Sheet on a deformable sphere: Wrinkle patterns suppress curvature-induced delamination

The adhesion of a stiff film onto a curved substrate often generates elastic stresses in the film that eventually give rise to its delamination. Here we predict that delamination of very thin films can be dramatically suppressed through tiny, smooth deformations of the substrate, dubbed here “wrinklogami,” that barely affect the macro-scale topography. This “prolamination” effect reflects a surprising capability of smooth wrinkles to suppress compression in elastic films even when spherical or other doubly curved topography is imposed, in a similar fashion to origami folds that enable construction of curved structures from an unstretchable paper. We show that the emergence of a wrinklogami pattern signals a nontrivial isometry of the sheet to its planar, undeformed state, in the doubly asymptotic limit of small thickness and weak tensile load exerted by the adhesive substrate. We explain how such an “asymptotic isometry” concept broadens the standard usage of isometries for describing the response of elastic sheets to geometric constraints and mechanical loads.

Figure 1

(a) A thin adhesive film delaminates from a glass ball of radius $R$ when its area ($\sim W^2$) exceeds a critical fraction ${\varphi }_{\text{rig}}$ [Eq. (1)] of the surface area of the ball ($\sim R^2$). Before delamination, longitudes of the film acquire an average strain $\sim {(W/R)}^2$. (b),(c) Top and side views of an ultrathin PS film floating on a curved liquid surface (see [8]). The film remains attached to the surface, and the joint deformation consists of flattening of the liquid portion beneath the film (c) and a periodic array of radial wrinkles (b). Courtesy of H. King and N. Menon. (d) Schematic figure of our model system: a thin disk (yellow, light gray) of Young's modulus $E_f$, thickness $t$, and radius $W$ is attached to a spherical substrate (blue, dark gray) of radius $R\gg W$, by the substrate-vapor surface tension $\gamma$ that pulls on the film's edge. (For simplicity, we assume $\gamma \approx \Gamma$, where $\Gamma$ is the adhesion energy; see Sec. 2b2.) The resistance of the substrate to deformation of its spherical shape is modeled through a Winkler's stiffness $K$ (a ball of $N$ springs, each with an effective constant $4\pi R^{2}K/N$). The generalization to the case of an isotropic solid with Young's modulus $E_s$ is discussed in Sec. 5b.

Figure 2

(a) Phase diagram for the adhesion of thin films on curved substrates. Regime I (green, light gray): Soft substrates are deformed significantly, suppressing elastic strain and thus delamination at all values of the coverage fraction $\varphi$. Regime II (red, medium gray): Rigid substrates are undeformed, and the film is severely strained. For a given point in regime II we illustrate the delamination process upon increasing $\varphi$. Regime III (blue, dark gray): For substrates with intermediate stiffness, both their macro-scale deformation and strain in the laminated film are suppressed by forming fine wrinkles for ${\varphi }_{\text{wr}}<\varphi <{\varphi }_{\text{def}}$. Delamination occurs at $\varphi ={\varphi }_{\text{def}}\gg {\varphi }_{\text{rig}}$ [Eqs. (54b) and (56b)]. In the “no lamination” regime (amber, below lower horizontal dashed line), the energetic cost of bending precludes any adhesion. The boundaries between regimes II/III and III/I are described, respectively, by the relations: $K_{\text{rig}}\sim \Gamma /t^2$ [Eq. (51)] and $K_{\text{soft}}\sim \left(E_{f}t\right)/R^2$ [Eq. (52)]. Logarithmic scales are used for the axes, making the phase boundaries appear as straight lines. (b) An analogous phase diagram for an elastic substrate with modulus $E_s$. The lines that separate regimes II/III and III/I, are, respectively, $E_{s,\text{rig}}\sim {\gamma }^{3/4}{E}_f^{1/4}/t^{3/4}$ [Eq. (57)] and $E_{s,\text{soft}}\sim {\gamma }^{1/2}{E}_f^{1/2}{t}^{1/2}/R$ [Eq. (61)]. (c) Varying the substrate stiffness $K$, we plot the predicted evolution of the system for a fixed adhesion energy $\Gamma$ and film's thickness $t$ as the coverage fraction $\varphi$ increases. The blue (dotted) curve is described by Eqs. (54a) and (56a), and the black (solid) curve is described by Eqs. (54b) and (56b). Delamination occurs above the black (solid) curve, wrinkling occurs between the blue (dotted) curve and black (solid) curve, and large substrate deformations occur to the left of the cyan (vertical dashed) line. (d) An analogous diagram to (c) for an elastic substrate with modulus $E_s$.

