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# Ribbon curling via stress relaxation in thin polymer films

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved December 30, 2015 (received for review July 26, 2015)

## Significance

The forming of thin film structures through differential stress relaxation is an important engineering design tool to create flexible objects such as rolls, spirals, and origamis. We exploit the everyday process inspired from gift wrapping of curling an initially straight ribbon by pulling it over a blade to probe the mechanical shear response of thin polymer sheets. Experiments show that curling occurs over a limited range of loads applied to the ribbon, with the curl radius reaching a maximum at intermediate loads. A theoretical model reveals several patterns of irreversible yielding across the ribbon, and the dependence of curl radius on pulling speed shows that stress relaxes dynamically as the ribbon passes over the blade.

## Abstract

The procedure of curling a ribbon by running it over a sharp blade is commonly used when wrapping presents. Despite its ubiquity, a quantitative explanation of this everyday phenomenon is still lacking. We address this using experiment and theory, examining the dependence of ribbon curvature on blade curvature, the longitudinal load imposed on the ribbon, and the speed of pulling. Experiments in which a ribbon is drawn steadily over a blade under a fixed load show that the ribbon curvature is generated over a restricted range of loads, the curvature/load relationship can be nonmonotonic, and faster pulling (under a constant imposed load) results in less tightly curled ribbons. We develop a theoretical model that captures these features, building on the concept that the ribbon under the imposed deformation undergoes differential plastic stretching across its thickness, resulting in a permanently curved shape. The model identifies factors that optimize curling and clarifies the physical mechanisms underlying the ribbon’s nonlinear response to an apparently simple deformation.

Bend a ribbon over a scissor blade by pressing it firmly down with your thumb and pull the ribbon over the blade. This is the commonplace method for curling ribbons for decorative gift wrapping. But what is the mechanism by which ribbon coils are produced? How does the coil depend on the speed of pulling, the shape of the blade, and the tension in the ribbon? We address these questions using a combination of experiments and theory, showing how curling arises via a plastic deformation that is regulated by both spatial and temporal effects.

A familiar example of curvature generation in thin materials is the bimetallic strip (or thermocouple), for which curvature arises from differential thermal expansion of two adherent layers of elastic material; the strip bends on heating but recovers its initially straight configuration on cooling to its original temperature (1). Some biological materials exploit the same principle, but with expansion driven by water fluxes: Reversible differential expansion arises in bilayer plant tissues such as the anther (2), pine cone scale (3), or wet paper (4), whereas irreversible differential cell expansion drives the bending of roots and shoots (5).

In contrast to these examples, a ribbon is a homogeneous material, at least before curling. Further, the curling deformation is permanent, pointing to the fact that a part of the material undergoes yield during the deformation process. Accordingly, the experiments we report below reveal that a threshold load must be applied to induce curling, whereas excessive loading may prevent curling or even tear the ribbon completely. Similar plastic deformation is inadvertently applied by rolling up paper scrolls for storage, resulting in curled edges that have plagued Chinese scrolls for centuries (6); however, the generation of this widthwise curvature is negligible in our narrow ribbons. Unlike prior studies of the bending of elastoplastic beams under a stationary transverse load (7, 8), our study addresses dynamic stress relaxation effects and shows how curling is modulated by an axial load. Our work also differs from studies of bending of soft viscoplastic threads, for which yield surfaces are orthogonal, rather than parallel, to the thread axis (9).

Somewhat surprisingly, our experiments show that the maximum ribbon curvature is typically generated at an intermediate load. We develop a theoretical model to explain the nonmonotonic relationship between curvature and load. The central idea is that as a material element of ribbon passes onto the blade and is forced into a configuration with high curvature it undergoes yield in a region near its outermost surface; however, as the ribbon element leaves the blade, there can be a further deformation involving irreversible stretching of the ribbon close to its inner surface. The former deformation promotes curling, and the latter reduces it. Curvature is modulated further by the pulling speed, which determines how the transit time of the element over the blade compares to the stress relaxation time of the material. Our model captures the key elements of this robust phenomenon and demonstrates how the curling process provides insight to the shear response of thin, stiff sheets of polymer that yield under relatively low loads.

