Continuity and discontinuity of kirigami's high-extensibility transition: A statistical-physics viewpoint

Recently, kirigami's high extensibility has been understood as a transition in the force-elongation curve. In this Rapid Communication, we consider a model, which modifies our previous model, to show a striking analogy between the present theory and the Landau theory of continuous thermodynamic transitions, if we regard a rotation angle and elongation of kirigami as the order parameter and the inverse temperature, respectively. The present study points out the importance of the distinction between the discontinuity and continuity of the high-extensibility transition in an elementary kirigami structure, and shows that the mechanical response of kirigami can be understood using the tools of statistical physics, which have been proved to be useful in many fields of physics.

Figure 1

(a) Geometry of a simplified kirigami specimen. (b), (c) Two typical examples of the force $F$ as a function of the elongation $\mathrm{\Delta }$ in the vicinity of the transition point, one exhibiting an abrupt (discontinuous) transition and the other exhibiting a continuous transition. The former is obtained from a ridged polystyrene plate of Young's modulus $E=3.1$ GPa ($h=0.2$ mm, $w=40$ mm, $d=2$ mm), while the latter from a soft elastomer sheet of $E=7.9$ MPa ($h=1$ mm, $w=30$ mm, $d=5$ mm).

Figure 2

Kirigami's geometry under a given elongation $\delta =\mathrm{\Delta }/\left(2N\right)$. In the figure, the vectors $\widehat{x}, \widehat{y}, \widehat{t},$ and $\widehat{n}$ are unit vectors, defined in the text, and used to show the plane on which the corresponding illustration is drawn. (a) Front view of an extended kirigami. (b) Side view. (c) Magnified side view. (d) The unit of kirigami containing the points $A, A^{\prime }, C$, and $C^{\prime }$ seen from the direction $\widehat{n}$. (e) The same unit seen from the direction of $\widehat{t}$.

Figure 3

Free energy $\tilde{U}$ as a function of $\theta$ for various $\delta$ at $h/d=0.1$.

Figure 4

(a) The value ${\theta }^{*}$ of “the order parameter” $\theta$ predicted by the theories as a function of “the inverse temperature” $\delta$ for $h/d=0.1$, demonstrating that the present and previous models are continuous and discontinuous, respectively. (b) Normalized free energy minimized with respect to $\theta$ as a function of $\delta$ and (c) Normalized force-elongation curve for $h/d=0.1$.