Figure 3

A schematic diagram comparing the various normalized energies upon increasing the coverage fraction $\varphi$: the energy of axisymmetric configuration, $u^{\mathrm{axi}}$, is described by the blue (thin black) parabola; the energetic cost of adhesion, $\Gamma /E_{f}t$, is described by the black (thick) horizontal line; and the two components of the wrinkle energy, $u^{\mathrm{wr}}=u^{\mathrm{dom}}+u^{\mathrm{sub}}$ [Eq. (33)], where $u^{\mathrm{dom}}\sim w_{\mathrm{surf}}$ is the tensile work done by the adhesive substrate pulling on the film [thin red (gray) line], which overrides the energy stored in the compression-free stress field [see Eqs. (48) and (50)], and $u^{\mathrm{sub}}$ is the energetic cost of bending and substrate deformation associated with the formation of wrinkles [Eq. (49b), thick red (gray) line]. A wrinkled state emerges at values of $\varphi >{\varphi }_{wr}$, where the axisymmetric state is compressive near the edge and the combination of bending and substrate deformation cost is sufficiently small. The energy component $u^{\mathrm{sub}}$ is plotted for parameter regimes II, III-A, and III-B. In regime II, $u^{\mathrm{wr}}\approx u^{\mathrm{sub}}\gg u^{\mathrm{axi}}$ for all coverages $\varphi$, and delamination occurs at $\varphi ={\varphi }_{\mathrm{rig}}$ when $u^{\mathrm{axi}}$ exceeds $\Gamma /E_{f}t$. In regime III-A $u^{\mathrm{wr}}\approx u^{\mathrm{sub}}$, wrinkling sets in at $\varphi ={\varphi }_{\text{wr}}^A$, and delamination occurs at $\varphi ={\varphi }_{\text{def}}^A$. In regime III-B, $u^{\mathrm{wr}}\approx u^{\mathrm{dom}}$ [thin red (gray) line], wrinkles emerge at ${\varphi }_{\text{wr}}^B$, and delamination is suppressed until a large coverage fraction ${\varphi }_{\text{def}}^B\sim O\left(1\right)$ is reached. The nonzero intercept of the energy $u^{\mathrm{axi}}$ reflects the (small) bending energy of a film with macro-scale curvature $1/R$.

Figure 4

Plots of the shape $\zeta \left(r\right)$ and the stresses ${\sigma }_{rr}\left(r\right),{\sigma }_{\theta \theta }\left(r\right)$ in the unwrinkled, axisymmetric configuration for a fixed value of the confinement $\alpha \approx 17$ and a few representative values of $\tilde{K}$: $\tilde{K}=1$ [red (solid gray) line], $\tilde{K}=10$ [green (dashed gray) line], $\tilde{K}=100$ [blue (solid black) line], and $\tilde{K}=\infty$ (dotted black line), in which case the vertical displacement of the plate is exactly ${\zeta }_{\text{sph}}$ and the stress is given by the analytic solution, Eqs. (27).

Figure 5

The wrinkled shape of a laminated film [yellow (light gray)] on a spherical substrate [blue (dark gray)]. The out-of-plane undulations of the film and the attached substrate are accompanied by in-plane oscillations of the perimeter of the film, which are of the same periodicity but with amplitude reduced by a factor of $W/R$. The amplitude of the wrinkles has been exaggerated to highlight their shape.

Figure 6

Plots of the shape $\zeta \left(r\right)$ and the stresses ${\sigma }_{rr}\left(r\right),{\sigma }_{\theta \theta }\left(r\right)$ in the wrinkled configuration for a fixed value of the confinement $\alpha \approx 17$ and a few representative values of $\tilde{K}$: $\tilde{K}=1$ [red (solid gray) lines], $\tilde{K}=10$ [green (dashed gray) lines], $\tilde{K}=100$ [blue (solid black) lines], and $\tilde{K}=\infty$ (dotted black lines), in which case the (nonoscillatory) vertical displacement of plate is exactly ${\zeta }_{\text{sph}}$ and the stress is given by the analytic solution, Eqs. (36, 37, 38).

