Wrinkling of a thin circular sheet bonded to a spherical substrate

Abstract

We consider a disc-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of the sheet plus a (Winkler-type) substrate energy. Treating the thickness of the sheet h as a small parameter, we determine the leading-order behaviour of the energy as h tends to zero, and we give (almost matching) upper and lower bounds for the next-order correction. Our analysis of the leading-order behaviour determines the macroscopic deformation of the sheet; in particular, it determines the extent of the wrinkled region, and predicts the (non-trivial) radial strain of the sheet. The leading-order behaviour also provides insight about the length scale of the wrinkling, showing that it must be approximately independent of the distance r from the centre of the sheet (so that the number of wrinkles must increase with r). Our results on the next-order correction provide insight about how the wrinkling pattern should vary with r. Roughly speaking, they suggest that the length scale of wrinkling should not be exactly constant—rather, it should vary slightly, so that the number of wrinkles at radius r can be approximately piecewise constant in its dependence on r, taking values that are integer multiples of h^−a with .

This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications’.

1. Introduction

We model the wrinkling of a disc-shaped elastic sheet bonded to a compliant sphere, as shown schematically in figure 1. The source of the wrinkling is easy to understand: if we assume for a moment that the sheet is inextensible in the radial direction and that the centre of the disc is attached to the north pole, then each circle |x|=r is approximately mapped to the circle S_r on the sphere at distance r from the north pole. As the arclength of S_r is less than 2πr, circles |x|=r must wrinkle to avoid (large) compression. Roughly, the typical slope of the wrinkling is determined by the geometry of the sphere (the contrast between 2πr and |S_r|), while the wavelength is determined by competition between the bending energy (which prefers coarse, large-amplitude wrinkling) and the substrate energy (which prefers small deformations, hence fine, low-amplitude wrinkling).

Figure 1.

Circular sheet on a ball. (Online version in colour.)

The preceding account is oversimplified. Our sheet is not inextensible, and we permit it to slip along the sphere. By stretching slightly in the radial direction the sheet can reduce the energetic cost of wrinkling, as the circle |x|=r is then approximately mapped to a circle on the sphere slightly longer than S_r. As we will explain in due course, the macroscopic deformation of our sheet is determined by the competition between membrane effects (which prefer less stretching) and the energetic cost of wrinkling (which prefers more stretching).

The behaviour of thin elastic sheets experiencing compression due to geometric effects has recently received a lot of attention. Without attempting a comprehensive review, let us mention studies concerning a sheet on a deformable sphere [1–3]; indentation of a pressurized ball [4]; indentation of a floating sheet [2,5]; wrinkling of a stamped plate [6]; and crystalline sheets on curved surfaces [7,8]. Among these references, paper [1] deserves special note, because (as we explain in §2) our model is particularly close to the one considered there.

It is well known that with increasing compression a thin elastic sheet undergoes an instability (like Euler buckling), the onset of which is well understood using linear analysis (this is the so-called ‘near-threshold’ (NT) regime). As the compression increases one enters a different, ‘far-from-threshold’ (FT) regime [9], in which predictions from the linear theory cease to be valid. In contrast with the NT regime, in the FT regime the sheet (almost) completely releases the compressive stresses by deforming out of plane (e.g. by wrinkling). The wrinkling wavelength is then set by a competition between the bending resistance (which prefers long wavelengths) and mechanisms favouring short wavelengths (e.g. tension, curvature along the wrinkles, and adhesion to a substrate). The natural goals in the FT regime are to predict the wavelength of wrinkles (by deriving a so-called ‘local λ-law’ [2,10]) and/or to predict the macroscopic deformation of the sheet. These goals are the primary focus of many of the papers cited above [1,2,4,5,7,9].

While our goal in this paper is very similar, there is an unexpected twist compared with the aforementioned work. There the energy consists of a dominant part which decides the macroscopic deformation, and a subdominant part which controls the scale of the wrinkling. Put differently, in the limit of vanishing thickness the wrinkling does not cost any energy (because the energetic contribution from wrinkling is subdominant), and the macroscopic deformation of the sheet can be obtained via tension-field theory (in mathematical language: by minimizing a relaxed functional). By contrast, in the problem we consider the cost of wrinkling is comparable to other terms in the dominant energy; as a result one cannot use tension-field theory or solve a relaxed problem to predict the macroscopic deformation of the sheet. Instead, one must minimize an effective functional, in which the elastic energy of radial tension competes with the (substrate + bending) energy of circumferential wrinkling. As the energetic cost of wrinkling contributes to the leading-order term in the energy, minimization of the effective energy determines (at least approximately) the length scale of wrinkling at radius r. Our problem has this character because we consider a stiff elastic substrate, quite different in character from the liquid substrate considered in [5] and stiffer than the relatively compliant Winkler foundations considered in [1,2,4,7] (see §2 for more about this).

