Fractonic View of Folding and Tearing Paper: Elasticity of Plates Is Dual to a Gauge Theory with Vector Charges

We offer a fractonic perspective on a familiar observation—a flat sheet of paper can be folded only along a straight line if one wants to avoid the creation of additional creases or tears. Our core underlying technical result is the establishment of a duality between the theory of elastic plates and a fractonic gauge theory with a second rank symmetric electric field tensor, a scalar magnetic field, a vector charge, and a symmetric tensor current. Bending moment and momentum of the plate are dual to the electric and magnetic fields, respectively. While the flexural waves correspond to the quadratically dispersing photon of the gauge theory, a fold defect is dual to its vector charge. Crucially, the fractonic condition constrains the latter to move only along its direction, i.e., the fold’s growth direction. By contrast, fracton motion in the perpendicular direction amounts to tearing the paper.

Figure 1

Schematic of an elastic plate of thickness $h$. The deformation of the plate is described by the vertical (along the $z$ direction) displacement $w(x_1,x_2)$ of the midplane [Eq. (11)].

Figure 2

(a) Tear defect located at the origin with $w(x_1,w_2)=(1/2\pi )\arctan (x_2/x_1)$ and tear defect density (eqn. (16)) $\tau ={\delta }^{(2)}(x)$ that satisfies Eq. (14) with $a=1$. (b) Fold defect located at the origin with $w(x_1,x_2)=(1/2\pi )[x_1\ln (|x|)-x_2\arctan (x_2/x_1)]$ and defect density ${\zeta }_{\alpha }=-{\delta }_{\alpha 2}{\delta }^{(2)}(x)$ that satisfies Eq. (17) with ${\psi }_{\alpha }=-{\delta }_{\alpha 2}$. For the tear defect (a) the slopes ${\theta }_{\alpha }={\partial }_{\alpha }w$ and curvatures $R_{\alpha \beta }={\partial }_{\alpha }{\partial }_{\beta }w$ are smooth everywhere (except the origin), while for fold defect (b) only the latter applies. Function arctan returns a value in $[-\pi ,\pi ]$, with ${\lim }_{ϵ\to 0}\arctan (-\epsilon /-1)=-\pi$.

Figure 3

Configuration obtained by transporting a fold defect with strength ${\psi }_{\alpha }=-{\delta }_{\alpha 2}$ from $x_2=-\infty$ along the $x_2$ axis to the origin. Displacement $w(x_1,x_2)=(1/2\pi )[x_1\ln (|x|)+x_2\arctan (x_1/x_2)]$ satisfies Eq. (17) and its curvatures are identical to Fig. 2.