Growth-induced delamination of an elastic film adhered to a cylinder

Abstract

We study the delamination induced by the growth of a thin adhesive sheet from a cylindrical, rigid substrate. Neglecting the deformations along the axis of the cylinder, we treat the sheet as a one-dimensional flexible and compressible ring, which adheres to the substrate by capillary adhesion. Using the calculus of variations, we obtain the equilibrium equations and in particular arrive at a transversality condition involving in a non-trivial way the curvature of the substrate, the extensibility of the ring and capillary adhesion. By numerically solving the equilibrium equations, we show that delamination by growth occurs through a discontinuous transition from the fully adherent solution to the partially delaminated one. The shape of the delaminated part can take the form either of a ruck, with a small slope, or a fold, with a large slope. Furthermore, in the weak adhesion regime, complete delamination may occur. We construct the phase diagram between the different solutions in the parameter space. In the quasi-incompressible limit, numerical results are also supported by asymptotic calculations both in the strong and weak adhesion regimes.

Keywords

Delamination snap-through buckling capillary adhesion transversality condition

1. Introduction

In everyday experience, the peeling of adhesive films from both hard and soft, curved substrates is a fairly recurring event. Although, for example, the formation of a wrinkle on a bottle label has trivial consequences, the delamination of thin films becomes a more significant issue for coating technologies such as stretchable electronics [1 –3]. Thus, understanding the mechanics of adhesive sheets becomes crucial for applications, but also provides an interesting mathematical paradigm in which mechanics of slender bodies, adhesion properties and substrate geometry meet [4].

Delaminated blisters originate in compressed films [5, 6]. This compression may be due to external compressive forces applied to the edge of the sheet, or to the growth of the sheet when it is confined, or to thermal dilation.

Delamination blister formation is a classic problem, in pioneering works the analysis was limited to blisters with a small slope in the realm of small deformations theory. However, in many new applications, this limitation is inappropriate and many subsequent studies modeled thin films as nonlinear Euler’s Elasticae [7 –9], allowing large deflections to be handled. Moreover, the occurrence of delamination blister has been studied extensively over the last two decades in a variety of situations with both inextensible and extensible rods adhered to flat adhesive substrates [10 –15], considered either rigid or soft.

Delamination also occurs when flexible rings are confined within a small container, whether rigid or flexible, both in the presence or absence of adhesion [10, 16 –19]. In these cases, the curvature of the confining wall affects the equilibrium shape by promoting the regions of adhesion, acting as an effective adhesion.

In an inextensible sheet, the transition to a delaminated shape from a flat or concave substrate requires an infinite compressive force [20]. In contrast, if a small compression of the sheet is allowed, the compressive force remains finite and delamination occurs via a discontinuous transition to a blister of finite amplitude [12, 13, 17].

In this work, we study the delamination of a compressible thin film that adheres to the outer surface of a cylinder by capillary action. The aim of the paper is to extent results obtained for a compressible adhesive sheet on a flat substrate to a circular substrate [13, 18], in the same time it generalizes the problem of a compressible film in contact with a curved surface [17] in presence of capillary adhesion. Extensibility and adhesion play a role in defining the two key characteristic lengths of the model, namely, $ℓ_{b}$ that relates bendability to extensibility, and the elastocapillary length $ℓ_{ec}$ that relates bendability to adhesion. We show how the physical features of the transition depend on the values of $ℓ_{b}, ℓ_{ec}$ and the radius of the cylinder $r$ and derive the phase diagrams for different values of the ratios $ℓ_{b} / r$ and $ℓ_{ec} / r$ .

By comparing the stored energies of the adhered state, the small-slope ruck (Figure 1(c)), and the fold configuration (Figure 1(d)), we determine the possible behaviors of growing thin sheet. In the nearly-incompressible case, when the adhesion is sufficiently strong, i.e., $ℓ_{b} << ℓ_{ec} << r$ , we find that the film delaminates into a small-slope ruck, which then grows and finally turns into a large-slope fold, when $ℓ_{b} = 10^{- 5 / 2} r$ , while the film delaminates directly into a large slope fold, when $ℓ_{b} = 10^{- 3 / 2} r$ . Finally, when the adhesion is weak $(ℓ_{ec} >> r)$ , we only observe a transition to a ruck which then develops to a completely delaminated solution (Figure 1(b)) as the film grows.