As already mentioned, the minimum of our effective functional determines the macroscopic deformation and the limiting energy as the thickness . But more: it gives a lower bound for the energy E_h when h is positive. (For the precise definition of E_h, see (2.1) below.) The amount by which E_h exceeds the minimum of the effective functional is informative; therefore, the estimation of this excess energy is a major focus of our work. Our main mathematical result, theorem 2.1, provides upper and lower bounds on the excess energy, showing (very roughly speaking) that it is approximately linear in h.

Our estimate of the excess energy has implications for the fine-scale structure of the wrinkling. This is because E_h includes the cost of changing the wrinkling pattern as a function of the distance r from the centre of the sheet, whereas the effective functional ignores this cost. Indeed, the effective functional estimates the length scale of wrinkling by balancing the azimuthal bending against the substrate term; this leads to the conclusion (found also in [1]) that the scale of the wrinkling should be approximately proportional to h (independent of r). It follows that the number of wrinkles at |x|=r should increase approximately linearly with r. Our analysis of the upper bound on the excess energy shows that the length scale of wrinkling should not be exactly constant—rather, it should vary a bit, so that the number of wrinkles at radius r can be approximately piecewise constant in its dependence on r, taking values that are integer multiples of h^−a with a≈1/2.

The picture that emerges has a lot of symmetry: the macroscopic deformation (determined by minimizing the effective energy) involves radial tension, and the number of wrinkles at radius r is an approximately (but not exactly) linear function of r. This symmetry is a conclusion, not a hypothesis, of our analysis. It is of course crucial that our sheet is disc shaped, and that the sphere is a body of revolution around the axis determined by the centre of the sheet.

It is not a new idea that energy scaling laws can be used to identify the wrinkled region and to explain the local length scale of wrinkling. The best-understood examples are problems where the geometry is simple (typically flat) and the direction of wrinkling is fixed by some source of uniaxial tension (e.g. a stretched annular sheet [11,12] or a hanging drape [13]). Problems involving biaxial compression are less well understood, though there has been progress in special cases (e.g. the shape of a blister in a compressed thin film [14–16], and the herringbone pattern seen in a compressed thin film bonded to a compliant substrate [17]). The problem considered here involves compression, but its geometry is rather controlled due to the presence of the substrate and the use of von Kármán theory.

The paper is organized as follows. Section 2 presents our model, states our main mathematical results (theorem 2.1) and provides further discussion about their implications. Sections 3 and 4 prove the lower-bound half of theorem 2.1. The argument relies on certain properties of the minimizers of some one-dimensional calculus of variations problems closely related to our effective functional. Section 3 states the required properties in proposition 3.1 then uses them to prove the bound, while §4 provides the proof of proposition 3.1. Finally, §5 proves the upper-bound half of theorem 2.1. This is done by identifying an explicit wrinkling pattern (varying appropriately with r) with relatively small excess energy. While the pattern given there is not the energy minimizer (we do not solve an Euler–Lagrange equation), it provides an indication about how a wrinkling pattern should look in order to achieve the minimum energy scaling law.

2. The model and the main results

We consider a thin elastic sheet of circular shape with thickness h>0 and radius r₀>0, which sits on an elastic ball of radius R≫r₀. The energy of the system has three terms: the membrane energy of the sheet, which measures deviation from the deformation being an isometry; the bending energy of the sheet, which penalizes curvature; and a substrate energy, which prefers the sheet to be sphere shaped. For the membrane term, we use a Föppl–von Kármán model (taking Poisson’s ratio equal to zero for simplicity); for the substrate term, we use a Winkler foundation. Focusing on the energy per unit thickness and normalizing by Young’s modulus of the sheet, our elastic energy functional is

2.1

Here, Ω denotes the disc of radius r₀, centred at the origin; u and ξ are the in-plane and out-of-plane displacements of the sheet, and e(u)=(∇u+(∇u)^T)/2 denotes the linear strain associated with u. The non-dimensional constant α_s (which we assume is strictly positive) determines the relative stiffness of the substrate compared with that of the film. As the substrate term involves only ξ and not u, our model requires that the sheet conform to the sphere, but permits it to slide along the sphere. (The standard Föppl–von Kármán bending term would be ; we have dropped the factor for notational simplicity. This simplification does not change the problem significantly, although it affects the precise form of the effective functional.)

Our energy (2.1) is very similar to the one considered by Hohlfeld & Davidovitch in [1]. There are, however, two significant differences: (i) their energy has an additional term, representing surface tension, which induces a state of (small) radial tension; (ii) they focus on the limit of ‘asymptotic isometry’, which is achieved when both the surface tension and our α_s tend to 0. Our situation is different, because we take α_s to be non-zero (and fixed) as . As a result, our sheet does not achieve asymptotic isometry (despite the absence of surface tension); rather, it is in a state of radial tension. To make the comparison with [1] more explicit: their substrate term is —the same as ours except that the coefficient is K/E_fh instead of α_sh⁻². (Here, E_f is the elastic modulus of the sheet and K is a constant determining the stiffness of their substrate.) Thus, their analysis (sending while holding K fixed) corresponds, in our notation, to considering α_s=K/E_fh. This is not the regime we consider; rather, our α_s is fixed and positive as .

