Session L52: Focus Session: Extreme Mechanics - Origami, Creasing, and Folding

Extreme Folding

Our understanding of the mathematics and algorithms behind paper folding, and geometric folding in general, has increased dramatically over the past several years. These developments have found a surprisingly broad range of applications. In the art of origami, it has helped spur the technical origami revolution. In engineering and science, it has helped solve problems in areas such as manufacturing, robotics, graphics, and protein folding. On the recreational side, it has led to new kinds of folding puzzles and magic. I will give an overview of the mathematics and algorithms of folding, with a focus on new mathematics and sculpture.   

Geometry in the mechanics of origami

We present a mechanical model for curved fold origami in which the bending energies of developable regions are balanced with a phenomenological energy for the crease. The latter energy comes into play as a source of geometric frustration, allowing us to study shape formation by prescribing crease patterns. For a single fold annular configuration, we show how geometry forces a symmetry breaking of the ground state by increasing the width of the ribbon. We extend our model to study multiple fold structures, where we derive geometrical constraints that can be written as recursive relations to build the surface from valley to mountain, and so on. We also suggest a mechanical model for single vertex folds, mapping this problem to an elastica on the sphere.   

Photo-Origami -- Using Light to Bend, Fold, and Buckle

We describe photo-origami, a method to program spatially- and temporally-variable mechanical, chemical, and optical fields into a polymer that enable controllable, sequenced, macroscopic bending and folding to create three-dimensional structures. We combine mechanical and optical stimuli to locally rearrange the polymer's network topology which allows us to program a residual stress state into the film; upon release of mechanical constraints, we realize a wide variety of desired shapes. We demonstrate, through a combination of theory, simulation-based design, synthesis, and experiment, the operative phenomena and capabilities of photo-origami. We demonstrate architectures that rely on bending, folding, instabilities, and post-buckling behavior to achieve their three-dimensional form, starting from a flat sheet. We also describe a theory that couples the hereditary nature of photophysics, chemistry, and large-deformation mechanics and enables simulations of the fabricated structures that are in good agreement with the experiments.   

Pleated and Creased Structures

The strategic placement of curved folds on a paper annulus produces saddle-shaped origami. These exotic geometries resulting from simple design processes motivate our development of a computational tool to simulate the stretching, bending and folding of thin sheets of material. We seek to understand the shape of the curved origami figure by applying the computational tool to simulate a thin annulus with single or multiple folds. We aim to quantify the static geometry of this simplified model in order to delineate methods for actuation and control of similar developable structures with curved folds. Miura-ori pattern is a periodic pleated structure defined in terms of 2 angles and 2 lengths. The unit cell embodies the basic element in all non-trivial pleated structures - the mountain or valley folds, wherein four folds come together at a single vertex. The ability of this structure to pack and unpack with a few degrees of freedom leads to their use in deployable structures such as solar sails and maps, just as this feature is useful in insect wings, plant leaves and flowers. We probe the qualitative and quantitative aspects of the mechanical behavior of these structures with a view to optimizing material performance.   

Hierarchical Stress Focusing in Elastic Ridge

A crumpled or confined elastic sheet contains many stress-focusing structures and singularities, primarily ridges and vertices, which may contain much of the strain. We seek to determine the degree and quality of stress focusing within the geometry of a single ridge. Previous work on the ridge assumes the asymptotic limit of infinitely sharp vertices. However, in a physically realistic sheet any vertex or intersection of ridges will naturally have a finite radius of curvature greater than the sheet's thickness. We simulate these more physically realistic boundary conditions in a ridge using Surface Evolver.   

Stress focusing and collapse of a thin film under constant pressure

Thin elastic sheets and shells are prone to focus stress when forced, due to their near inextensibility. Singular structures such as ridges, vertices, and folds arising from wrinkles, are characteristic of the deformation of such systems. Usually the forcing is exerted at the boundaries or at specific points of the surface, in displacement controlled experiments. On the other hand, much of the phenomenology of stress focusing can be found at micro and nanoscales, in physics and biology, making it universal. We will consider the post-buckling regime of a thin elastic sheet that is subjected to a constant normal distributed force. Specifically, we will present experiments made on thin elastoplastic sheets that collapse under atmospheric pressure. For instance, in vacuum-sealing technology, when a flat plastic bag is forced to wrap a solid volume, a series of self-contacts and folds develop. The unfolded bag shows a pattern of scars whose structure is determined by the geometry of the volume and by the exact way it stuck to its surface, by friction. Inspired by this everyday example we study the geometry of folds that result from collapsing a hermetic bag on regular rigid bodies.   

