Elastic Ribbon Research Articles

Elastic ribbons in bubble columns: When elasticity, capillarity, and gravity govern equilibrium configurations.

Taking advantage of the competition between elasticity and capillarity has proven to be an efficient way to design structures by folding, bending, or assembling elastic objects in contact with liquid interfaces. Elastocapillary effects often occur at scales where gravity does not play an important role, such as in microfabrication processes. However, the influence of gravity can become significant at the desktop scale, which is relevant for numerous situations including model experiments used to provide a fundamental physics understanding, working at easily accessible scales. We focus here on the case of elastic ribbons placed in two-dimensional bubble columns: by introducing an elastic ribbon inside the central soap films of a staircase bubble structure in a square cross-sectioncolumn, the deviation from Plateau's laws (capillarity-dominated case dictating the shape of usual foams) can be quantified as a function of the rigidity of the ribbon. For long ribbons, gravity cannot be neglected. We provide a detailed theoretical analysis of the ribbon profile, taking into account capillarity, elasticity, and gravity. We compute the total energy of the system and perform energy minimization under constraints, using Lagrangian mechanics. The model is then validated via a comparison with experiments with three different ribbon thicknesses.

MERCI: Mixed curvature-based elements for computing equilibria of thin elastic ribbons

Thin elastic ribbons represent a class of intermediary objects lying in-between thin elastic plates and thin elastic rods. Although the two latter families of thin structures have received much interest from the Computer Graphics community over the last decades, ribbons have seldom been considered and modelled numerically so far, in spite of a growing number of applications in Computer Design. In this paper, starting from the reduced developable ribbon models [Sadowsky 1929; Wunderlich 1962] recently popularised in Soft Matter Physics, we propose a both accurate and efficient algorithm for computing the statics of thin elastic ribbons. Inspired by the super-clothoid model for thin elastic rods, our method relies on compact ribbon elements whose normal curvature varies linearly with respect to arc length s , while their geodesic torsion is quadratic in s . In contrast however, for the sake of efficiency, our algorithm avoids building a fully reduced kinematic chain and instead treats each element independently, gluing them only at the final solving stage through well-chosen bilateral constraints. Thanks to this mixed variational strategy, which yields a banded Hessian, our algorithm recovers the linear complexity of low-order models while preserving the quadratic convergence of curvature-based models. As a result, our approach is scalable to a large number of elements, and suitable for various boundary conditions and unilateral contact constraints, making it possible to handle challenging scenarios such as confined buckling experiments or Möbius bands with contact. Remarkably, our mixed algorithm proves an order of magnitude faster compared to Discrete Elastic Ribbon models of the literature while achieving, in a few seconds only, high accuracy levels that remain out of reach for such low-order models. Additionally, our numerical model can incorporate various ribbon energies, including the RibExt model for quasi-developable ribbons recently introduced in Physics [Audoly and Neukirch 2021], which allows to transition smoothly between a rectangular Kirchhoff rod and a (developable) Sadowsky ribbon. Our numerical scheme is carefully validated against demanding experiments of the Physics literature, which demonstrates its accuracy, efficiency, robustness, and versatility. Our Merci code is publicly available at https://gitlab.inria.fr/elan-public-code/merci for the sake of reproducibility and future benchmarking.

Computational model of twisted elastic ribbons.

We develop an irregular lattice mass-spring model to simulate and study the deformation modes of a thin elastic ribbon as a function of applied end-to-end twist and tension. Our simulations reproduce all reported experimentally observed modes, including transitions from helicoids to longitudinal wrinkles, creased helicoids and loops with self-contact, and transverse wrinkles to accordion self-folds. Our simulations also show that the twist angles at which the primary longitudinal and transverse wrinkles appear are well described by various analyses of the Föppl-von Kármán equations, but the characteristic wavelength of the longitudinal wrinkles has a more complex relationship to applied tension than previously estimated. The clamped edges are shown to suppress longitudinal wrinkling over a distance set by the applied tension and the ribbon width, but otherwise have no apparent effect on measured wavelength. Further, by analyzing the stress profile, we find that longitudinal wrinkling does not completely alleviate compression, but caps the magnitude of the compression. Nonetheless, the width over which wrinkles form is observed to be wider than the near-threshold analysis predictions: the width is more consistent with the predictions of far-from-threshold analysis. However, the end-to-end contraction of the ribbon as a function of twist is found to more closely follow the corresponding near-threshold prediction as tension in the ribbon is increased, in contrast to the expectations of far-from-threshold analysis. These results point to the need for further theoretical analysis of this rich thin elastic system, guided by our physically robust and intuitive simulation model.