We choose to work in dimensional variables: as u, ξ and h have the dimensions of length, our energy (2.1) has dimension length². However, the problem would not be significantly different if we worked in non-dimensional variables, replacing x by x/R, etc. The non-dimensional version of the energy looks the same as (2.1) except that the domain is a ball of radius r₀/R, the factor in front of |∇∇ξ|² is (h/R)², and the substrate term is .

This model can be criticized on the ground that the Winkler substrate term is somewhat idealized. A more realistic model of a film bonded to a uniform elastic ball would replace our Winkler term with a non-local expression involving the H^1/2 norm of the surface displacement (see for example [17] or §3 of [18]). This suggests replacing our Winkler term by

2.2

(which scales once again like length²). In fact, our analysis of the lower bound would also work for this non-local substrate term. However, our analysis of the upper bound requires estimating the energy of an explicitly given deformation, which would be much more difficult for a non-local model. Moreover, the use of (2.2) would not eliminate another key idealization, namely that the sheet is free to slip along the substrate. In addition, our goal is to consider a thought-experiment not a physical experiment: how does geometry induce wrinkling, when a thin elastic sheet is required to conform to a sphere? With these considerations in mind, we take the view that the Winkler model used in (2.1) is appropriate for our purposes.

In fact, Winkler-type substrate terms have been used to model many experiments. In [1,2,4,7] (see also [5]), such a term arises from the gravitational potential of the fluid below the sheet. There is, however, an important difference: in that work the prefactor scales as h⁻¹, whereas in (2.1) it scales as h⁻². By considering one-dimensional examples, one sees that in our setting the cost of wrinkling (determined by optimizing the length scale, based on competition between the bending and substrate terms) is O(1) (half way between h² and h⁻²); this is why it contributes to the leading-order energy. In [1,2,4,7], by contrast, the different scaling makes the cost of wrinkling o(1) (so it is subdominant).

We turn now to a discussion of our results. They involve (i) identification of an effective functional F₀, whose minimum determines the radial strain in the sheet, the approximate length scale of wrinkling and the limiting behaviour of the minimum elastic energy () as ; and (ii) upper and lower bounds for the excess energy, defined as the difference between and .

To describe the effective functional, we must first discuss the energetic cost of wrinkling. As explained in the Introduction, wrinkling is a way for a circle |x|=r to fit into less space. We shall show that the elastic cost of such a circle ‘shrinking by amount −η’ is well approximated by W_rel(η), where

2.3

More precisely, W_rel(η) estimates (with very small error) the cost of fitting a circle of length 1−η into a unit interval. If η>0 (or slightly negative), it is optimal to deform the curve uniformly in plane; in this regime the cost is entirely membrane energy and W_rel(η)=η². However, if η is negative enough, it is better to deform the curve out of plane (to waste arclength by wrinkling so to speak); in this regime, the cost is obtained by optimizing the length scale of the wrinkling, and the resulting formula is linear in η. The function W_rel(η) is continuously differentiable, though its second derivative is discontinuous at the boundary between the two regimes (figure 2).

Figure 2.

The graph of W_rel(η), superimposed on the parabola η². In the region corresponding to wrinkling ,W_rel(η) is linear; for values of η corresponding to an unwrinkled state, W_rel(η)=η². (Online version in colour.)

As noted early in the Introduction, we expect the sheet to be in a state of radial tension, because stretching a bit in the radial direction reduces the energetic cost of wrinkling. The effective functional F₀ captures this effect, by keeping (only) the energy due to radial stretching and our estimate for the cost of wrinkling

2.4

Here, is a real-valued function of one variable (r=|x|), constrained by the boundary condition . We shall show that minimization of this one-dimensional variational problem provides a good estimate for the radial deformation of the sheet. In addition, it provides a lot of information about the length scale of the wrinkling at radius r (via the analysis that led to W_rel).

Assuming that the sheet actually wrinkles (i.e. assuming that the excess length of the outer circles is large enough to induce wrinkling rather than compression), the cost of changing the length scale of wrinkling from circle to circle does not enter the leading-order energy; rather, it contributes to the principal correction (the excess energy). As we are interested in how the wrinkling pattern changes with r, it is crucial to understand the principal correction. In fact, it seems to be of order h (more precisely: we have a lower bound that is linear in h, and an upper bound that is almost linear in h).

Our main mathematical result is the following characterization of the leading-order energy and the principal correction.

Theorem 2.1. —

Assume α_s∈(0,2⁻⁸(r₀/R)⁴), and let ϵ be the excess energy, defined by

Then there exist constants h₀>0, c₀>0 and , all depending on α_s,r₀ and R, such that for any h∈(0,h₀) the excess energy ϵ satisfies

where the correction grows slower than t^α for any α>0.

