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We consider a disc-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of...
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Citations
... Since then there have been many successful applications to phenomena in materials science. In the case of thin elastic sheets, there have been notable results for sheets wrinkling under tension [BK14], under compression [BCDM02], or when attached to a substrate [KN13,BK17]. ...
We consider a thin elastic sheet with a finite number of disclinations in a variational framework in the F\"oppl-von K\'arm\'an approximation. Under the non-physical assumption that the out-of-plane displacement is a convex function, we prove that minimizers display ridges between the disclinations. We prove the associated energy scaling law with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet. One of the key estimates in the proof that we consider of independent interest is a generalization of the monotonicity property of the Monge-Amp\`ere measure.
... Can such incompatibilities be used to design and control complex wrinkled surfaces at will? Wrinkles have been in the limelight for their theoretical importance in understanding geometric nonlinearities in elasticity [13][14][15][16][17][18][19][20] and also for their practical significance in emerging engineering applications such as lithography-free micropatterning [21][22][23][24] . Yet, despite decades of study, a general predictive theory of confinement-induced wrinkling remains elusive. ...
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.
... We use x ⊗ y to denote the outer product of x and y, andν for the outwards-pointing unit normal vector at ∂ . Our formula for the energy is directly analogous to the one used in [74] to study the wrinkling of an internally pressurized spherical shell, as well as the one used in [11,24,34] to study the wrinkling of a flat disc on a spherical substrate; it is a geometrically linearized version of the energy used in [1] for general floating shells. Here, to fix ideas, we focus on the setup of a weakly curved shell on a planar liquid substrate, noting that our analysis can be adapted to the more general setup of a weakly curved shell on a weakly curved substrate. ...
... As remarked above, we are not the first to make such a simplification in the study of elastic patterns: other authors including those of [11,24,34,74] have used geometrically linear models as well. The picture that has emerged is that, whereas the quantitative predictions of such models can only be asymptotically correct, their qualitative predictions often reflect those of a more nonlinear model. ...
... In particular, the almost minimizers of E b,k,γ must exhibit an (approximately) azimuthally symmetric response, of course subject to the assumptions at the start of Section 1.2 under which our -convergence results hold. Before moving on, we note that a similar result can be proved for the case of a flat disc attached to a weakly curved spherical substrate-a model problem that has been the focus of much previous research, including at least [11,24,34]. The conclusion is that optimal μ are uniquely determined, absolutely continuous, and parallel toê θ ⊗ê θ with a density as above. ...
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 \(\Gamma \)-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 problem for the effective displacement of the shell. It asks, in a linearized way, to cover up a maximum area with a length-shortening map to the plane. Convex analysis yields a boundary value problem characterizing the accompanying patterns via their defect measures. Partial uniqueness and regularity theorems follow from the method of characteristics on the ordered part of the shell. In this way, we can deduce from the principle of minimum energy the leading order features of stamped elastic shells.
... Thin elastic membranes form complex wrinkle patterns when put on substrates of different shapes [1][2][3][4][5][6][7]. Such patterns continue to receive attention across science and engineering [8][9][10][11][12][13]. ...
... This is the two-dimensional version of the lower curve in Fig. 1c, and we justify it in the SI. Applying (5) gives ...
Thin elastic membranes form complex wrinkle patterns when put on substrates of different shapes. Such patterns continue to receive attention across science and engineering. This is due, in part, to the promise of lithography-free micropatterning, but also to the observation that similar patterns arise in biological systems from growth. The challenge is to explain the patterns in any given setup, even when they fail to be robust. Building on the theoretical foundation of [Tobasco, to appear in Arch. Ration. Mech. Anal., arXiv:1906.02153], we derive a complete and simple rule set for wrinkles in the model system of a curved shell on a liquid bath. Our rules apply to shells whose initial Gaussian curvatures are of one sign, such as cutouts of saddles and spheres. They predict the surprising coexistence of orderly wrinkles alongside disordered regions where the response appears stochastic, which we confirm in experiment and simulation. They also unveil the role of the shell's medial axis, a distinguished locus of points that we show is a basic driver in pattern selection. Finally, they explain how the sign of the shell's initial curvature dictates the presence or lack of disorder. Armed with our simple rules, and the methodology underlying them, one can anticipate the creation of designer wrinkle patterns.
... Many of these rich and complex morphologies stem from a basic consideration: A sufficiently thin sheet prefers to minimize costly stretching deformations in favor of low-energy bending. For sheets constrained to planar or gently curved topographies, wrinkles are an effective method for relaxing compressive stresses while minimizing out-ofplane displacements [13][14][15][16]. Wrinkles can even allow an initially planar sheet to hug the contour of a doubly curved geometry, such as a sphere or saddle, with negligible stretching [17][18][19][20][21][22]. ...
