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Programming mechanical metamaterials using origami tessellations
Journal Pre-proof Programming mechanical metamaterials using origami tessellations Y.L. He, P.W. Zhang, Z. You, Z.Q. Li, Z.H. Wang, X.F. Shu PII:
S0266-3538(19)31342-9
DOI:
https://doi.org/10.1016/j.compscitech.2020.108015
Reference:
CSTE 108015
To appear in:
Composites Science and Technology
Received Date: 14 May 2019 Revised Date:
25 December 2019
Accepted Date: 17 January 2020
Please cite this article as: He YL, Zhang PW, You Z, Li ZQ, Wang ZH, Shu XF, Programming mechanical metamaterials using origami tessellations, Composites Science and Technology (2020), doi: https://doi.org/10.1016/j.compscitech.2020.108015. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Author statement Manuscript title: Programming mechanical metamaterials using origami tessellations I have made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; And I have drafted the work or revised it critically for important intellectual content; And I have approved the final version to be published; And I agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. All persons who have made substantial contributions to the work reported in the manuscript, including those who provided editing and writing assistance but who are not authors, are named in the Acknowledgments section of the manuscript and have given their written permission to be named. If the manuscript does not include Acknowledgments, it is because the authors have not received substantial contributions from nonauthors. Zhihua Wang and Xuefeng Shu
1
Programming mechanical metamaterials using origami
2
tessellations
3
Y.L. He1,2, P.W. Zhang3, Z. You4, Z.Q. Li1,2, Z.H. Wang1,2,∗ X.F. Shu1,2,**
4
1
Institute of Applied Mechanics, College of Mechanical and Vehicle Engineering, Taiyuan University
5 6
of Technology, Taiyuan 030024, China. 2
Shanxi Key Laboratory of Material Strength and Structural Impact, Taiyuan University of Technology,
7
Taiyuan 030024, China
8
3
9
College of management & engineering, Shanxi University of Finance and Economics, Taiyuan 030006, China
10
4
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
11
Abstract
12 13
Origami structures, whose mechanical properties can be tuned by programing
14
the crease network, have been sought in application including DNA origami
15
nanorobots and deployable space structures. However, the existing researches mainly
16
placed on the cylindrical origami tubes or the periodic cellular metamaterials, which
17
are based on the stacking of individual 3D origami units. Here, we present a novel
18
mechanical metamaterial, which can be constructed via using the curved-crease
19
origami (CCO), with tunable Poisson’s ratio and stiffness. We further analytically and
20
experimentally demonstrate that the Poisson's ratio of these structures exhibits some
∗
Corresponding authors. Tel.: +86 351 6018560 E-mail address: [email protected], [email protected]
1
21
new properties by altering the length to width ratio of the structures. In addition, we
22
verify the tunable stiffness of the CCO-based metamaterials by changing the patterns
23
of these structures. Our approach can be used to design and construct the next
24
generation of the CCO-based metamaterials with tunable mechanical properties for
25
engineering applications.
26 27
Keywords: Origami structures, Tunable Poisson’s ratio, Tunable stiffness.
28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 2
43 44
Origami, an ancient handcraft art, has stimulated curiosity and research
45
interest of scientists from different disciplines and fields, not only because origami
46
patterns can be found in the nature1-4, but also because it shows some exotic
47
properties that exist large potential for engineering application5-11. Meanwhile,
48
origami-inspired metamaterials provide the possibility for scientific communities to
49
manipulate the physical properties by reprogramming the crease and tessellating the
50
structures into an ideal pattern. These properties have been verified, including tunable
51
Poisson’s ratio and stiffness12-17, tunable chirality18 and thermal expansion19,
52
programmable collapse20 and curvature21, self-foldability22,23 and self-locking24-26,
53
high stiffness-to-weight27, multi-stability28 and multi-bit memory29. One good case is
54
that the multi-stable stacked-origami30 can be used to control wave propagation31.For
55
example, the sound absorption and noise reduction can be achieved by the design of
56
the origami structures. And origami structure also can be used to manipulate the
57
electromagnetic response18 by transforming the geometrical configurations. Previous
58
studies reported that reentrant origami structure28 exhibited tunable negative Poisson’s
59
ratio and structural bistability simultaneously. What’s more, the negative stiffness and
60
ultra-stiffness origami cellular solids20 have been exploited to program collapsible
61
mechanical metamaterials. Furthermore, the lattice structures32 of such metamaterial
62
have been manufactured by 3D-Printer or Schematic Drawing of the Pre-Folding
63
Technique33,34. In addition, a set of basic theories of the origami structure were
64
reported. For example, the geometric folding algorithms35 that can be used to 3
65
manufacture any 3D structure with the least creases. The mechanical characterization
66
of creased sheets36 and the elastic theory of the folding paper37 provide a platform for
67
designing and analyzing the origami structures. However, the theories were mainly
68
used on the reconfigurable architected materials38 or the periodic origami
69
structures24,25,39,40 which were constructed in the ordered unit-cells. The mechanical
70
behavior of CCO-based metamaterials, which are constructed in different strategies,
71
has rarely been investigated.