As the minimum of the effective functional gives the leading-order elastic energy as , it is important to note that its minimizer can be made entirely explicit. In fact, assuming as in theorem 2.1 that α_s<2⁻⁸(r₀/R)⁴ and using the Euler–Lagrange equation for F₀, it is straightforward to compute that

2.5

with . The first interval (0,r_w) in (2.5) corresponds to the unwrinkled region, while the second interval (r_w,r₀) corresponds to the wrinkled region. The condition α_s<2⁻⁸(r₀/R)⁴ is equivalent to r_w<r₀, i.e. it ensures that the wrinkled region is non-trivial.

To explain the relationship between our effective functional F₀ and the elastic energy E_h, we note that F₀ is closely related to the functional F_h, defined for by

2.6

Indeed, the term prefers , and differs from by a term that is O(h²). This rather formal calculation will be justified by proposition 3.1, which shows (among other things) that . Thus (by the triangle inequality), we can replace by in the definition of the excess energy. The functional F_h, in turn, is a lower bound for the elastic energy. This will be explained in §3; briefly, it follows from Plancherel’s formula ( represents the azimuthal average of the radial displacement, while represents the azimuthal average of ξ+|x|²/2R). Incidentally, when wrinkling is not expected (i.e. if the minimizer of F_h satisfies for r∈(0,r₀)), the arguments in §3 show that the radial extension of minimizes E_h, so that .

To explain how the minimizer of the effective functional provides information about the length scale of wrinkling, we must say more about our analysis of the energetic cost of wrinkling (the calculation behind W_rel, defined by (2.3)). Its starting point is an expression for

integrated over the circle |x|=r (for the precise definition see (3.6)). It depends not only on η (the argument of W_rel) but also on the normal displacement of the sheet (at radius r), viewed as a function of θ. The formula (2.3) is obtained by optimizing the displacement. But this calculation gives more than just the optimum; in particular, it reveals the extra energy W_r−W_rel associated with a non-optimal choice of the wrinkling. The exact expression is (3.14), but the main point is this: if a_k(r) is the kth Fourier coefficient of ξ(r,⋅) then W_r−W_rel is at least

This suggests wrinkling with frequency k such that , i.e. , a choice that makes the length scale of wrinkling independent of r and proportional to h. (Essentially, the same calculation can be found in [1].) Our conclusions about the length scale of wrinkling are not a surprise. Indeed, as this length scale is set by the competition between the bending and substrate terms, it should not depend on r. Because the wrinkles are arranged radially, it follows that the number per circle (the wavenumber |k|) should increase linearly with r.

The preceding calculation considers each circle separately. But we have already mentioned that E_h is strictly above the minimum of F₀, because the radial variation of the wrinkling pattern costs additional (‘excess’) energy, whose magnitude is at least of order h. So our identification of an ‘optimal’ k should not be taken literally. Rather, we expect a wrinkling pattern such that W_r−W_rel is of order h. Thus, we expect that should be at most of order h^1/2 on the support of a_k.

In §5, we provide an explicit wrinkling pattern whose excess energy is approximately linear in h. The observation in the last paragraph—that the active k at radius r can be a certain distance from —turns out to be crucial. As changing the wrinkling pattern costs elastic energy, the successful construction avoids changing it too often. Very roughly speaking, the active k is a function of r which approximates the linear function but takes only values that are integer multiples of h^−a with . An account of why such a construction seems necessary is provided at the beginning of §5.

One feature of our story is a little bit surprising. A wrinkled surface resists curvature in the direction parallel to the wrinkles. In some recent studies of wrinkling, this effect plays a crucial role in determining the local length scale [2,19]. Our problem is different from the ones in those studies; the key differences (as noted earlier) are that (i) the coefficient of our substrate term scales like α_sh⁻² with α_s fixed and positive as , and therefore (ii) the leading-order behaviour in our setting is not obtained from tension-field theory or a relaxed problem, but rather by our effective functional. It is nevertheless natural to wonder at what order the effect of curvature is felt as . As our upper and lower bounds on the leading-order correction (the scaling of with respect to h) do not match, our results are consistent with the possibility that curvature effects might play a role at this order. As for the length scale of wrinkling, while our results show that it is approximately constant (this is required to get the correct leading-order behaviour), its small deviations from constancy might be influenced by effects that do not enter our analysis, such as the extra cost of wrinkling in a curved environment.

3. The lower bound

In the rest of the paper, will stand for ≤C, where C>0 is a generic constant depending on α_s, r₀, R. (The implicit constant is often dimensional.)

The radial symmetry of the domain Ω and of the substrate allows for convenient representation of the energy (2.1) in the radial coordinates (r,θ)

3.1

Motivated by the fact that in thin sheets (i.e. h≪1 and so α_sh⁻²≫1) the quantity ξ+|x|²/2R should be very small, instead of ξ we consider w defined by

Note that for any f∈L²(0,2π), we have

3.2

where . We see that for we have

3.3

Defining , we use the definition of w and (3.2) twice to write

3.4

Plugging (3.3) and (3.4) into (3.1), we get that

3.5

where ,

3.6

and

3.7

To show that ϵ≥c₀h, we need to gather some properties of the functionals F_h.