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 polymer films confined to liquid interfaces with spherical, hyperbolic, and cylindrical geometries, which are complemented by experiments on macroscopic membranes inflated with gas. We use measurements across this wide range of geometries, boundary conditions, and length scales to quantify several robust morphological features of the crumpled phase, and we build an empirical phase diagram for crumple formation that disentangles the competing effects of curvature and compression. Our results suggest that crumples are a generic microstructure that emerge at large curvatures due to a competition of elastic and substrate energies.
... Many of these rich and complex morphologies stem from a basic consideration: A sufficiently thin sheet prefers to minimize costly stretching deformations in favor of low-energy bending. For sheets constrained to planar or gently curved topographies, wrinkles are an effective method for relaxing compressive stresses while minimizing out-of-plane displacements [13][14][15][16]. Wrinkles can even allow an initially planar sheet to hug the contour of a doubly-curved geometry, such as a sphere or saddle, with negligible stretching [17][18][19][20][21][22]. ...
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 polymer films confined to liquid interfaces with spherical, hyperbolic, and cylindrical geometries, which are complemented by experiments on macroscopic membranes inflated with gas. We use measurements across this wide range of geometries, boundary conditions, and lengthscales to quantify several robust morphological features of the crumpled phase, and we build an empirical phase diagram for crumple formation that disentangles the competing effects of curvature and compression. Our results suggest that crumples are a generic microstructure that emerge at large curvatures due to a competition of elastic and substrate energies.
... The relaxed energy density W * for a sheet with zero thickness (a "membrane") is a function of its effective strain, which vanishes on bi-axially compressed states and is otherwise strictly positive. When applied to the study of tension-driven wrinkling of thin elastic sheets [11,25], one finds that the extent of the un-wrinkled region is determined as well as the direction of the wrinkles by the solution of a relaxed problem of the form ...
... The use of a more nonlinear model ("geometrically nonlinear" as in [1] or "fully nonlinear" as in [11]) would of course yield more accurate results, but would require several significant mathematical advances beyond the ones achieved here. As remarked above, we are not the first to make such a simplification in the study of elastic pattern formation: many other authors including those of [13,26,38,69] have used geometrically linear models as well. ...
How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns formed by thin shells floating on a water bath, we develop a rigorous method via $\Gamma$-convergence for answering this question to leading order in the shell's thickness and other small parameters. The experimentally observed patterns involve regions of well-defined wrinkling alongside other "disordered" regions where the local features of the patterns can, for thin enough shells, depend on the history of the experiment. Our goal is to explain the appearance and lack thereof of such "wrinkling domains". Rescaling by the energy of a typical wrinkling pattern, we derive a limiting area problem that asks (in an appropriately linearized way) to cover up as much area as possible in the plane with a length-shortening map. Convex analysis yields a boundary value problem characterizing optimal patterns via their effective strains or, what is equivalent, their "defect measures". Optimal defect measures are in general non-unique. Nevertheless, in some cases their restrictions to certain regions are uniquely determined and solution formulas exist. In this way, we can deduce from the principle of minimum energy the formation of the observed wrinkling domains.
... In these works, elastic sheets bonded to elastic substrates are modelled using a variational form of the von Kármán plate equations (which can be derived from full 3D nonlinear elasticity [27]) and identifies wrinkling as the result of competition between minimisation of the nonconvex membrane energy and the regularising bending energy. In particular, scaling laws of the energy with respect to thickness of the elastic sheet were identified and it was demonstrated that this fitted with characteristic wrinkling patterns seen in the physical world ( [28] and [29]). While these studies provide precise estimates, they have a limited (and well acknowledged) range of validity regarding properties of the displacement field of the plate which cannot capture some of the phenomenology we see in thick, multi-layered elastic media. ...
Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects. Namely, we consider the effect of added pressure, surface tension, an upper substrate and fibres. We obtain analytical estimates of the instability in the two asymptotic regimes of long and short wavelengths.
This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.
... They show how the properties of wrinkled shells can be understood well beyond the onset of instability by exploiting the large wavenumber of instability and, further, that the scale of instability observed in this highly wrinkled state varies spatially. The spatial variation in wrinkle patterns caused by geometrical incompatibility is studied using rigorous mathematical methods by Bella & Kohn [9]. For their problem, Bella and Kohn show that the wrinkle wavelength is largely constant in space, but varies slightly to avoid the energetic cost of changing the number of wrinkles, which would be required to keep the wavelength precisely constant. ...
Classical studies in engineering usually analyse the possibility of instability as a predictive tool that is used in order to avoid the onset of material failure in structural applications: how should one design a water tower so that it will not collapse when filled? Recently, a multidisciplinary
