Introduction

Mechanical metamaterials, the man-made materials with mechanical properties mainly defined by their structures instead of the properties of each component, recently have attracted great attention1,2,3,4. Origami, creating three-dimensional (3D) structures from two-dimensional (2D) sheets through a process of folding along creases, provides an interesting source for designing mechanical metamaterials and has been transformed by mathematicians, scientists and engineers to utilize the folded objects' deformability and compactness in applications ranging from space exploration (e.g., a foldable telescope lens5), to automotive safety (e.g., airbags6), biomedical devices (e.g., heart stent7) and extremely foldable and stretchable electronics8,9. Notable progress has been made in the area of origami theory including tree theory10 and its corresponding computer program11, folding along creases12,13,14 and geometric mechanics of a periodic origami pattern15. Among classes of origami patterns, a particular one, namely rigid origami, in which the faces between the creases remain rigid during folding/unfolding and only the creases deform, is different from most origami patterns that require faces bending or partial crumpling to make many-step folds. Idealized rigid origami possesses one of the most obvious advantages of origami in terms of deformation, i.e., the deformation is completely realized by the folding/unfolding at the creases and does not involve any deformation at the rigid faces4. The geometric characteristics, such as the necessary condition around a single vertex for rigid origami16,17 have been studied and a computer simulator for rigid origami18 exists. There have also been made limited efforts to study the structural characteristics of one particular rigid origami, namely Miura-ori19, as a mechanical metamaterial, with the main focus on the negative Poisson's ratio15,20, though these properties can be more rigorously examined. It is noticed that the existing studies are mainly focused on periodic origami patterns (e.g., Miura-ori); however, non-periodic origami patterns as mechanical metamaterials have not gained attention yet, partially due to the difficulties in theoretical analysis and modeling. To span a much wider spectrum of using rigid origami as mechanical metamaterials, we report a systematic study of two rigid origami folding patterns, not only the periodic Miura-ori but also a non-periodic Ron Resch folding21 using combined analytical and numerical approaches. Specifically, we rigorously address the commonly mistaken in-plane Poisson's ratio of Miura-ori, which was believed to be always negative but we show here that it can be positive as well and its physical interpretation. The ubiquitous non-local interactions between vertices of rigid origami patterns are captured through a non-local finite element approach and the compressive buckling resistance of a Ron Resch tube is studied, which inspires a theoretical and experimental study of the load bearing capability of the Ron Resch pattern. The result witnesses the superb load bearing capability of this Ron Resch pattern. Based on the approaches in this paper, mechanical properties of different rigid origami patterns, both periodic and non-periodic ones, can be readily studied.

Results

Unit cell and the whole pattern of a Miura-ori

Figure 1a illustrates a Miura-ori (n1, n2) in its folded state, containing n1 ( = 11) vertices in x1 direction and n2 ( = 11) vertices in x2 direction, with x3 as the out-of-plane direction. Its corresponding planar state is shown in Fig. 1b. The geometry of a Miura-ori is defined by many identical rigid parallelogram faces (with four gray ones highlighted in Fig. 1b) linked by edges that can be folded into “mountain” and “valley” creases. The Miura-ori is a periodic structure and its unit cell is shown in Fig. 1c, where the four parallelograms are identical with the short sides of length a, the long sides of length b and the acute angle β[0°,90°]. Since the necessary condition for rigid origami16,17 states that there are n − 3 degrees of freedom, where n is the number of edges at one vertex, Miura-ori with n = 4 has only one degree of freedom. Therefore, if the shape of a parallelogram face is prescribed, i.e. β, a and b are given, one parameter ϕ[0°,2β], the projection angle between two ridges, can be used to characterize the folding of the unit cell of Miura-ori, with ϕ = 2β for the planar state and ϕ = 0° for the completely collapsed state. The size of the unit cell is l = 2bsin(ϕ/2), and , in x1, x2 and x3 directions, respectively. It is noted that the length of the “tail” bcos(ϕ/2) is not considered in the unit cell15. The periodicity of this pattern only requires two dihedral angles α1[0°,180°] and α2[0°,180°] to characterize the geometry (Fig. 1c), which are given by

