Folding of Tubular Waterbomb

Abstract

Origami has recently emerged as a promising building block of mechanical metamaterials because it offers a purely geometric design approach independent of scale and constituent material. The folding mechanics of origami-inspired metamaterials, i.e., whether the deformation involves only rotation of crease lines (rigid origami) or both crease rotation and facet distortion (nonrigid origami), is critical for fine-tuning their mechanical properties yet very difficult to determine for origami patterns with complex behaviors. Here, we characterize the folding of tubular waterbomb using a combined kinematic and structural analysis. We for the first time uncover that a waterbomb tube can undergo a mixed mode involving both rigid origami motion and nonrigid structural deformation, and the transition between them can lead to a substantial change in the stiffness. Furthermore, we derive theoretically the range of geometric parameters for the transition to occur, which paves the road to program the mechanical properties of the waterbomb pattern. We expect that such analysis and design approach will be applicable to more general origami patterns to create innovative programmable metamaterials, serving for a wide range of applications including aerospace systems, soft robotics, morphing structures, and medical devices.

Full text

Research Article
Folding of Tubular Waterbomb
Jiayao Ma,
1,2
Huijuan Feng ,
1,2
Yan Chen,
1,2
Degao Hou,
1,2
and Zhong You
2,3
1
Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, 135 Yaguan Road,
Tianjin 300350, China
2
School of Mechanical Engineering, Tianjin University, 135 Yaguan Road, Tianjin 300350, China
3
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
Correspondence should be addressed to Yan Chen; yan_chen@tju.edu.cn and Zhong You; zhong.you@eng.ox.ac.uk
Received 16 January 2020; Accepted 23 March 2020; Published 10 April 2020
Copyright © 2020 Jiayao Ma et al. Exclusive Licensee Science and Technology Review Publishing House. Distributed under a
Creative Commons Attribution License (CC BY 4.0).
Origami has recently emerged as a promising building block of mechanical metamaterials because it offers a purely geometric
design approach independent of scale and constituent material. The folding mechanics of origami-inspired metamaterials, i.e.,
whether the deformation involves only rotation of crease lines (rigid origami) or both crease rotation and facet distortion
(nonrigid origami), is critical for fine-tuning their mechanical properties yet very difficult to determine for origami patterns with
complex behaviors. Here, we characterize the folding of tubular waterbomb using a combined kinematic and structural analysis.
We for the first time uncover that a waterbomb tube can undergo a mixed mode involving both rigid origami motion and
nonrigid structural deformation, and the transition between them can lead to a substantial change in the stiffness. Furthermore,
we derive theoretically the range of geometric parameters for the transition to occur, which paves the road to program the
mechanical properties of the waterbomb pattern. We expect that such analysis and design approach will be applicable to more
general origami patterns to create innovative programmable metamaterials, serving for a wide range of applications including
aerospace systems, soft robotics, morphing structures, and medical devices.
1. Introduction
Mechanical metamaterials are artificially designed structures
that offer extreme and unusual, yet useful, mechanical prop-
erties determined by their structural and geometric configu-
rations rather than only intrinsic material properties of
their composing elements. Conventional mechanical meta-
materials are often formed by quasi-1D rods or links [1–6].
Recently, origami has emerged as a promising building block
of mechanical metamaterials with versatile functionalities
and programmability [7–17], due to its capability of trans-
forming a 2D crease pattern into a complex 3D sculpture,
purely geometric traits independent of both scale and con-
stituent materials, and ease of manufacturing [18, 19].
The mechanical properties of an origami metamaterial
are primarily determined by its folding mechanics. When
rigid patterns such as Miura-ori [7, 8] are utilized, the facets
do not stretch or bend but only rotate about the creases, and
the metamaterials behave like a kinematic mechanism during
folding. On the other hand, nonrigid ones [15, 16] enable
simultaneous crease rotation and facet distortion, resulting
in structural deformation of the metamaterials. Difference
in folding mechanics leads to distinct mechanical properties
such as stiffness [15]. Here, we report a tubular waterbomb
pattern with a transition between rigid origami motion and
structural deformation, making it possible for programming
the behavior of the derived metamaterial.