Figure 1.

Schematic representation of the film profiles: (a) fully adhered configuration, (b) fully delaminated, (c) ruck, and (d) fold.

The article is organized as follows. In section 2, we posit the energy functional of the model from which we derive the equilibrium equations and the boundary conditions in section 3. The variational approach provides a novel transversality condition at the detachment point, which combines the intrinsic lengths in a non-trivial way. In section 4, we discuss the main results of our theory, and compare the numerical results with the asymptotic approximations, as derived in Appendix 1. Finally, in section 5, we draw the conclusions.

2. Energy functional

We consider an infinite cylinder of radius $r$ , covered with a tape of natural length $L$ , naturally straight. We assume the tape to be compressible, i.e., its length may change when the tape is deformed. Furthermore, the length of the rod can change as the rod grows, which corresponds to varying $L$ .

The geometry of the curve is described by the position vector $r (S) = [x (S), y (S)]$ , where $S$ is the referential arc-length. The unit tangent and unit normal to the curve are given by $τ (S) = [\cos θ (S), \sin θ (S)]$ and $ν (S) = [- \sin θ (S), \cos θ (S)]$ , where $θ (S)$ is the counter-clockwise angle between the $x - axis$ and the tangent. Since we consider extensible rods, it is convenient to introduce also the local stretch $λ (S) : = ∥ r^{'} (S) ∥$ . Furthermore, we denote with $e_{α}$ the unit tangent to the circular substrate forming an angle $α$ with the $x - axis$ (see Figure 2).

Figure 2.

Schematic representation of the elastic film in contact with the outer surface of a cylinder of radius $r$ . We assume mirror symmetry with respect to the $y - axis$ . The free part of the curve is parameterised by values of the referential arc length $S \in [0, \bar{S})$ , where $\bar{S}$ corresponds to the detachment point.

We derive the equilibrium equations, and the boundary conditions for the partial delaminated solution, by imposing stationarity of the energy functional which comprises two parts: the energy of the detached region $W_{f}$ and the energy of the adhered region $W_{a}$ , where

W_{f} = \int_{0}^{\bar{S}} w_{f} d S, W_{a} = \int_{\bar{S}}^{\frac{L}{2}} w_{a} d S

(1)

with

w_{f} = k {(θ^{'})}^{2} + b (λ - 1)^{2} + 2 n \cdot (r^{'} - λ τ),

(2a)

w_{a} = k \frac{λ^{2}}{r^{2}} + b (λ - 1)^{2} + 2 n \cdot (r^{'} - λ e_{α}) - 2 λ Δ γ .

(2b)

Both $w_{f}$ and $w_{a}$ account for a bending term, proportional to the bending rigidity $k$ , and a strain energy, proportional to the compression modulus $b$ . The quantity

ℓ_{b} : = \sqrt{\frac{k}{b}}

defines a characteristic length. Since $b = EA$ and $k = EI$ , being $E$ is the elastic modulus of the material, $A$ is the area of the film cross-section, and $I$ is the second moment of area of the sheet cross-section, the intrinsic length $ℓ_{b}$ turns out to be of the order of the strip thickness.

The last term in equation (2a) enforces the constraints

x^{'} = λ \cos θ, y^{'} = λ \sin θ,

(3)

which translate the assumption that the rod is extensible but unshearable. The vector $n$ is a Lagrange multiplier representing the rod’s internal force [21].

The last term in the energy density (2b) is an adhesion-promoting term, proportional to the adhesion area through the constant $Δ γ$ , the sheet-substrate adhesion energy density. Thus, a further intrinsic characteristic length can be defined

ℓ_{ec} : = \sqrt{\frac{k}{Δ γ}},

(4)

which is called the elastocapillary length.

3. Equilibrium equations

Let us consider $h, η$ , and $u$ be the variations of $θ, λ$ , and $r$ , respectively. Within the free region, the variation $h, η$ , and $u$ are arbitrary, so that we obtain

\begin{matrix} δ W_{f} = \int_{0}^{\bar{S}} [(\frac{\partial w_{f}}{\partial θ} - {(\frac{\partial w_{f}}{\partial θ^{'}})}^{'}) h + \frac{\partial w_{f}}{\partial λ} η \\ - {(\frac{\partial w_{f}}{\partial r^{'}})}^{'} \cdot u] d S + {(\frac{\partial w_{f}}{\partial θ^{'}})}_{{\bar{S}}^{-}} {\bar{h}}^{-} + {(\frac{\partial w_{f}}{\partial r^{'}})}_{{\bar{S}}^{-}} \cdot {\bar{u}}^{-} + {\bar{w}}_{f}^{-} δ \bar{S}, \end{matrix}

(5)

where bar above the variables denotes that these variables are calculated in $S = \bar{S}$ . A superscript “minus” (respectively, “plus”) means that the corresponding function is evaluated as $S$ approaches $\bar{S}$ from the left (respectively, right).