and equal 180° for the planar state and 0° for the completely collapsed state. When the whole structure of a Miura-ori is put in an imaginary box with the dashed lines as the boundaries (Fig. 1a), the dimension of the whole Miura-ori is then given by

and thus the imaginary volume occupied by this Miura-ori is given by

Apparently even the Miura-ori is periodic, its size in x2 direction (i.e. W) is not proportional to its counterpart for the unit cell, w, due to the existence of the “tail” with length bcos(ϕ/2). Consequently, it is not accurate to use the unit cell to study the size change of a whole Miura-ori (e.g., Poisson's ratio), particularly for smaller patterns (e.g., used in15,20).

In-plane Poisson's ratio of Miura-ori

In-plane Poisson's ratio of Miura-ori is believed to be negative from intuitive observations and as testified by some theoretical studies using the unit cell (Fig. 1c)15,20. An accurate mean to define the Poisson's ratio is to use the size of a whole Miura-ori, instead of using the unit cell. Specifically, the in-plane Poisson's ratio ν12 is defined as , where and are the infinitesimal strains in x1- and x2-directions, respectively. Using equation (2), the in-plane Poisson's ratio ν12 is obtained as

where η = a/b. Another in-plane Poisson's ratio ν21 is just the reciprocal of ν12. Figure 2a shows the contour of ν12 as a function of angle ϕ and a combination parameter (n2 − 1)ηcosβ. Clearly, ν12 can be negative or positive, which is different from commonly observed negative in-plane Poisson's ratio. The boundary separating the negative and positive regimes of ν12 is defined by vanishing the denominator of ν12, i.e., (n2 − 1)ηcosβ = cos2(ϕ/2). At this boundary, ν12 flips between positive and negative infinities; thus ν12[−∞,+∞]. For one scenario, where n2 = 5 (small pattern), η = 1/2, β = 78.5° and thus (n2 − 1)ηcosβ < 1, ν12 is positive for ϕ[0,101.5°] and changes to negative for ϕ[101.5°,2β]. For another scenario, n2 = 13 (large pattern), , β = 45° and thus (n2 − 1)ηcosβ > 1, ν12 stays negative, as reported by others using the unit cell15,20; and the Miura-ori becomes an auxetic material. Similar analysis can be applied on the out-of-plane Poisson's ratios. Details can be found in the Supplementary Information, Section “Out-of-plane Poisson's Ratio”.

Figure 2b provides an intuitive explanation of the sign change in the in-plane Poisson's ratio ν12. For the specific example with n1 = n2 = 5, η = 1/2 and β = 78.5°, the size of this Miura pattern in x1 direction, L, decreases monotonically from the planar state to the collapsed state, which is pictorially shown in the three insets for ϕ = 157°( = 2β), ϕ = 140° and ϕ = 20° with L1 > L2 > L3. In contrast to L, the respective size of this pattern in the x2 direction, W, does not change monotonically with the angle ϕ. From the planar state to the collapsed state, W1 > W2 but W2 < W3, which gives ν12 < 0 when L and W change in the same direction and ν12 > 0 when L and W change in the opposite direction. The non-monotonic change of W is due to the “tail” term bcos(ϕ/2) in equation (2), which was missed in previous studies15,20. As shown in the Supplementary Information, Section “Change of Length W” for more details, the two terms in W (equation (2)) dominate at different stage of folding.

In addition to the negative and positive in-plane Poisson's ratio of the Miura-ori, the ranges of Poisson's ratios, specifically, ν12[−∞,+∞], ν13[0,∞], ν23[−∞,∞] (see Supplementary Information, Section “Out-of-plane Poisson's Ratio”) are also fascinating if the range of Poisson's ratio for common materials is considered as the reference, i.e., ν[−1,0.5]. Now we interpret these fascinating phenomena in terms of shear and bulk modulus of Miura-ori.

