Wrinkles and folds in a fluid-supported sheet of finite size

Wrinkling composite sheets

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet.

Wrinkling Instability in Unsupported Epithelial Sheets

We investigate the elasticity of an unsupported epithelial monolayer and we discover that unlike a thin solid plate, which wrinkles if geometrically incompatible with the underlying substrate, the epithelium may do so even in the absence of the substrate. From a cell-based model, we derive an exact elasticity theory and discover wrinkling driven by the differential apico-basal surface tension. Our theory is mapped onto that for supported plates by introducing a phantom substrate whose stiffness is finite beyond a critical differential tension. This suggests a new mechanism for an autonomous control of tissues over the length scale of their surface patterns.

Packing of elastic rings with friction

We study the deformations of elastic filaments confined within slowly shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of uniform bending stiffness. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. There follows a sequence of deformations that is a mixture of spiralling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the − 3 / 2 and − 2 powers, respectively. With non-zero friction, the elastic energy follows a similar scaling but with a prefactor up to eight times larger, due to delayering and bending with a range of small curvatures. For friction coefficients as large as 1, the deformations are qualitatively similar with and without friction at the outer wall. Above 1, the wall friction case becomes dominated by buckling near the wall.

Asymptotic softness of a laterally confined sheet in a pressurized chamber

Elastohydrodynamic models, that describe the interaction between a thin sheet and a fluid medium, have been proven successful in explaining the complex behavior of biological systems and artificial materials. Motivated by these applications we study the quasistatic deformation of a thin sheet that is confined between the two sides of a closed chamber. The two parts of the chamber, above and below the sheet, are filled with an ideal gas. We show that the system is governed by two dimensionless parameters, Δ and η, that account respectively for the lateral compression of the sheet and the ratio between the amount of fluid filling each part of the chamber and the bending stiffness of the sheet. When η≪1 the bending energy of the sheet dominates the system, the pressure drop between the two sides of the chamber increases, and the sheet exhibits a symmetric configuration. When η≫1 the energy of the fluid dominates the system, the pressure drop vanishes, and the sheet exhibits an asymmetric configuration. The transition between these two limiting scenarios is governed by a third branch of solutions that is characterized by a rapid decrease of the pressure drop. Notably, across the transition the energetic gap between the symmetric and asymmetric states scales as δE∼Δ^{2}. Therefore, in the limit Δ≪1 small variations in the energy are accompanied by relatively large changes in the elastic shape.

Drop Spreading and Confinement in Swelling-Driven Folding of Thin Films

Surface instabilities are a versatile method for generating three-dimensional (3D) surface microstructure. When an elastomeric film weakly bonded to a substrate is swollen with solvent, buckle delamination and subsequent sliding of the film on the substrate lead to the formation of tall, self-contacting, and permanent folds. This paper explores the mechanics of fold development when such folding is induced by placing a drop on the surface of the film. We show that capillary effects can induce a strong coupling between folding and drop spreading: as folds develop, they wick the solvent toward the periphery of the drop, further propagating radially aligned folds. Accordingly, a solvent drop spreads far more on films that are weakly adhered to the substrate. As drop size reduces and folding becomes increasingly confined, debonding propagates along the perimeter of the wetted region, thus leading to corral-shaped fold patterns. On the other hand, as drop size increases and confinement effects weaken, isotropically oriented folds appear at a spacing that reduces as swelling increases. The spacing between the folds and the size of the corrals are both determined by the extent to which a single fold relieves compressive stress in its vicinity by sliding. We develop a model for folding which explicitly accounts for the fact that folds must initiate with near-zero volume under the buckle. The model shows that folds can appear even at very low swelling if there are large pre-existing debonded regions at the film-substrate interface.