72
Here, we provide a strategy for the design of the 3D cellular metamaterials
73
based on the curved-crease origami (CCO) pattern. In order to provide reference for
74
the design of future origami-based cellular structures which based on the stitching and
75
cutting of the cellular metamaterials, we systematically explored the mechanical
76
response of the CCO-based cellular metamaterials. In addition, previous 3D
77
origami-based structures such as the cellular origami structure24,25 and origami-based
78
cylindrical structures27-29, which are often made up of individual tessellating repeat
79
unit cells along out-of-plane direction. Therefore, the mechanical behavior of these
80
structures is uniform throughout. Different to these structures, the CCO-based unit
81
cells have the rigid folding stage and the plastic deformation stage when the number
82
of the layer is more than three and the divide line of these two stages is the
83
self-locking. Therefore, the displacement-force curve of the structures has
84
multi-platforms as shown in the supplementary Figure 15, which may be an excellent
85
energy absorption equipment. In this paper, we only pay attention to the mechanical
86
properties of the origami structures during the folded motion. Thus, we firstly gave 4
87
the generation expression of the Poisson’s ratio and the stiffness in several
88
representative unit cells, closed tubes and the cellular metamaterials and verified that
89
our design strategy is an effective way to program the Poisson’s ratio and stiffness.
90
And we can judge change trend of the force-displacement curve from the contour map
91
of the stiffness during the folding motion. Therefore, this research paves the way for
92
the next generational cellular metamaterials which will bring the origami-inspired
93
metamaterials closer toward the engineering application.
94
Design strategy
95
To build the 3D origami-inspired programming cellular metamaterials, we
96
started by making up the CCO unit cells that were made up of two unsymmetrical
97
elements, whose geometrical patterns can be described by the length parameters
98 99
( ,
, ℎ ) and the original angles
(see Fig.1). We then proposed three methods
on the stacking sequences of the unit cells along the out-of-plane direction (see
100
Fig.2(a)). (i) We linked the two-layers unit cells with another CCO unit cells while
101
ensuring that the motions were compatible between them. The adjacent faces and the
102
holes between different unit cells remain aligned along the out-of-plane direction.
103 104
That is, the geometrical parameters ( ,
, ℎ ) are identical between the connected
faces. (ii) We connect two kinds of unit cells to form a closed tube (see Model a-2).
105
The patterns of these two kinds of the unit cells are identical when the flat rectangular
106
grids are excluded. For example, the tubes are connected by the four-layers and
107
five-layers unit cells. Moreover, the tubes are composed of two-layers and
108
three-layers unit cells also is a special example of strategy (i). To avoid confusion, we 5
≥ 2. (iii) We introduce the two-layers unit cells into
109
set the strategy (ii) meet with
110
the structure based on strategy (ii) (see Model a-3). Complementing these strategies,
111
we don’t introduce other types of unit cells in Model a-3, because these structures are
112
regarded as the derivatives of the above models.
113
What is more, there are two methods on the design of the tube along the
114
in-plane direction. Based on these two methods, the closed tubes and unclosed tubes
115
are constructed. For one thing, the tube is completely closed as shown in Fig.2(a). To
116
ensure that the elements can be connected together to form a closed tube, the curve
117
creases are compatible for the two overlapping faces (see Supplementary Fig.1(a,b)).