Miura-ori subjected to shear and hydrostatic deformation

To study the shear deformation that is non-uniform across the Miura-ori, we developed a numerical approach to characterize the geometric features of the Miura-ori, i.e., the non-local interactions between rigid faces. As shown in Fig. 1b, the vertex marked by the solid blue dot not only interacts with its nearest-neighboring vertices (marked by the solid red dots) through the rigid faces, but also its second-neighboring vertices (marked by the solid green dots) through dihedral angles. Thus the interactions between vertices are non-local. This non-local nature is ubiquitous in rigid origami and can be more complicated for other patterns, which can be further illustrated by the Ron Resch pattern21, detailed in the Supplementary Information, Section “Non-local Interactions in the Ron Resch Pattern” (e.g., Supplementary Fig. S3). We developed a non-local finite element based model and Fig. 1d shows the non-local element for Miura-ori. Details can be found in Methods.

Supplementary Figure S4 shows a deformed state of a (n1 = 13, n2 = 13) Miura-ori subjected to a finite shear force in the negative x1 direction. Here it is noticed that an initially periodic Miura-ori deforms non-uniformly under shear loading, which disables the definition of a shear modulus. It is seen that the Miura-ori responds in an opposite way to shear force. Specifically and clearly, the vertical lines tilt to the positive x1 direction, as the shear force is applied along the negative x1 direction. This opposite relationship is thus consistent with ν12[−∞,+∞].

The bulk modulus K of Miura-ori can be defined the same way as that in continuum mechanics to link the hydrostatic pressure p and the volumetric strain θ( = ε11 + ε22 + ε33),

Using the principle of superposition (details given in the Supplementary Information, Section “Bulk Modulus of Miura-ori”), the bulk modulus K is given by

where are the tangential moduli of the stress-strain curve. Using the work conjugate relation, stresses are expressed as , where Wtot = Utot/V is the elastic energy density with Utot given by Supplementary Eq. S2 and V given by equation (3). As shown in the Supplementary Information, Section “Range of Tensile and Bulk Modulus” for details, the tensile (E11, E22, E33) and bulk moduli (K) have a wide range of variation and some of them vary from 0 to infinity, such as K.

Ron Resch pattern and its buckling resistance

Next we study a non-periodic rigid origami folding, namely a Ron Resch pattern, using the developed non-local finite element approach. The Ron Resch pattern and its non-local elements are given in Supplementary Fig. S3. To illustrate the non-periodicity, several Ron Resch patterns (specifically, a Ron Resch dome, a tube and a stingray) have been studied and the histograms of the three dihedral angles β1, β2 and β3 are shown in Supplementary Fig. S5. It is obvious that the Ron Resch pattern is non-periodic and the importance of a universal numerical platform to study this type of rigid origami is thus apparent. We first study the buckling resistance of a Ron Resch tube (Fig. 3a). A Ron Resch tube in its folded state contains many equilateral triangles. As shown in the zoom-in details in Fig. 3a, the dihedral angles β1[0°,90°] and β1[90°,180°]. Because of the folded state, the centroids of these equilateral triangles form spikes pointing to the central axis of the Ron Resch tube as shown in the top view of Fig. 3a. The boundary condition for the axial compressive buckling is that one end of the tube is fixed and the other is subjected to a compressive force, which is the same as the Euler buckling. Figure 3b shows the compressive force normalized by kRR/b varies as the compressive strain increases and the insets show some characteristic snapshots at the compressive strains of 13%, 30% and 45% from left to right, respectively. Here kRR is the spring constant of the hinges for dihedral angles (detailed in the Supplementary Information, Section “Work Conjugate Relation – Stress and Moduli for Miura-ori”) and b is the size of the right triangles (Supplementary Fig. S3). It is interesting to find that buckling does not occur, which can be explained by the negative Poisson's ratio. Upon compression, the two dihedral angles β1 and β2 decrease, which lead to the further pushing the spikes towards the central axis of the tube. Thus the compression leads to a shorter tube with smaller radius due to the negative Poisson's ratio (the leftmost inset of Fig. 3b). Further compression leads to an even smaller tube radius (the middle inset of Fig. 3b). Eventually, the equilateral triangles form completely folded states, which is captured by β1 = 0°, β2 = 120° and results in a much smaller tube radius (the rightmost inset of Fig. 3b). At the completely folded state, the tube cannot be further compressed because of the rigidity. Thus, axial compressive force does not lead to the buckling of a Ron Resch tube.