It is worth noticing that $S = \bar{S}$ is not fixed and, hence, any variations at the detachment point should include a further contribute due to change of the curve length, so that

δ {\bar{r}}^{-} = {\bar{u}}^{-} + {\bar{λ}}^{-} τ^{-} δ \bar{S},

(6a)

δ {\bar{θ}}^{-} = {\bar{h}}^{-} + ({\bar{θ}}^{'})^{-} δ \bar{S} .

(6b)

The virtual displacement in the adhered region is purely tangential to the ring, i.e., $u = u_{α} e_{α}$ , where $e_{α}$ is the unit tangent to the substrate, so that

δ W_{a} = \int_{\bar{S}}^{\frac{L}{2}} [\frac{\partial w_{a}}{\partial λ} η - {(\frac{\partial w_{a}}{\partial r^{'}})}^{'} \cdot e_{α} u_{α}] d S - {(\frac{\partial w_{a}}{\partial r^{'}} \cdot e_{α})}_{{\bar{S}}^{+}} {\bar{u}}_{α}^{+} - {\bar{w}}_{a}^{+} δ \bar{S} .

(7)

Since the detachment point lies on a given curve (in our case a circumference), its virtual displacement can be due to either a longitudinal deformation of the rod (which occurs when the rod is stretched but material points in contact with the substrate do not change and $\bar{S}$ remains fixed) or to a change in the contact point (which can occur even if the rod is unstretched). Summing these two contributions, we get

δ {\bar{r}}^{+} = ({\bar{u}}^{+} \cdot e_{α} + {\bar{λ}}^{+} δ \bar{S}) e_{α} .

(8)

Note that, since the angle $π - \bar{θ}$ subtends half of the adhered arc, the length of the adhered curve can be expressed as

r (π - \bar{θ}) = \int_{\bar{S}}^{\frac{L}{2}} λ d S,

(9)

whence

r δ {\bar{θ}}^{+} = - \int_{\bar{S}}^{\frac{L}{2}} δ λ d S + {\bar{λ}}^{+} δ \bar{S} .

(10)

Using equations (6b) and (10), with the requirement that $δ {\bar{θ}}^{-} = δ {\bar{θ}}^{+}$ , we get

{\bar{h}}^{-} = (\frac{{\bar{λ}}^{+}}{r} - {({\bar{θ}}^{'})}^{-}) δ \bar{S} - \int_{\bar{S}}^{\frac{L}{2}} δ λ d S .

(11)

The requirement that the first term of $δ W_{f}$ vanishes for any arbitrary choice of $h (S)$ , $η (S)$ , and $u (S)$ , leads to equilibrium equations of the delaminated curve:

k θ^{''} + λ n_{y} \cos θ - λ n_{x} \sin θ = 0,

(12a)

n_{x} \cos θ + n_{y} \sin θ - b (λ - 1) = 0,

(12b)

n^{'} = 0,

(12c)

which have to be supplemented with the shape equation (3). We are considering the following Dirichlet conditions in $S = 0$ ,

x (0) = 0, θ (0) = 0,

(13)

so that also the corresponding variation fields, $u_{x}$ and $h$ , vanish in $S = 0$ . By contrast, we do not assume vanishing variation for the vertical component $u_{y}$ , so that we obtain the free boundary condition

n_{y} (0) = 0 .

(14)

This, together with equation (12c), yields $n_{y} (S) \equiv 0$ , and we can omit $n_{y}$ from equation (12).

A similar argument for the adhered region yields

n_{α} - b (λ - 1) + Δ γ + \frac{k}{r} (\frac{λ}{r} - ({\bar{θ}}^{-})^{'}) = 0,

(15a)

n^{'} \cdot e_{α} = 0 .