Analysis of cone-like singularities in twisted elastic ribbons

Twisting a thin elastic ribbon is known to produce a localised deformation pattern resembling a cone whose tip is located on the edge of the ribbon. Using the theory of inextensional ribbons, we present a matched asymptotic analysis of these singularities for ribbons whose width-to-length ratio w/ℓ≪1 is small. An inner layer solution is derived from the finite-w Wunderlich model and captures the fast, local variations of the bending and twisting strains in the neighbourhood of the cone-like region; it is universal up to a load intensity factor. The outer solution is given by the zero-w Sadowsky model. Based on this analysis, we propose a new standalone ribbon model that combines the Sadowsky equations with jump conditions providing a coarse-grained description of cone-like singularities, and give a self-contained variational derivation of this model. Applications to the Möbius band and to an end-loaded open ribbon are presented. Overall, the new model delivers highly accurate approximations to the solutions of the Wunderlich model in the limit w≪ℓ while avoiding the numerical difficulties associated with cone-like singularities.

Elastocapillary deformation of thin elastic ribbons in 2D foam columns.

The ability of liquid interfaces to shape slender elastic structures provides powerful strategies to control the architecture of mechanical self assemblies. However, elastocapillarity-driven intelligent design remains unexplored in more complex architected liquids – such as foams. Here we propose a model system which combines an assembly of bubbles and a slender elastic structure. Arrangements of soap bubbles in confined environments form well-defined periodic structures, dictated by Plateau's laws. We consider a 2D foam column formed in a container with square cross-section in which we introduce an elastomer ribbon, leading to architected structures whose geometry is guided by a competition between elasticity and capillarity. In this system, we quantify both experimentally and theoretically the equilibrium shapes, using X-ray micro-tomography and energy minimisation techniques. Beyond the understanding of the amplitude of the wavy elastic ribbon deformation, we provide a detailed analysis of the profile of the ribbon, and show that such a setup can be used to grant a shape to a UV-curable composite slender structure, as a foam-forming technique suitable to miniaturisation. In more general terms, this work provides a stepping stone towards an improved understanding of the interactions between liquid foams and slender structures.

Buckling pattern formation of the elastic ribbon by twist deformation

A thin elastic object frequently exhibits various pattern formations, such as wrinkles, facets, and loops, under twisting deformation. In the present study, we conduct experimental and numerical analysis for twist deformation of elastic ribbons with various boundary conditions. A sequential pattern transition has been observed from helicoid to local loop through triangular facets in the traction boundary condition. We also observed small oscillations in torque during the development of facets. Isogeometric analysis based on St.Venant-Kirchhoff hyperelastic material reproduced the experimental results.

Inducing a Curl with a Stretch

Patterning grooves into the surface of an elastic ribbon can cause the ribbon to curl into a tube shape when it is stretched.

Snap-through behaviors of a pre-deformed ribbon under midpoint loadings

Based on the geometrically nonlinear Kirchhoff rod theory, the snap-through behaviors of an asymmetrically clamped ribbon under midpoint loadings are explored through a numerical approach. The pre-compressed elastic ribbon would experience supercritical pitchfork bifurcation and transform into multiple stable/unstable patterns when subjected to lateral end translations. With its midpoint pushed through a prescribed path and beyond a threshold, the pre-deformed ribbon will jump to other inverted equilibrium configurations, which is known as snap-through buckling. Instead of the well-established snap-through behaviors in 2D bistable beams, we observe several different snap-through processes for the elastic ribbons in a 3D scenario. Exploiting the efficiency and robustness of our developed discrete model, a systematic parameter sweep is performed to quantify the snap-through processes of elastic ribbons in different boundary conditions and categories, and the results of which could provide a fundamental understanding of structural mechanics and further motivate the designs and applications of engineered systems.

