Asymptotic derivation of a higher-order one-dimensional model for tape springs.

We derive a one-dimensional model for tape springs. The derivation starts from nonlinear thin-shell theory and uses a dimension reduction technique that combines a centreline-based parametrization of the tape-spring midsurface with the assumption that the strain varies slowly along the length of the tape spring. The one-dimensional model is effectively a higher-order rod model: at leading order, the strain energy depends on the extensional, bending and twisting strains and is consistent with classical results from the literature; the two following orders are novel and capture the dependence of the strain energy on the strain gradients. The cross-sectional displacements are solved as part of the dimension reduction process, making the one-dimensional model asymptotically exact. We expect that the model will accurately and efficiently capture the deformations and instabilities in tape springs, including those involving highly localized deformations. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

A nonlinear beam model of photomotile structures.

Actuation remains a significant challenge in soft robotics. Actuation by light has important advantages: Objects can be actuated from a distance, distinct frequencies can be used to actuate and control distinct modes with minimal interference, and significant power can be transmitted over long distances through corrosion-free, lightweight fiber optic cables. Photochemical processes that directly convert photons to configurational changes are particularly attractive for actuation. Various works have reported light-induced actuation with liquid crystal elastomers combined with azobenzene photochromes. We present a simple modeling framework and a series of examples that study actuation by light. Of particular interest is the generation of cyclic or periodic motion under steady illumination. We show that this emerges as a result of a coupling between light absorption and deformation. As the structure absorbs light and deforms, the conditions of illumination change, and this, in turn, changes the nature of further deformation. This coupling can be exploited in either closed structures or with structural instabilities to generate cyclic motion.

Bending Response of a Book with Internal Friction.

We study the bending of a booklike system, comprising a stack of elastic plates coupled through friction. The behavior of this layered system is rich and nontrivial, with a nonadditive enhancement of the apparent stiffness and a significant hysteretic response. A dimension reduction procedure is employed to develop a centerline-based theory describing the stack as a nonlinear planar rod with internal shear. We consider the coupling between the nonlinear geometry and the elasticity of the stacked plates, treating the interlayer friction perturbatively. This model yields predictions for the stack's mechanical response in three-point bending that are in excellent agreement with our experiments. Remarkably, we find that the energy dissipated during deformation can be rationalized over 3 orders of magnitude, including the regimes of a thick stack with large deflection. This robust dissipative mechanism could be harnessed to design new classes of low-cost and efficient damping devices.

Shape-morphing architected sheets with non-periodic cut patterns.

We investigate the out-of-plane shape morphing capability of single-material elastic sheets with architected cut patterns that result in arrays of tiles connected by flexible hinges. We demonstrate that a non-periodic cut pattern can cause a sheet to buckle into three-dimensional shapes, such as domes or patterns of wrinkles, when pulled at specific boundary points. These global buckling modes are observed in experiments and rationalized by an in-plane kinematic analysis that highlights the role of the geometric frustration arising from non-periodicity. The study focuses on elastic sheets, and is later extended to elastic-plastic materials to achieve shape retention. Our work illustrates a scalable route towards the fabrication of three-dimensional objects with nonzero Gaussian curvature from initially-flat sheets.

Shape Transformations of Epithelial Shells.

Regulated deformations of epithelial sheets are frequently foreshadowed by patterning of their mechanical properties. The connection between patterns of cell properties and the emerging tissue deformations is studied in multiple experimental systems, but the general principles remain poorly understood. For instance, it is in general unclear what determines the direction in which the patterned sheet is going to bend and whether the resulting shape transformation will be discontinuous or smooth. Here these questions are explored computationally, using vertex models of epithelial shells assembled from prismlike cells. In response to rings and patches of apical cell contractility, model epithelia smoothly deform into invaginated or evaginated shapes similar to those observed in embryos and tissue organoids. Most of the observed effects can be captured by a simpler model with polygonal cells, modified to include the effects of the apicobasal polarity and natural curvature of epithelia. Our models can be readily extended to include the effects of multiple constraints and used to describe a wide range of morphogenetic processes.