Discussion

This paper paves the ways towards the study of interesting and unique geometric and mechanical properties of origami structures as mechanical metamaterials. It is expected that through a combination of this approach and multiphysics simulations (e.g., COMOSL Multiphysics), more interesting properties can be explored. For example, the negative response between shear force and deformation and infinite tensile and bulk modulus may lead to some unique sound and vibration behaviors. When integrated with functional materials on origami patterns with micrometer feature sizes (e.g., the size of its rigid faces), such as nanowires and two-dimensional materials, the foldability of the origami pattern would provide unique tunable metamateirals with intriguing optical, electrical and magnetic properties, which is in fact under pursue. When combined with applications, the analysis of origami as mechanical metamaterials can help to guide the development of origami based devices8,9. As all of these properties and applications are rooted from the way of folding, origami also provides a unique and powerful way on manufacturing. For example, plywood with unusually strong load bearing capability at the completed folded state can be manufactured in large-scale and low-cost by pre-creasing the wood panel based on the Ron Resch pattern. It is thus believed that origami may provide many interesting applications in science and engineering.

Methods

Non-local finite element method

Starting from the energy perspective, the elastic energy stored in a folded state is just the rotational energy at the creases since all the faces are rigid. If the creases are considered as elastic hinges, the elastic energy takes the quadratic term of the dihedral angels between creases. For the Miura-ori, the elastic energy can be written as

where and are the spring constants of the hinges for dihedral angles α1 and α2 for Miura-ori (superscript “Mo”), respectively; α1,eq and α2,eq are the corresponding dihedral angles for α1 and α2 at the undeformed state (or equivalently, just folded state); the summation runs over all dihedral angles. Similarly, the elastic energy can be readily constructed for the Ron Resch pattern,

where the superscript “RR” denotes the Ron Resch pattern and the subscripts have a similar meaning as explained for the Miura-ori. It is reasonable to take and , for paper folding (although in most machine-made papers, the fibers tend to run in one direction and so the hinge constants for edges running in different directions will be different).

Because the dihedral angles are completely determined by the coordinates of vertices in rigid origami, the elastic energy can also be expressed as a function of coordinates of vertices, i.e., Utotal = Utotal(x), where and xi is the position of a vertex i and N is the total number of the vertices. When the external load is applied at vertex i, the total potential energy is . The equilibrium state of a rigid origami corresponds to a state of minimum energy and can be given by

which needs to be solved to reach the equilibrium state of a rigid origami. There are many approaches that can be utilized to solve Supplementary Eq. S3, such as the conjugate gradient method and steepest descent method that just use the first derivatives of Πtotal, or the finite element method that uses both the first derivative (as the non-equilibrium force ) and the second derivatives (as the stiffness matrix ) of Πtotal. The governing equation for the finite element method is

where u = xx(0) is the displacement of the vertices with x(0) as the initial position of the vertices. For nonlinear systems, equation (10) is solved iteratively until the equilibrium characterized by the vanishing non-equilibrium force P = 0. For discrete vertices in rigid origami that has a great deal of similarity with atomic systems, the finite element method has been extended to capture the non-local interactions23,24.

There are two aspects to consider when the finite element method is used. Firstly, to calculate the non-equilibrium force P and stiffness matrix K, the elastic energy Utotal needs to be explicitly written as a function of vertex coordinates, which is detailed in the Supplementary Information, Section “Nonlinearity of the Elastic Energy with respect to the Coordinates of Vertices”. Therefore, iteration is needed. Secondly, non-local elements are required to capture the non-local interactions within a single element. For example, those nine vertices marked by blue, red and green dots in Fig. 1d form one non-local element for the Miura-ori, focusing on the central vertex marked by the blue dot. Similar non-local elements (i.e., all solid circles and open circles in Supplementary Fig. S3) are constructed for the Ron Resch pattern, as shown in Supplementary Fig. S3. It may be noticed that the definition of non-local elements depends on specific rigid origami patterns and in each pattern different elements are formed for different types of vertices. These non-local elements are implemented in the commercial finite element package, ABAQUS, via its user defined elements (UEL).