In origami, the waterbomb tube refers to an origami
structure made from a crease pattern obtained by tessellation
of the waterbomb bases. A typical waterbomb base is a six-
crease pattern with two colinear mountain creases and four
diagonal valley ones intersecting at a common vertex [20].
The base has been used to create many fascinating origami
objects including the structure that is the focus of this article
[21–23]. One of the most distinctive characteristics of the
waterbomb tube is that it has a negative Poisson’s ratio: when
compressed, both its length and radius get smaller. This has
led to a number of notable practical applications such as an
expandable medical stent graft [24], a transformable worm
robot [25], and a deformable robot wheel [26]. Recently,
the authors also obtained programmable stiffness and shape
modulation in the waterbomb tube using a bar-and-hinge
AAAS
Research
Volume 2020, Article ID 1735081, 8 pages
https://doi.org/10.34133/2020/1735081

numerical model [27]. Despite that, its precise folding behav-
iors and mechanical properties have remained ambiguous.
Therefore, in this paper, we aim to expose the exact folding
mechanics of the waterbomb tube by means of kinematics
and structural analysis.
2. Kinematic Modelling of Rigid Folding
Figure 1(a) illustrates the crease pattern of a waterbomb tube
defined by four independent geometric parameters—width
2a, sector angle α, and the number of bases longitudinally,
m, and circumferentially, n. When we join together the left
and right edges of the pattern, we can obtain a waterbomb
tube [21, 22]. We will illustrate the motion behavior of the
waterbomb tube with a representative model. First, we cre-
ate a waterbomb tube in the fully contracted configuration
(Figure 1(b) ①), where the facets in the middle row collide.
When we slightly expand the tube along its central axis, its
radius increases as well as the length, and a uniform radius
along the tube is obtained (we shall demonstrate later that
such a configuration always exists) (Figure 1(b) ②). With
further expansion, it develops a pineapple shape with clo-
sure at both ends (Figure 1(b) ③and ④). Subsequently, it
opens up its ends again (Figure 1(b) ⑤) and then regains
a uniform radius (Figure 1(b) ⑥). After this, the tube can
be only marginally deployed, and the change in shape is
hardly noticeable.
To determine the folding mechanics, we first build a kine-
matic model for the waterbomb tube. The pattern has three
distinct vertex groups: central vertex Aand edge vertices B
and C, the motion of which can be modeled as a spherical 6
Rlinkage with three degrees of freedom (DOFs) [28, 29].
As such, the tube is a network of these linkages, leading to a
multi-DOF system (details in the supplementary material, S2).
4 5 6
2a
𝛼
B
C
A
1 2 3
40 mm
m = 3
n = 6
(a)
(b)
Figure 1: The waterbomb origami pattern and model. (a) The waterbomb pattern formed by tessellating the waterbomb bases. Solid and
dashed lines represent mountain and valley creases, respectively. A typical base is shown in blue, which is placed side by side forming the
middle row. On the adjacent rows, the bases are shifted by half a base (red). Four geometric parameters: width of the base 2a, sector angle
α, the numbers of bases in the longitudinal direction m, and circumferential direction n—completely defining the pattern. A, B, and C are
the three groups of representative vertices. (b) Deployment of a card model of a waterbomb tube. The model was made from conventional
cards of 0.3 mm in thickness obtained from stationery stores. The geometric parameters were 2a=46mm,α=45
°,m=7, and n=6.
2 Research

Thus, we reduce the DOF by making the following assump-
tions of symmetry based on our observation in Figure 1(b).
First, the motion of the tube is symmetric about the equato-
rial plane that passes through the middle of the tube and
divides it into two identical top and bottom halves. Second,
all bases that are circumferentially placed in the same row
have identical folding behavior. Finally, each base moves in
a plane-symmetric way about a plane (presented as the red
dot-dash line in Figure 2(a)) passing through the two moun-
tain creases and the central axis of the tube. We first discuss
thecasewhenm(i.e., the number of rows in the tube) is
odd. Figure 2(a) presents a strip out of the origami pattern
forming a tube with an odd number of rows. The equatorial
plane passes through the center of the middle row defined as
row 0. According to the above assumptions, linkage A0is
symmetric about the equatorial plane and the plane passing
vertex A0and tube axis, which reduces its DOF to one. Sub-
sequently, the overall DOF of the tube becomes one, and its
motion, described by the respective dihedral angles of all the
linkages, can be found out through kinematic analysis
(details in the supplementary material, S3).
