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 anisotropically growing sheets in the classical Rayleigh-Ritz formalism. We illustrate the great potential in this approach by simulating the inflation of airbags, the buckling of a stretched cylinder, as well as the formation and scaling of wrinkles at free boundaries of growing sheets. Finally, we compare the folding of spatially confined sheets subject to growth and shrinking confinement to find that the two processes are equivalent.
We discuss shape profiles emerging in inhomogeneous growth of squeezed tissues. Two approaches are used simultaneously: i) conformal embedding of two-dimensional domain with hyperbolic metrics into the plane, and ii) a pure energetic consideration based on the minimization of the total energy functional. In the latter case the non-uniformly pre-stressed plate, which models the inhomogeneous two-dimensional growth, is analyzed in linear regime under small stochastic perturbations. It is explicitly demonstrated that both approaches give consistent results for buckling profiles and reveal self-si…
When a flexible filament is confined to a fluid interface, the balance between capillary attraction, bending resistance, and tension from an external source can lead to a self-buckling instability. We perform an analysis of this instability and provide analytical formulae that compare favorably with the results of detailed numerical computations. The stability and long-time dynamics of the filament are governed by a single dimensionless elastocapillary number quantifying the ratio between capillary to bending stresses. Complex, folded filament configurations such as loops, needles, and racquet…
Periodic wrinkling of a rigid capping layer on a deformable substrate provides a useful method for templating surface topography for a variety of novel applications. Many experiments have studied wrinkle formation during the compression of a rigid film on a relatively soft pre-strained elastic substrate, and most have focused on the regime where the substrate thickness can be considered semi-infinite relative to that of the film. As the relative thickness of the substrate is decreased, the bending stiffness of the film dominates, causing the bilayer to transition to either local wrinkling or a…
We theoretically explain the complete sequence of shapes of deflated spherical shells. Decreasing the volume, the shell remains spherical initially, then undergoes the classical buckling instability, where an axisymmetric dimple appears, and, finally, loses its axisymmetry by wrinkles developing in the vicinity of the dimple edge in a secondary buckling transition. We describe the first axisymmetric buckling transition by numerical integration of the complete set of shape equations and an approximate analytic model due to Pogorelov. In the buckled shape, both approaches exhibit a locally compr…
We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler-Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or globa…
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…
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.
Hydrostatically pressurized circular rings confined to two dimensions (or cylinders constrained to have only z-independent deformations) undergo Euler type buckling when the outside pressure exceeds a critical value. We perform a stability analysis of rings with arc-length dependent bending moduli and determine how weakened bending modulus segments affect the buckling critical pressure. Rings with a 4-fold symmetric modulation are particularly susceptible to collapse. In addition we study the initial post-buckling stages of the pressurized rings to determine possible ring folding patterns.
Bubbles made of smectic liquid crystals (LC) can display buckling and wrinkling. These deformations are induced in the course of the relaxation of an elongated bubble towards a sphere. The bubbles are created through the collapse of a catenoid LC film. Two mechanisms leading to wrinkles are reported: (a) Compression of the smectic film by outer forces, and (b) Invagination of the bubble shortly after formation. The flow on the bubble surface is analyzed, providing insight into how the outer forces act on the bubble and induce local surface deformation. It is found that the orientation of the w…
The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key role of branch-point (or "monkey-saddle") singularities in generating complex wrinkling patterns in isometric immersions, and show how they arise naturally from minimizing the elastic energy.
Motivated by the redrawing of hot glass into thin sheets, we investigate the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed non-uniform temperature field. We first determine the associated base flow by solving the long-timescale stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized stretching velocity $α$, and a dimensionless width of the heating zone $β$. This allows us to determine the conditions for the onset of an out-of-plane wrinkling instability stated in terms of an eigenvalue problem for a linear part…
Colloidal capsules can sustain an external osmotic pressure; however, for a sufficiently large pressure, they will ultimately buckle. This process can be strongly influenced by structural inhomogeneities in the capsule shells. We explore how the time delay before the onset of buckling decreases as the shells are made more inhomogeneous; this behavior can be quantitatively understood by coupling shell theory with Darcy's law. In addition, we show that the shell inhomogeneity can dramatically change the folding pathway taken by a capsule after it buckles.
We consider the axial compression of a thin sheet wrapped around a rigid cylindrical substrate. In contrast to the wrinkling-to-fold transitions exhibited in similar systems, we find that the sheet always buckles into a single symmetric fold, while periodic solutions are unstable. Upon further compression, the solution breaks symmetry and stabilizes into a recumbent fold. Using linear analysis and numerics, we theoretically predict the buckling force and energy as a function of the compressive displacement. We compare our theory to experiments employing cylindrical neoprene sheets and find rem…
We re-examine previous observations of folding kinetics of compressed lipid monolayers in light of the accepted mechanical buckling mechanism recently proposed [L. Pocivavsek et al., Soft Matter, 2008, 4, 2019]. Using simple models, we set conservative limits on a) the energy released in the mechanical buckling process and b) the kinetic energy entailed by the observed folding motion. These limits imply a kinetic energy at least thirty times greater than the energy supplied by the buckling instability. We discuss possible extensions of the accepted picture that might resolve this discrepancy.
