Two-Dimensional Mechanical Metamaterials with Unusual Poisson Ratio Behavior

We design two-dimensional mechanical metamaterials that may be deformed substantially at little or no energy cost. Examples of such deformable structures include assemblies of rigid isosceles triangles hinged at their corners on the macroscale and polymerized phenanthrene molecules forming porous graphene on the nanoscale. In these and in a large class of related structures, the Poisson ratio $\nu$ diverges for particular strain values. It also changes its magnitude and sign, and displays a shape-memory effect.

Figure 1

Deformations in a 2D assembly of rigid isosceles triangles. (a) Adjacent triangles with opening angle $\alpha$ and mutual orientation defined by the closing angle $\beta$, hinged as tip to corner, forming the primitive unit cell. The height $x_0$ of the triangle and the length $y_0$ of its base define the horizontal and vertical length scales. (b) Snapshots of the assembly of $\alpha ={120}^{\circ }$ triangles for different values of $\beta$. The conventional rectangular unit cell is twice the size of the primitive unit cell. (c) Contour plot of the Poisson ratio ${\nu }_{xy}=-(dy/y)/(dx/x)$ as a function of $\alpha$ and $\beta$. The dotted red line highlights the behavior of the assembly of $\alpha ={120}^{\circ }$ triangles. (d) Poisson ratio ${\nu }_{xy}$ as a function of $\beta$ in the $\alpha ={120}^{\circ }$ system. (e) Changes in the scaled width $x/x_0$ and height $y/y_0$ of the conventional unit cell for $\alpha ={120}^{\circ }$ caused by change of the angle $\beta$.

Figure 2

Deformations in porous graphene, a phenanthrene-based 2D mechanical metamaterial. (a) Structure of the ${\mathrm{C}}_{14}{\mathrm{H}}_{10}$ phenanthrene molecule and its relation to the $\alpha ={120}^{\circ }$ isosceles triangle in Fig. 1. (b) Equilibrium structure of 2D porous graphene consisting of polymerized phenanthrene molecules with $\beta ={70}^{\circ }$. Saturating hydrogen atoms are shown by the lighter and smaller spheres. Changes in the scaled width $x/x_0$ (c) and height $y/y_0$ (d) of the conventional unit cell in the assembly of triangles and porous graphene as a function of the closing angle $\beta$. (e) Poisson ratio ${\nu }_{xy}$ in porous graphene as a function of $\beta$. (f) Strain energy in the ${\mathrm{C}}_{56}{\mathrm{H}}_{28}$ conventional unit cell as a function of $\beta$. The dashed and dotted lines connecting data points for porous graphene in (c)–(f) are guides for the eye.

Figure 3

Electronic structure of porous graphene, a phenanthrene-based 2D mechanical metamaterial, based on DFT PBE calculations. (a) Band structure of the equilibrium structure with $\beta ={70}^{\circ }$ obtained with the rectangular ${\mathrm{C}}_{56}{\mathrm{H}}_{28}$ unit cell. High-symmetry points in the rectangular Brillouin zone are shown in the inset. (b) Fundamental band gap $E_g$ as a function of the angle $\beta$. DOS, density of states.

Figure 4

Changes in the scaled width $x/x_0$ and height $y/y_0$ of the conventional unit cell for different values of the opening angle $\alpha$ as a function of the closing angle $\beta$. The relevant quantities are defined in Fig. 1.

Figure 5

Deformations in a 2D assembly of rigid equilateral triangles. (a) Adjacent triangles with mutual orientation defined by the closing angle $\beta$, hinged at the corners, forming the primitive unit cell. The triangle height $x_0$ and the length $y_0$ of its base define the horizontal and vertical length scales. (b) Snap shots of the assembly of triangles for different values of $\beta$. The conventional unit cells of width $x$ and height $y$ are indicated.