Consider a particular example with m=3,n=6, and
α=45
°. As shown in Figure 2(b), φ0,1is defined as the
dihedral angle between two triangular facets that pass the
top mountain crease in linkage A0. Taking φ0,1as the
input, we can obtain the other five dihedral angles of the
linkage by kinematic analysis. Subsequently, we can deter-
mine all dihedral angles in the tube by using these known
dihedral angles as inputs for adjacent linkages. To depict
the extent of deployment of the tube, we define θas the
folding angle between the two largest triangular facets of a
base on row 0 and the following equation holds: θ=φ0,1.
The nondimensional radii of the vertices, r/a, are plotted
against θin Figure 2(c), together with five representative
configurations of the tube during deployment. We can
draw three conclusions from the result. First, there exist
two particular configurations, II with θ=65:88°and IV
with θ= 144°. On those configurations, the radii of all
the vertices Biand Cibecome equal and so do those of
vertices Ai(i=0,1), see the red dots on the curves. In
other words, all the bases take the same geometric form
at either configuration, resulting in a tube of a uniform
radius. Thus, we prove theoretically that it is possible to con-
struct a uniform waterbomb tube out of rigid sheet materials,
provided that the amount of prefolding is correct. Second,
the rigid-foldable range of the tube is bounded by two values
of θ,θmin and θmax. The lower bound θmin =60
°corresponds
to the compactly folded configuration I: ϕB0,4=0, i.e., two tri-
angular facets on both sides of the common crease B0C‐1
overlap entirely, whereas the upper bound θmax = 147:96°is
associated with the most expanded configuration V in which
the upper sides of the bases on row 1 form a regular hexagon
with a side length 2a. The supplementary material (S3.B.1)
provides detailed derivations on how both bounds are
obtained. Third, the uniform radius and the bound config-
urations divide the tube deployment process with distinct
shapes, i.e., a pineapple shape with the largest radius
attained at crease B0C‐1in row 0, such as configuration
III with θ= 120°, when 65:88°<θ< 144°, and a dogbone
shape with the smallest radius reached at crease B0C‐1,
when 60°≤θ<65:88°and 144°<θ≤147:96°.
3. Mechanism-Structure-Mechanism Transition
When we increase the number of rows, existing rows will
retain their motion in the original tube with m=3 and drive
concurrently the motion of newly added ones. However, the
upper sides of bases on the end rows will form an n-sided reg-
ular polygon earlier, resulting in the termination of the
motion. For example, if we increase mto 7, θmin remains to
be 60
°
while θmax decreases to 144.24
°
. Figure 3(a) presents
the motion sequence of the tube, where configurations I
and V correspond to the minimum and maximum values of
θ, respectively. Figure 3(b) plots the dihedral angles ϕBi,4
between adjacent bases of row i(i=0,1,2,3) against θ.
Figure 3(c) gives the radii of the vertices Ai,Bi, and Ci,rAi ,
rBi, and rCi ,vs.θ. As in the case of m=3, the curves intersect
at two points (marked by red dots), which indicates that the
tube also has a uniform radius at configurations II and IV
when m=7. Comparing it to the tube with m=3,itis
found that the values of θat these two configurations
are exactly the same. In fact, it can be proven kinemati-
cally that the uniform radius configurations are indepen-
dent of the number of rows (supplementary material,
S3.B.2). We can intuitively imagine that when all bases
in the tube are in identical shape, more rows can be added
to the tube in a geometrically compatible way.
