# Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

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Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved December 14, 2015 (received for review November 4, 2015)

## Significance

Thin elastic sheets buckle and wrinkle to relax compressive stresses. Wrinkling metrologies have recently been developed as noninvasive probes of mechanical environment or film properties, for instance in biological tissues or textiles. This work proposes and experimentally tests a prediction for the local wavelength of wrinkles in nonuniform curved topographies.

## Abstract

Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles—their wavelength—is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film.

Wrinkles emerge in response to confinement, allowing a thin sheet to avoid the high energy cost associated with compressing a fraction *λ*, of wrinkles reflects a balance between two competing effects: the bending resistance, which favors large wavelengths, and a restoring force that favors small amplitudes of deviation from the flat, unwrinkled state. Two such restoring forces are those due to the stiffness of a solid foundation or the hydrostatic pressure of a liquid subphase (Fig. 1*A*). Cerda and Mahadevan (1) realized that a tension in the sheet can give rise to a qualitatively similar effect (Fig. 1*B*) and thereby proposed a universal law that applies in situations where the wrinkled sheet is nearly planar and subjected to uniaxial loading:*E* the Young’s modulus, *t* the sheet’s thickness, and **1** is appealing in its simplicity, but it applies only for patterns that are effectively one-dimensional. In particular, it does not apply when the stress varies spatially or when there is significant curvature along the wrinkles.

Here, we study two experimental settings in which these limitations are crucial: (*i*) indentation of a thin polymer sheet floating on a liquid, which leads to a horn-shaped surface with negative Gaussian curvature, and (*ii*) a circular sheet attached to a curved liquid meniscus with positive Gaussian curvature. In both cases, wrinkle patterns live on a curved surface, show spatially varying wavelengths, and are limited in spatial extent. The extent of finite wrinkle patterns in a variety of such 2D situations has recently been addressed (6, 8⇓⇓–11) and was found to depend largely on external forces and boundary conditions. However, a general prescription for the internal structure of the pattern (i.e., the wavelength and any spatial dependence) has been lacking.

Our work leads to two central insights: that the curvature of the subphase gives rise to a new stiffness of geometric origin (which dominates **1**] is sufficient to describe the spatial variation of wrinkle wavelengths. These insights allow us to implement the law [**1**] for a spatially varying *x*)^{2} is proportional to the fractional length **2**] together with [**1**], which we call the “local *λ* law,” greatly expands the quantitative description of wrinkle patterns.

## Theory

We derive the local *λ* law in Eqs. **1** and **2** by considering the setup depicted in Fig. 1*C*: a rectangular sheet of thickness *t* and length *L* attached to a deformable, cylindrical substrate of radius *R*. Although this idealized system is not studied here experimentally (and a real cylinder may not actually buckle in the orderly way shown in Fig. 1*C*), it provides a simple, pedagogic framework in which to consider the various types of stiffnesses that govern the wrinkle wavelength.

For simplicity, we assume the Winkler model, where the substrate responds linearly to a deflection from its rest shape, and use the Föppl–von Kármán (FvK) equations for the mechanical equilibrium of the sheet. Here the sheet can be described using planar coordinates *y* axis parallel to the cylinder axis. The shape of the sheet

Now consider the effect of a compression along the cylinder axis (*T* in the azimuthal direction (*λ* in the *y* direction about *f* and *λ* vanish individually.

The formation of wrinkles enables a complete relaxation of compressive and shear stresses. As **6**] remains finite (in an expansion of the FvK equations in powers of the wrinkle amplitude, *f*, subjected to the slaving condition [**5**]), as does the mean profile of the sheet,

The next order in the FT expansion, as described in *Supporting Information*, yields corrections to the stress tensor at **7**] can be understood by substituting this stress component into the first FvK equation [**3**], where it gives rise to a new force that is proportional to *f*: an entirely new source of stiffness.

In detail, the linearized normal force balance [**3**] reads**5**, admits a solution for any *λ*. Inspection of Eq. **8** reveals the mechanism underlying wrinkle formation. As in Euler buckling, a destabilizing compressive force (*λ*). However, Eq. **8** reveals three other types of stabilizing forces: the tension along wrinkles (*λ*, by [**5**]). This competition leads to the wavelength selection expressed in Eq. **1**.