118
In addition, there is a special closed tube (see Model c-4) whose crease patterns are
119
shown in the Supplementary Fig.3, and its three basic orthographic views are shown
120
in the Supplementary Fig.13. What’s more, the structure is deployable along the
121
in-plane direction (see the Supplementary Movie 1). And the crease patterns need
122
match the combination method of the tube as Supplementary Fig. (3,4) shown. For
123
another thing, the tube is unclosed as shown in the Fig.2(b). Moreover, there are the
124
holes in the unclosed metamaterials which is beneficial to reduce the accumulation of
125
moisture. These unclosed metamaterials will be investigated in detail in the next work.
126
Two elements or unit cells are attached by the sheet with the same shape of the two
127
overlapping planes. These closed tubes and unclosed tubes have the rigid folding
128
stage and the plastic deformation stage, which is an improvement of the previous
129
closed-loop tubes27-29. Considering the space-filling, the patterns of the tube and the
130
space-filling elements should be compatible. Finally, the tubes could be expanded or 6
131
filled by the elements to form the cellular metamaterials which can be actively altered
132
into numerous patterns by changing the assembly and the size of elements.
133
Furthermore, the space-filling assemblies of cellular metamaterials could form the
134
next generation cellular metamaterials as shown in Fig.2(d), providing a strategy for
135
the design of deployable next generation cellular metamaterials.
136
Geometry of unit cells and tube
137
In order to calculate the geometrical relationship of the CCO origami unit cells
138
during its folding motion, we take out a half model of the CCO unit cells, which
139
corresponds to an element that has been folded along the crease. The lengths of
140
L and H correspond to the displacement as shown in Fig.1(b). The dihedral angle and
141
the angle between lines were given in Fig. 1(b). And the corresponding geometrical
142
relationship in the 3D unit cells was given in the Supplementary Fig. (8, 9). The
143
mathematical expression of the geometrical relationship was described in the
144
Supplemental Information 2 in detail (That is, geometry of the elements and unit cells
145
based
146 147
on
CCO
∈ 0, ⁄2 ,
150
,
supplemental
∈ 0,
tan
materials).
Noted
=!
. #$
,
where the
%
the
angle
=
=
was given as follow (1) at the self-locking state
in the unit cells. Therefore, we can obtain the variation range of arccos (!
that
could be obtained by the given geometry.
In addition, we can obtain the expression
151 152
>
the
Moreover, the relationship between the angle
148 149
in
, ,
⁄!
%)
≤
%
≤
%
%
is the natural dihedral angles in the undeformed state. 7
as follow (2)
153 154
The Poisson’s ratio of the unit cells and closed tubes
155 156 157 158 159 160
In order to investigate the “Poisson’s ratio” of the CCO cells in different directions, we define them according to the previous study28. It was given as follow
+,- = −
(/-/-)
(/,/,)
, +,1 = −
(/1 ⁄1) (/, ⁄, )
+,2 = −
(/2/2)
(/,/,)
(3)
Here, we only give the general expression. The specific expression of
Poisson’s ratio +,- , +,1
+,2 have been given in the Supplemental information
3.2.
161
To verify the analytical solutions, we manufacture three groups prototypes of the
162
CCO unit cells using the sheets (See the Supplementary Information 3.1 for details).
163
Accordingly, we measure the dimensions of the length
164
some height intervals of H, as we gradually compress the CCO unit cells by the
165
experiment set up along the out-of-plane direction. As the Fig. 3 shown, an acrylic
166
plate with the equal size grids is placed on the top surface of prototypes and the plate
167
is fixed at a certain height. The other two acrylic plates are fixed on the bottom, which
168
are used to fix the scissor-type lifting platform. This allows the upper plate of the
169
scissor-type lifting platform to transitional motion in vertical direction. The position
170
of the upper plate of the scissor-type lifting can be controlled by the knob. The height
171
of the prototype could be measured by digital display caliper. The level gauge is used
172
to ensure that the upper acrylic plate is parallel to the top plate of the lifting platform.
173
In addition, the camera is fixed above the upper acrylic plates to capture the digital
174
image of the prototypes’ cross-sectional area. Based on the pictures from camera, we 8
, ,L,H and the angle 2ξ in
175 176
obtain the length parameters ( ,
, 4) and the angle 2 in different height (5) by
the Image processing software. We compare the Poisson’s ratio that is measured with
177
the analytical solutions which is calculated from the Eq. (3). They are in good
178
agreement between theoretical and experimental results.