Swelling effects on localized adhesion of an elastic ribbon.

We investigate the adhesion mechanism between an elastic strip of vinylpolysiloxane bent in a racquet-like shape, and a thick elastomeric substrate with the aim to understand how local swelling modifies adhesion. Using a modified loop-tack adhesion test, we place a droplet of silicone oil in between the two materials, vary the dwell time and measure the force required to separate the two interfaces. The experiments are then compared with an analytical model that describes how the critical peel force is modified as the interfacial surface energy changes over time. Our study reveals that in certain circumstances swelling can enhance adhesion. More specifically, strong adhesion is obtained when most of the droplet is absorbed by the solid. By contrast, when the droplet remains at the interface a small adhesive force is measured.

Twist-Induced Snapping in a Bent Elastic Rod and Ribbon.

Snapping of a slender structure is utilized in a wide range of natural and manmade systems, mostly to achieve rapid movement without relying on musclelike elements. Although several mechanisms for elastic energy storage and rapid release have been studied in detail, a general understanding of the approach to design such a kinetic system is a key challenge in mechanics. Here we study a twist-driven buckling and fast flip dynamics of a geometrically constrained ribbon by combining experiments, numerical simulations, and an analytical theory. We identify two distinct types of shape transitions: A narrow ribbon snaps, and a wide ribbon forms a pair of localized helices. We construct a phase diagram and explain the origin of the boundary, which is determined largely by the geometry. We quantify the effects of gravity and clarify the timescale dictating the rapid flipping. Our study reveals the unique role of geometric twist-bend coupling in the fast dynamics of a thin constrained structure, which has implications for a wide range of biophysical and applied physical problems.

The Biomechanics of Lesion Formation in Endolymphatic Hydrops: Single and Double Hit Mechanisms.

The vestibular membranes of the cochlea and saccule are subject to two simultaneous constraints as they deform in endolymphatic hydrops. Boundary tethers impose a bulge-type constraint during pressure-induced transverse membrane displacement, while inherent elasticity imposes a stretch-type constraint during stress-induced longitudinal membrane distention. The aim of this study is to reconcile the effect of these dual constraints on membrane deformation. It is hypothesized that it is the interaction of these constraints that determines whether a stable membrane configuration can be achieved or progression to endolymphatic hydrops will occur. Reissner's membrane was modeled as a flat elastic ribbon that was bound along its lateral edges and subject to trans-mural pressure. The bulge and stretch constraints on membrane deformation were formulated mathematically. A graphic solution of the constraint functions was used to examine the nature of the interaction and determine how pressure and elasticity influence the hydropic process. The graphic analysis shows how bulge and stretch phenomena interact to achieve an equilibrium point that satisfies both physical requirements. Nominal values of pressure and elasticity are projected to result in a stable membrane equilibrium in the precritical zone with the modest isolated increases in either parameter alone compatible with stability. However, a sufficiently large increase in either pressure or elasticity alone can constitute a single hit mechanism to exceed the critical point and destabilize the membrane. Moreover, simultaneous modest increases in both pressure and elasticity, neither of which would be sufficient in its own right, can be additive and constitute a double hit mechanism to destabilize the membranes as well. Finally, extreme values of pressure and elasticity that fail to intersect imply that no solution is feasible and that the affected membranes will fail immediately. Sufficiently large increases in either endolymphatic pressure or membrane elasticity alone can destabilize the membranes and constitute single hit mechanisms for inducing hydrops. Combined moderate increases in both trans-mural pressure and membrane elasticity can also destabilize the membranes and constitute a double hit mechanism for hydrops induction.