We define the energy density *λ* by identifying the leading terms of the energy associated with the restoring forces in [**8**], and using Eq. **5**,**2**, was obtained by replacing the relevant tension-field terms in Eq. **8** [namely, the *x*; namely, **2**. Importantly, this derivation assumes that the wavelength varies sufficiently slowly in space so that the energetic cost of gradients in the wavelength, *λ* law, Eqs. **1** and **2**.

Following ref. 1, we note that the three terms that compose **2**, correspond to distinct types of stiffness, associated with the substrate, the exerted tension, and the curvature along the wrinkles. By analogy to the substrate stiffness *B* that are not discussed in ref. 1. First, the tension-induced stiffness,

Before proceeding to discuss specific examples, let us note that although the tension-induced stiffness *λ* law are often characterized by the existence of a tensile direction (*λ*. Although Eq. **1** may be relevant also for more complex types of wrinkle patterns [e.g., under biaxial compression (20) or depressurizing a shell with a stiff core (21)], confinement of sheets in the absence of an imposed tension often leads to patterns with deep folds or stress-focusing zones (22⇓–24), rather than to the oscillatory wrinkles described by Eqs. **1** and **2** and manifested in the following experimental examples.

## Indentation of a Floating Sheet

To test the local *λ* law, we study the indentation of a thin polystyrene (PS) sheet (thickness ^{3}. The sheet is poked from beneath by a rod with a spherical tip of radius 0.79 mm. The deformation is observed by two cameras that capture the side and top views of the sheet. The indentation height *δ* is changed by a translation stage and is measured with an accuracy of 50 μm.

The combination of loads due to the indentation height *δ* at the center (*γ* that pulls the edge of the sheet (*A*–*D*). In ref. 11, tension-field theory was used to predict the macroscale axially symmetric shape *ζ*.) The wrinkle pattern is governed by the dimensionless indentation height, *t* and a factor of 2 in *E* and *F*. The sheet returns to being flat over the scale

The shape *f* in this polar geometry, from which we compute the tension-induced stiffness *Supporting Information*). These stiffnesses, together with **1** and **2**.

For **1** yields*t*. Fig. 3*A* shows the experimentally measured wrinkle wavelength at a fixed radial distance *B* shows not only a collapse of the data with the predicted (curvature-dominated) scaling relation, *t*-independent prefactor **10**.

For smaller values of indentation height, the data deviate from curvature-dominated behavior. This is in agreement with the local *λ* law, which predicts that *B* that include all three terms in

In Fig. 4 we plot the number of wrinkles, *r*.] Results are shown for a wide range of *t* and *A* and *B*) and *C*). The colored curves show the prediction from Eqs. **1** and **2**, whereas the black curve is obtained by approximating

As we saw in Fig. 3, *K*_{curv} is dominant also for other *r* as the indentation increases. Close to the inner boundary of the wrinkled zone, the tension-induced stiffness, *r*, the wrinkled sheet is almost planar, and the dominant stiffness is due to the substrate; we then expect *m* with radial distance (26):

Approaching the edge of the film, there is a substantial increase in *B*). (Fig. 4*C* presents data from large sheets; here the edge fell outside the illuminated region, so the edge cascade was not visible). The local effect of the liquid meniscus or other boundary forces (18, 28⇓–30) are not accounted for in Eqs. **1** and **2**.

Finally, Figs. 3 and 4 also include data at large values of the indentation height where crumples and folds appear in the sheet. In contrast to the purely wrinkled state where the shape undulates around an axially symmetric profile *F*). This surprising observation echoes recent studies on a related system (24).

## A Sheet on a Drop

To test the generality of the local *λ* law, we study wrinkling of a circular PS sheet in another geometry: a liquid surface with positive Gaussian curvature. This is experimentally realized by placing the sheet on (*i*) an air–water meniscus (as in ref. 8) or (*ii*) a water drop in oil (dodecane or silicone oil) (24) and controlling the curvature *R* of the water meniscus.

The dimensionless confinement, *A*. The wrinkled zone grows as *α* increases. In Fig. 5 *B* and *C*, *Insets* show the number of wrinkles, *α*.

A tension-field solution to the FvK equation was found (8), using the assumption of small slopes, valid for *m*^{2}*f*(*r*)^{2}.