## Results

### Experimental Results.

A schematic side-view diagram of the experimental apparatus is shown in Fig. 1*A*. A polymer ribbon (10) of thickness *C*) (*SI Appendix*).

Images of the experimental ribbon configurations over each blade are shown in Fig. 1*B* for loads in the range 50 g

Dimensional permanent curvature measurements are shown using symbols in Fig. 2*A*, as a function of axial load applied to the ribbon for each of the four blades. The data corresponding to experiments with the sharp and *A*). The four sets of experimental data also suggest that a critical load needs to be exceeded in order for the ribbon to curl, and this critical load increases significantly with reduction in blade curvature. Hence, the modest loads required to bend the ribbon over the sharp and 1-mm blades meant that the ribbon geometry varied with load over the entire range investigated (Fig. 1*B*), whereas the larger loads required to bend the ribbon over the *B* indicate that the curvature of the ribbon curl decreases monotonically with linear pulling speed.

To inform comparison with the theoretical model, the material properties of the PVC ribbon were measured with uniaxial tensile tests performed using an Instron 3345 (L2957) universal testing system. The Young’s modulus *C* as a function of applied stress, where each symbol indicates experiments performed on an individual ribbon sample. The strain rate is approximately zero below a critical yield stress and increases approximately linearly above this threshold. A linear fit to the data above the plastic yield threshold in Fig. 2*C* gave a viscosity coefficient of *C*, small but measurable plastic strain rates were also found below this estimate of the yield stress for applied stress values larger than 32 MPa. Finally, tensile stress relaxation experiments performed by rapidly ramping the strain imposed on a sample ribbon to a fixed value and recording the reduction in stress that followed (*SI Appendix*) yielded maximum rates of plastic stress relaxation that increased weakly up to 1.8 MPa/s at 1.6% imposed strain and then more sharply and approximately linearly to 8 MPa/s for 4.4% imposed strain, relaxing to stresses in the range 28–36 MPa, with evidence of yield taking place even at strains below 1%. In summary, the ratio

### Physical Interpretation.

Fig. 3 illustrates the mechanism that we propose to explain ribbon curling. The ribbon’s complex constitutive properties are idealized by treating it as an isotropic elastic/viscoplastic material with a yield stress *i*) rises smoothly while the ribbon is off the blade, (*ii*) adopts the curvature of the blade while in contact with it, and then (*iii*–*v*) falls once off the blade (Fig. 3, *Top*), giving rise to stress distributions across the ribbon illustrated in Fig. 3 for low and high loads (cases A and B, respectively). Although the ribbon adopts a steady shape, material elements experience a time-varying curvature as they pass over the blade. The off-blade curvature distributions (*i* and *iii*–*v*) are regulated by a balance between the ribbon’s bending resistance (which we assume is unaffected by any plastic deformation) and the imposed axial load. We assume that, as it passes over the blade, the ribbon element experiences a transverse strain gradient proportional to its instantaneous curvature, stretching the ribbon on its outer surface and compressing it (relatively) at its inner surface. Thus, upstream of the blade (*i*), the curvature-induced strain induces a transverse gradient of stress through an initial elastic response, which acts in addition to the axial loading on the ribbon. Where the stress exceeds *i* and *ii* in Fig. 3); initially, this takes place close to the ribbon’s outer surface. As the stress in the yielded region relaxes, the yielded region widens in order that the ribbon element can support a constant net axial load. Passing off the blade, the stress in the yielded region relaxes toward *iii* in Fig. 3).