Furthermore, the tube undergoes nonrigid folding within
a region of θbetween θ=90:72
°
at configuration III
L
and
θ= 128:52°at III
R
(supplementary material, S3.B.3). This
is clearly demonstrated by the intermediate configuration
III in Figure 3(a), where the central vertices A3of the
bases on the two end rows collide with each other. The
reason is that the dihedral angle ϕB3,4<0 in this range,
indicates interference among the facets on row 3, which
is not permitted in rigid origami. This conclusion is fur-
ther supported by the fact that rA3 <0 in the same range
as shown in Figure 3(c). Therefore, the tube has rigid ori-
gami motion only within two distinct regions of θ:60°≤
θ≤90:72°and 128:52°≤θ≤144:24°. At configurations III
L
and III
R
, the ends of the cylinder are closed, and the tube
becomes a concealed volume. It has been shown that
structural deformation is required for any change in a
concealed volume [30]. Thus, in order to move a physical
tube specimen from one rigid-foldable range to the other,
the component sheet material has to deform, and the tube
works as a structure instead of a mechanism. This kind of
folding behavior is referred to as mechanism-structure-
mechanism transition.
The mechanism-structure-mechanism transition leads to
a dramatic variation in stiffness for waterbomb-based struc-
tures and metamaterials. Within the rigid origami regime,
the tube has a low stiffness determined by the torsional stiff-
ness of the creases. When entering the structural range, the
stiffness will be significantly increased due to facet deforma-
tion taken place in the tube for its shape change. This feature
is demonstrated through a structural analysis of a tube using
the finite element method. The model had identical n=6,
3Research

𝜃
E
I: 𝜃 = 60° II: 𝜃 = 65.88° III: 𝜃 = 120° IV: 𝜃 = 144° V: 𝜃 = 147.96°
r/a
0
1
2
(a) (b)
(c)
III III VIV
𝜃
B2
B0
C0A0
B1
C2
C1
A1
A2
B(m–1)/2
C(m–3)/2
C(m–1)/2
Row 1
Row 2
Row (m – 1)/2
(End row)
E′
B′0
E′
Row 0
Symmetric plane
A(m–1)/2
B(m–3)/2
C0
A0
E
B0
C–1
C–1
A0
O0
B0
C0
C0
C1
B1
B0
A0A1
90°
180°/n
𝜙B0,4
𝜑0,1
𝜙B0,4
C′–1
B′0
𝜃
Figure 2: Rigid foldability of the waterbomb tube with an odd m. (a) Top half of a longitudinal strip in a waterbomb tube. E‐E′is the equator
of the tube. Vertices above the equator are marked as Ai,Bi, and Ci, while those below are marked as A‐i,B‐i, and C‐i, in which 0≤i≤ðm‐1Þ/2.
(b) The front view of a waterbomb tube with the equatorial row 0 and rows immediately adjacent to it and the top view of the equatorial row.
One of the bases on the equatorial row is shown in blue. O0is the center of the tube. (c) Variation of nondimensional radii r/aof vertices Ai,Bi,
and Ci(i=0,1) of a tube with m=3,n=6, and α=45
°with respect to the folding angle θand five representative configurations I to V of the
tube in front and top views. The corresponding angles θare listed below the configurations. Also see SM Movie S1.
4 Research

m=7, and α=45
°with that in Figure 3(a), and the facets
were set to be 2 orders stiffer than the creases to distinguish
the deformation of these two components [31–33] (supple-
mentary material, S4). The simulation started from θ= 130°
near configurations III
R
and terminated at θ=88
°just
beyond configuration III
L
. The folding process of the tube
is shown in Figure 3(e), from which physical contact and
deformation of the facets at the ends are clearly seen at
I II III IV
i = 0
i = 1
i = 2
i = 3
1
0
–1
1
2
–1
r/a
III III IV
0
0
3
12
Energy (J)
6
I III IV
9
Mechanism
Structure
(a)
(b) (c) (d)
(e) (f)
Dierence
150°
90°
30°
–30°
0°
IIILIIIL
IIIRIIIR
IIIL
IIIR
IIILIII
IIIR
50°
45.46°
44.63°
𝛼 = 40° 55°
45°
35°
𝜃 = 130°𝜃 = 88°
𝛼
60°
90°
120°
150°
45°
I: 𝜃 = 60° II: 𝜃 = 65.88° IIIL: 𝜃 = 90.72° IIIR: 𝜃 = 128.52° IV: 𝜃 = 144° V: 𝜃 = 144.24°III: 𝜃 = 120°
𝜙Bi,4
A0A1
A2
A3
B0
B1
B2
B3
C0
C1
C2
C3
rA3/a
𝜃𝜃
𝜃
𝜃
x102
Figure 3: Mechanism-structure-mechanism transition of longer tubes with an odd m. (a) Front and top views of the tube with m=7,n=6,
and α=45
°deploying from configurations I to V. The corresponding folding angles θare listed below the deployment sequence. The tube is