(15b)

Assuming a constant $λ$ in the adhered region, we deduce from equation (15a) that also $n_{α}$ is constant. Hence, equation (15b) implies that the normal (or radial) component of $n$ vanishes.

In order to study the remaining boundary terms, we assume the continuity conditions $δ {\bar{θ}}^{+} = δ {\bar{θ}}^{-}$ and $δ {\bar{r}}_{α}^{-} = δ {\bar{r}}_{α}^{+}$ , where $δ {\bar{r}}_{α}^{\pm} : = δ {\bar{r}}^{\pm} \cdot {\bar{e}}_{α}$ . Thus, the request that the coefficient of $δ {\bar{r}}_{α}$ vanishes yields the continuity condition of the internal force tangential component

{\bar{n}}^{-} \cdot {\bar{e}}_{α} = {\bar{n}}^{+} \cdot e_{α},

(16)

while the request that the $δ \bar{S}$ coefficient vanishes leads to

w_{f} (S^{-}) - w_{a} (S^{+}) + {\frac{\partial w_{f}}{\partial θ^{'}} |}_{\bar{S}} (\frac{{\bar{λ}}^{+}}{r} - {({\bar{θ}}^{'})}^{-}) + {\bar{n}}_{α}^{+} {\bar{λ}}^{+} - {\bar{n}}_{α}^{-} {\bar{λ}}^{-} = 0 .

(17)

Combining equations (12b), (15a), and (16), we derive ${\bar{λ}}^{-}$ and we replace the result into equation (17) so getting the transversality condition at the detachment point

({\bar{λ}}^{+} - r ({\bar{θ}}^{'})^{-})^{2} - \frac{2 r^{2} {\bar{λ}}^{+}}{ℓ_{ec}^{2}} + \frac{ℓ_{b}^{2}}{ℓ_{ec}^{4} r^{2}} [ℓ_{ec}^{2} ({\bar{λ}}^{+} - r ({\bar{θ}}^{'})^{-}) - r^{2}]^{2} = 0 .

(18)

We note that in the context of the mechanics of elastic rods, this condition has been a subject of interest for many years. A comprehensive treatment of this transversality condition for elastic rods can be found in the works of O’Reilly [22, 23], where it is derived as a balance of “material momentum” and it is considered as an independent balance law in addition to the balance of forces and moments. As such, our equation (18) can be viewed as an implicit manifestation of a balance of “material momentum”. However, we prefer here a more direct approach and derive it from a classical variational principle. Unlike the cases classically studied, our condition (18) takes into account adhesion, extensibility, and curvature of the container, the combined effect of which, to the best of our knowledge, has never been explicitly considered before.

Furthermore, at $S = \bar{S}$ , the following boundary conditions hold

x (\bar{S}) = r \sin \bar{θ}, y (\bar{S}) = r (1 - \cos \bar{θ}) .

(19)

Note that, since in the adhered region $λ (S) = {\bar{λ}}^{+}$ , equation (9) yields

\bar{θ} = π - \frac{{\bar{λ}}^{+}}{r} (\frac{L}{2} - \bar{S}) .

(20)

3.1. Known limiting cases

In the limit of vanishing capillary adhesion $(ℓ_{ec} / r \to \infty)$ , the transversality condition (18) yields (see the literature [17])

{\bar{λ}}^{+} - r ({\bar{θ}}^{'})^{-} = 0

(21)

By contrast, in the inextensible case $(ℓ_{b} = 0, {\bar{λ}}^{+} = 1)$ , equation (18) reads (see also the literature [10, 16, 24])

{(1 - r {({\bar{θ}}^{'})}^{-})}^{2} - \frac{2 r^{2}}{ℓ_{ec}^{2}} = 0 .

(22)

Finally, the flat limit ( $\bar{S} / r \to 0$ and $ℓ_{ec} / r \to 0$ ) provides (see the literature [11, 12])

[({\bar{θ}}^{'})^{-}]^{2} - 2 \frac{{\bar{λ}}^{+}}{ℓ_{ec}^{2}} + \frac{ℓ_{b}^{2}}{ℓ_{ec}^{4}} = 0 .

(23)

3.2. Summary of the equations

For the reader’s convenience, we report here the main equations, the unknown fields, and the boundary conditions of our problem.