Figure 7

(a) The behavior of the laminated states in the bendability-confinement plane (${\epsilon }^{-1},\alpha$). Here we assume that the sheet is sufficiently thin such that the bendability is high ($\epsilon \ll 1$), and consider a fixed, small value of the deformability parameter ${\tilde{K}}^{-1}\ll 1$. The horizontal red (thick gray) line $\alpha ={\alpha }_c\left(\tilde{K}\right)$ corresponds to the critical confinement below which the tensile (adhesive) force $\gamma$ exerted at the edge of the sheet is sufficiently strong that the sheet is under pure tension. The vertical blue (thick gray) line ${\epsilon }^{-1}\approx \tilde{K}\gg 1$ (where $\epsilon ={\epsilon }_g$) corresponds to the minimal bendability value for which the energetic cost $u^{\mathrm{sub}}$ of bending and substrate deformation makes the wrinkled state energetically favorable in comparison to the axisymmetric (compressed, unwrinked) state. The thin gray curve separates two parts of the parameter regime at which a wrinkle pattern is energetically favorable for the laminated state. Above this curve, the energy $u^{\mathrm{sub}}$ [Eq. (49b)] of bending and substrate deformation governs the energy $u^{\mathrm{wr}}$ of the wrinkled state. Below the thin gray curve, bendability is sufficiently large and the wrinkle energy is governed by the energy $u^{\mathrm{dom}}$ [Eq. (48)], associated with the work done by the tensile load at the film's edge. (b) Replotting the diagram in panel (a), where we replace the bendability and confinement parameters ${\epsilon }^{-1}$ and $\alpha$ with the pristine dimensionless parameters of thickness $\tilde{t}=t/R$ and coverage fraction $\varphi ={(W/R)}^2$, and consider fixed, small values of the deformability parameter ${\tilde{K}}^{-1}\ll 1$ and of the mechanical tension ${\delta }_m\ll 1$. The solid gray curves correspond to the respective curves in panel (a). The black curve marks the threshold ${\varphi }_{\text{def}}$ above which delamination is energetically favorable, and the vertical dashed orange (gray) line marks the maximal value of $\tilde{t}$, above which the film does not delaminate. The green (gray) double arrows show the parameter regimes that we call, respectively, II, III-A, and III-B (regime I, which corresponds to sufficiently soft substrate, i.e., $\tilde{K}\ll 1$, is not shown in this figure). If $\sqrt{{\delta }_m/\tilde{K}}<\tilde{t}<\sqrt{{\delta }_m}$, the system is at regime II, where the energetically favorable laminated state is axisymmetric (compressed, unwrinkled), and delamination becomes favorable for $\varphi >{\varphi }_{\text{rig}}$ [Eq. (1)]. If $\tilde{t}<\sqrt{{\delta }_m/\tilde{K}}$, the system is at parameter regime III, where the film becomes wrinkled at $\varphi >{\varphi }_{\text{wr}}$ and delamination becomes energetically favorable for $\varphi >{\varphi }_{\text{def}}$, where ${\varphi }_{\text{wr}},{\varphi }_{\text{def}}$ are given by Eqs. (55) and (56).

Figure 8

Diagrams analogous to Fig. 7, where the spherical substrate is assumed to be an isotropic solid of Young's modulus $E_s$ (instead of stiffness $K$). The two parameter axes are ${\tilde{t}}^{-1}$ and $\varphi$, as in Fig. 7, and the parameters with small, fixed values are now ${\delta }_m=\gamma /E_{f}t$ [as in Fig. 7] and the ratio $(E_s/E_f)$ between the Young's moduli of the substrate and the film. Rather than a single diagram, we find it easier to plot here three separate diagrams [47], for three characteristic values of $E_s/E_f$, which correspond, respectively, to (a) regime II; (b) regime III-A; and (c) regime III-B. Similarly to Fig. 7, regime I, which corresponds to sufficiently soft substrate, $E_s<E_{s,\text{soft}}$, is not shown here.

Figure 9

A schematic diagram that depicts the two distinct types of asymptotic isometries in problems of class (C), in which a film of thickness $t$ is subjected to a fixed geometric constraint and a weak tensile load $\gamma$ on its boundary. The horizontal and vertical axes are measures for the thickness and the exerted tensile load, respectively, for a given geometric constraint. (The actual, dimensionless measures are $B/Y{W}^2$ and $\gamma /Y$). Below a curve ${\gamma }^{*}\left(t\right)$ we expect the emergence of a wrinklogami pattern, where the isometry is approached asymptotically by homogenous suppression of all stress components [blue (dark gray) trajectory: “route 2 to isometry” in our classification]. Above the curve ${\gamma }^{*}\left(t\right)$, we expect the shape to be approximated as a perturbation to real isometric transformation (i.e., developable or piecewise-developable map) of a 2D film, which often involves the formation of stress-focusing zones at ridges and vertices [red (gray) trajectory: “route 1 to isometry” in our classification].