Volume-constrained deformation of a thin sheet as a route to harvest elastic energy

Thin sheets exhibit rich morphological structures when subjected to external constraints. These structures store elastic energy that can be released on demand when one of the constraints is suddenly removed. Therefore, when adequately controlled, shape changes in thin bodies can be utilized to harvest elastic energy. In this paper, we propose a mechanical setup that converts the deformation of the thin body into a hydrodynamic pressure that potentially can induce a flow. We consider a closed chamber that is filled with an incompressible fluid and is partitioned symmetrically by a long and thin sheet. Then, we allow the fluid to exchange freely between the two parts of the chamber, such that its total volume is conserved. We characterize the slow, quasistatic, evolution of the sheet under this exchange of fluid, and derive an analytical model that predicts the subsequent pressure drop in the chamber. We show that this evolution is governed by two different branches of solutions. In the limit of a small lateral confinement we obtain approximated solutions for the two branches and characterize the transition between them. Notably, the transition occurs when the pressure drop in the chamber is maximized. Furthermore, we solve our model numerically and show that this maximum pressure behaves nonmonotonically as a function of the lateral compression.

Delamination of open cylindrical shells from soft and adhesive Winkler's foundation

The interaction between thin elastic films and soft-adhesive foundations has recently gained interest due to technological applications that require control over such objects. Motivated by these applications we investigate the equilibrium configuration of an open cylindrical shell with natural curvature κ and bending modulus B that is adhered to soft and adhesive foundation with stiffness K. We derive an analytical model that predicts the delamination criterion, i.e., the critical natural curvature, κ_{cr}, at which delamination first occurs, and the ultimate shape of the shell. While in the case of a rigid foundation, K→∞, our model recovers the known two-states solution at which the shell either remains completely attached to the substrate or completely detaches from it, on a soft foundation our model predicts the emergence of a new branch of solutions. This branch corresponds to partially adhered shells, where the contact zone between the shell and the substrate is finite and scales as ℓ_{w}∼(B/K)^{1/4}. In addition, we find that the criterion for delamination depends on the total length of the shell along the curved direction, L. While relatively short shells, L∼ℓ_{w}, transform continuously between adhered and delaminated solutions, long shells, L≫ℓ_{w}, transform discontinuously. Notably, our work provides insights into the detachment phenomena of thin elastic sheets from soft and adhesive foundations.

Tunable wrinkling of thin nematic liquid crystal elastomer sheets

Instabilities in thin elastic sheets, such as wrinkles, are of broad interest both from a fundamental viewpoint and also because of their potential for engineering applications. Nematic liquid crystal elastomers offer a new form of control of these instabilities through direct coupling between microscopic degrees of freedom, resulting from orientational ordering of rodlike molecules, and macroscopic strain. By a standard method of dimensional reduction, we construct a plate theory for thin sheets of nematic elastomer. We then apply this theory to the study of the formation of wrinkles due to compression of a thin sheet of nematic liquid crystal elastomer atop an elastic or fluid substrate. We find the scaling of the wrinkle wavelength in terms of material parameters and the applied compression. The wavelength of the wrinkles is found to be nonmonotonic in the compressive strain due to the presence of the nematic. Finally, due to soft modes, the critical stress for the appearance of wrinkles can be much higher than in an isotropic elastomer and depends nontrivially on the manner in which the elastomer was prepared.

Modeling the behavior of inclusions in circular plates undergoing shape changes from two to three dimensions

Growth of biological tissues and shape changes of thin synthetic sheets are commonly induced by stimulation of isolated regions (inclusions) in the system. These inclusions apply internal forces on their surroundings that, in turn, promote 2D layers to acquire complex 3D configurations. We focus on a fundamental building block of these systems, and consider a circular plate that contains an inclusion with dilative strains. Based on the Föppl-von Kármán (FvK) theory, we derive an analytical model that predicts the 2D-to-3D shape transitions in the system. Our findings are summarized in a phase diagram that reveals two distinct configurations in the post-buckling region. One is an extensive profile that holds close to the threshold of the instability, and the second is a localized profile, which preempts the extensive solution beyond the buckling threshold. While the former solution is derived as a perturbation around the flat configuration, assuming infinitesimal amplitudes, the latter solution is derived around a buckled state that is highly localized. We show that up to vanishingly small corrections that scale with the thickness, this localized configuration is equivalent to that expected for ultra-thin sheets, which completely relax compressive stresses. Our findings agree quantitatively with direct numerical minimization of the FvK energy. Furthermore, we extend the theory to describe shape transitions in polymeric gels, and compare the results with numerical simulations that account for the complete elastodynamic behavior of the gels. The agreement between the theory and these simulations indicates that our results are observable experimentally. Notably, our findings can provide guidelines to the analysis of more complicated systems that encompass interaction between several buckled inclusions.

