Abstract
We describe mechanical metamaterials created by folding flat sheets in the tradition of origami, the art of paper folding and study them in terms of their basic geometric and stiffness properties, as well as load bearing capability. A periodic Miuraori pattern and a nonperiodic Ron Resch pattern were studied. Unexceptional coexistence of positive and negative Poisson's ratio was reported for Miuraori pattern, which are consistent with the interesting shear behavior and infinity bulk modulus of the same pattern. Unusually strong load bearing capability of the Ron Resch pattern was found and attributed to the unique way of folding. This work paves the way to the study of intriguing properties of origami structures as mechanical metamaterials.
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Introduction
Mechanical metamaterials, the manmade materials with mechanical properties mainly defined by their structures instead of the properties of each component, recently have attracted great attention^{1,2,3,4}. Origami, creating threedimensional (3D) structures from twodimensional (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 lens^{5}), to automotive safety (e.g., airbags^{6}), biomedical devices (e.g., heart stent^{7}) and extremely foldable and stretchable electronics^{8,9}. Notable progress has been made in the area of origami theory including tree theory^{10} and its corresponding computer program^{11}, folding along creases^{12,13,14} and geometric mechanics of a periodic origami pattern^{15}. 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 manystep 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 faces^{4}. The geometric characteristics, such as the necessary condition around a single vertex for rigid origami^{16,17} have been studied and a computer simulator for rigid origami^{18} exists. There have also been made limited efforts to study the structural characteristics of one particular rigid origami, namely Miuraori^{19}, as a mechanical metamaterial, with the main focus on the negative Poisson's ratio^{15,20}, though these properties can be more rigorously examined. It is noticed that the existing studies are mainly focused on periodic origami patterns (e.g., Miuraori); however, nonperiodic 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 Miuraori but also a nonperiodic Ron Resch folding^{21} using combined analytical and numerical approaches. Specifically, we rigorously address the commonly mistaken inplane Poisson's ratio of Miuraori, which was believed to be always negative but we show here that it can be positive as well and its physical interpretation. The ubiquitous nonlocal interactions between vertices of rigid origami patterns are captured through a nonlocal 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 nonperiodic ones, can be readily studied.
Results
Unit cell and the whole pattern of a Miuraori
Figure 1a illustrates a Miuraori (n_{1}, n_{2}) in its folded state, containing n_{1} ( = 11) vertices in x_{1} direction and n_{2} ( = 11) vertices in x_{2} direction, with x_{3} as the outofplane direction. Its corresponding planar state is shown in Fig. 1b. The geometry of a Miuraori 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 Miuraori 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 origami^{16,17} states that there are n − 3 degrees of freedom, where n is the number of edges at one vertex, Miuraori 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 Miuraori, 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 x_{1}, x_{2} and x_{3} directions, respectively. It is noted that the length of the “tail” bcos(ϕ/2) is not considered in the unit cell^{15}. The periodicity of this pattern only requires two dihedral angles α_{1}∈[0°,180°] and α_{2}∈[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 Miuraori is put in an imaginary box with the dashed lines as the boundaries (Fig. 1a), the dimension of the whole Miuraori is then given by
and thus the imaginary volume occupied by this Miuraori is given by
Apparently even the Miuraori is periodic, its size in x_{2} 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 Miuraori (e.g., Poisson's ratio), particularly for smaller patterns (e.g., used in^{15,20}).
Inplane Poisson's ratio of Miuraori
Inplane Poisson's ratio of Miuraori 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 Miuraori, instead of using the unit cell. Specifically, the inplane Poisson's ratio ν_{12} is defined as , where and are the infinitesimal strains in x_{1} and x_{2}directions, respectively. Using equation (2), the inplane Poisson's ratio ν_{12} is obtained as
where η = a/b. Another inplane Poisson's ratio ν_{21} is just the reciprocal of ν_{12}. Figure 2a shows the contour of ν_{12} as a function of angle ϕ and a combination parameter (n_{2} − 1)ηcosβ. Clearly, ν_{12} can be negative or positive, which is different from commonly observed negative inplane Poisson's ratio. The boundary separating the negative and positive regimes of ν_{12} is defined by vanishing the denominator of ν_{12}, i.e., (n_{2} − 1)ηcosβ = cos^{2}(ϕ/2). At this boundary, ν_{12} flips between positive and negative infinities; thus ν_{12}∈[−∞,+∞]. For one scenario, where n_{2} = 5 (small pattern), η = 1/2, β = 78.5° and thus (n_{2} − 1)ηcosβ < 1, ν_{12} is positive for ϕ∈[0,101.5°] and changes to negative for ϕ∈[101.5°,2β]. For another scenario, n_{2} = 13 (large pattern), , β = 45° and thus (n_{2} − 1)ηcosβ > 1, ν_{12} stays negative, as reported by others using the unit cell^{15,20}; and the Miuraori becomes an auxetic material. Similar analysis can be applied on the outofplane Poisson's ratios. Details can be found in the Supplementary Information, Section “Outofplane Poisson's Ratio”.