Creasy modeling of a compressed elastic surface

Compression of an elastic layer attached to a rigid substrate leads to nucleation and growth of creases. We explore crease formation by a numerical model allowing control of compressive strain, anisotropy and bulk modulus. We address questions on arrangement and geometry of creases and model also the stabilizing effect of surface tension at small scales.   

Creasing instability of elastomers under uniaxial compression

Soft polymers placed under compressive stress can undergo an elastic creasing instability in which sharp folds spontaneously form on the free surfaces. This process may play an important role in contexts as diverse as brain morphogenesis, failure of tires, and electrical breakdown of soft polymer actuators, but our understanding of this instability is still quite limited. We describe a simple experimental system to study creasing of thin elastomer films under uniaxial compression. The equilibrium depths, spacings and shapes of creases are characterized and found to show excellent agreements with numerical results. Further, we use this system to explore the important roles played by surface energy and adhesion in the onset and hysteretic behavior of creases.   

Sulcus formation in a compressed elastic half space

When a block of rubber, biological tissue or other soft material is subject to substantial compression, its surfaces undergo a folding instability. Rather than having a smooth profile, these folds contain cusps and hence have been called creases or sulcii rather than wrinkles. The stability of a compressed surface was first investigated by Biot (1965), assuming the strains associated with the instability were small. However, the compression threshold predicted with this approach is substantially too high. I will introduce a family of analytic area preserving maps that contain cusps (and hence points of infinite strain) that save energy before the linear stability threshold even at vanishing amplitude. This establishes that there is a region before the linear stability threshold is reached where the system is unstable to infinitesimal perturbations, but that this instability is quintessentially non-linear and cannot be found with linear strain elasticity.   

Compression induced folding of a sheet: An integrable system

The apparently intractable shape of a fold in a compressed elastic film lying on a fluid substrate is found to have an exact solution. Such systems buckle at a nonzero wave vector set by the bending stiffness of the film and the weight of the substrate fluid. Our solution describes the entire progression from a weakly displaced sinusoidal buckling to a single large fold that contacts itself. The pressure decrease is exactly quadratic in the lateral displacement. We demonstrate a subtle connection to the sine-Gordon problem, which reveals a new symmetry in the folding phenomenon.   

Wrinkles or creases in a bi-layer structure

Wrinkles and creases are different modes of instability. In this work, we try to answer for a bi-layer structure with different modulus ratios and thickness ratios of the film and substrate whether wrinkles or creases form first when the bi-layer is under uniform compression. The onset of wrinkles corresponds to a bifurcation point, and we use the linear perturbation method to analyze the critical strain for the onset of wrinkles. Since the initiation of creases is autonomous, we directly apply the critical condition for crease initiation in a half space calculated by the finite element method in the literature to the situation of a bi-layer structure with finite thickness. By comparing the critical strains for the formation of wrinkles and creases under different modulus and thickness ratios, a phase diagram of the formation of wrinkles or creases is obtain. Although the critical strains for both wrinkle and crease initiation depend on the state of strain, remarkably the phase diagram is independent of the state of strain. As a result, creases tend to set in for more compliant and thicker films, while wrinkles tend to set in for stiffer and thinner films. Especially, when the modulus ratio of the film and substrate is smaller than 1.67, creases always form earlier than wrinkles, no matter what the thickness ratio is. We further verify the result experimentally by compressing a bi-layer of polymers with different modulus and thickness ratios.   

Wrinkles and Folds in Ultra-Thin Polymer Films

Wrinkles and folds are observed in many biological systems during morphogenesis processes. However, the mechanics of how these wrinkles and folds form are not completely understood. Studying the mechanics of wrinkles and folds will not only provide us with fundamental insights of nonlinear deformation processes but also allow for the fabrication of unique patterned surfaces that can be controlled reversibly. In this study, we examine wrinkles and folds of polystyrene films of thickness ranging from 5 nm to 180 nm attached to uniaxially-strained polydimethylsiloxane substrates. The strain is released incrementally to apply increasing compressive strain to the attached film. The wavelength and the amplitude of local out-of-plane deformation are measured as global compression is increased to distinguish between different buckling modes. The transition from wrinkling to folding is observed by tracking the statistics of amplitude distribution sampled across a large lateral area, and a critical strain map is constructed to observe how film thickness affect the resulting buckling modes.   

Relaxation mechanisms in the unfolding of thin sheets

When a thin sheet is crumpled, creases form in which plastic deformations are localized. Here we study experimentally the relaxation process of a single fold in a thin sheet subjected to an external strain. The unfolding process is described by a quick opening at first, and then a progressive slow relaxation of the crease. In the latter regime, the necessary force needed to open the folded sheet at a given displacement is found to decrease logarithmically in time, allowing its description through an Arrhenius activation process. We accurately determine the parameters of this law and show its general character by performing experiments on both Mylar and paper sheets.