Exact solutions for the wrinkle patterns of confined elastic shells
Ian Tobasco, Yousra Timounay, Desislava Todorova, Graham C. Leggat, Joseph D. Paulsen, Eleni Katifori
备注：
This version of the article has been accepted for publication, after peer review and is subject to Springer Nature's AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://doi.org/10.1038/s41567-022-01672-2
Complex textured surfaces occur in nature and industry, from fingerprints to lithography-based micropatterns. Wrinkling by confinement to an incompatible substrate is an attractive way of generating reconfigurable patterned topographies, but controlling the often asymmetric and apparently stochastic wrinkles that result remains an elusive goal. Here, we describe a new approach to understanding the wrinkles of confined elastic shells, using a Lagrange multiplier in place of stress. Our theory reveals a simple set of geometric rules predicting the emergence and layout of orderly wrinkles, and explaining a surprisingly generic co-existence of ordered and disordered wrinkle domains. The results agree with numerous test cases across simulation and experiment and represent an elementary geometric toolkit for designing complex wrinkle patterns.
Thin elastic sheets supported on compliant media form wrinkles under lateral compression. Since the lateral pressure is coupled to the sheet's deformation, varying it periodically in time creates a parametric excitation. We study the resulting parametric resonance of wrinkling modes in sheets supported on semi-infinite elastic or viscoelastic media, at pressures smaller than the critical pressure of static wrinkling. We find distinctive behaviors as a function of excitation amplitude and frequency, including (a) a different dependence of the dynamic wrinkle wavelength on sheet thickness compar…
Thin elastic sheets and membranes are known to wrinkle when they are stretched -- the associated physics is highly non-linear. The mechanics of thin films that exhibit unusual behavior upon stretching, when they possess auxetic structure, i.e. when their apparent Poisson's ratio is negative, is presented here. Wrinkling is now suppressed within the bulk of auxetic films when tensioned, whereas localized creases confined to the clamps, that decay away exponentially, appear. These edge wrinkles are characterized for their amplitude and wavelength experimentally, theoretically, and computationall…
We study the deformation of a fluctuating crystalline sheet confined between two flat rigid walls as a simple model for layered solids where bonds among atoms {\it within} the same layer are much stronger than those {\it between} layers. When subjected to sufficiently high loads in an appropriate geometry, these solids deform and fail in unconventional ways. Recent experiments suggest that configurations named {\it ripplocations}, where a layer folds backwards over itself, are involved. These structures are distinct and separated by large free energy barriers from smooth {\it ripples} of the a…
We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asympto-tic solutions for thin sectors. When t…
We investigate the mechanical response of a compressed monolayer of large and dense particles at a liquid-fluid interface: a granular raft. Upon compression, rafts first wrinkle; then, as the confinement increases, the deformation localizes in a unique fold. This characteristic buckling pattern is usually associated to floating elastic sheets and as a result, particle laden interfaces are often modeled as such. Here, we push this analogy to its limits by comparing the first quantitative measurements of the raft morphology to a theoretical continuous elastic model of the interface. We show that…
Thin solids often develop elastic instabilities and subsequently complex, multiscale deformation patterns. Revealing the organizing principles of this spatial complexity has ramifications for our understanding of morphogenetic processes in plant leaves and animal epithelia, and perhaps even the formation of human fingerprints. We elucidate a primary source of this morphological complexity - an incompatibility between an elastically-favored "micro-structure" of uniformly spaced wrinkles and a "macro-structure" imparted through the wrinkle director and dictated by confinement forces. Our theory …
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…
We report on the modeling of the dynamics of confined lipid membranes. We derive a thin film model in the lubrication limit which describes an inextensible liquid membrane with bending rigidity confined between two adhesive walls. The resulting equations share similarities with the Swift-Hohenberg model. However, inextensibility is enforced by a time-dependent nonlocal tension. Depending on the excess membrane area available in the system, three different dynamical regimes, denoted as A, B and C, are found from the numerical solution of the model. In regime A, membranes with small excess area …