In our experiments, the substrate stiffness due to the gravity of the drop is negligible because the sheet’s radius (and the deformation of liquid it induces) is smaller than the capillary length. Hence, according to Eqs. **1** and **2**, the wavelength *α* increases, *λ* law with

Fig. 5 shows our measurements of *λ* law, Eqs. **1** and **2**, with no fitting parameters, is especially good at large values of the confinement *α* in Fig. 5*B*. Fig. 5*C* shows quantitative deviations from the prediction, which may be due to the liquid meniscus at the free edge of the sheet; surface tension is larger in Fig. 5*C* than in Fig. 5*B*, as denoted in the figure legend. The wrinkling cascade due to the liquid meniscus, which causes *C*). These observations, along with what was noted in the previous section for indentation, suggest that the boundary cascade may be more complicated in curved geometries than in a flat geometry, where the cascade dies exponentially with a penetration length

Another common feature between this geometry and the indentation experiment is an instability at a finite, large value of the relevant confinement parameter (*α*) in which the sheet becomes decorated with crumples (8). Nonetheless, the above prediction for the number of wrinkles

## Discussion

We have shown excellent agreement between the prediction of the local *λ* law, Eqs. **1** and **2**, and experimental measurements of the spatially varying wrinkle wavelength in two different geometries: one with negative and one with positive Gaussian curvature. This agreement illustrates the key role played by the geometric stiffness, *λ* law in relatively complex scenarios.

A similar type of geometric stiffness, which is determined by the underlying curvature rather than by the exerted loads, is known to govern the (unwrinkled) response of intrinsically curved elastic shells to loads (19). To demonstrate the geometric link between shells and sheets, consider the uniaxial compression of a cylindrical shell: An ordered pattern of diamond-like blocks is observed (31), whose characteristic size is proportional to the geometric mean of the radius (*R*) and thickness (*t*) of the shell, **1** by substituting **1** and **2** compose the first attempt to describe the combined effect of the geometric stiffness,

Notwithstanding the experimental evidence for the local *λ* law, Eqs. **1** and **2**, its validity is limited to situations in which the spatial variation of the wavelength **9**. Going beyond this, one might expect situations in which gradients in the wrinkle wavelength are explicitly penalized via *C*, the tension lines are parallel to (*n* wrinkles to *λ* law and experiments suggests that, in circumstances that remain to be understood, the effect of

## Materials and Methods

### Film Preparation.

We made polymer films by spin-coating dilute solutions of polystyrene (

### Wrinkle Analysis.

We performed a custom automated analysis (adapted from ref. 8) of the top-view images to measure wavelength, *λ*, as a function of radial coordinate in the wrinkle patterns. To reduce noise, image intensity was first averaged over small intervals along the radial coordinate. We then filtered the signal in the *θ* coordinate, to eliminate long-wavelength components due to uneven lighting. Finally, an autocorrelation was performed at each radius, which gave a decaying sinusoidal signal. The wrinkle wavelength was determined as twice the distance to the first autocorrelation trough. When crumples or folds were present, angular sectors lying between these structures were analyzed in the same fashion.

## Curvature-Induced Stiffness: The General Setting

### The Far-from-Threshold Expansion.

In the main text, we explained how the macroscale features of the wrinkle pattern, namely, the asymptotic shape, **6** (main text), are obtained from the FvK equations as the leading order in the far-from-threshold (FT) expansion. This leading-order analysis is also known as “tension-field theory.” Here we explain how the FT expansion around the tension-field limit gives rise to the **7** and **8** of the main text, respectively). These results were key in our derivation of the local *λ* law (Eqs. **1** and **2** in the main text).

The FT expansion is singular in the sense that both the wrinkle’s amplitude and wavelength approach zero as the film thickness vanishes. However, the ratio of amplitude to wavelength is held at a constant, finite value as this limit is taken; the ratio’s value is determined by the slaving condition (Eq. **5** in main text). The actual (control) small parameter here is the inverse “bendability,” **11** of main text), which vanishes for a given exerted tensile strain in the limit of vanishing thickness. However, because the wrinkle amplitude *f* vanishes as *f* as the small parameter in the FT expansion. [Note that the FT expansion is markedly different from the small-amplitude expansion, assumed in classical postbuckling theory; this expansion is a perturbation to the compressed, unwrinkled state of the sheet and is not subject to the slaving condition, Eq. **5** of the main text (16).]