completely concealed between III
L
and III
R
. Also see SM Movie S2. (b) Variation of dihedral angles ϕBi,4 vs.θ. The red curve shows that
ϕB3,4 <0between III
L
and III
R
. (c) The nondimensional radii r/aof vertices Ai,Bi, and Ci(i=0,1,2,3)vs.θ, in which the red curve shows
rA3<0 between III
L
and III
R
. (d) Relationship among nondimensional radius of vertices A3(rA3/a), θ, and α. Some values of αare listed
alongside their corresponding curves. The shaded plane is where rA3=0. Blue solid lines are for rA3>0 and grey dashed lines for rA3<0
(where physical interference happens). (e) Front and top views of the waterbomb tube undergoing structural deformation. At
configurations III
R
to III
L
, the tube forms a concealed volume. Any in-between configuration, e.g., configuration III, requires structural
deformation, as presented in the close up figure where the vertices are squeezed and distorted by each other. (f ) Elastic strain energy vs.θ
when the tube is modeled as a mechanism (in grey) and as a structure (in black) and their difference (in red).
5Research

configuration III. The elastic strain energy of the tube is plot-
ted in Figure 3(f). To manifest the effect of facet deformation,
afictional mechanism motion was also simulated for the tube
by allowing the facets to freely penetrate into each other. In
the mechanism mode, most energy is stored in the creases
and distributed almost linearly. In the structural mode, both
facets and creases deform. The energy difference between the
two acts as an indicator of the level of deviation from rigid
origami motion. The larger it is, the more extra deformation
is required to enable the tube to move. It is also worth point-
ing out that the energy difference is not strictly zero at config-
uration III
L
where the tube resumes rigid folding. This is due
to some localized residual deformation near the vertices in
contact in the structural mode, without affecting the global
folding behaviour of the tube. Compared with the fictional
mechanism mode, a 110% higher maximum strain energy
is required to fold the tube through the structural range, indi-
cating a larger stiffness. The precise stiffness variation
depends on the sheet material of the tube, but when that is
known, it is possible to predict the external loading that will
cause the tube to move from one rigid origami range to the
other. Notice that the stiffness property of this tubular
n
0°
30°
60°
90°
120°
𝜃
𝜃 = 22.5° 𝜃 = 54.43° 𝜃 = 88.82° 𝜃 = 120.75°
150°
180° 𝜃max
𝜃min
5101520
40 mm
(a)
(b)
Figure 4: Waterbomb tubes with an odd mand varying n. (a) Rigid foldable range of tubes with m=7,α=45
°, and nfrom 5 to 20. The red
lines indicate that the tube is in the structural range. (b) Deployment of a tube with m=7,α=45
°, and n=16and a card model with identical
geometry and 2a=46mm. The corresponding folding angles θare listed below the deployment sequence. The rigid folding range of the tube
is 22:5°≤θ≤120:75°. At the end of rigid folding, the yellow and purple triangular facets hit each other, causing interference in the shaded
area. Also see SM Movie S3.
6 Research

waterbomb can also be revealed with the bar-and-hinge
model such as the Merlin code [31] and the Origami Contact
Simulator [34], or the smooth fold model [35].
4. Programmability of Folding Mechanics
Using the established kinematic model, we can program the
existence and range of the mechanism-structure-mechanism
transition by varying geometric parameters of the pattern.
Take m=7and n=6as an example. The radius of the vertex
A3and rA3/avs.θfor various αis presented in Figure 3(d),
where we can find the transition occurs when 44:63°<α<
45:46°.Whenα≥45:46°, the tube conducts a pure rigid ori-
gami motion from θmin to θmax during deployment. Since
θmin and θmax are also related to α, the motion range shrinks
with α.Ifα≤44:63°, the tube experiences only one curtailed
rigid origami range due to physical interference. For instance,
when α=40
°, the rigid motion range is 142:97°≤θ≤146:54°.
