Geometric stress focusing, e.g. in a crumpled sheet, creates point-like vertices that terminate in a characteristic local crescent shape. The observed scaling of the size of this crescent is an open question in the stress focusing of elastic thin sheets. According to experiments and simulations, this size depends on the outer dimension of the sheet, but intuition and rudimentary energy balance indicate it should only depend on the sheet thickness. We address this discrepancy by modeling the observed crescent with a more geometric approach, where we treat the crescent as a curved crease in an isometric sheet. Although curved creases have already been studied extensively, the crescent in a crumpled sheet has its own unique features: the material crescent terminates within the material, and the material extent is indefinitely larger than the extent of the crescent. These features together with the general constraints of isometry lead to constraints linking the surface profile to the crease-line geometry. We construct several examples obeying these constraints, showing finite curved creases are fully realizable. This approach has some particular advantages over previous analyses, as we are able to describe the entire material without having to exclude the region around the sharp crescent. Finally, we deduce testable relations between the crease and the surrounding sheet, and discuss some of the implications of our approach with regards to the scaling of the crescent size.
We investigate the dissipative mechanisms exhibited by creased material sheets when subjected to mechanical loading, which comes in the form of plasticity and relaxation phenomena within the creases. After demonstrating that plasticity mostly affects the rest angle of the creases, we devise a mapping between this quantity and the macroscopic state of the system that allows us to track its reference configuration along an arbitrary loading path, resulting in a powerful monitoring and design tool for crease-based metamaterials. Furthermore, we show that complex relaxation phenomena, in particula…
A creased thin disk is generally bistable since the crease could be pushed through to form a stable cone-like inverted state with an elastic singularity corresponding to the vertex of the conical surface. In a recent study, we found that this bistability could be destroyed by removing the singularity through cutting a hole around the vertex, depending on the size and shape of the hole. This paper extends our recent work and is based on the following observations in tabletop models of creased disks with circular holes: (i) reducing the circumference of the creased disk by removing an annular se…
Thin sheets that are forced at their boundaries develop a variety of shapes aimed at minimising elastic energy by curving spontaneously in ways that break the symmetry of the sheet and the forcing. Characterising such buckling generally requires detailed analysis. Here, we follow a complementary approach, applicable to axisymmetric systems. It aims to predict the qualitative nature of any buckling using only the simple axisymmetric geometry of the initial system. We view an axisymmetric deformation according to the displacement it imposes on the radial lines making up the sheet. Thus, e.g., in…
Hand crumpled paper balls involve intricate structure with a network of creases and vertices, yet show simple scaling properties, which suggests self-similarity of the structure. We investigate the internal structure of crumpled papers by the micro computed tomography (micro-CT) without destroying or unfolding them. From the reconstructed three dimensional data, we examine several power laws for the crumpled square sheets of paper of the sizes $L=50\sim 300$ mm, and obtain the mass fractal dimension $D_M = 2.7\pm 0.1$ by the relation between the mass and the radius of gyration of the balls, an…
Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put …
Liquid crystal elastomers (LCEs) can undergo large reversible contractions along their nematic director upon heating or illumination. A spatially patterned director within a flat LCE sheet thus encodes a pattern of contraction on heating, which can morph the sheet into a curved shell, akin to how a pattern of growth sculpts a developing organism. Here we consider, theoretically, numerically and experimentally, patterns constructed from regions of radial and circular director, which, in isolation, would form cones and anticones. The resultant surfaces contain curved ridges with sharp V-shaped c…
While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability is captured by a constraint on the metric within the bulk. Naively one would then guess that this constraint is enough to ensure that the deformations determining the boundary conditions on these edges respect the isometry constraint. If matters were this simple, unfortunately, it would imply unbalanced torques (as well as forces) along the edge unless manife…