For the free part $S \in [0, \bar{S})$ , in the absence of vertical loads on the free curve, $n_{y} (S) = 0$ , so that the balance equations read

k θ^{''} - λ n_{x} \sin θ = 0,

(24a)

n_{x}^{'} = 0,

(24b)

x^{'} = λ \cos θ,

(24c)

y^{'} = λ \sin θ,

(24d)

where the stretch $λ (S)$ is calculated as

λ = 1 + \frac{n_{x}}{b} \cos θ .

(25)

The only unknown in the adhered region is the constant stretch ratio ${\bar{λ}}^{+}$ , which can be calculated from the continuity of the tangential component $n_{α}$ . Combining equations (15a), (16), and (25), we obtain

{\bar{λ}}^{+} = \frac{b r^{2}}{k + b r^{2}} (λ ({\bar{S}}^{-}) + \frac{Δ γ}{b} + \frac{k}{r} θ^{'} ({\bar{S}}^{-})) .

(26)

Finally, we also have the identity (9)

\bar{θ} = π - \frac{{\bar{λ}}^{+}}{r} (\frac{L}{2} - \bar{S}) .

(27)

Overall, we have four $S - dependent$ unknowns $x (S)$ , $y (S)$ , $θ (S)$ , and $θ^{'} (S)$ , and two constant unknowns $\bar{S}$ , $n_{x}$ . From these, we can reconstruct the stretch ratio $λ (S)$ in the free part (from equation (25)), the stretch ratio ${\bar{\bar{λ}}}^{+}$ in the adhered region (from equation (26)), and the detachment angle $\bar{θ}$ (from equation (27)).

The boundary conditions are

x (0) = 0, θ (0) = 0, θ (\bar{S}) = \bar{θ}

(28a)

x (\bar{S}) = r \sin \bar{θ}, y (\bar{S}) = r (1 - \cos \bar{θ}) .

(28b)

Finally, the transversality condition (18) yields the additional equation at the boundary, necessary to close the problem.

4. Results

We study the problem for different values of the parameter $L$ , which mimics a growth process, and for various values of the characteristic lengths $r, ℓ_{b}$ , and $ℓ_{ec}$ , which depend on the geometric and constitutive features of the system.

We will consider three different sets of possible solutions, namely (see Figure 1):

(a) The completely adhered solution. This equilibrium configuration is a circumference of radius $r$ , fully in contact with the substrate. The tape is uniformly strained/compressed, with

λ_{adh} = 2 π \frac{r}{L} .

(29)

(b) The fully delaminated solution is represented by a circumference, with no contact point with the substrate. This solution is admitted only if $R \geq r$ , where $R = L / (2 π)$ .

(c), (d)Partially delaminated configurations, where the detached region is a symmetric blister of referential length $2 \bar{S}$ . The shape of this blister depends on $R / r$ , on the constitutive parameters of the tape and on the capillary adhesion $Δ γ$ . We distinguish between two partially delaminated solutions: a ruck (see Figure 1(c)), i.e., a small-slope delamination where opposite sides of the rod do not touch each other, and a fold (see Figure 1(d)), in which the rod undergoes large-slope deformation with self-contact.

We have studied the problem by numerically solving equations (24) with the corresponding boundary conditions (28a) and (18), using the MATLAB function bvp4c, for several values of the parameters. A first feature of the solutions is that, given $ℓ_{b}$ , the detachment angle $\bar{θ}$ as a function of the excess-length $ϵ$ ,

ϵ : = \frac{L - 2 π r}{2 π r},

(30)

has two characteristic trends that depend on the value of $ℓ_{ec}$ . For weak capillary-strength, i.e., large values of $ℓ_{ec} / r, \bar{θ}$ is a monotonic increasing function of $ϵ$ , so that the growth leads to an increasing detachment of the tape until it induces a complete delamination when $\bar{θ} = π$ . By contrast, when $ℓ_{ec} / r$ is small, in a regime of strong adhesion, $\bar{θ}$ is non-monotonic and shows a maximum value (see Figure 3). Figure 3 also shows that there is a limiting value for the excess-length $ϵ$ below which only the adhered solution is possible. Above this threshold, the buckled solution comprises two branches (see also Figures 6 and 7) and correspondingly for one value of $ϵ$ we can observe two buckled solutions having different sizes and different detachment angles $\bar{θ}$ . After an initial increase in delamination for small $ϵ$ , the adhered region starts to grow again and the solution evolves from ruck to fold. Figures 4 and 5 show how the maximum detachment angle, ${\bar{θ}}_{\max}$ , and the corresponding $ϵ ({\bar{θ}}_{\max})$ , depend on $ℓ_{e c}$ .

