We address the fully-developed wrinkle pattern formed upon stretching a Hookean, rectangular-shaped sheet, when the longitudinal tensile load induces transverse compression that far exceeds the stability threshold of a purely planar deformation. At this "far from threshold" parameter regime, which has been the subject of the celebrated Cerda-Mahadevan (CM) model, the wrinkle pattern expands throughout the length of the sheet and the characteristic wavelength of undulations is much smaller than its width. Employing Surface Evolver simulations over a range of sheet thicknesses and tensile loads we elucidate the theoretical underpinnings of the far-from-threshold framework in this set-up. We show that the evolution of wrinkles comes in tandem with collapse of transverse compressive stress, rather than vanishing transverse strain, such that the stress field approaches asymptotically a compression-free limit, describable by tension field theory. We compute the compression-free stress field by simulating a Hookean sheet that has finite stretching modulus but no bending rigidity, and show that this singular limit encapsulates the geometrical nonlinearity underlying the amplitude-wavelength ratio of wrinkle patterns in physical, highly bendable sheets, even though the actual strains may be so small that the local mechanics is perfectly Hookean. Finally, we revisit the balance of bending and stretching energies that gives rise to a favorable wrinkle wavelength, and study the consequent dependence of the wavelength on the tensile load as well as the thickness and length of the sheet.
A controlled surface wrinkling pattern has been widely used in diverse applications, such as stretchable electronics, smart windows, and haptics. Here, we focus on hexagonal wrinkling patterns because of their great potentials in realizing anisotropic and tunable friction and serving as a dynamical template for making non-flat thin films through self-assembling processes. We employ large-scale finite element simulations of a bilayer neo-Hookean solid (e.g., a film bonded on a substrate) to explore mechanical principles that govern the formation of hexagonal wrinkling patterns and strategies fo…
The wrinkle pattern exhibited upon stretching a rectangular sheet has attracted considerable interest in the "extreme mechanics" community. Nevertheless, key aspects of this notable phenomenon remain elusive. Specifically -- what is the origin of the compressive stress underlying the instability of the planar state? what is the nature of the ensuing bifurcation? how does the shape evolve from a critical, near-threshold regime to a fully-developed pattern of parallel wrinkles that permeate most of the sheet? In this paper we address some of these questions through numerical simulations and anal…
Thin films buckle easily and form wrinkled states in regions of well defined size. The extent of a wrinkled region is typically assumed to reflect the zone of in-plane compressive stresses prior to buckling, but recent experiments on ultrathin sheets have shown that wrinkling patterns are significantly longer and follow different scaling laws than those predicted by standard buckling theory. Here we focus on a simple setup to show the striking differences between near-threshold buckling and the analysis of wrinkle patterns in very thin films, which are typically far from threshold.
We discuss mechanical buckling instabilities of a rigid film under compression interacting repulsively with a substrate through a thin fluid layer. The buckling occurs at a characteristic wavelength that increases as the 1/4th power of the bending stiffness, like a gravitational instability studied previously by Milner et al. However, the potential can affect the characteristic buckling wavelength strongly, as predicted by Huang and Suo. If the potential changes sufficiently sharply with thickness, this instability is continuous, with an amplitude varying as the square root of overpressure. We…
We analyze the buckling of a rigid thin membrane floating on a dense fluid substrate. The interplay of curvature and substrate energy is known to create wrinkling at a characteristic wavelength $λ$, which localizes into a fold at sufficient buckling displacement $Δ$. By analyzing the regime $Δ<<λ$, we show that wrinkles are unstable to localized folding for {\em arbitrarily small} $Δ$. After observing that evanescent waves at the boundaries can be energetically favored over uniform wrinkles, we construct a localized Ansatz state far from boundaries that is also energetically favored. The resul…
We investigate the elastic buckling of a triangular prism made of a soft elastomer. A face of the prism is bonded to a stiff slab that imposes an average axial compression. We observe two possible buckling modes which are localized along the free ridge. For ridge angles $ϕ$ below a critical value $ϕ^\star\approx 90^\circ$ experiments reveal an extended sinusoidal mode, while for $ϕ$ above $ϕ^\star$ we observe a series of creases progressively invading the lateral faces starting from the ridge. A numerical linear stability analysis is set up using the finite-element method and correctly predict…
When a spherical elastic capsule is deflated, it first buckles axisymmetrically and subsequently loses its axisymmetry in a secondary instability, where the dimple acquires a polygonal shape. We explain this secondary polygonal buckling in terms of wrinkles developing at the inner side of the dimple edge in response to compressive hoop stress. Analyzing the axisymmetric buckled shape, we find a compressive hoop stress with parabolic stress profile at the dimple edge. We further show that there exists a critical value for this hoop stress, where it becomes favorable for the membrane to buckle o…
It is well known that an annular sheet could wrinkle as a result of axisymmetric tensile loads applied at the edges. In this system, regions under compression appear due to Poisson effect in the azimuthal direction yielding an incompatible excess in length, so that the membrane has to buckle out of the plane. Then, radial wrinkles emerge following the direction of the tensile principal stress. This so called Lamé configuration has been widely used as theoretical and experimental paradigms for wrinkling instabilities. In this work, we explore the consequences of changing the geometry of this mo…
We investigate with experiments the twist induced transverse buckling instabilities of an elastic sheet of length $L$, width $W$, and thickness $t$, that is clamped at two opposite ends while held under a tension $T$. Above a critical tension $T_λ$ and critical twist angle $η_{tr}$, we find that the sheet buckles with a mode number $n \geq 1$ transverse to the axis of twist. Three distinct buckling regimes characterized as clamp-dominated, bendable, and stiff are identified, by introducing a bendability length $L_B$ and a clamp length $L_{C}(<L_B)$. In the stiff regime ($L>L_B$), we find that …