Figure 2b provides an intuitive explanation of the sign change in the inplane Poisson's ratio ν_{12}. For the specific example with n_{1} = n_{2} = 5, η = 1/2 and β = 78.5°, the size of this Miura pattern in x_{1} 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 L_{1} > L_{2} > L_{3}. In contrast to L, the respective size of this pattern in the x_{2} direction, W, does not change monotonically with the angle ϕ. From the planar state to the collapsed state, W_{1} > W_{2} but W_{2} < W_{3}, which gives ν_{12} < 0 when L and W change in the same direction and ν_{12} > 0 when L and W change in the opposite direction. The nonmonotonic change of W is due to the “tail” term bcos(ϕ/2) in equation (2), which was missed in previous studies^{15,20}. 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 inplane Poisson's ratio of the Miuraori, the ranges of Poisson's ratios, specifically, ν_{12}∈[−∞,+∞], ν_{13}∈[0,∞], ν_{23}∈[−∞,∞] (see Supplementary Information, Section “Outofplane 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 Miuraori.
Miuraori subjected to shear and hydrostatic deformation
To study the shear deformation that is nonuniform across the Miuraori, we developed a numerical approach to characterize the geometric features of the Miuraori, i.e., the nonlocal interactions between rigid faces. As shown in Fig. 1b, the vertex marked by the solid blue dot not only interacts with its nearestneighboring vertices (marked by the solid red dots) through the rigid faces, but also its secondneighboring vertices (marked by the solid green dots) through dihedral angles. Thus the interactions between vertices are nonlocal. This nonlocal nature is ubiquitous in rigid origami and can be more complicated for other patterns, which can be further illustrated by the Ron Resch pattern^{21}, detailed in the Supplementary Information, Section “Nonlocal Interactions in the Ron Resch Pattern” (e.g., Supplementary Fig. S3). We developed a nonlocal finite element based model and Fig. 1d shows the nonlocal element for Miuraori. Details can be found in Methods.
Supplementary Figure S4 shows a deformed state of a (n_{1} = 13, n_{2} = 13) Miuraori subjected to a finite shear force in the negative x_{1} direction. Here it is noticed that an initially periodic Miuraori deforms nonuniformly under shear loading, which disables the definition of a shear modulus. It is seen that the Miuraori responds in an opposite way to shear force. Specifically and clearly, the vertical lines tilt to the positive x_{1} direction, as the shear force is applied along the negative x_{1} direction. This opposite relationship is thus consistent with ν_{12}∈[−∞,+∞].
The bulk modulus K of Miuraori can be defined the same way as that in continuum mechanics to link the hydrostatic pressure p and the volumetric strain θ( = ε_{11} + ε_{22} + ε_{33}),
Using the principle of superposition (details given in the Supplementary Information, Section “Bulk Modulus of Miuraori”), the bulk modulus K is given by
where are the tangential moduli of the stressstrain curve. Using the work conjugate relation, stresses are expressed as , where W_{tot} = U_{tot}/V is the elastic energy density with U_{tot} 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 (E_{11}, E_{22}, E_{33}) 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 nonperiodic rigid origami folding, namely a Ron Resch pattern, using the developed nonlocal finite element approach. The Ron Resch pattern and its nonlocal elements are given in Supplementary Fig. S3. To illustrate the nonperiodicity, 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 β_{1}, β_{2} and β_{3} are shown in Supplementary Fig. S5. It is obvious that the Ron Resch pattern is nonperiodic 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 zoomin details in Fig. 3a, the dihedral angles β_{1}∈[0°,90°] and β_{1}∈[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 k^{RR}/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 k^{RR} is the spring constant of the hinges for dihedral angles (detailed in the Supplementary Information, Section “Work Conjugate Relation – Stress and Moduli for Miuraori”) 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 β_{1} and β_{2} 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 β_{1} = 0°, β_{2} = 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.