Figure 3.

Detachment angle $\bar{θ}$ versus the excess-length $ϵ$ , with $ℓ_{ec} = 0.1 r, ℓ_{b} = 10^{- 5 / 2} r$ . Red dashed line shows the asymptotic approximation as given in equation (59).

Figure 4.

Maximum detachment angle as a function of $ℓ_{ec}$ , with $ℓ_{b} = 10^{- 5 / 2} r$ . Red dashed line shows the asymptotic approximation as given in equation (62).

Figure 5.

Maximum excess-length at detachment as a function of $ℓ_{ec}$ , with $ℓ_{b} = 10^{- 5 / 2} r$ . Red dashed line shows the asymptotic approximation as given in equation (61).

We have explored the cases $ℓ_{b} = 10^{- 3 / 2} r$ , and $ℓ_{b} = 10^{- 5 / 2} r$ , that correspond to very thin films. For these values, the typical observed solutions have a delaminated part with a much longer length than $ℓ_{b}$ , which implies that their compression energy is completely relaxed, i.e., ${\bar{λ}}^{+} \approx 1$ . Hence, we can approximate the solution with that of a Euler’s elastica and take advantage of the its closed-form solutions and their asymptotic approximations (see Appendix 1 for details).

Figures 6 and 7 show the comparison between the energies associated with each solution branch as a function of $ϵ$ , for $ℓ_{b} = 10^{- 3 / 2} r$ and two values of the elastocapillary length, namely, $ℓ_{ec} = 0.1 r$ and $ℓ_{ec} = r$ . The stored energy for the adhered configuration is easily obtained by substituting equation (29) in equation (1),

\frac{W_{adh}}{k / r} = - \frac{2 π r^{2}}{ℓ_{ec}^{2}} + \frac{π}{1 + ϵ} (1 + \frac{r^{2} ϵ^{2}}{ℓ_{b}^{2}}),

(31)

and is shown as a red solid line. The energy associated with the partially delaminated solutions is instead calculated with a numerical simulation, drawn in blue in Figures 6 and 7. It is interesting to notice that the delaminated solution comprises two branches. The upper branch, with higher energy, corresponds to smaller rucks, whereas larger rucks correspond to the lower branch. For sufficiently small values of $ϵ$ , only the adhered solution is found. Finally, the dashed black line in Figure 7 shows the energy of the folded solutions.

Figure 6.

Energy profiles with $ℓ_{b} = 10^{- 3 / 2} r$ , and $ℓ_{ec} = r$ . Black dotted line shows the energy of the inextensible case (Euler’s elastica).

Figure 7.

Energy profiles with $ℓ_{b} = 10^{- 3 / 2} r$ , and $ℓ_{ec} = 0.1 r$ . Black dashed line shows the energy in the folded configuration, the transition is determined by the intersection of the red line (adhered solution) with the blue/black line (corresponding respectively to the ruck/fold solutions). For this value of the parameters we observe a direct transition from the adhered solution to the fold.

In Figure 6, we also report the energy of the delaminated solution in the inextensible case (Euler’s elastica), calculated as in equation (48). As expected, when the compression reaches a critical value, the tape buckles so to relax its internal stress and compression. Thus, the buckled solution is, to a good approximation, determined by minimizing the bending energy, while the compression energy can be neglected. It is then natural to assume that the solution with $λ = 1$ , which correctly provides the buckled shape of the beam and its energy, is also correct in the presence of a moderate adhesion potential.

From the numerical analysis with the two chosen values of $ℓ_{b}$ , we identify different regions of the parameter space $(ℓ_{ec} / r, ϵ)$ , shown in Figures 8 and 9, and characterized by the global minimum of the energy. The boundaries of these regions are calculated as intersections between the energy branches of the various solutions, and identify the critical values of $ϵ$ as a function of $ℓ_{ec}$ . For example, in Figures 6 and 7, we show the intersection between the adhered and the delaminated energy branches, corresponding to a transition from the adhered solution to a ruck (in Figure 6) or to a fold (in Figure 7). Therefore, for sufficiently small values of $ℓ_{ec}$ , we observe that patterns of large-slope, fold-like structures may emerge directly from a uniformly compressed state, rather than grow gradually from small-slope rucks. This is in agreement with the results of Davidovitch and Démery [13], where the authors study the delamination from a flat rigid substrate under axial compression.