Passage of the element further off the blade leads to a reduction in curvature and hence in the transverse strain gradient. Thus, via an initial elastic response, there is a corresponding reduction in the transverse stress gradient; this may be visualized as counterclockwise pivoting of the stress distribution about the ribbon’s midline. If the yielded region remains confined to the upper half of the ribbon (*iv* and *v* in case A, Fig. 3), then no further yielding occurs. However, if the yielded region penetrates into the lower half of the ribbon, pivoting of the stress lowers the stress near the outer wall but creates a new zone near the ribbon’s midline where the stress exceeds *iv* in case B, Fig. 3). This, in turn, induces a second phase of stress relaxation, involving widening of the central yielded region (*v* in case B, Fig. 3), and further irreversible elongation of the ribbon; this process is promoted by further curvature reduction as the ribbon element straightens out. Ultimately, the yielded region extends to the inner surface of the ribbon, reducing the gradient of irreversible strain. When unloaded, the curvature of the ribbon element is determined by the overall gradient of the net plastic strain; this gradient grows as the ribbon yields near its outer surface (*v* in case A, Fig. 3) but falls if there is additional yielding near the inner surface (*v* in case B, Fig. 3).

The minimal load required to induce a curl (Fig. 2*A*) can therefore be associated with the threshold required to induce yield at the ribbon’s outer surface; the increase of curvature with load is associated with thickening of this yielded region (*i*–*iii* in case A, Fig. 3), and the reduction of curvature with load at higher load is due to the compensating yield near the inner surface (*iv* and *v* in case B, Fig. 3). The reduction of curvature with pulling speed (Fig. 2*B*) arises because the ribbon element has limited time in which to undergo stress relaxation while on the blade. Curling is maximized by driving the on-blade yield surface to the ribbon centerline (but not beyond), and by ensuring the ribbon moves slowly enough for the stress to relax fully before leaving the blade.

We can use experimentally measured parameters to estimate the load required to induce curling. We represent the load as an axial stress *A*) but does not explain why the load leading to maximum curling falls for sharper blades. To address such questions, we now turn to a quantitative model, summarized briefly below and explained more fully in *Materials and Methods* and *SI Appendix*.

### Model Predictions.

Model predictions are shown using lines in Fig. 2*A*. The model predicts that curling takes place for loads satisfying*A*) and the maximum beam curvature

The predicted ribbon curvature is composed of two curves, one rising with load and the second falling (Fig. 2*A*). On the rising curve, yield is confined to the upper half of the ribbon cross-section; on the falling curve, yield extends into the lower half of the cross-section. The threshold between the curves depends on geometric and material parameters and the speed at which the ribbon passes over the blade. Choosing values of *SI Appendix*), it was not possible to identify a single parameter set appropriate for all four blades, reflecting limitations of the constitutive model. The minimum load for curling is closely related to the assumed yield stress (see Eq. **1**); experimental data show that the sharp blade induces yielding at a lower effective yield stress, which is reflected by choosing a lower value of *SI Appendix*, Fig. S1) show stress relaxation occurring more rapidly for larger strains, we adopt a smaller relaxation time for the experiments with sharper blades.

Lines on Fig. 2*B* show how the curvature is predicted to fall with increasing speed for a fixed load, using parameters for the 1-mm blade. The rate of decay of curvature with speed is captured reasonably well, and the model confirms that greater curvature may generally be achieved at lower speeds (and higher loads) by allowing for complete stress relaxation in the upper half of the ribbon. Although not evident in the experimental data, the model predicts that this effect may be offset at very low speeds (to the left of the kink in predicted curves), where the yield surface penetrates the lower half of the ribbon: In this case the model suggests that slightly greater curvatures can be achieved under lower loads.

The model predicts net axial elongation (in addition to curling) that undergoes a transition from modest to steep increase with load at approximately the load required for maximum curvature (Fig. 2*A*). Hence, net axial elongation is most significant along the falling part of the curvature–load curve. The experimental data confirm this prediction (*SI Appendix*, Fig. S2).

## Discussion

Perhaps the most surprising feature of the experimental data reported here is the nonmonotonic dependence of curvature on load, showing that the applied load must be carefully tuned to maximize permanent ribbon curvature when using a blade of given radius. The load applied to the ribbon serves multiple purposes: It wraps the ribbon over the blade, forcing it to curve; it elevates the axial stress in the ribbon toward the yield stress; and it regulates the pattern of plastic deformation across the cross-section of the ribbon. When the ribbon is curved, stretching of the ribbon at its outer surface may be sufficient to induce plastic deformation locally. This deformation is applied to a length of ribbon by running the ribbon over the blade, at a speed that is sufficiently slow for part of the ribbon’s cross-section to stretch irreversibly. If the stress relaxes while the ribbon is in a curved configuration, then straightening of the ribbon as it leaves the blade elevates the stress on the inner surface of the ribbon. If the load is sufficiently great, this can induce further plastic deformation, reducing the transverse strain gradients that lead to permanent curvature.