Load bearing capability of a Ron Resch plate
This intriguing buckling resistance phenomenon motivates a further study of the load bearing capability of the Ron Resch pattern. The compressive load applied on top of a Ron Resch dome leads to a completely compact and flat state (namely, a Ron Resch plate), where the equilateral triangles collapse to threefold structures with β_{1} = 0°, β_{2} = 120° and β_{3} = 90° (Fig. 3c). Figure 3d shows the striking load bearing capability of a Ron Resch dome folded from a single sheet of 20lb copy paper: a 32.4 lb load is carried by a Ron Resch plate with actual mass 4.54 g. This remarkable capability is mainly a result of the folded structure, not the material properties of the paper, which suggests that origami can produce exceptional mechanical metamaterials. Figure 3e shows snapshots of the bottom of Ron Resch plate when 3lb load (left panel) and 32.4lb load are applied (right panel). It is found that at the failure point (where the 32.4lb load is applied), the tips of the threefold structures are flattened and instability occurs. To compare the load bearing capability of a Ron Resch plate with threefold supporting structures and commonly seen sixfold ones that are used in airplane wings, the buckling analysis is conducted to compute their critical compressive loads P_{cr} by using finite element package ABAQUS (details are given in the Supplementary Information, Section “Buckling Analysis of a Ron Resch Plate and a SixFold Supporting Structure”). Figures 3f and 3g show the first buckling modes for the Ron Resch plate and the sixfold structure. By assigning the same geometric parameters (including thickness and height of the support) and material properties (including elastic modulus and Poisson's ratio), P_{cr} of the Ron Resch plate is about 50% larger than that of the sixfold structure. Though the sixfold structure has higher symmetry to increase P_{cr} in a linear fashion (i.e., P_{cr} ~ order of symmetry), the decreasing height of the support for the Ron Resch plate from the center to the surroundings increases P_{cr} in a quadratic fashion (i.e., P_{cr} ~ 1/height^{2}), which endows a higher load bearing capability of the Ron Resch plate. This result suggests that generalized Ron Resch patterns with higher order symmetry^{22} would have even greater load bearing capability.
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 twodimensional 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 devices^{8,9}. 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 largescale and lowcost by precreasing 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
Nonlocal 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 Miuraori, the elastic energy can be written as
where and are the spring constants of the hinges for dihedral angles α_{1} and α_{2} for Miuraori (superscript “Mo”), respectively; α_{1,eq} and α_{2,eq} are the corresponding dihedral angles for α_{1} and α_{2} 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 Miuraori. It is reasonable to take and , for paper folding (although in most machinemade 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., U_{total} = U_{total}(x), where and x_{i} 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 Π_{total}, or the finite element method that uses both the first derivative (as the nonequilibrium force ) and the second derivatives (as the stiffness matrix ) of Π_{total}. The governing equation for the finite element method is
where u = x − x^{(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 nonequilibrium 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 nonlocal interactions^{23,24}.
There are two aspects to consider when the finite element method is used. Firstly, to calculate the nonequilibrium force P and stiffness matrix K, the elastic energy U_{total} 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, nonlocal elements are required to capture the nonlocal interactions within a single element. For example, those nine vertices marked by blue, red and green dots in Fig. 1d form one nonlocal element for the Miuraori, focusing on the central vertex marked by the blue dot. Similar nonlocal 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 nonlocal elements depends on specific rigid origami patterns and in each pattern different elements are formed for different types of vertices. These nonlocal elements are implemented in the commercial finite element package, ABAQUS, via its user defined elements (UEL).
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Acknowledgements
We acknowledge the seed grant support from the Office of Associate Dean for Research at Ira A. Fulton School of Engineering and Office of Knowledge Enterprise and Development, Arizona State University.
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C.L., D.K. and H.J. carried out and designed experiments and analysis. C.L., D.K. and H.J. wrote the paper. C.L., D.K., G.K., H.Y. and H.J. commented on the paper.
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Lv, C., Krishnaraju, D., Konjevod, G. et al. Origami based Mechanical Metamaterials. Sci Rep 4, 5979 (2014). https://doi.org/10.1038/srep05979
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DOI: https://doi.org/10.1038/srep05979
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