Adhesive tape loop

TL;DR

The paper addresses the stability of self-adhering slender loops formed by bending a straight adhesive strip into a loop with an overlap. It combines PDMS-based experiments with a scaling argument and a planar Kirchhoff-rod model to map a state space defined by the non-dimensional adhesion strength $G_c L^2 / K$ and the normalized overlap $\Delta/L$, predicting both equilibrium shapes and the conditions under which equilibrium is possible. Key contributions include a four-region planar rod formulation with an equivalent-overlap representation, validation against experiments showing three distinct loop states, and identification of limiting equilibria with explicit expressions for $G_c L^2 / K$ as $\Delta/L$ approaches 0 or 0.5. The work offers a practical route to infer self-adhesion strength from geometry and lays groundwork for exploring more complex twisted or Mobius configurations in sticky soft materials.

Abstract

We present an experimental and theoretical study of the mechanics of an \emph{adhesive tape loop}, formed by bending a straight rectangular strip with adhesive properties, and prescribing an overlap between the two ends. For a given combination of the adhesive strength and the extent of the overlap, the loop may unravel, it may stay in equilibrium, or open up quasi-statically to settle into an equilibrium with a smaller overlap. We define the state space of an adhesive tape loop with two parameters: a non-dimensional adhesion strength, and the extent of overlap normalized by the total length of the loop. We conduct experiments with adhesive tape loops fabricated out of sheets of polydimethylsiloxane (PDMS) and record their states. We rationalize the experimental observations using a simple scaling argument, followed by a detailed theoretical model based on Kirchhoff rod theory. The predictions made by the theoretical model, namely the shape of the loops the states corresponding to equilibrium, show good agreement with the experimental data. Our model may potentially be used to deduce the strength of self-adhesion in sticky soft materials by simply measuring the smallest overlap needed to maintain a tape loop in equilibrium.

Figures (6)

• Figure 1: Typical experimental equilibrium loop shapes. a.-d. A loop made from Sylgard 184 that is first in a marginal state, then (as the overlap is reduced) finds itself in a more circular shape. e.-h. A loop made from Ecoflex which begins in a marginal state and progresses to a nearly circular shape as the overlap is reduced.
• Figure 2: (a) Schematic of a typical adhesive tape loop with overlap. (b) A free body diagram of the two overlapping regions $A$ and $C$. (c) A tape loop statically equivalent to the loop shown in a. with the overlapping regions $A$ and $C$ replaced by the shaded region with a modified constitutive relation.
• Figure 3: (a) Overlay of the boundary of tape loop obtained by image processing and computed centerline (black) with the experiment, (b) Comparison of the centerline prediction from theory with experiment and (c) Comparison of the curvature plots obtained from theory and experiment for a single sample.
• Figure 4: A comparison of experiments with the theory in the state space of an adhesive tape loop. The theoretical curves trace the points of limiting equilibria for a loop of a given normalized thickness $t/L$, and divide the state space such that the points above the curve admit equilibrium configurations, while the points below a curve do not. The scatter of points are experimental data, where the green circles and the red triangles represent loops that stay in equilibrium and the ones that unravel, respectively. The purple squares denote loops that opened up from the imposed state and settled in a different equilibrium with a smaller $\Delta$. The numerical value $t/L=0.03$ represents the maximum value of length normalized thickness used in the experiments. The dashed blue curve represents the limiting equilibria of a tape loop with zero thickness. For such a curve, $G_c L^2/K\rightarrow 3\pi^2/2$ as $\Delta\rightarrow 0$. The dashed magenta curve correspond to the scaling \ref{['Eqn:scaling1']}.
• Figure 5: A subset of 'Equilibrium with overhang' loops from Fig 4. is shown here. These loops were initially imposed with an overlap $\Delta_{initial}$ which are represented as orange diamonds and subsequently opened up and settled into equilibria represented by the purple squares. The black curves represent the theoretically predicted path going from one state to another, whose explicit parameterisation is given by $\{(\Delta-x)/(L-x), G_c(L-x)^2/K\}$. The dotted black line is a trace of this curve starting from a point chosen such that $G_c L^2/K\rightarrow 3\pi^2/2$ as $\Delta\rightarrow 0$.