From discrete to continuum models of three-dimensional deformations in epithelial sheets.

Epithelial tissue, in which cells adhere tightly to each other and to the underlying substrate, is one of the four major tissue types in adult organisms. In embryos, epithelial sheets serve as versatile substrates during the formation of developing organs. Some aspects of epithelial morphogenesis can be adequately described using vertex models, in which the two-dimensional arrangement of epithelial cells is approximated by a polygonal lattice with an energy that has contributions reflecting the properties of individual cells and their interactions. Previous studies with such models have largely focused on dynamics confined to two spatial dimensions and analyzed them numerically. We show how these models can be extended to account for three-dimensional deformations and studied analytically. Starting from the extended model, we derive a continuum plate description of cell sheets, in which the effective tissue properties, such as bending rigidity, are related explicitly to the parameters of the vertex model. To derive the continuum plate model, we duly take into account a microscopic shift between the two sublattices of the hexagonal network, which has been ignored in previous work. As an application of the continuum model, we analyze tissue buckling by a line tension applied along a circular contour, a simplified set-up relevant to several situations in the developmental contexts. The buckling thresholds predicted by the continuum description are in good agreement with the results of stability calculations based on the vertex model. Our results establish a direct connection between discrete and continuum descriptions of cell sheets and can be used to probe a wide range of morphogenetic processes in epithelial tissues.

Liquid ropes: a geometrical model for thin viscous jet instabilities.

Thin, viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitchlike patterns including meanders, W patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasistatic geometrical model which captures the patterns, consisting of three coupled ordinary differential equations for the radial deflection, the orientation, and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the belt speed.

Furrow constriction in animal cell cytokinesis.

Cytokinesis is the process of physical cleavage at the end of cell division; it proceeds by ingression of an acto-myosin furrow at the equator of the cell. Its failure leads to multinucleated cells and is a possible cause of tumorigenesis. Here, we calculate the full dynamics of furrow ingression and predict cytokinesis completion above a well-defined threshold of equatorial contractility. The cortical acto-myosin is identified as the main source of mechanical dissipation and active forces. Thereupon, we propose a viscous active nonlinear membrane theory of the cortex that explicitly includes actin turnover and where the active RhoA signal leads to an equatorial band of myosin overactivity. The resulting cortex deformation is calculated numerically, and reproduces well the features of cytokinesis such as cell shape and cortical flows toward the equator. Our theory gives a physical explanation of the independence of cytokinesis duration on cell size in embryos. It also predicts a critical role of turnover on the rate and success of furrow constriction. Scaling arguments allow for a simple interpretation of the numerical results and unveil the key mechanism that generates the threshold for cytokinesis completion: cytoplasmic incompressibility results in a competition between the furrow line tension and the cell poles' surface tension.

Instant fabrication and selection of folded structures using drop impact.

A drop impacting a target cutout in a thin polymer film is wrapped by the film in a dynamic sequence involving both capillary forces and inertia. Different 3D structures can be produced from a given target by slightly varying the impact parameters. A simplified model for a nonlinear dynamic Elastica coupled with a drop successfully explains this shape selection and yields detailed quantitative agreement with experiments. This first venture into the largely unexplored dynamics of elastocapillary assemblies opens up the perspective of mass production of 3D packages with individual shape selection.

Solid drops: large capillary deformations of immersed elastic rods.

Under the effect of surface tension, a blob of liquid adopts a spherical shape when immersed in another fluid. We demonstrate experimentally that soft, centimeter-size elastic solids can exhibit a similar behavior: when immersed into a liquid, a gel having a low elastic modulus undergoes large, reversible deformations. We analyze three fundamental types of deformations of a slender elastic solid driven by surface stress, depending on the shape of its cross section: a circular elastic cylinder shortens in the longitudinal direction and stretches transversally; the sharp edges of a square based prism get rounded off as its cross sections tend to become circular; and a slender, triangular based prism bends. These experimental results are compared to analysis and nonlinear simulations of neo-Hookean solids deformed by surface tension and are found to be in good agreement with each other.

