Discrete symmetries control geometric mechanics in parallelogram-based origami

Abstract

Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson’s ratio Miura-ori origami crease pattern. Here, we develop a formalism for linear compatibility that enables explicit investigation of the interplay between geometric symmetries and functionality in origami crease patterns. We apply this formalism to a particular class of periodic crease patterns with unit cells composed of four arbitrary parallelogram faces and establish that their mechanical response is characterized by an anticommuting symmetry. In particular, we show that the modes are eigenstates of this symmetry operator and that these modes are simultaneously diagonalizable with the symmetric strain operator and the antisymmetric curvature operator. This feature reveals that the anticommuting symmetry defines an equivalence class of crease pattern geometries that possess equal and opposite in-plane and out-of-plane Poisson’s ratios. Finally, we show that such Poisson’s ratios generically change sign as the crease pattern rigidly folds between degenerate ground states and we determine subfamilies that possess strictly negative in-plane or out-of-plane Poisson’s ratios throughout all configurations.

Unexpected bending behavior of architected 2D lattice materials

“…Our result defies the common belief and is attributed to the competitive interaction between axial torsion and out-of-plane bending of the beams. Note that similar irrelevance of bending curvature to IPR has been seen in some origami with specific crease patterns (46)(47)(48), but both the structure and mechanisms are different. To capture the role of axial torsion of the beams, a continuum model based on Cosserat elasticity (49,50) is developed to describe the unusual bending behavior of the 2D lattices.…”

Modelling planar kirigami metamaterials as generalized elastic continua

“…The same approach applies to the more general family of parallelogram-slit kirigami, and goes beyond its mechanisms to capture its soft modes [47]. Certain generalized Miura-Ori origami patterns have been similarly coarse-grained in [33,35]. We focus on the class of rhombi-slit kirigami here because we think it strikes a balance between breadth and simplicity.…”

“…Recently, a third approach has emerged from the idea of finding effective, coarse-grained models for the deformations of origami and kirigami [29][30][31][32][33][34][35]. In this approach, the metamaterial is modelled as an effectively continuum object, with the aim of capturing the collective response of its cells through averaging.…”

Geometric mechanics of hybrid origami assemblies combining developable and non-developable patterns

“…Future work could verify the in-plane Poisson's ratio experimentally in a similar manner to [19] and compare with theoretical and numerical results. Poisson's ratio in bending could also be analytically determined as has been done with the Miura-ori [22], eggbox [23], morph [14] and other tessellated patterns [24,25].…”