Numerical simulations of thin sheets undergoing large deformations are computationally challenging. Depending on the scenario, they may spontaneously buckle, wrinkle, fold, or crumple. Nature's thin tissues often experience significant anisotropic growth, which can act as the driving force for such instabilities. We use a recently developed finite element model to simulate the rich variety of nonlinear responses of Kirchhoff-Love sheets. The model uses subdivision surface shape functions in order to guarantee convergence of the method, and to allow a finite element description of anisotropical…
The buckling of elastic bodies is a common phenomenon in the mechanics of solids. Wrinkling of membranes can often be interpreted as buckling under constraints that prohibit large amplitude deformation. We present a combination of analytic calculations, experiments, and simulations to understand wrinkling patterns generated in a bilayer membrane. The model membrane is composed of a flexible spherical shell that is under tension and that is circumscribed by a stiff, essentially incompressible strip with bending modulus B. When the tension is reduced sufficiently to a value σ, the strip forms wr…
Instability patterns of rolling up a sleeve appear more intricate than the ones of walking over a rug on floor, both characterized as uniaxially compressed soft-film/stiff-substrate systems. This can be explained by curvature effects. To investigate pattern transitions on a curved surface, we study a soft shell sliding on a rigid cylinder by experiments, computations and theoretical analyses. We reveal a novel post-buckling phenomenon involving multiple successive bifurcations: smooth-wrinkle-ridge-sagging transitions. The shell initially buckles into periodic axisymmetric wrinkles at the thre…
We present a combined theoretical and experimental study of the buckling of a thin film wrapped around a sphere under the action of capillary forces. A rigid sphere is coated with a wetting liquid, and then wrapped by a thin film into an initially cylindrical shape. The equilibrium of this cylindrical shape is governed by the antagonistic effects of elasticity and capillarity: elasticity tends to keep the film developable while capillarity tends to curve it in both directions so as to maximize the area of contact with the sphere. In the experiments, the contact area between the film and the sp…
Twisted ribbons subjected to a tension exhibit a remarkably rich morphology, from smooth and wrinkled helicoids, to cylindrical or faceted patterns. These shapes are intimately related to the instability of the natural, helicoidal symmetry of the system, which generates both longitudinal and transverse stresses, thereby leading to buckling of the ribbon. In this paper, we focus on the tessellation patterns made of triangular facets. Our experimental observations are described within an "asymptotic isometry" approach that brings together geometry and elasticity. The geometry consists of paramet…
I consider the problem of a thin membrane on which a metric has been prescribed, for example by lithographically controlling the local swelling properties of a polymer thin film. While any amount of swelling can be accommodated locally, geometry prohibits the existence of a global strain-free configuration. To study this geometrical frustration, I introduce a perturbative approach. I compute the optimal shape of an annular, thin ribbon as a function of its width. The topological constraint of closing the ribbon determines a relationship between the mean curvature and number of wrinkles that pr…
This paper presents an experimental investigation on pattern formation and evolution in bilayered tubular organs using swelling deformation of polydimethylsiloxane (PDMS) and aims at supplying a thorough comparison with theoretical and finite element results. To create a twin model in modelling and simulation, the shear modulus in the incompressible neo-Hookean material is estimated via uni-axial tensile and pure shear tests. Five bilayered tubes with different material or geometrical parameters are fabricated. Swelling experiments are carried out for these samples in an individual experimenta…
We propose theoretical and numerical analyses of smart bending deformation of a dielectric-elastic bilayer in response to a voltage, based on the nonlinear theory of electro-elasticity and the associated linearized incremental field theory. We reveal that the mechanism allowing the bending angle of the bilayer can be tuned by adjusting the applied voltage. Furthermore, we investigate how much can the bilayer be bent before it loses its stability by buckling when one of its faces is under too much compression. We find that the physical properties of the two layers must be selected to be of the …
The localized loading of an elastic sheet floating on a liquid bath occurs at scales from a frog sitting on a lily pad to a volcano supported by the Earth's tectonic plates. The load is supported by a combination of the stresses within the sheet (which may include applied tensions from, for example, surface tension) and the hydrostatic pressure in the liquid. At the same time, the sheet deforms, and may wrinkle, because of the load. We study this problem in terms of the (relatively weak) applied tension and the indentation depth. For small indentation depths, we find that the force--indentatio…
We investigate the finite bending and the associated bending instability of an incompressible dielectric slab subject to a combination of applied voltage and axial compression, using nonlinear electro-elasticity theory and its incremental version. We first study the static finite bending deformation of the slab. We then derive the three-dimensional equations for the onset of small-amplitude wrinkles superimposed upon the finite bending. We use the surface impedance matrix method to build a robust numerical procedure for solving the resulting dispersion equations and determining the wrinkled sh…
We investigate the out-of-plane shape morphing capability of single-material elastic sheets with architected cut patterns that result in arrays of tiles connected by flexible hinges. We demonstrate that a non-periodic cut pattern can cause a sheet to buckle into three-dimensional shapes, such as domes or patterns of wrinkles, when pulled at specific boundary points. These global buckling modes are observed in experiments and rationalized by an in-plane kinematic analysis that highlights the role of the geometric frustration arising from non-periodicity. The study focuses on elastic sheets, and…