Influence of Stratum Corneum on the entire skin mechanical properties, as predicted by a computational skin model.

BACKGROUND/PURPOSE: Skin mechanical properties are globally well described. The aim of this paper is to evaluate, by means of a numerical model, the influence of Stratum Corneum (SC) on skin folding resulting from an in-plane compression. METHODS: A computational skin model was developed where skin is divided into three layers (SC, epidermis and upper dermis, and deep dermis) of different thicknesses and elastic moduli. Skin surface deformation, consecutive to the application of a given surface compression, was computed by minimizing the mechanical energy of the multi-layered tissue. Influence of SC thickness and elastic modulus on skin buckling is presented. RESULTS: Varying both SC thickness and elastic modulus has a marked influence on both wavelength and amplitude of the skin's surface folds. These two parameters display a logarithmic variation versus SC elastic modulus. CONCLUSION: Although representing about one hundredth of the total skin thickness, SC has a marked influence on the skin mechanical properties.

The surprising dynamics of a chain on a pulley: lift off and snapping.

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. Here, we consider the dynamics of the chain when one of the weights is removed and, thus, one end is pulled with constant acceleration. This simple change has dramatic consequences for the ensuing motion: at a finite time, the chain 'lifts off' from the pulley, and the free end subsequently accelerates faster than the end that is pulled. Eventually, the chain undergoes a dramatic reversal of curvature reminiscent of the crack or snap, of a whip. We combine experiments, numerical simulations and theoretical arguments to explain key aspects of this dynamical problem.

An introduction to the mechanics of the lasso.

Trick roping evolved from humble origins as a cattle-catching tool into a sport that delights audiences all over the world with its complex patterns or 'tricks'. Its fundamental tool is the lasso, formed by passing one end of a rope through a small loop (the honda) at the other end. Here, we study the mechanics of the simplest rope trick, the Flat Loop, in which the rope is driven by the steady circular motion of the roper's hand in a horizontal plane. We first consider the case of a fixed (non-sliding) honda. Noting that the rope's shape is steady in the reference frame rotating with the hand, we analyse a string model in which line tension is balanced by the centrifugal force and the rope's weight. We use numerical continuation to classify the steadily rotating solutions in a bifurcation diagram and analyse their stability. In addition to Flat Loops, we find planar 'coat-hanger' solutions, and whirling modes in which the loop collapses onto itself. Next, we treat the more general case of a honda that can slide due to a finite coefficient of friction of the rope on itself. Using matched asymptotic expansions, we resolve the shape of the rope in the boundary layer near the honda where the rope's bending stiffness cannot be neglected. We use this solution to derive a macroscopic criterion for the sliding of the honda in terms of the microscopic Coulomb static friction criterion. Our predictions agree well with rapid-camera observations of a professional trick roper and with laboratory experiments using a 'robo-cowboy'.

Shape of an elastic loop strongly bent by surface tension: experiments and comparison with theory.

When a very flexible wire is dipped into a soapy solution, it collapses onto itself. We consider the regions of high curvature where the wire folds back onto itself, enclosing a capillary film. The shapes of these end loops are measured in experiments using soap films and compared to a known similarity solution. The sizes of these structures provide a simple and reliable way to measure surface tension.

Fragmentation of rods by cascading cracks: why spaghetti does not break in half.

When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this end leads to a burst of flexural waves, whose dynamics are described by a self-similar solution with no adjustable parameters. These flexural waves locally increase the curvature in the rod, and we argue that this counterintuitive mechanism is responsible for the fragmentation of brittle rods under bending. A simple experiment supporting the claim is presented.