR sheet or tube subject to the y-axis tension.

Elasticity of the Hierarchical NPR Tubes
Based on the Euler beam theory, Wang et al. [17] derived the equivalent elastic parameters of the STs (positive Poisson's ratio) from that of the arm tubes and verified the results through finite element simulations. The Young's modulus E was substituted with the parameter E to describe the equivalent modulus of the CNT and STs, in which  is the thickness to diameter ratio of these thin hollow cylinder tubes. Similarly, in the following, we analytically study the elastic properties of the hierarchical NPR tubes shown in Figure 2a.

The Level 1 NPR Tube
We start by analyzing the level 1 hierarchical NPR sheet and tube under uniaxial tension p in the direction y, in which the fundamental unit (level 0) is a solid nanorod or a thin hollow cylinder, such as a CNT (Figure 2a). From the force diagram of the representative junction shown in Figure 2b it is evident from the structure periodicity that: in which   1 M is the bending moment and   0 l is the arm length.
If the fundamental unit (level 0) is a solid nanorod, we have: I are its diameter, cross section area and inertia moment. If the fundamental unit (level 0) is a thin hollow cylinder, such as a CNT, denoting its equivalent thickness as   0 t , we have [17]: I are its equivalent diameter, cross section area, inertia moment and is the thickness-to-diameter ratio.
In order to study the level 1 NPR sheet and tube, we analyze the first level representative junction ( Figure 2b). Its lengths along the x-axis and y-axis are elongations along the two directions can be obtained through structural analysis; we find: in which   0 E is the Young's modulus of the fundamental unit (level 0). Then, the equivalent strains along the two directions can easily be calculated as: for the solid nanorod 3 sin 3 2 for the thin hollow cylinder 6 1 1 2sin  24  1 sin   1 1 2sin  2 3  cos  4 for the solid nanorod 1 sin 1 1 2sin 1 3 cos for the thin hollow cy 1 sin is the aspect ratio of the level 0 arms. Thus, the equivalent Poisson's ratio   1  is calculated as: for the solid nanorod 6 1 1 2sin 2 3 cos 1 1 s i n sin 2 3 for the thin hollow cylinder 6 1 1 2sin 1 3 cos The level 1 NPR sheet with size n times along the y-axis (Figure 2a). We treat it as a plate with equivalent thickness (1) t , Young's modulus   1 E and Poisson's ratio   1  . By rolling the level 1 NPR sheet in the direction y along the longitudinal axis, the level 1 NPR tube is thus obtained. From the equivalence between the circumference of the level 1 NPR tube and the width   1 x L of the level 1 NPR sheet, it is easy to calculate the equivalent diameter (1) d of the level 1 NPR tube: Then, the slenderness ratio (1)  and the thickness-to-diameter ratio (1)  become: 1 sin 2 cos where (1) l is the length of the level 1 NPR tube. Except for the NPR tubes with very small diameters, the slight change of angles between the arms due to rolling can be ignored. Thus, the results obtained for the level 1 NPR sheet are easily extended to the level 1 NPR tube. Accordingly, the total deformation   1 y L  along the length direction of the level Thus, the tensional rigidity   1 y k of the level 1NPR tube is: Then, the axial rigidity of the level 1 NPR tube can be obtained as: That is to say: 1 sin for the solid nanorod 1 1 2sin 2 3 cos 1 sin for the thin hollow cylinder 1 1 2sin 1 3 cos  is the equivalent cross section area of the level 1 NPR tube.
The bending rigidity of the level 1 NPR tube can be expressed as [17]: Or, if the fundamental unit (level 0) is the thin hollow cylinder, we have:

The Level N NPR Tube
The equivalent elastic parameters of any level N ( 1 N  ) NPR tube can be recursively derived by repeating the analysis reported in Section 2.2.1.
About the Poisson's ratio of the level N tube, similar to Equation (7), if the fundamental unit (level 0) is the solid nanorod:  18) or if the fundamental unit (level 0) is the thin hollow cylinder: is the slenderness ratio of the arms of the Nth level NPR tube. With respect to the axial rigidity     N N E A of the level N NPR tube, if the fundamental unit (level 0) is the solid nanorod, from Equation (14) it is easy to obtain that: is the slenderness ratio of the arms of the ith level NPR tube. Similar to Equation (15), the bending rigidity     N N E I of the level N NPR tube is: is the equivalent diameter of the level N NPR tube.
It is also easy to get that, if the fundamental unit (level 0) is the solid nanorod:  Finally, for the effect of the hierarchical level N on the axial rigidity     N N E A , the following parameters are considered: The related results are displayed in Figure 5. We can see that axial rigidity    

Conclusions
A new kind of hierarchical tube with a negative Poisson's ratio is proposed in this paper. The equivalent elastic properties of the NPR hierarchical tubes under small deformations are derived through the Euler beam theory. The results show that both the angles between the arms and the slenderness ratio of the arms have great influences on the equivalent modulus, axial rigidity and Poisson's ratio of the hierarchical NPR tube and can thus be tuned to match the requirements of a specific application. Under longitudinal axial tension, all levels of the negative Poisson's ratio hierarchical tubes will expand in the transverse directions rather than shrink. Using these NPR tubes as reinforced fibers in composite materials can result in a higher resistance to fiber pullout and, thus, provides new strategies for the design of bio-inspired fiber reinforced composites with superior toughness. It should be noted that the theory in this paper is limited to the hierarchical NPR tubes with slender arms in small deformations.