Emergence of active nematics in chaining bacterial biofilms

Growing tissue and bacterial colonies are active matter systems where cell divisions and cellular motion generate active stress. Although they operate in the non-equilibrium regime, these biological systems can form large-scale ordered structures. How mechanical instabilities drive the dynamics of active matter systems and form ordered structures are not well understood. Here, we use chaining Bacillus subtilis, also known as a biofilm, to study the relation between mechanical instabilities and nematic ordering. We find that bacterial biofilms have intrinsic length scales above which a series of mechanical instabilities occur. Localized stress and friction drive buckling and edge instabilities which further create nematically aligned structures and topological defects. We also observe that topological defects control stress distribution and initiate the formation of sporulation sites by creating three-dimensional structures. In this study we propose an alternative active matter platform to study the essential roles of mechanics in growing biological tissue.

Microfluidic probing of the complex interfacial rheology of multilayer capsules

Encapsulation of chemicals using polymer membranes enables control of their transport and delivery for applications such as agrochemistry or detergency. To rationalize the design of polymer capsules, it is necessary to understand how the membranes' mechanical properties control the transport and release of the cargo. In this article, we use microfluidics to produce model polymer capsules and study in situ their behavior in controlled divergent flows. Our model capsules are obtained by assembling polymer mono and hydrogen-bonded bilayers at the surface of an oil droplet in water. We also use microfluidics to probe in situ the mechanical properties of the membranes in a controlled divergent flow generated by introducing the capsules through a constriction and then in a larger chamber. The deformation and relaxation of the capsules depend on their composition and especially on the molecular interactions between the polymer chains that form the membranes and the anchoring energy of the first layer. We develop a model and perform numerical simulations to extract the main interfacial properties of the capsules from the measurement of their deformations in the microchannels.

Wrinkling and folding patterns in a confined ferrofluid droplet with an elastic interface

A thin elastic membrane lying on a fluid substrate deviates from its flat geometry on lateral compression. The compressed membrane folds and wrinkles into many distinct morphologies. We study a magnetoelastic variant of such a problem where a viscous ferrofluid, surrounded by a nonmagnetic fluid, is subjected to a radial magnetic field in a Hele-Shaw cell. Elasticity comes into play when the fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes a gel-like elastic layer. A perturbative linear stability theory is used to investigate how the combined action of magnetic and elastic forces can lead to the development of smooth, low-amplitude, sinusoidal wrinkles at the elastic interface. In addition, a nonperturbative vortex sheet approach is employed to examine the emergence of highly nonlinear, magnetically driven, wrinkling and folding equilibrium shape structures. A connection between the magnetoelastic shape solutions induced by a radial magnetic field and those produced by nonmagnetic means through centrifugal forces is also discussed.

Fluid-supported elastic sheet under compression: Multifold solutions

The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.

Filiform corrosion as a pressure-driven delamination process

Filiform corrosion produces long and narrow trails on various coated metals through the detachment of the coating layer from the substrate. In this work, we present a combined experimental and theoretical analysis of this process with the aim to describe quantitatively the shape of the cross-section, perpendicular to the direction of propagation, of the filaments produced. For this purpose, we introduce a delamination model of filiform corrosion dynamics and show its compatibility with experimental data where the coating thickness has been varied systematically.