Proposition 3.1. —

Under the condition α_s<2⁻⁸(r₀/R)⁴, the functional F_h, defined in (2.6), admits a unique minimizer . Moreover, denoting , there exists C=C(α_s,r₀,R) such that

3.8

and

3.9

where and is defined in (2.5). Finally, for any non-negative B∈L¹(0,r₀) and any such that

we have

3.10

The proof of proposition 3.1 will be given in §4. As R_h≥0, relation (3.5) and the fact that W_r(η,ξ)≥W_rel(η), together with (3.10) applied with , imply a lower bound

which using (3.8) can be postprocessed into

3.11

where we recall that and . To obtain the desired lower bound for the l.h.s. in (3.11), for r∈(0,r₀) let be the Fourier coefficients of w(r,⋅). To simplify notation, we will drop the r-dependence in a_k(r). Then,

The lower bound ϵ≥c₀h will be a consequence of the following lemma.

Lemma 3.2. —

Let δ>0, and let ϱ₀,ϱ₁∈(0,r₀) be such that and

3.12

Then we have a lower bound of the form

3.13

where l:=ϱ₁−ϱ₀.

Condition (3.12) expresses the fact that the sheet is expected to wrinkle at circles r=ϱ_i,i=0,1.

Proof of lemma 3.2. —

Recall that l=ϱ₁−ϱ₀ and let . As explained in §2, k_i is the optimal wrinkling wavenumber at position ϱ_i, i.e. the one which minimizes the energetic cost of wrinkling. By reorganizing the definition of W_r, we get

where represents the arclength wasted by wrinkling. As in all the quantities only the modulus of k appears, to simplify the notation we assume that a_k=0 for k<0. From the definition of W_rel, we get for the following identity:

3.14

If either σ(ϱ₀)≤δ or σ(ϱ₁)≤δ, then the conclusion of the lemma follows easily. Indeed, in this case we can write (3.12) as with for i=0 or i=1; it follows that , which implies the conclusion of the lemma. So we may assume for the rest of the proof that σ(ϱ₀)≥δ and σ(ϱ₁)≥δ.

Let . We consider two cases. If the energy at ϱ₀ is not concentrated near the optimal wavenumber k₀, meaning

3.15

then by (3.14)

In the other case, we have

where we used an elementary inequality . To estimate the first term on the r.h.s., we observe that (k/ϱ₀)²≤2(k/ϱ₁)² and that for |k−k₀|<K we have . Hence,

For the second term, we observe that for |k−k₀|≤K we have , and so

Summing these two and using that yields

which concludes the proof of the lemma. ▪

Proof of the lower-bound half of theorem , i.e. the estimate ≥. —

Let I be any interval of length such that I⊂[(2r_w+r₀)/3,(r_w+2r₀)/3] (such intervals exist, provided that the constant h₀ in the statement of theorem 2.1 is chosen sufficiently small). The condition I⊂[(2r_w+r₀)/3,(r_w+2r₀)/3] ensures that I stays away from the inner and outer boundaries of the wrinkled region (which are at r_w and r₀, respectively). One verifies using (2.5) the existence of a constant C₂>0, such that for all r∈I (and all such intervals I). It follows that . Hence at each r∈I, we have either or . Writing I=(a,b), let us define M₀⊂(a,(2a+b)/3), M₁⊂((a+2b)/3,b) to be maximal sets such that for r∈M₀∪M₁

3.16

We distinguish two cases. If |M₀|<|I|/6 or |M₁|<|I|/6, the complement I∖(M₀∪M₁) has measure at least |I|/6 and inside this complement (3.16) is false, and so by the above considerations . Using the triangle inequality, this gives , which then implies , which in turn after integration yields .

In the other case, i.e. if both |M₀|≥|I|/6 and |M₁|≥|I|/6, for i=0,1 we can choose , and apply lemma 3.2 to get

where the fact that implies that the r.h.s. in the previous estimate is at least of order h. Combining the two cases, we get

Finally, we use estimate (3.11) together with the fact that and r∼1 inside [2r_w+r₀)/3,(r_w+2r₀)/3], and the estimate on the difference , and cover at least half of [2r_w+r₀)/3,(r_w+2r₀)/3] with disjoint intervals like the interval I considered above to get the desired lower bound . ▪

4. Proof of proposition 3.1.

We begin by showing that F_h achieves its minimum. Indeed, replacing the membrane term in (2.6) by its positive part—that is, replacing the first term under the integral by —makes the problem convex. This modified problem achieves its minimum by the direct method of the calculus of variations. We claim that its minimizer also minimizes the original functional F_h. This follows from the fact that for the expression must be non-negative almost everywhere (as otherwise one can easily modify to obtain a competitor with strictly smaller energy). Heuristically this is clear, because the relation says simply that the sheet is in tension in the radial direction. This argument also shows that the minimizer is unique, because the modified functional is everywhere convex, and strictly convex whenever .