Figure 8.

Phase diagram with $ℓ_{b} = 10^{- 5 / 2} r$ . Dashed lines report the asymptotic results as given in equation (64) (dashed red line) and equation (68) (dashed blue line).

Figure 9.

Phase diagram with $ℓ_{b} = 10^{- 3 / 2} r$ . Dashed lines report the asymptotic results as given in equation (64) (dashed red line) and equation (68) (dashed blue line).

As can be seen from the phase diagrams in Figures 8 and 9, and from equations (64) and (68), the asymptotic approximations of $ϵ_{cr}$ have different critical exponents in the cases of strong or weak adhesion, $ℓ_{ec} << r$ or $ℓ_{ec} >> r$ respectively.

Furthermore, Figures 8 and 9 report also the transition lines from ruck to fold, that, for a fixed $ℓ_{ec} / r$ , correspond to the smallest $ϵ$ at which a self-contact of the film occurs.

Finally, the continuous line in the upper right part of the Figures 8 and 9 represents the critical threshold for the transition from buckling to complete delamination. This identifies the value of $ϵ$ at which $\bar{θ}$ reaches $π$ . In the inextensible case, this threshold is given in equation (42) of Napoli and Goriely [25]:

ϵ_{cr}^{(del)} = \frac{\sqrt{2}}{ℓ_{ec} / r - \sqrt{2}} .

(32)

In the two compressible cases treated, the numerical results are not significantly different from equation (32).

5. Conclusions

We have studied the delamination induced by the growth of a compressible thin elastic sheet adhering to the outside of a cylindrical surface, due to capillary adhesion. The quasi-incompressibility assumption is consistent with very thin sheets, where $ℓ_{b}$ is of the order of the sheet thickness.

We find that, due to the simultaneous presence of substrate curvature, capillary adhesion, and compressibility, the transversality condition, i.e. the additional boundary condition which serves to close the problem with unknown detachment length, is new and not obvious. This boundary condition, obtained through the classical methods of calculus of variations, reduces to expressions already known in the literature in special cases. This formula can be easily generalized to the case where the film covers the interior of the cylindrical surface instead of the exterior.

Our analysis is based on the numerical resolution of a nonlinear boundary value problem. The nonlinearity of the problem results in multiple solutions whose energies are subsequently compared with each other. We assume that the branch of minimum energy is the one that is actually observed in a possible experiment.

We can summarize the results as follows. If initially the sheet length is equal to the circumference of the cylinder, the sheet adheres perfectly. During growth, there is a first stage in which the sheet, while compressing, still adheres completely to the substrate, thanks to capillary adhesion. Above a certain compression threshold, the sheet undergoes a snap-buckling, partially detaching and forming a ruck, thus releasing its compression energy. The delaminated part is effectively stretch-free. This allows us to use Euler’s elastica model to approximated the solutions.

Euler’s elastica offers the enormous benefit of being integrable and admitting closed-form solutions in terms of elliptic functions. Given the complexity of these functions, the obtained solutions are not immediately interpretable. However, it is possible to develop asymptotic expansions of these solutions in terms of the excess-length $ϵ$ . In the general case, the coefficients of this asymptotic expansions are obtained from solving transcendental equations which implicitly involve the characteristic lengths $ℓ_{b}, ℓ_{ec}$ , and $r$ . However, in both the strong adhesion limit $ℓ_{ec} << r$ and the weak adhesion limit $ℓ_{ec} >> r$ , it is possible to explicitly find these coefficients. This allows us to find useful asymptotic laws that relates the geometrical and material features of the problem. The success of these asymptotic results could be very useful in interpreting experimental results or in helping to design targeted experiments. For example, we are able to capture the critical thresholds for the onset of ruck solutions (see equations (64) and (68)), in perfect agreement with numerical simulations.

Footnotes

Appendix 1 Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the by the MIUR Project PRIN 2020, “Mathematics for Industry 4.0”, Project No. 2020F3NCPX, and also conducted under the auspices of the GNFM-INdAM.

ORCID iD

Gaetano Napoli

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