We demonstrate a novel technique to fabricate mechanically tunable slippery surfaces using one dimensional (anisotropic) elastic wrinkles. Such wrinkles show tunable topography (amplitude) on the application of mechanical strain. Following Nepenthes pitcher plants, lubricating fluid infused solid surfaces show excellent slippery behavior for test liquid drops. Therefore combining the above two i.e. infusing suitable lubricating fluid on elastic wrinkles would enable us to fabricate mechanically tunable slippery surfaces. Completely stretched (flat) wrinkles have uniform coating of lubricating …
We consider the point-indentation of a pressurized elastic shell. It has previously been shown that such a shell is subject to a wrinkling instability as the indentation depth is quasi-statically increased. Here we present detailed analysis of this wrinkling instability using a combination of analytical techniques and finite element simulations. In particular, we study how the number of wrinkles observed at the onset of instability grows with increasing pressurization. We also study how, for fixed pressurization, the number of wrinkles changes both spatially and with increasing indentation dep…
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…
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…
A thin, elastic sheet floating on the surface of a liquid bath wrinkles when poked at its centre. We study the onset of wrinkling as well as the evolution of the pattern as indentation progresses far beyond the wrinkling threshold. We use tension field theory to describe the macroscopic properties of the deformed film and show that the system passes through a host of different regimes, even while the deflections and strains remain small. We show that the effect of the finite size of the sheet ultimately plays a key role in determining the location of the wrinkle pattern, and obtain scaling rel…
We show that a viscoelastic thin sheet driven out of equilibrium by active structural remodelling develops a rich variety of shapes as a result of a competition between viscous relaxation and activity. In the regime where active processes are faster than viscoelastic relaxation, wrinkles that are formed due to remodelling are unable to relax to a configuration that minimises the elastic energy and the sheet is inherently out of equilibrium. We argue that this non-equilibrium regime is of particular interest in biology as it allows the system to access morphologies that are unavailable if restr…
We model the nonlinear response of a lubricated contact composed of a two-dimensional lipid membrane immersed in a simple fluid between two parallel flat and porous walls under shear. The nonlinear dynamics of the membrane gives rise to a rich dynamical behavior depending on the shear velocity. In quiescent conditions (i.e., absence of shear), the membrane freezes into a disordered labyrinthine wrinkle pattern. In the presence of shear, we find four different regimes depending on the shear rate. Regime I. For small shear, the labyrinthine pattern is still frozen, but exhibits a small drift whi…
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 …
The wrinkling of thin elastic objects provides a means of generating regular patterning at small scales in applications ranging from photovoltaics to microfluidic devices. Static wrinkle patterns are known to be governed by an energetic balance between the object's bending stiffness and an effective substrate stiffness, which may originate from a true substrate stiffness or from tension and curvature along the wrinkles. Here we investigate dynamic wrinkling, induced by the impact of a solid sphere onto an ultra-thin polymer sheet floating on water. The vertical deflection of the sheet's centre…
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 rod-like 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 …
We investigate impact of a sphere onto a floating elastic sheet and the resulting formation and evolution of wrinkles in the sheet. Following impact, we observe a radially propagating wave, beyond which the sheet remains approximately planar but is decorated by a series of radial wrinkles whose wavelength grows in time. We develop a mathematical model to describe these phenomena by exploiting the asymptotic limit in which the bending stiffness is small compared to stresses in the sheet. The results of this analysis show that, at a time $t$ after impact, the transverse wave is located at a radi…
Universal wrinkling dynamics of a sheet on viscous liquid
Université de Mons（蒙斯大学）；University of Oxford（牛津大学）；École Normale Supérieure de Lyon（里昂高等师范学院）；ESPCI Paris（巴黎高等物理化工学院）；Université libre de Bruxelles (ULB)（布鲁塞尔自由大学）
A novel bilayer is introduced, consisting of a stiff film adhered to a soft substrate with patterned holes beneath the film and substrate interface. To uncover the transition of surface patterns, two dimensional plane strain simulations are performed on the defected bilayer subjected to uniaxial compression. Although the substrate is considered as the linear elastic material, the presence of defects can directly trigger the formation of locally ridged and then folding configurations from flat surface with a relatively small compressive strain. It is followed by the coexisting phases of folds a…