At risk of redundancy, we sketch another proof that the minimizer exists, by applying the direct method of the calculus of variations directly to F_h. Indeed, as h is strictly positive the functional F_h controls the second derivative of . Therefore, for a minimizing sequence would converge strongly, and this can be used to pass to the limit in the first term.

To obtain information about the form of , we first set w=0 and minimize F_h(v,0), which is the same as minimizing F₀. This can be done by guessing that (0,r₀) splits into two non-trivial intervals: (0,r_w), where , meaning in this region the sheet will not wrinkle (hence we use that W_rel(η)=η² here), and (r_w,r₀), where the sheet wrinkles. Solving the resulting ordinary differential equation leads to the explicit form (2.5) for the minimizer of F₀.

Using the optimality properties for and , we get that and . In addition, for any smooth test function φ we have for some function g

We can use (2.5) to get an explicit formula for g, and in particular to show that g is bounded by some constant C_g. Here by DF_h=0 and D_vF_h=0, we mean that the Gateaux derivative (i.e. directional derivative) of the functional F_h vanishes for every direction (v,w) and (v,0), respectively.

Recall that (see (2.5)) and define . Then using a simple identity for of the form

4.1

we get using Taylor’s theorem with the remainder in the integral form (here we use that W_rel′ is absolutely continuous)

4.2

We have already observed that , , and also . We add the previous two relations and use these facts about DF_h to obtain

4.3

First, observe that both σ_h≥0 and σ₀≥0—the first relation comes from the Euler–Lagrange equation while the second can be derived either by direct computation or through the Euler–Lagrange equation. Next, we observe that W_rel′′≥0 a.e., which follows from the convexity of W_rel and can also be explicitly computed. Hence, using that σ_h≥0,σ₀≥0,W_rel′′≥0 we see that all the terms on the l.h.s. of (4.3) are non-negative, in particular , which by Hölder’s inequality implies , which in turn shows that the r.h.s. in (4.3) is at most of order h².

As the r.h.s. in (4.2) is bounded by the l.h.s. in (4.3), we see that

Using that , we get that . The trivial fact that (which holds as is the minimum over a larger class of test functions) permits us to conclude (3.8).

The r.h.s. in (4.3) being also implies , which by an interpolation inequality implies . We recall the definition of σ_h,σ₀, and see that implies . As W_rel(η) is uniformly convex in the regime , meaning in the region where lies well within this regime we get control in L² on the difference , by a standard embedding of W^1,2 into L^p in , for any we get , and (3.9) follows. In addition, as for we have , we also see that when r∈J.

To obtain the formula for σ_h, we observe that for r∈(r_w,r₀) together with the above bound on implies, assuming h is small enough, that for r∈((2r_w+r₀)/3,r₀), which in turn gives for r∈((2r_w+r₀)/3,r₀). Solving the Euler–Lagrange equation together with the corresponding boundary condition σ(r₀)=0, we immediately obtain the formula for σ_h.

To prove (3.10), using (4.1) again we get for any B∈L¹(0,r₀) and any

where the last term is non-negative by the convexity of W_rel. Using interpolation, we get that , which then implies . Arguing as above, we can show that for small (but not too small) values of r. Combining this with the strict positivity of W_rel′′, we deduce for such r the smallness (in terms of e) of the L² norm of . This can be upgraded using a Sobolev inequality to get . The proof of proposition 3.1 is now complete.

5. The upper bound

To prove the upper bound, it is sufficient to define for each h∈(0,h₀) a deformation (u_h,w_h) with the property . Indeed, combining this with (3.8) gives the desired upper bound (with a possibly different c₁ in the definition of κ). As we do not attempt to get the optimal κ, it is enough to do this only for some discrete set of h, which has 0 as its limit point and has the property that neighbouring h differ at most by a factor of 2. In practice, we will eventually restrict our attention to values of h such that h^δ−1/2 is an integer, where the value of δ is close to 0. To simplify the notation, we drop the subscript h from the deformation to be constructed, writing (u,w) rather than (u_h,w_h).

Our construction is guided by the proof of the lower bound. The basic idea is to modify the minimizer of F_h by wrinkling where necessary, and estimate the increase in the energy due to wrinkling. Using (3.14), the amount of arclength we need to waste at each circle (together with the optimal length scale of wrinkling) can be read off from . Nevertheless, learning from the proof of the lower bound we anticipate that we should not really use wrinkling with the optimal period, as it will be costly to change this period too often—B(r) would then be too large (in fact of order O(1)). Anyway, as we explained in §2, it is not necessary for the wavenumber k to take the ‘optimal’ value at radius r. Rather, what matters is that be at most of order h^1/2. We shall achieve this by making the wrinkling modes appear/disappear over length scales of order h^1/2. It takes some time to motivate the construction. We shall explain the key ideas in two passes.