179
The result of the Poisson’s ratio of three -layers unit cells was given in Fig.3,
180
which has different initial angles and length-width ratio in different directions. The
181
other results of Poisson’s ratio of CCO unit cells have been given in the
182
Supplementary Fig.10 such as the angle difference between
183 184
and
in the
four-layers unit cells. Here, Fig. 4(a) shows the Poisson’s ratio +,2 of the CCO unit cells in the cases of α=55°, α=65°, α=75° and length-width ratio is
:
= 2 ∶ 1. The
185
corresponding analytical contour plot, which is a function of the continuous α and the
186
is shown in Fig. 4(b). The Fig.4 (c, d)
187 188 189 190 191 192 193 194 195 196
folding ratio defined as (90° −
% )⁄90°,
indicates the change of Poisson’s ratio in different length-width ratio, while the initial
angle α = 65° remains unchanged. We find that the CCO unit cells emerge the positive in-plane Poisson’s ratio +,2 at initial angle α = 65°, when the length-width ratio of
:
and :
= 1 ∶ 3 . Previous research reported that the zigzag-based origami14-16
is above approximately 3:1 or below approximately 1:3. Meanwhile,
the Poisson’s ratio +,2 are identical when the length-width ratio
:
=3∶1
produced the positive in-plane Poisson’s ratio at initial angle above α = 75°, but the
fact that positive Poisson’s ratio produces at a smaller angle by changing the
length-width ratio have not been reported. In addition, the Poisson’s ratio +,- of the
CCO unit cells become negative when the initial angle α is above approximate 9
197 198 199 200 201 202
75° (see the Supplementary Fig.11(a)). Therefore, the conventional 2D Miura-ori also has multiple sign flips of the Poisson’s ratio just like the reentrant origami-based
metamaterial28. Figure 3 (e, f) indicate that the Poisson’s ratio +,- of the CCO unit cells changes from positive to negative, when the length-width ratio
:
is below
approximately 2:3 at the initial angle α = 65°. Besides, the Poisson’s ratio +,1 of the CCO unit cells has a transient transition between the positive and negative, when :
is below approximately 1:2 at the initial angle α = 65°
203
the length-width ratio
204
as Fig. 4 (g, h) shown. These unique features of the CCO unit cells have not been
205 206 207 208 209 210
reported in the previous research. By solving d = 0 and d4 = 0, we could obtain the analytical expression of the transition between positive and negative for the Poisson’s ratio +,2 , +,- . We put the concrete expression of d = 0 as follow:
2
#$ 2
%
+ ∑%N
$C
$C
DEF GH K IFJ% LH MFJ%K GH
⋅
DEF LH PQ% RH
IDEF K RH MDEF K LH
=0 (4)
It is plotted with a white dashed curve in Fig.4 (f).
211
In addition, Fig.5 shows the effect of the angle difference on the Poisson’s
212
ratios of the four-layers CCO unit cells with different initial angles. Meanwhile, the
213
effect of the different initial angle and length to width ratio on the Poisson’s ratio of
214 215 216 217 218
the four-layers CCO unit cells, whose angle difference is S = 10∘ , is given in the Supplementary Fig.10. Compare the Fig.5 (a) and (c) with the Fig. 5 (b) and (d), we
can find that the transform of the Poisson’s ratio +,- , +,2 between negative and positive disappear when the different angle is S = 15∘ . This is because that the self-locking emerge at a smaller folding ratio as the angle different increases. That is 10
219 220
consistent with the expression tan −
=!
∙ #$
,
=
=
and ∆α =
. Compare with the Supplementary Fig.10 (a) and (b), the positive Poisson’s
221
ratio and the negative Poisson’s ratio will produce at a smaller initial angle in the Fig
222
5. (c) and (a). They indicate the transform between the negative Poisson’s ratio and
223
positive Poisson’s ratio at smaller initial angle as the angle difference decreases.
224 225
To further explore the properties of the cellular metamaterial, we investigate
226
the Poisson’s ratio of the Model c-4 and the closed tube which is the basic framework
227
of the cellular metamaterials. The theoretical expression of Poisson’s ratio is given in
228
the Supplementary Information 3.3.