Delamination of a thin sheet from a soft adhesive Winkler substrate

A uniaxially compressed thin elastic sheet that is resting on a soft adhesive substrate can form a blister, which is a small delaminated region, if the adhesion energy is sufficiently weak. To analyze the equilibrium behavior of this system, we model the substrate as a Winkler or fluid foundation. We develop a complete set of equations for the profile of the sheet at different applied pressures. We show that at the edge of delamination, the height of the sheet is equal to sqrt[2]ℓ_{c}, where ℓ_{c} is the capillary length. We then derive an approximate solution to these equations and utilize them for two applications. First, we determine the phase diagram of the system by analyzing possible transitions from the flat and wrinkled to delaminated states of the sheet. Second, we show that our solution for a blister on a soft foundation converges to the known solution for a blister on a rigid substrate that assumed a discontinuous bending moment at the blister edges. This continuous convergence into a discontinuous state marks the formation of a boundary layer around the point of delamination. The width of this layer relative to the extent length of the blister, ℓ, scales as w/ℓ∼(ℓ_{c}/ℓ_{ec})^{1/2}, where ℓ_{ec} is the elastocapillary length scale. Notably, our findings can provide guidelines for utilizing compression to remove thin biofilms from surfaces and thereby prevent the fouling of the system.

Packing loops into annular cavities

The continuous packing of a flexible rod in two-dimensional cavities yields a countable set of interacting domains that resembles nonequilibrium cellular systems and belongs to a new class of lightweight material. However, the link between the length of the rod and the number of domains requires investigation, especially in the case of non-simply connected cavities, where the number of avoided regions emulates an effective topological temperature. In the present article we report the results of an experiment of injection of a single flexible rod into annular cavities in order to find the total length needed to insert a given number of loops (domains of one vertex). Using an exponential model to describe the experimental data we quite minutely analyze the initial conditions, the intermediary behavior, and the tight packing limit. This method allows the observation of a new fluctuation phenomenon associated with instabilities in the dynamic evolution of the packing process. Furthermore, the fractal dimension of the global pattern enters the discussion under a novel point of view. A comparison with the classical problems of the random close packing of disks and jammed disk packings is made.

Localization in an idealized heterogeneous elastic sheet

Localized deformation is ubiquitous in many natural and engineering materials as they approach failure, and a significant effort has been made to understand localization processes with simple continuum models. Real materials are much more commonly heterogeneous but it is unclear exactly how heterogeneity affects outcomes. In this work we study the response of an idealized heterogenous elastic sheet on a soft foundation as it is uniaxially compressed. The patterned surface layers are created by selective ultraviolet/ozone treatment of the top surface of a polydimethylsiloxane (PDMS) sample using a TEM grid as a mask. By controlling the exposure time of UV/O₃, samples ranging from continuous thin films to sets of isolated small plates were created. We find that patterned regions noticeably localize while bulk regions appear as uniform wrinkles, and that local and global strains depend on the pattern pitch, exposure levels and the treatment protocol. Remarkably, various responses can be modeled using well-understood theory that ignores pattern details aside from the small distance between the adjacent boundaries and the local value of strain.

Pattern transitions in a compressible floating elastic sheet

Compressible thin layers floating on a liquid surface develop wrinkled and folded patterns under lateral pressure.

The compression of a heavy floating elastic film

We study the effect of film density on the uniaxial compression of thin elastic films at a liquid-fluid interface. Using a combination of experiments and theory, we show that dense films first wrinkle and then fold as the compression is increased, similarly to what has been reported when the film density is neglected. However, we highlight the changes in the shape of the fold induced by the film's own weight and extend the model of Diamant and Witten [Phys. Rev. Lett., 2011, 107, 164302] to understand these changes. In particular, we suggest that it is the weight of the film that breaks the up-down symmetry apparent from previous models, but elusive experimentally. We then compress the film beyond the point of self-contact and observe a new behaviour dependent on the film density: the single fold that forms after wrinkling transitions into a closed loop after self-contact, encapsulating a cylindrical droplet of the upper fluid. The encapsulated drop either causes the loop to bend upward or to sink deeper as the compression is increased, depending on the relative buoyancy of the drop-film combination. We propose a model to qualitatively explain this behaviour. Finally, we discuss the relevance of the different buckling modes predicted in previous theoretical studies and highlight the important role of surface tension in the shape of the fold that is observed from the side-an aspect that is usually neglected in theoretical analyses.

