4.7 Supplementary Materials
4.7.6 Numerical investigation of patterns behavior
4.7.6.1 Numerical calculation of the number of DOFs of the patterns
In the bar-framework analysis, compatibility equation is to relate the nodal displacements d to the bar extensions e via compatibility matrix C as follows
Cd e (4.30) From the above equation, the nullspace of the compatibility matrix provides the solution in which the bars do not extend. To model rigid origami behavior, we need to add an angular
constraint to the compatibility matrix whose nullspace can provide the nodal displacements d for which the facets do not bend either. The angular constraint can be written in terms of the dihedral fold angles between triangulated facets connected by added fold lines [10, 23]. Hence,
facet d
J d (4.31) where
is the dihedral fold angle between two adjoining triangulated facets and Jfacet is the Jacobian of the angular constraint considered for the triangulated facets intersecting by added fold lines. Therefore, the augmented compatibility matrix is as followsfacet
C C
J (4.32) Accordingly, the number of internal infinitesimal mechanisms (i.e., the number of independent DOFs) can be obtained from the expression
3 ( ) 6
m j rank C (4.33)
in which j is the number of joints (i.e., the number of vertices). In the above relation, the 6 DOFs related to the rigid-body motions of 3D structures are excluded.
We used the above relation to obtain the number of DOFs for the patterns considering rigid origami behavior. The results are justified based on the geometry of the patterns as well as existence of the implicit formation of the structure of the Miura-ori cells with one-DOF mechanism as described in Section 4.7.2.
Figure 4.15: Behavior of the sheet of BCH3 under bending and the results of eigen-value analysis of a 3 by 3 pattern of BCH3. (a) Sheet of BCH3 deforms into a saddle-shaped under bending which is typical behavior for materials having a positive Poisson’s ratio.
(b) Twisting, (c) saddle-shaped and (d) rigid origami behavior (planar mechanism) of a 3 by 3 pattern of BCH3 (with a=1; b=2; 60 ).
(b)
(C)
(d) (a)
Figure 4.16: Behavior of a sheet of the pattern shown in Figure 4.3(c) under bending and results of eigen-value analysis of a 2 by 3 sheet of the pattern. (a) The sheet deforms into a saddle-shaped under bending (i.e., typical behavior seen in materials having a positive Poisson’s ratio). (b) Twisting, (c) saddle-shaped from two different views and (d) rigid origami behavior (planar mechanism) of a 2 by 3 pattern shown in Figure 4.3(c) (with a=1; b=2; 60 ).
(b)
(c)
(d) (a)
Figure 4.17: Behavior of a sheet of the pattern shown in Figure 4.3(d) under bending and the results of eigen-value analysis of a 2 by 3 sheet of the pattern. (a) The sheet deforms into a saddle-shaped under bending, i.e. a typical behavior seen in materials having a positive Poisson’s ratio. (b) Twisting, (c) saddle-shaped from two different views and (d) rigid origami behavior (planar mechanism) of a 2 by 3 pattern shown in Figure 4.3(d) (with a=1; b=2; 60 ).
(b)
(c)
(d) (a)
5 Tuning the Miura-ori Properties by Dislocating the Zigzag Strips
This chapter is adapted from an article authored by Maryam Eidini, in review [94].
Abstract
The Japanese art of turning flat sheets into three dimensional intricate structures, origami, has inspired design of mechanical metamaterials. Mechanical metamaterials are artificially engineered materials with uncommon properties. Miura-ori is a remarkable origami folding pattern with metamaterial properties and a wide range of applications. In this study, by dislocating the zigzag strips of a Miura-ori pattern along the joining ridges, we create a class of one-degree of freedom (DOF) cellular mechanical metamaterials.
The resulting configurations are based on a unit cell in which two zigzag strips surround a hole with a parallelogram cross section. We show that dislocating zigzag strips of the Miura-ori along the joining ridges preserves and/or tunes the outstanding properties of the Miura-ori. The introduced materials are lighter than their corresponding Miura-ori sheets due to the presence of holes in the patterns. Moreover, they are amenable to similar modifications available for Miura-ori which make them appropriate for a wide range of applications across the length scales.
Keywords: Miura-ori; Poisson’s ratio; auxetic; metamaterial; origami; kirigami; zigzag;
herringbone.
5.1 Introduction
Miura-ori, a zigzag/herringbone-base origami folding pattern, has attracted substantial attention
in science and engineering for its remarkable properties [71, 95, 96, 97, 98]. The exceptional mechanical properties of the Miura-ori [97, 39], the ability to produce its morphology as a self-organized buckling pattern [71, 95] and its geometric adaptability [99, 100] has made the pattern suited for applications spanning from metamaterials [97] to fold-core sandwich panels [101].
Moreover, Miura-ori is a mechanical metamaterial with negative Poisson's ratio for a wide range of its geometric parameters [65, 78]. Mechanical metamaterials are artificially engineered materials with unusual material properties arising from their geometry and structural layout. In-plane Poisson’s ratio is defined as the negative ratio of transverse to axial strains. Poisson’s ratios of many common isotropic elastic materials are positive, i.e., they expand transversely when compressed in a given direction. Conversely, when compressed, materials with negative Poisson's ratio or auxetics contract in the directions perpendicular to the applied load. Discovery and creating of auxetic materials has been of interest due to improving the material properties of auxetics [89, 102, 103, 104]. Auxetic behavior may be exploited through rotating rigid and semi-rigid units [105, 106], chiral structures [107, 108], reentrant structures [109, 110, 111], elastic instabilities in switchable auxetics [112, 113], creating cuts in materials [114], and in folded sheet materials [97, 65]. The latter is the concentration of the current research.
Research studies have shown that the herringbone geometry leads to auxetic properties in folded sheet materials [97, 65] and textiles [115, 116], and its morphology arises in biological systems [75, 76, 117]. Due to possessing unprecedented deformability, the herringbone structure fabricated by bi-axial compression, has been also used in deformable batteries and electronics [118, 98, 119].
Kirigami, the art of paper cutting, has been applied in science and engineering as three dimensional (3D) core cellular structures and solar cells among others [64, 120, 121]. The current research expands on a recent study by Eidini and Paulino [65] where origami folding has been combined with cutting patterns to create a class of cellular metamaterials. In the present study, we use the concept of the Poisson’s ratio of a one-DOF zigzag strip (i.e.,z tan2) [65] which provides inspiration to tune and/or preserve the properties of the Miura-ori. In this regard, by dislocating the zigzag strips of the Miura-ori pattern along the joining fold lines, we create a novel class of metamaterials. The resulting configurations are based on a one-DOF unit cell in which two zigzag strips surround a hole with a parallelogram cross section.