Experiments characterizing the material properties of the ribbon under elongation demonstrate surprisingly complex constitutive properties that we have not attempted to represent in full detail, choosing instead to work within the framework of a relatively simple (quasi-one-dimensional elastic–viscoplastic) constitutive model. Our semiquantitative predictions are sufficient to provide the physical insight needed to rationalize ribbon curling, during which extensional, shear, and viscous effects interact. Our model discounts frictional effects that may induce heating or surface deformations; these may contribute to curling in other circumstances.

The experimental protocol described here offers insights into material properties under shear of thin materials that are stiff but that yield at relatively low loads. The yield stress and relaxation time can be hard to define unambiguously for the polymer materials that often constitute ribbons, even in simple extensional tests. However, estimated geometric, material, and dynamic parameters (

## Materials and Methods

### Model Description.

A full description of the mathematical derivation and solution of the model can be found in *SI Appendix*. The following highlights the model’s key aspects.

To allow physical insight, our model seeks to capture the essential features of the experiment using a minimal number of parameters. We impose a strain field on a ribbon element as it moves from state to state and compute the resulting stress field. We calculate strain profiles by modeling the ribbon as an Euler–Bernoulli beam that is subject to an applied load and the constraint that it wraps around the blade for a portion of its length. This yields a curvature profile that is uniform on the blade and decays over a distance *Top*). The ribbon element experiences a transverse strain gradient, induced by the imposed curvature, superimposed on a transversely uniform axial strain. As shown in *SI Appendix*, the ribbon is found to be in point contact with the blade at low loads (when *SI Appendix*, Figs. S4–S6.

Given the imposed strain, we compute the stress by modeling the ribbon as a elastic/perfectly viscoplastic (Bingham–Maxwell) material, parametrized by a stiffness *C* and *SI Appendix*), to transfer precisely to the more complex shear deformations associated with the curling experiment. Instead, we use the measured transit speed

To model plastic deformation, we nondimensionalize lengths by the ribbon thickness *h*, directed from the inner to the outer wall of an element of ribbon, the axial strain distribution is assumed to take the form

where *h* appears as a parameter in Eq. **2**; there is plastic deformation wherever

As shown in *SI Appendix*, this system can be reformulated as an integro-differential equation for

which was solved numerically up until a time at which the stress had fully relaxed. Once the load is removed from the ribbon, each element relaxes to form a coil with the ribbon’s centerline having a constant equilibrium curvature

to give

Solutions of the numerical model are shown in Fig. 3 and *SI Appendix*, Figs. S7–S10. These are conveniently categorized by values of the dimensionless curling number, **1**) lies at **4** can be simplified to yield an analytic approximation for the rising part of the curve relating equilibrium curvature

where *SI Appendix* and *SI Appendix*, Fig. S11). In this limit the ribbon is predicted to be entirely in point contact with the blade for

## Acknowledgments

We thank H. Cass and A. Crowe for preliminary experiments, E. Häner for help with image analysis, and D. Pihler-Puzović for measurements of Young’s modulus. B.C. thanks A. M. Klales; P. Hine for preliminary experiments; and V. Vitelli, V. N. Manoharan, and L. Mahadevan for preliminary discussions. We are grateful to a referee for valuable suggestions relating to our numerical solution scheme. This work was supported by an Addison–Wheeler Fellowship (to C.P.) and Engineering and Physical Sciences Research Council Grant EP/J007927/1.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: anne.juel{at}manchester.ac.uk.

Author contributions: C.P., B.C., O.E.J., and A.J. designed research; C.P., J.M., B.C., O.E.J., and A.J. performed research; C.P., J.M., O.E.J., and A.J. analyzed data; and C.P., O.E.J., and A.J. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1514626113/-/DCSupplemental.

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