The folding behavior of the waterbomb tube can also be
programmed by varying the number of bases in a row. To
demonstrate this, consider m=7and α=45
°,butnchanging
from 4 to 20. We find from the theoretical model that rigid
folding occurs only when n≥5. The upper and lower limits,
θmin and θmax, are plotted against nin Figure 4(a), together
with the transitional structural range (highlighted by red lines)
if it exists. The result indicates that the mechanism-structure-
mechanism transition consistently appears when n≥6.How-
ever, when nis relatively large, n≥14 in the current geometric
setup, the mechanism motion range in the neighborhood of
θmax becomes very narrow, and practically, the tube can be
considered reaching a stable configuration rather than another
mechanism motion range. Another interesting phenomenon
when nis large is that the tube can form a spherical shape that
was used as an origami wheel [36] or artificial muscles [37].
The deployment process of such a tube with m=7,α=45
°,
and n=16is presented in Figure 4(b). Our analysis shows that
when θreaches 120.75
°
, the two triangles in yellow and purple,
respectively, meet and overlap at theshaded area, whichceases
the rigid folding process before the tube ends are closed.
So far, we have only discussed the situation of mbeing odd.
Similar behavior exists when mis even. The equatorial row of
the tube no longer exists in this case, and the equatorial plane
passes through the midpoints of the creases linking vertices B0
and C0. However, the tube remains a single DOF system as the
two directly adjacent rows above and below the equatorial
plane must behave the same under symmetrical assumptions,
and they subsequently drive the motion of the remaining rows.
Moreover, the tube also has a pair of uniform radius configu-
rations identical to its odd row counterpart with the same n
and α, albeit mdiffers (supplementary material, S3.C). It
means that the uniform radius configurations of the tube are
solely decided by parameters αand n. It is not related to m.
5. Conclusions
We have uncovered the true folding mechanics of the tubular
waterbomb and its dependence on pattern geometric param-
eters. Through a rigorous kinematic analysis, we have dem-
onstrated that some waterbomb tubes are capable of rigid
origami motion, whereas others will experience what we refer
to as a mechanism-structure-mechanism transition. And a
structural analysis has revealed a significant increase in
stiffness when the tube transforms into the structural range.
Furthermore, we have derived theoretically the correlation
between the occurrence and range of the mechanism-
structure-mechanism transition and geometric parameters
of the pattern, making accurate programming of the
mechanical properties readily achievable. Thus, this work
can not only facilitate the development of mechanical meta-
materials making use of the intriguing properties of the
tubular waterbomb but also provide an analysis framework
for novel programmable metamaterials with wide engineer-
ing applications such as soft robotics, morphing structures,
and medical devices. To adopt our analytical model where
symmetry assumption was made in these applications, the
key is to maintain the symmetry of the pattern during
motion. Considering that the thickness of the sheets in
the waterbomb pattern cannot be ignored in physical appli-
cations, the analytical model should be adjusted to the
thick-panel origami model [23]. The motion symmetry
would be satisfied automatically by structural constraints
introduced by the panel thickness.
Conflicts of Interest
The authors declare no competing financial interests.
Authors’Contributions
Jiayao Ma and Huijuan Feng contributed equally to this work.
J. Ma, Y. Chen, and Z. You initiated and directed the research.
H. Feng and Y. Chen carried out the kinematic analysis. J. Ma
and D. Hou carried out the numerical simulation. All the
authors prepared all the figures and supplementary informa-
tion.J.Ma,H.Feng,Y.Chen,andZ.Youwrotethemanuscript.
Acknowledgments
This work was supported by the National Natural Science
Foundation of China (Projects 51825503, 51721003, and
51575377) and the Air Force Office of Scientific Research
(FA9550-16-1-0339). During the course of this research, Z.
You was appointed as a visiting professor at Tianjin University.
Supplementary Materials
Supplementary 1. Supplementary Information: derivations,
figures, and equations.
Supplementary 2. Movie S1: animation of Figure 2(c).
Supplementary 3. Movie S2: animation of Figure 3(a).
Supplementary 4. Movie S3: animation of Figure 4(b).
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8 Research