Elastic membranes in confinement

An elastic membrane stretched between two walls takes a shape defined by its length and the volume of fluid it encloses. Many biological structures, such as cells, mitochondria and coiled DNA, have fine internal structure in which a membrane (or elastic member) is geometrically 'confined' by another object. Here, the two-dimensional shape of an elastic membrane in a 'confining' box is studied by introducing a repulsive confinement pressure that prevents the membrane from intersecting the wall. The stage is set by contrasting confined and unconfined solutions. Continuation methods are then used to compute response diagrams, from which we identify the particular membrane mechanics that generate mitochondria-like shapes. Large confinement pressures yield complex response diagrams with secondary bifurcations and multiple turning points where modal identities may change. Regions in parameter space where such behaviour occurs are then mapped.

Effect of the Polydispersity of a Colloidal Drop on Drying Induced Stress as Measured by the Buckling of a Floating Sheet

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 average particle sizes that the stress is effectively dominated by the larger particles of the distribution and not by the average particle size.

The effect of shear and confinement on the buckling of particle-laden interfaces

In this paper, we investigate the buckling of an air-water interface populated by lycopodium powder particles using a specially designed Langmuir trough with side walls that deformed affinely with the particle-laden interface in order to minimize the effect of shear during compression. Confinement effects from the side walls were studied by systematically reducing the width of the Langmuir trough and measuring the buckling wavelength. For interfaces wider than 20 mm, the bulk wavelength was found to be independent of interface width. Due to the presence of contact line friction along the sidewall, the amplitude and wavelength of the wrinkles near the side walls were found to be reduced by a factor two compared with the bulk. A cascade in wavelength was observed as one moved from the center of the particle-laden interface towards the sidewalls similar to what has been observed for thin floating polymer films. For interface widths less than 20 mm, the wavelength of the wrinkles in the bulk was found to decrease eventually approaching the wavelength measured along the side walls. The wavelength at the walls was not affected by confinement. At large compressive strains, a transition from wrinkles to folds was observed. These regions of strain localization formed as a train of folds shortly after the onset of wrinkling and grew in amplitude with increasing compression. Confinement was also found to have an impact on folding. To study the impact of shear during interface compression, a series of objects including circular cylinders and rectangular prisms were placed through the center of the particle-laden interface before compression. These objects enhanced wrinkling and folding upstream of the object, eliminated wrinkling and folding in a broad region downstream of the object, and realigned the wrinkles along the side of the immobile obstacles where shear strains were maximum.

Curvature Softening and Negative Compressibility of Gel-Phase Lipid Membranes

We show that gel-phase lipid membranes soften upon bending, leading to curvature localization and a negative compressibility. Using simulations of two very different lipid models to quantify shape and stress-strain relation of buckled membranes, we demonstrate that gel phase bilayers do not behave like Euler elastica and hence are not well described by a quadratic Helfrich Hamiltonian, much unlike their fluid-phase counterparts. We propose a theoretical framework which accounts for the observed softening through an energy density that smoothly crosses over from a quadratic to a linear curvature dependence beyond a critical new scale [Formula: see text](-1). This model captures both the shape and the stress-strain relation for our two sets of simulations and permits the extraction of bending moduli, which are found to be about an order of magnitude larger than the corresponding fluid phase values. We also find surprisingly large crossover lengths [Formula: see text], several times bigger than the bilayer thickness, rendering the exotic elasticity of gel-phase membranes more strongly pronounced than that of homogeneous compressible sheets and artificial metamaterials. We suggest that such membranes have unexpected potential as nanoscale systems with striking materials characteristics.

Equilibrium shapes of planar elastic membranes

Using a rod theory formulation, we derive equations of state for a thin elastic membrane subjected to several different boundary conditions-clamped, simply supported, and periodic. The former is applicable to membranes supported on a softer substrate and subjected to uniaxial compression. We show that a wider family of quasistatic equilibrium shapes exist beyond the previously obtained analytical solutions. In the latter case of periodic membranes, we were able to derive exact solutions in terms of elliptic functions. These equilibria are verified by considering a fluid-structure interaction problem of a periodic, length-preserving bilipid membrane modeled by the Helfrich energy immersed in a viscous fluid. Starting from an arbitrary shape, the membrane dynamics to equilibrium are simulated using a boundary integral method.