The stability of a tape loop

Abstract

If a thin sticky sheet is bent into a loop such that the two ends contact and overlap a small amount, will the loop remain closed? On one hand, there is mechanical energy stored in the bending of the material. On the other, there is interfacial energy created by the small amount of overlap (adhesion). Clearly, if the adhesion is high the loop remains closed and if the adhesion is low the loop will open. In order to find the limits of stability we employ Kirchhoff’s rod theory to model the planar equilibria of a closed loop of a strip obtained by overlapping its two terminal ends. The overlapping region is taken to be held together by an adhesive of constant strength. The reactive force density exerted by the adhesive to prevent detachment of the overlap is assumed to be purely normal to the centerlines of the inner and outer segments. Using appropriate jump conditions across the boundaries of the overlap, we compute the effective modulus of overlapping region, and reduce the entire system to a closed loop formed out of a strip with non-uniform thickness. We verify our predictions with experiments utilizing various polydimethylsiloxane elastomers. Ultimately, the technique could form the basis of a simple adhesion test for thin sticky films.