First pass. Our first pass is unsuccessful, but still informative. As noted above, we propose to use just choices of k that are integer multiples of h^−1/2, changing from one k to the next on a length scale of order h^1/2. As the amplitudes of the modes change over scale h^1/2, one finds after some calculation that B(r) is of order h (which is acceptable). But what does it mean to ‘change from one k to the next’? The obvious (though ultimately unsuccessful) idea is to use ‘building blocks’, as done for example in [13,20,21]. The building block between two radii (say, r₁ and r₂) would have single-mode wrinkling at the two extremes (r=r₁ and r=r₂) and a suitable interpolation in the middle. A standard approach to this interpolation would be to take , where k₁,k₂ are the optimal choices at r₁ and r₂, respectively, and f=1,g=0 near r₁ while f=0,g=1 near r₂. Recall that k₁,k₂∼h⁻¹, while, by our choice of r₁ and r₂, |k₁−k₂|∼h^−1/2. To make the second term in the ‘error’ R_h (see (3.7)) negligible one needs to choose −∂_θu_θ to be approximately the deviation of (∂_θw)²/2r from its average. Computing , we see that the first two terms are of order 1 with a period of order h (as k₁,k₂∼h⁻¹), and therefore their contribution to u_θ (i.e. after integrating once in θ) will be of order h. However, the remaining term is problematic. Indeed, it can be written as . The (k₁+k₂) term is not harmful by the same argument as above, however the (k₁−k₂) term is problematic. In fact, it provides a term of order h^1/2 in the expression for u_θ; this occurs because (k₁−k₂)∼h^−1/2 whereas k₁, k₂ and k₁+k₂ are all of order h⁻¹. Moreover, as w changes with r on length scale h^1/2, so should u_θ. Thus, one expects ∂_ru_θ∼1. This makes the third term in R_h (the cross term) of order 1—much too large.

In summary: we need a better answer for what it should mean to ‘change from one k to the next’.

Second pass. To get started, we need some notation. We will consider an ansatz (u,w) of the form and , where for each r∈(0,r₀) we require . Then using (3.5), we get that

5.1

where recall that , and ξ(r,θ)=w(r,θ)−r²/2R. Using (3.14), we know that W_r−W_rel completely vanishes in the tensile region (i.e. where and w_osc=0) and will be small also in the rest of the domain (i.e. in the wrinkled region) provided that both is approximately equal to and is small in the support of a_k. The first condition says that the wrinkling wastes the right amount of arclength, while the second condition says the wrinkling should be near the optimal frequency .

The key issue is how to avoid the problematic term that appeared in u_θ in the first pass. To explain the idea, we pretend for the moment that the frequencies k are allowed to be real valued, not just integers. Let us define , where m is a non-negative smooth mask (specifically: a non-negative function supported in such that ). The function A(r) modulates w and should be chosen so that the wrinkled profile wastes the right amount of arclength. For such w we compute . As before we want to compute the integral (in θ) of this quantity, which has the form . Focusing on the latter ‘troublesome’ integral involving the term , we see that while its value is not small it does not change in r (except for the dependence on A), which can be seen by the change of variables in the double integral. Therefore, the contribution to u_θ from this part will be r-independent, and so ∂_ru_θ will not be too large. For discrete frequencies, this problematic quantity will become h-periodic in r; the periodicity can be used to show that it is almost constant with a very small derivative.

The argument just sketched almost works. Unfortunately, it does not quite work, because when the r-derivative of u_θ falls on the A(r)-term one seems to need that u_θ itself is small—which is unfortunately not true. To overcome this difficulty, the argument presented below includes a further tweak—it uses only frequencies that are multiples of h^δ−1/2 for some δ>0.

In the rest of this section, we use the preceding ideas to give an honest proof of the upper bound. To get started, we fix a small δ>0 and we require from now on that all the constants be independent of δ. For r∈(0,r₀) and θ∈[0,2π), we define with

where m should as before be a smooth non-negative mask, A will be chosen later and N:=h^δ−1/2 has without loss of generality an integer value. As we will need estimates on derivatives of m, we make a particular choice if and m=0 elsewhere.

To estimate the excess energy, we now estimate term by term the r.h.s. of (5.1). Using the definitions and properties of W_r and W_rel (see (3.6) and (3.14)), we estimate the difference of the third and fourth term in (5.1)

5.2

5.3

where . To estimate (5.3), we observe that the function vanishes for and nearby, and by the support condition for m we see that (5.3) is bounded from above by A²(r)h^δh^−δ[(h^1/2+δ/r)h^−δ]²=A²(r)h/r². As we will eventually choose A such that and A will be supported away from the origin, we see that we will have . To simplify the notation, from now on we will assume that α_s=1 (in the general case all the subsequent constants might depend also on α_s).