To avoid repeated references to the main text, let us summarize here the main equations that we discuss in *Supporting Information*; we then refer to them with new labels: The normal force balance on the sheet is

The ratio between the amplitude and wavelength of the wrinkles is determined by the slaving condition*L* the natural length of the sheet in the direction of confinement. (Note that a small-slope approximation has been used on the right hand side of [**S3**].) Finally, the asymptotic stress field (tension-field limit) is**S2**], of the asymptotic shape **S4**), and *f*. The major result of the following calculation is that the contribution *f*, and the proportionality constant scales with the curvature **S1**.

### Correction to the Stress Field.

The most fundamental assumption of tension-field theory, and thereby the FT expansion, is that the elastic energy of the sheet (and an attached deformable substrate) can be also expanded into a series,*λ*).

For an asymptotically planar shape [i.e., **S6**. In particular, the additional energies due to the wrinkles themselves are negligible in comparison with

We now show that when the asymptotic shape is curved along the wrinkle’s direction, **S6**, implies**S7**, it is sufficient to consider the effect of the shear stress

Recalling the Hookean stress–strain relationship,**S3**, implies **S8**] that **S6**, making tension-field theory an invalid limit of the wrinkled sheet in the high-bendability limit. We conclude from this argument that, instead, the last term in Eq. **S8** must be precisely canceled by the in-plane contribution *.*,*λ* in the **S9**] (and eliminate the energetically expensive shear stress) is if**S11**] that the existence of a nonzero curvature along the wrinkle direction, **S7**]. In the absence of this curvature [i.e.,

### Normal Force Balance.

Let us consider now the normal force balance (first FvK equation), Eq. **S1**. We expand it around the tension-field limit by substituting for the shape and stress their respective expressions in Eq. **S2** and Eqs. **S4** and **S11**. The leading terms in the expansion appear at

At the next order, **S12**], the first term results from the bending force (because the curvature of wrinkles in the oscillatory direction

Two important features of Eq. **S12**, which have been emphasized previously (10, 16), are (*i*) the normal force resulting from the coupling between shear stress and curvature is weak, **S12**, and (*ii*) although the compressive component of the stress tensor **S12**), whereas all other terms in Eq. **S12** are positive, representing the various stabilizing forces that all favor a flat, unwrinkled state.

For the purpose of this paper, the most important aspect of Eq. **S12** is the existence of the normal force **S12**,

## Calculation of K eff for Indentation-Induced Wrinkling

The indentation-induced wrinkling of a floating film illustrates the importance of the curvature-induced stiffness. In this problem, wrinkles are observed from an inner position, *c* is a constant of integration. The curvature-induced stiffness is then given by*m* and *r*) is as given by Eq. **S16**, rather than by replacing λ with **5**.

Bringing the three stiffnesses (*λ* law (Eqs. **1** and **2** of the main text), we find that the number of wrinkles may be written^{˜} denotes nondimensionalization of a quantity by the vertical scale

In the main text we considered the case **S17**] that**10** of the main text.

To produce the theoretical curves in Figs. 3*B* and 4 of the main text, the value of **S14**], valid in the wrinkled region, at

## Acknowledgments

This work was supported by the Keck Foundation (J.D.P., T.P.R., N.M., and B.D.), NSF-DMR Grant 120778 (to H.K. and N.M.), NSF-Materials Research Science and Engineering at University of Massachusetts Amherst (instrumentation facilities), NSF CAREER Award DMR-11-51780 (to E.H., Z.Q., and B.D.), ERC StG 637334 (to D.V.), and a fellowship from the Simons Foundation Award 305306 (to B.D.).

## Footnotes

↵

^{1}J.D.P. and E.H. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: bdavidov{at}physics.umass.edu, menon{at}physics.umass.edu, or dominic.vella{at}maths.ox.ac.uk.

Author contributions: J.D.P., E.H., H.K., T.P.R., N.M., D.V., and B.D. designed research; J.D.P., E.H., H.K., J.H., Z.Q., T.P.R., N.M., D.V., and B.D. performed research; J.D.P. analyzed data; J.D.P., N.M., D.V., and B.D. wrote the paper; J.D.P. and H.K. designed experimental setups, conducted experiments, and developed software; E.H. conceived the idea of the paper and developed theory; J.H. designed experimental setups and conducted experiments; N.M. designed experimental setups and developed software; Z.Q. performed calculations; T.P.R. designed experimental setups; and D.V. and B.D. developed theory and performed calculations.

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.1521520113/-/DCSupplemental.

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