Local mechanical description of an elastic fold

Abstract

To go beyond the simple model for the fold as two flexible surfaces or faces linked by a crease that behaves as an elastic hinge, we carefully shape and anneal a crease within a polymer sheet and study its mechanical response. First, we carry out an experimental study that consists on recording both the shape of the fold in various loading configurations and the associated force needed to deform it. Then, an elastic model of the fold is built upon a continuous description of both the faces and the crease as a thin sheet with a non flat reference configuration. The comparison between the model and experiments yields the local fold properties and explains the significant differences we observe between tensile and compression regimes. Furthermore, an asymptotic study of the fold deformation enables us to determine the local shape of the crease and identify the origin of its mechanical behaviour.The process of folding solves the necessity to reduce the space occupied by a large slender object, while keeping the possibility to recover its original shape. Examples of this mechanism are common in nature such as leaves which grow folded inside buds before blooming [1][2][3] or insects that fold their wings inside shells on ground and deploy them on flight [4][5][6]. To add to their mandatory use, folds allow for shaping on demand three-dimensional structures starting from a two-dimensional thin plate [7].Two extreme examples of such objects are crumpled paper [8,9] and origami. The latter is literally the art of folding paper with a specific pattern into sculptures such as the famous paper crane. While origami can be pleasing to the eye, the selection of specific folding patterns generates original properties for the resulting object such as an apparent negative Poisson ratio [10]. When coupled with the elasticity of the constituent material, origami-based metamaterials are able to generate structures with a wide scope of programmable mechanical properties [11][12][13][14] and shapes [15][16][17]. The apparent scalability of origami allows for applications that range from small scales with the folding of DNA strands [18] or tunable microscopic origami machines [19] to large scales with biological systems [20,21] or the transport of deployable structures in space [22].Before studying origami patterns made of a complex network of folds, one should first decipher the behaviour of their most fundamental element: a single fold. Frequently, a fold is defined as two planar surfaces linked by a hinge-like crease setting an angular discontinuity in the structure. From there, different assumptions are made on its properties. If one considers a pure geometrical approach, the faces are assumed to be rigid panels while the crease angle is a degree of freedom of the system. In this case, self-avoidance and kinematic constrains impose relations between the different crease angles of a given origami structure [10,23] that constraint the corresponding degrees of freedom. While this model is fine for origami-like systems with rigid struct...

Foldable cones as a framework for nonrigid origami

“…The functions g i , which depend on the frame vectors at both sides of the crease, account for the elastic energy stored in the i-th crease. For simplicity, we consider point-like creases, although the model can be generalized to extended creases where a crease is a localized regions with a given natural curvature, as shown in reference [25]. The variation of functional (3) yields a set of n ordinary differential equations given by (see Appendix A)…”

Curvature Tuning in Folded Strips Through Hyperstatic Applied Rotations

“…When ϕ 0 = 0, there is no fold; when ϕ 0 < 0 the fold is called a mountain and when ϕ 0 > 0 the fold is called a valley (Figure 1). The linear relationship between bending moment and dihedral angle was showed experimentally in Lechenault et al (2014) and Jules et al (2019) to hold for a wide range of dihedral angles. In their experiments, Lechenault and co-workers used the setup in Figure 3.…”

Non-Euclidean origami

“…The branch splitting has significant consequences for the energy and stability landscapes of non-Euclidean 4-vertices. When we model the hinge elasticity with torsional springs and assume the plates are rigid, the energy of a vertex is given by [3,4,17,20,20,21,[45][46][47]:…”