Cylinder morphology of a stretched and twisted ribbon.

A rich zoology of morphologies emerges from a simple stretched and twisted elastic ribbon. Despite a lot of interest, all the observed shapes are not quantitatively described. This is the case of the cylindrical shape that prevails at large tension and twist, which emerges from a transverse buckling instability of the helicoid. Here, we propose a simple description of this cylindrical shape. By comparing its energy to the energy of other configurations, helicoidal and facetted, we are able to determine its location on the tension-twist phase diagram. The theoretical predictions are in good quantitative agreement with the experimental results and complement previous results from linear stability analysis.

The Wrinkling of a Twisted Ribbon

Recent experiments by Chopin and Kudrolli (Phys Rev Lett 111:174302, 2013) showed that a thin elastic ribbon, when twisted into a helicoid, may wrinkle in the center. We study this from the perspective of elastic energy minimization, building on recent work by Chopin et al. (J Elast 119(1–2):137–189, 2015) in which they derive a modified von Karman functional and solve the relaxed problem. Our main contribution is to show matching upper and lower bounds for the minimum energy in the small-thickness limit. Along the way, we show that the displacements must be small where we expect that the ribbon is helicoidal, and we estimate the wavelength of the wrinkles.

Anomalously Soft Non-Euclidean Springs

In this work we study the mechanical properties of a frustrated elastic ribbon spring-the non-Euclidean minimal spring. This spring belongs to the family of non-Euclidean plates: it has no spontaneous curvature, but its lateral intrinsic geometry is described by a non-Euclidean reference metric. The reference metric of the minimal spring is hyperbolic, and can be embedded as a minimal surface. We argue that the existence of a continuous set of such isometric minimal surfaces with different extensions leads to a complete degeneracy of the bulk elastic energy of the minimal spring under elongation. This degeneracy is removed only by boundary layer effects. As a result, the mechanical properties of the minimal spring are unusual: the spring is ultrasoft with a rigidity that depends on the thickness t as t^{7/2} and does not explicitly depend on the ribbon's width. Moreover, we show that as the ribbon is widened, the rigidity may even decrease. These predictions are confirmed by a numerical study of a constrained spring. This work is the first to address the unusual mechanical properties of constrained non-Euclidean elastic objects.

Structural mechanics and helical geometry of thin elastic composites.

Helices are ubiquitous in nature, and helical shape transition is often observed in residually stressed bodies, such as composites, wherein materials with different mechanical properties are glued firmly together to form a whole body. Inspired by a variety of biological examples, the basic physical mechanism responsible for the emergence of twisting and bending in such thin composite structures has been extensively studied. Here, we propose a simplified analytical model wherein a slender membrane tube undergoes a helical transition driven by the contraction of an elastic ribbon bound to the membrane surface. We analytically predict the curvature and twist of an emergent helix as functions of differential strains and elastic moduli, which are confirmed by our numerical simulations. Our results may help understand shapes observed in different biological systems, such as spiral bacteria, and could be applied to novel designs of soft machines and robots.

Buckling of Naturally Curved Elastic Strips: The Ribbon Model Makes a Difference

We analyze the stability of naturally curved, inextensible elastic ribbons. In experiments, we first show that a loop formed using a metallic strip can become unstable if its radius is larger than its natural radius of curvature (undercurved case): the loop then folds onto itself into a smaller, multiply-covered loop. Conversely, a multi-covered, overcurved metallic strip can unfold dynamically into a circular configuration having a lower covering index. We analyze these instabilities using a one-dimensional mechanical model for an elastic ribbon introduced recently (Dias and Audoly in J. Elast., 2014), which extends Sadowsky’s developable elastic ribbon model in the presence of natural curvature. Combining linear stability analyses and numerical computations of the post-buckled configurations, we classify the equilibria of the ribbon as a function of the ratio of its natural curvature to its actual curvature. Our ribbon model is formulated in close analogy with classical rod models; this allows us to adapt classical stability methods for rods to the case of a ribbon. The stability of a ribbon is found to differ significantly from that of an anisotropic rod: we attribute this difference to the fact that the tangent twisting modulus of a ribbon can be negative, in contrast to what is possible in the well-studied case of linearly elastic rods. The specific stability properties predicted by the curved ribbon model are confirmed by a finite element analysis of cylindrical shells having a small height-to-radius ratio.

