Nonuniform growth and topological defects in the shaping of elastic sheets

Interaction of a defect with the reference curvature of an elastic surface

The morphological response of two-dimensional curved elastic sheets to an isolated defect (dislocation/disclination) is investigated within the framework of Föppl-von Kármán shallow shell theory. The reference surface, obtained as a shell configuration in the absence of defect and external forces, accordingly has a finite non-zero curvature. The interaction of the defect with the curvature of the reference surface is emphasized through the problem of defect driven buckling of an elastic sheet. Detailed bifurcation diagrams, including the post-buckling deformation behaviour, are plotted for several combinations of defect types, reference curvatures, and boundary conditions. A pitchfork bifurcation is obtained when the reference surface is flat irrespective of the defect type and boundary condition. For curved reference surfaces there are some cases where the pitchfork bifurcation persists and others where it does not. The varied response demonstrates the rich interaction of the defects with the curvature of the reference surface.

The MoSeS dynamic omnigami paradigm for smart shape and composition programmable 2D materials

The properties of 2D materials can be broadly tuned through alloying and phase and strain engineering. Shape programmable materials offer tremendous functionality, but sub-micron objects are typically unachievable with conventional thin films. Here we propose a new approach, combining phase/strain engineering with shape programming, to form 3D objects by patterned alloying of 2D transition metal dichalcogenide (TMD) monolayers. Conjugately, monolayers can be compositionally patterned using non-flat substrates. For concreteness, we focus on the TMD alloy MoSe[Formula: see text]S[Formula: see text]; i.e., MoSeS. These 2D materials down-scale shape/composition programming to nanoscale objects/patterns, provide control of both bending and stretching deformations, are reversibly actuatable with electric fields, and possess the extraordinary and diverse properties of TMDs. Utilizing a first principles-informed continuum model, we demonstrate how a variety of shapes/composition patterns can be programmed and reversibly modulated across length scales. The vast space of possible designs and scales enables novel material properties and thus new applications spanning flexible electronics/optics, catalysis, responsive coatings, and soft robotics.

Distortion-controlled isotropic swelling: numerical study of free boundary swelling patterns

Modern fabrication tools have now provided a number of platforms for designing flat sheets that, by virtue of their nonuniform growth, can buckle and fold into target three-dimensional structures. Theoretically, there is an infinitude of growth patterns that can produce the same shape, yet almost nothing is understood about which of these many growth patterns is optimal from the point of view of experiment, and few can even be realized at all. Here, we ask the question: what is the optimal way to design isotropic growth patterns for a given target shape? We propose a computational algorithm to produce optimal growth patterns by introducing cuts into the target surfaces. Within this framework, we propose that the patterns requiring the fewest or shortest cuts produce the best approximations to the target shape at finite thickness. The results are tested by simulation on spherical surfaces, and new challenges are highlighted for surfaces with both positive and negative Gaussian curvatures.

Grayscale gel lithography for programmed buckling of non-Euclidean hydrogel plates

Shape programmable materials capable of morphing from a flat sheet into controlled three dimensional (3D) shapes offer promise in diverse areas including soft robotics, tunable optics, and bio-engineering. We describe a simple method of 'grayscale gel lithography' that relies on a digital micromirror array device (DMD) to control the dose of ultraviolet (UV) light, and therefore the extent of swelling of a photocrosslinkable poly(N-isopropyl acrylamide) (PNIPAm) copolymer film, with micrometer-scale spatial resolution. This approach allows for effectively smooth profiles of swelling to be prescribed, enabling the preparation of buckled 3D shapes with programmed Gaussian curvature.

Geometry and mechanics of thin growing bilayers

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness that describes a sheet in terms of both thickness and plate shape. We find that the mean curvature of the isometric state is three fourths the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.

Disclinations, e-cones, and their interactions in extensible sheets

We investigate the nucleation, growth, and spatial organization of topological defects with a ribbon shaped elastic sheet which is stretched and twisted. Singularities are found to spontaneously arrange in a triangular lattice in the form of vertices connected by stretched ridges that result in a self-rigidified structure. The vertices are shown to be negative disclinations or e-cones which occur in sheets with negative Gaussian curvature, in contrast with d-cones in sheets with zero-Gaussian curvature. We find the growth of the wrinkled width of the ribbon to be consistent with a far-from-threshold approach assuming a compression-free base state. The system is found to show a transition from a regime where the wavelength is given by the ribbon geometry, to where it is given by its elasticity as a function of the ratio of the applied tension to the elastic modulus and cross-sectional area of the ribbon.

Shape transformations of soft matter governed by bi-axial stresses

Rational design of the programmable soft matter requires understanding of the effect of a complex metric on shape transformations of thin non-Euclidean sheets. In the present work, we explored experimentally and using simulations how simultaneous or consecutive application of two orthogonal perturbations to thin patterned stimuli-responsive hydrogel sheets affects their three-dimensional shape transformations. The final shape of the sheet is governed by the metric, but not the order, in which the perturbations are applied to the system, and is determined by the competition of small-scale bidirectional stresses. In addition, a new, unexpected transition from a planar state to an equilibrium helical shape of the hydrogel sheet is observed via a mechanism that is yet to be explained.

Making the cut: lattice kirigami rules

In this Letter we explore and develop a simple set of rules that apply to cutting, pasting, and folding honeycomb lattices. We consider origami-like structures that are extrinsically flat away from zero-dimensional sources of Gaussian curvature and one-dimensional sources of mean curvature, and our cutting and pasting rules maintain the intrinsic bond lengths on both the lattice and its dual lattice. We find that a small set of rules is allowed providing a framework for exploring and building kirigami—folding, cutting, and pasting the edges of paper.