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…
How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via $Γ$-convergence for answering this question to leading order in the shell's thickness and other small parameters. The observed patterns involve "ordered" regions of well-defined wrinkles alongside "disordered" regions whose local features are less robust; as little to no tension is applied, the preference for order is not a priori clear. Rescaling by the energy of a typical pattern, we derive a limiting variational probl…
Elastomeric materials when sliding on clean and rough surfaces generate wrinkles at the interface due to tangential stress gradients. These interfacial folds travel along the bottom of elastomer as surface detachment waves to facilitate the apparent sliding motion of elastomer. At very low sliding speed compared to elastic surface waves, the process is dominated by surface adhesion and relaxation effects, and the phenomenon is historically referred to as Schallamach waves. We report in this letter the observation of fast-traveling intersonic detachment waves exceeding the Rayleigh and shear wa…
By reinforcing the interaction energy of the liquid with respect to the surface using total wetting boundary conditions, the response of liquids to mechanical shear stress is stronger and exhibits at sub-millimeter scale elastic properties. This study extends here to liquids with strong electrostatic interactions such as ionic liquids and paramagnetic liquids. We show that it is also possible to identify and measure a non-zero low-frequency shear elasticity at sub-millimeter scale. The observation of mesoscopic elastic properties in liquids generally considered as viscous away from any phase t…
Investigations of the elastic behavior of bulk elastomers have traditionally proceeded on the basis of classical rubber elasticity, which regards chains as thermally fluctuating but disregards the thermal fluctuations of the cross-links. Here, we consider an incompressible and flat elastomer film of an axisymmetric shape confined between two large hard co-planar substrates, with the axis of the film perpendicular to the substrates. We address the impact that thermal fluctuations of the cross-links have on the free energy of elastic deformation of the system, subject to the requirement that the…
In this paper we present a review on our recent experimental investigations into the phase behavior of the deeply cooled water confined in a nanoporous silica material, MCM-41, with elastic neutron scattering technique. Under such strong confinement, the homogeneous nucleation process of water is avoided, which allows the confined water to keep its liquid state at temperatures and pressures that are inaccessible to the bulk water. By measuring the average density of the confined heavy water, we observe a likely first-order low-density liquid (LDL) to high-density liquid (HDL) transition in the…
Thin elastic solids are easily deformed into a myriad of three-dimensional shapes, which may contain sharp localized structures as in a crumpled candy wrapper, or have smooth and diffuse features like the undulating edge of a flower. Anticipating and controlling these morphologies is crucial to a variety of applications involving textiles, synthetic skins, and inflatable structures. Here we show that a "wrinkle-to-crumple" transition, previously observed in specific settings, is a ubiquitous response for confined sheets. This unified picture is borne out of a suite of model experiments on poly…
Cell-induced wrinkling patterns on soft substrates
Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所）；Institute of Bioengineering, School of Life Sciences, École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院生命科学学院生物工程研究所）；Division of Bioengineering, Graduate School of Engineering Science, Osaka University（大阪大学工学研究科生物工程学系）
Confinement of filamentary objects is ubiquitous in numerous biological, medical, and engineering scenarios. Quantitatively determining the mechanical interaction between flexible filaments and surface confinement is particularly challenging due to the unknown contact force induced by elasticity interacting with geometric constraints. Here, we consider a simplified model of confined filamentary object: an elastic curve under surface confinement. Local force and moment balance equation incorporating the role of contact is utilized to derive the contact force exerted by surface on the confined e…
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…
Ultrasonic experiments allow one to measure the elastic modulus of bulk solid or fluid samples. Recently such experiments have been carried out on fluid-saturated nanoporous glass to probe the modulus of a confined fluid. In our previous work [J. Chem. Phys., (2015) 143, 194506], using Monte Carlo simulations we showed that the elastic modulus $K$ of a fluid confined in a mesopore is a function of the pore size. Here we focus on modulus-pressure dependence $K(P)$, which is linear for bulk materials, a relation known as the Tait-Murnaghan equation. Using transition-matrix Monte Carlo simulation…