Roadmap to the Morphological Instabilities of a Stretched Twisted Ribbon

We address the mechanics of an elastic ribbon subjected to twist and tensile load. Motivated by the classical work of Green (Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 154(882):430, 1936; 161(905):197, 1937) and a recent experiment (Chopin and Kudrolli in Phys. Rev. Lett. 111(17):174302, 2013) that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon. Different types of instabilities appear in various “corners” of this 4D parameter space, and are addressed through distinct types of asymptotic methods. Our theory employs three instruments, whose concerted implementation is necessary to provide an exhaustive study of the various parameter regimes: (i) a covariant form of the Foppl–von Karman (cFvK) equations to the helicoidal state—necessary to account for the large deflection of the highly-symmetric helicoidal shape from planarity, and the buckling instability of the ribbon in the transverse direction; (ii) a far from threshold (FT) analysis—which describes a state in which a longitudinally-wrinkled zone expands throughout the ribbon and allows it to retain a helicoidal shape with negligible compression; (iii) finally, we introduce an asymptotic isometry equation that characterizes the energetic competition between various types of states through which a twisted ribbon becomes strainless in the singular limit of zero thickness and no tension.

“Wunderlich, Meet Kirchhoff”: A General and Unified Description of Elastic Ribbons and Thin Rods

The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich's model) and the limit of small width (Sadowksky's model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Fr\'enet-Serret's frame, the present work unifies the description of thin ribbons and thin rods.

Helicoids, Wrinkles, and Loops in Twisted Ribbons

We investigate the instabilities of a flat elastic ribbon subject to twist under tension and develop an integrated phase diagram of the observed shapes and transitions. We find that the primary buckling mode switches from being localized longitudinally along the length of the ribbon to transverse above a triple point characterized by a crossover tension that scales with ribbon elasticity and aspect ratio. Far from threshold, the longitudinally buckled ribbon evolves continuously into a self-creased helicoid with focusing of the curvature along the triangular edges. Further twist causes an anomalous transition to loops compared with rods due to the self-rigidity induced by the creases. When the ribbon is twisted under high tension, transverse wrinkles are observed due to the development of compressive stresses with higher harmonics for greater width-to-length ratios. Our results can be used to develop functional structures using a wide range of elastic materials and length scales.

Superhelical Architecture of the Myosin Filament-Linking Protein Myomesin with Unusual Elastic Properties

Author SummaryThe contraction and relaxation cycles of active muscles generate substantial mechanical forces, both axially and radially, that place extraordinary stress on the molecular structures within the muscle fibers. These forces are sensed and buffered by unusually long and elastic filament proteins with highly repetitive domain structures. Myomesin is one such repetitive filament protein that is thought to form bridges between the main contractile filaments of the muscle, providing the muscle structure with resistance in the radial dimension. To investigate how the repetitive structure of myomesin contributes to muscle elasticity, we determined the overall architecture of its complete repetitive domain array using a combination of four complementary structural biology methods. Our study reveals a long, dimeric tail-to-tail filament structure folded into an irregular superhelical coil arrangement of almost identical domain modules separated by short linkers. When we applied tension to these myomesin filaments, we found they could stretch to about 2.5 times their original length by unfolding these linkers, and then return to their original state when the tension was removed. Our findings explain how myomesin might adapt its overall length in response to the changing dimensions of the contracting and relaxing muscle, so acting as a highly elastic ribbon that maintains the overall structural organization of the muscle fibers. More generally, these findings demonstrate how repetitive domain modules, such as those in myomesin, can provide elasticity to highly organized biological structures.