229 230 231 232 233 234
Fig.6 (a, b) indicate that the experimental and the analytical results are
excellent agreement. Fig.6(a) shows the Poisson’s ratio +,2 is monotonically increasing in the case of α=55°, α=65° and α=75°. The Poisson’s ratio +WX of the Model c-4 (see Fig.6 (b)) is monotonically increasing in the fold motion, while the
Poisson’s ratio +WY undergoes the change from positive to negative, which is similar to the previous research39-41. Fig.6 (c) indicates that the combination of the two-layers
235
and three-layers CCO unit cells has the same trend with the single three-layers CCO
236
unit cells, but the size of Poisson’s ratio is different.
237
The stiffness of the metamaterials
238
Now, we investigate the stiffness of the closed tubes and the cellular structures
239
to verify the programming stiffness of these CCO-based metamaterials. Base on the
240
previous study14-16, the structure is regarded as the rigid plane attached together along 11
241 242 243 244
the crease lines which are viewed as a torsional spring. The ZG is the torsional spring constant of the per unit length along the crease line. Therefore, the potential energy of the closed tubes and cellular structures can be expressed as: [
245
246 247 248 249 250 251 252
253 254
∏ =U +Ω
Ω %l = − ∫
θ
θ0
f l%
d l% dθ dθ '
'
(5)
Where \ is the elastic energy which stored in the elastic hinges. The ]^_ is
the work of the external force `^_ along the a_ . The a_ is the diagonal of the cross-section along in-plane direction in the close-loop tube, which corresponds the displacement b c of the unit cells.
Based on the first-order variation, the external force `^_ at equilibrium can be
obtained as following equation:
f%l = f D1F1 =
dU dθ dl% dθ
(6)
Taking a derivative of above equations, we obtaioned K %l = K D1F1 =
df %l df l% dθ = d l% d l% d θ
(7)
255
Here, we only give the general expression of the in-plane stretching stiffness.
256
The concrete expression of the dimensionless stiffness of the closed tube and the
257
cellular metamaterials is given in the Supplementary Information 4.
258
The stiffness becomes negative when the initial angle is up to one point as
259
shown in Fig.7 (a, b). Compared the contour line of the Fig. 7 (a, b), the closed tube
260
will generate the negative stiffness at a smaller initial angle α than the cellular 12
261
structure. The dimensionless stretching stiffness of cellular structure is higher than the
262
closed tube as shown in Fig 7. (c). Here, the magnitude of stiffness depends on the
263 264
absolute value. The trend of d^_⁄ZG is a monotonic decreasing during stretching motion. But the magnitude of stiffness is not monotonic decrease. Moreover, it
265
decreases sharply firstly and then slowly changes. That is, there is less change on the
266
negative stiffness. Meanwhile, the point, where the cellular stiffness and the tubes
267
stiffness cross, may be not very important in Fig.7. It is only caused by the different
268
change rate of the cellular stiffness and the tube stiffness at different folding ratio. In
269
addition, we can determine the change trend of the force displacement curve from the
270 271 272 273
stiffness contour plot d^_⁄ZG . For example, the closed tube is in monotonically decreasing state which corresponds to α ≤ 45° (see Fig.7(a)). However, the tube produces the negative stiffness when the initial angle α is above 45°. Moreover,
these stiffness count plots provide a quantitative tool for designing the stiffness of
274
structures based on the engineering requirements.