Thin sheets that are forced at their boundaries develop a variety of shapes aimed at minimising elastic energy by curving spontaneously in ways that break the symmetry of the sheet and the forcing. Characterising such buckling generally requires detailed analysis. Here, we follow a complementary approach, applicable to axisymmetric systems. It aims to predict the qualitative nature of any buckling using only the simple axisymmetric geometry of the initial system. We view an axisymmetric deformation according to the displacement it imposes on the radial lines making up the sheet. Thus, e.g., in…
We study the stress developed during the drying of a colloidal drop of silica nanoparticles. In particular, we use the wrinkling instability of a thin floating sheet to measure the net stress applied by the deposit on the substrate and we focus on the effect of the particle polydispersity. In the case of a bidisperse suspension, we show that a small number of large particles substantially decreases the expected stress, which we interpret as the formation of lower hydrodynamic resistance paths in the porous material. As colloidal suspensions are usually polydisperse, we show for different avera…
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…
Certain surfactant monolayers at the water-air interface have been found to undergo, at a critical surface pressure, a dynamic instability involving multiple long folds of micron width. We exploit the sharp monolayer translations accompanying folding events to acquire, using a combination of fluorescence microscopy and digital image analysis, detailed statistics concerning the folding dynamics. The motions have a broad distribution of magnitudes and narrow, non-Gaussian distributions of angles and durations. The statistics are consistent with the occurrence of cooperative cascades of folds, im…
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…
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 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…
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip and disk geometries. By finding explicit piecewise smooth solutions of hyperbolic Monge-Ampere equations on a strip, we show there exist periodic isometric immersions of hyperbolic surfaces in the small slope regime. We extend these solutions to exact isometric immersions through …
Numerically simulating deformations in thin elastic sheets is a challenging problem in computational mechanics due to destabilizing compressive stresses that result in wrinkling. Determining the location, structure, and evolution of wrinkles in these problems have important implications in design and is an area of increasing interest in the fields of physics and engineering. In this work, several numerical approaches previously proposed to model equilibrium deformations in thin elastic sheets are compared. These include standard finite element-based static post-buckling approaches as well as a…
Growth-induced pattern formations in curved film-substrate structures have attracted extensive attentions recently. In most existing literature, the growth tensor is assumed to be homogeneous or piecewise homogeneous. In this paper, we aim at clarifying the influence of a growth gradient on pattern formation and pattern evolution in bilayered tubular tissues under plane-strain deformation. In the framework of finite elasticity, a bifurcation condition is derived for a general material model and a generic growth function. Then we suppose that both layers are composed of neo-Hookean materials. I…
We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such surface may modelize a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It first occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry, by folding along portions of meridians. We could exhibit unifying master curves for the relative volume variation at which first and secon…
The apparently intractable shape of a fold in a compressed elastic film lying on a fluid substrate is found to have an exact solution. Such systems buckle at a nonzero wavevector set by the bending stiffness of the film and the weight of the substrate fluid. Our solution describes the entire progression from a weakly displaced sinusoidal buckling to a single large fold that contacts itself. The pressure decrease is exactly quadratic in the lateral displacement. We identify a complex wavevector whose magnitude remains invariant with compression.
Predicting the large-amplitude deformations of thin elastic sheets is difficult due to the complications of self-contact, geometric nonlinearities, and a multitude of low-lying energy states. We study a simple two-dimensional setting where an annular polymer sheet floating on an air-water interface is subjected to different tensions on the inner and outer rims. The sheet folds and wrinkles into many distinct morphologies that break axisymmetry. These states can be understood within a recent geometric approach for determining the gross shape of extremely bendable yet inextensible sheets by extr…
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…
Shells are ubiquitous thin structures that can undergo large nonlinear elastic deformations while exhibiting combined modes of bending and stretching, and have profound modern applications. In this paper, we have proposed a new Isogeometric formulation, based on classical Koiter nonlinear shell theory, to study instability problems like wrinkling and buckling in thin shells. The use of NURBS-basis provides rotation-free, conforming, higher-order spatial continuity, such that curvatures and membrane strains can be computed directly from the interpolation of the position vectors of the control p…
Thin rigid sheets floating on a liquid substrate appear, for example, in coatings and surfactant monolayers. Upon uniaxial compression the sheet undergoes transitions from a compressed flat state to a periodic wrinkled pattern to a localized folded pattern. The stability of these states is determined by the in-plane elasticity of the sheet, its bending rigidity, and the hydrostatics of the underlying liquid. Wrinkles and folds, and the wrinkle-to-fold transition, were previously studied for incompressible sheets. In the present work we extend the theory to include finite compressibility. We an…