In order to make (5.2) small, we would like to choose a value of A(r) such that if the r.h.s. is positive (wrinkled region), and A(r)=0 if it is non-negative (non-wrinkled region). We have introduced as a proxy (a less oscillating approximation) for σ, because we do not want A to oscillate on scale h^δ, which would be inevitable if we defined A using . The advantage of using instead of σ to define the value of A is that in this case A² is as smooth as . In addition, as we will also need control on derivatives of A (and not only A²), we cut off A on scale h near the transition between the flat and wrinkled region (where derivatives of A would be singular)

with η being a smooth cut-off for in (i.e. η(t)=1 if t>2 and η(t)=0 if t<1). While without the cut-off η the derivative A′(r) would blow-up like (r−r_w)^1/2, due to this cut-off we see that , and similarly .

To estimate (5.2), we will use the following simple observation: for any smooth compactly supported function f and any there exists C, which depends on the support of f, such that for any t∈(0,1) and any shift we have

5.4

Indeed, for n=0 this holds more generally for the mth derivative of f, as implies

where . Moreover, for any m≥1, we have

and so by induction we get (5.4) for any derivative of f. In addition, as the previous chain of equalities (except for the first one) holds also for m=0, we get (5.4) also for f itself, which finishes the proof of (5.4).

This observation, applied to f=m² with n≥1/δ, yields after a change of variables the estimate

where in the last step we used an estimate for . Using the triangle inequality, we have . As the first term on the r.h.s. vanishes if r−r_w∉(0,2h) and is O(1) inside this interval, the previous estimate on implies that (5.2) is bounded by (C_m/δ)^(C_m′/δ)h.

Next, we turn to . From now on we will repeatedly use the fact that, due to the support condition on m, we have m[h^δk−rh^−1/2]≠0 only for at most h^−δ values of k, and . Owing to this, and also as , we see that , where the last h^−1/2 comes from the derivative in r. Hence in the wrinkled region and B=0 otherwise, and so .

We now estimate the remainder R_h, which consists of five terms

To do the estimates we first define u_θ and u_r

To estimate T₁, we use the definition of w_osc, together with the support condition on m and the fact that , to see that and . Therefore, we immediately see that . Using , we get , which by previous estimates on w_osc and ∂_rw_osc combined with the estimate on (see (3.9)) implies . Using the definition of u_θ, we see that , which together with the fact that and the estimate implies .

To estimate |T₃|, we first focus on |∂_r(u_θ/r)|, by dealing with the + and − part separately. For the + part, we have

where here and below we use an abbreviation m[…]m[…]=m[h^δk−rh^−1/2]m[h^δl−rh^−1/2]. Because the summation is performed over h^−2δ pairs of k,l, , and , the first half . The second half is different as the derivative falls on m, which introduces an additional factor h^−1/2, leading to the estimate . Altogether, we see that . For the − part, we have

Using the arguments above and N=h^{−(1/2−δ)}, the first half . This idea would not be enough for the second half, because the additional factor h^−1/2 would ruin the estimate. Instead, for fixed θ let us consider and observe that by a change of variables f(r) is h^1/2+δ-periodic. Moreover, we see that for any we have |f⁽ⁿ⁾|≤(C_mn)^(C_m′n)h^(1−n)/2, where we used estimates on m⁽ⁿ⁾ and the fact that each derivative introduces a factor h^−1/2. Let us now fix n≥1, and observe that f⁽ⁿ⁾ has to have mean zero (due to periodicity of f⁽ⁿ⁻¹⁾), in particular there exists t∈(0,h^1/2+δ) such that f⁽ⁿ⁾(t)=0. Using the bound on f⁽ⁿ⁺¹⁾ and first-order Taylor expansion, we get for any s∈(0,h^1/2+δ) that . We can now iterate such an estimate, and, as in each step we improve the exponent by δ, after 1/(2δ) steps we get that |f′|≤(C_mn)^(C_m′n)h^1/2 with n∼1/δ, which then leads to an estimate |∂_r(u_θ,−/r²)|≤(C_m/δ)^(C_m′/δ)h^1/2−2δ. The estimate |∂_r(u_θ/r)|≤(C_m/δ)^(C_m′/δ)h^1/2−2δ follows immediately.

Using that and the definition of u_r, we see that

As , which we get by direct computation, this in combination with the estimate on ∂_rw_osc and yields , and so .

To deal with T₄, we observe that , which then using bounds on A′ and A′′ can be estimated by Ch^−(1+δ)/2, which together with the additional h² factor gives .

Finally, we see that , and immediately follows. Altogether, we have shown that the excess energy is bounded by (C_m/δ)^(C_m′/δ)h^1−4δ. By choosing , we obtain an estimate , and the proof of the upper bound is complete.

With this choice of δ, we also observe that a simple estimate |∂_θw_osc|≤Ch^−δ/2 turns into . Hence, although the slopes are not uniformly bounded in h, they explode with a rate slower than any power of h.

Authors' contributions

Both authors conceived of and designed the study. P.B. led the investigation of the upper and lower bounds. Both authors drafted and approved the manuscript.

Competing interests

We declare we have no competing interests.

Funding

Support is gratefully acknowledged from NSF grant nos. DMS-1311833 and OISE-0967140 (R.V.K.), and from DFG grant no. BE 5922/1.1 (P.B.).