275
Discussion and conclusion
276
In this work, we introduce an effective strategy for the design of programming
277
mechanical metamaterial with tunable Poisson’s ratio and stiffness by the space-filling
278
of the CCO unit cells, which paves a way for the design of the next generation cellular
279
metamaterials based on the space-filling tessellations. Our study indicates that the
280
Poisson’s ratio could be changed by designing the patterns of the closed tube which
281
are the basic framework of the cellular metamaterials. We theoretically and
282
experimentally prove the transform of Poisson’s ratio from negative to positive at a 13
283
smaller initial angle (
284
negative and positive. Simultaneously, the changes of Poisson’s ratio between
285
negative and positive disappear when the angle difference of the four-layers CCO unit
286
cells reaches about 15 degrees (see Supplementary Fig.13)In addition, the stiffness of
287
the cellular metamaterials is tunable by using different assemblies of the CCO unit
288
cells as mentioned in the article or changing the space-filling tessellations of the CCO
289
elements in the rigid body motion stage. And the mechanical responses of these
290
strategies on the elastic-plastic deformation stage will be given in the next work. The
291
stiffness count plots provide a quantitative tool for designing the structures based on
292
the engineering requirements. What’s more, our strategy provides an effective
293
approach for assembling the elements into a cellular structure. The elements are
294
adjoined together by face, which is beneficial to ensure the integrity of the structure
295
and the coordination of deformation. Simultaneously, the pattern of the tubes is
296
programming, and we have given three kinds of strategies in this study. Furthermore,
297
many next generation cellular metamaterials can be manufactured via using different
298
assemblies of the cellular structures.
) and the transient transition of Poisson’s ratio between the
299 300 301 302 303 304 14
305 306
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27. Zhai, Z., Wang, Y. & Jiang, H. Origami-inspired, on-demand deployable and
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collapsible mechanical metamaterials with tunable stiffness. Proc. Natl. Acad. Sci.
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28. Yasuda, H. & Yang, J. Reentrant Origami-Based Metamaterials with Negative Poisson's Ratio and Bistability. Phys. Rev. Lett. 114, 185502 (2015). 29. Yasuda, H., Tachi, T., Lee, M. & Yang, J. Origami-based tunable truss structures for non-volatile mechanical memory operation. Nat. Commun. 8, 962, (2017). 17
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34. Elsayed, E. A. & Basily, B. B. A continuous folding process for sheet materials. Int. J. Mater. Prod. Tec. 21, 217-238 (2008). 35. Demaine, E. D. & O’Rourke, J. Geometric Folding Algorithms. Linkages, Origami, Polyhedra, (Cambridge University press, 2007). 36. Lechenault, F., Thiria, B. & Adda-Bedia, M. Mechanical response of a creased sheet. Phys. Rev. Lett. 112, 244301 (2014). 37. Brunck, V., Lechenault, F., Reid, A. & Adda-Bedia, M. Elastic theory of origami-based metamaterials. Phys. Rev. E 93, 033005 (2016). 38. Overvelde, J. T., Weaver, J. C., Hoberman, C. & Bertoldi, K. Rational design of reconfigurable prismatic architected materials. Nature 541, 347-352 (2017). 39. Kamrava, S., Mousanezhad, D., Ebrahimi, H., Ghosh, R. & Vaziri, A. 18
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Origami-based cellular metamaterial with auxetic, bistable, and self-locking
394
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395 396
40. Mousanezhad, D., Kamrava, S. & Vaziri, A. Origami-based Building Blocks for Modular Construction of Foldable Structures. Sci. Rep. 7, 14792 (2017).
397
41. Wang, F., Gong, H., Chen, X. & Chen, C. Q. Folding to Curved Surfaces: A
398
Generalized Design Method and Mechanics of Origami-based Cylindrical
399
Structures. Sci. Rep. 6 (2016)
400
Acknowledgements We thank Dr Y. Chen and J.Y. Ma at the Tianjin University for
401
technical support. This work was supported by the National Nature Science
402
Foundation of china (Grant Nos. 11772217, 11572214, 11802198 and 11772216), the
403
“1331 project” fund, Key Innovation Teams of Shanxi Province, the Top Young
404
Academic Leader of Shanxi and Opening foundation for state key laboratory for
405
Strength and vibration of Mechanical structures.
406
Author contributions Y.L. He, P.W. Zhang, Z.H. Wang and X.F. Shu proposed and
407
designed the research; Y.L. He, P.W. Zhang and Z. You design and fabricate the
408
models; Y.L. He and Z.Q. Li performed the experiment; Y.L. He, Z.H. Wang and X.F.
409
Shu prepared for manuscript.
410
Competing financial interests: The authors declare no competing financial interests.
411
19
412 413
Figure 1| The elements and unit cells of the CCO
414
the CCO unit cells on the sheet (see Fig. 1(a)), we can obtain two half model of the
415
CCO unit cells. The Fig.1(b) is one of the two half model of two-layers or
416
three-layers unit cells. In addition, the CCO unit cells (see Fig.1(c)) are consist of two
417
half model which are obtained from the crease pattern of the CCO unit cells.
20
Based on the crease pattern of
418
21
419
Figure 2 | Design strategies to construct 3D cellular metamaterials
420
Space-filling and tessellation of the unit cells are used to construct the cellular
421
metamaterials. The closed tubes and the unclosed tubes are constructed basing on the
422
strategy (i)-(iii) (see Model (a-1,2,3) and Model (b-1,2⋯6)). We then manufacture
423
the cellular metamaterials which are the space-filling of the tubes using the elements
424
or unit cells. In addition, the last column of the Fig.2 (a, b, c) corresponds the
425
self-locking stage of the cellular metamaterials. Using this strategy, the next
426
generation of cellular metamaterials are constructed (see Fig.2(d)). Finally, the
427
folding motion of the Model c-4 and the other closed cellular metamaterials will be
428
shown in the Supplementary Movie 1 and Supplementary Movie 2 respectively.
429
The folding motion of the unclosed cellular metamaterials, which follow the strategy
430
(i)-(iii) will be given in Supplementary Movie 3-5 respectively.
431
22
432 433
Figure 3 | The experimental setup. This experimental equipment is used to measure
434
the Poisson’s ratio.
435
23
436 24
437
Figure 4 | Poisson’s ratio changes of Three-layers CCO unit cells The geometric
438
parameters are
439
dividing line between the positive and negative Poisson’s ratio. The contour plot of
440
the Poisson’s ratio represents the function of initial angle and the folding ratio in
441
Fig.4(b). The changes of Poisson's ratio with length-width ratio and folding ratio are
442
In
443
ℎ = 20gg and
+
= 60gg. The white dashed line is the
shown in the Fig.3 (d), (f) and (h). The folding ratio is defined as (90° −
% )/90°.
addition, the initial folding ratio is 1%. The line of the Poisson's ratio is very close
444
(especially the part which corresponds to the experiment) in Fig. 4(a, c, e and f).
445
Therefore, the error bars overlap together are difficult to distinguish. To reflect the
446
error more clearly, we only give the error of one group of experimental data.
447
25
448 449 450 451
Figure 5 | The effect of angle different on Poisson’s ratio of the four-layers unit cells.
Fig. 5 shows the Poisson’s ratio of the four-layers CCO unit cells in the same
length parameters ( ,
, ℎ ) with different angles (∆α = 5° and 26
∆α = 15° ).
452 453 454 455
Figure 6 | Poisson’s ratio of closed tube
The number of the tube with two layers
and three layers are p = 1 and q = 2 respectively in Fig 6. (a)-(b). And the height
of every layer is 15mm. Three basic orthographic views are given in the
456
Supplementary Fig. 12 and 13, which correspond to Fig. 6 (a, b) respectively. Note
457
that the characteristic length correspond to the size of the model as is shown in
458
Supplementary Fig (1,2). In addition, the initial folding ratio is 1%.
459
27
460 461 462 463
Figure 7 | The in-plane stiffness of closed tubes and cellular metamaterials
Z^ /ZG represents the in-plane dimensionless stiffness. The write dash indicates the transform of the stiffness between the positive and negative. The initial folding angle =0° . The other initial folding angle j
464
is
465
S1-S3. The concrete length of the closed tube and the cellular metamaterials are given
466
in the based elements. (see the Supplementary Fig. 1).
467
28
and
could be obtained by equation
1
2 3
Figure 1| The elements and unit cells of the CCO
4
the CCO unit cells on the sheet (see Fig. 1(a)), we can obtain two half model of the
5
CCO unit cells. The Fig.1(b) is one of the two half model of two-layers or
6
three-layers unit cells. In addition, the CCO unit cells (see Fig.1(c)) are consist of two
7
half model which are obtained from the crease pattern of the CCO unit cells.
1
Based on the crease pattern of
8
9
10
2
11 12
Figure 2 | Design strategies to construct 3D cellular metamaterials
13
Space-filling and tessellation of the unit cells are used to construct the cellular
14
metamaterials. The closed tubes and the unclosed tubes are constructed basing on the
15
strategy (i)-(iii) (see Model (a-1,2,3) and Model (b-1,2⋯6)). We then manufacture
16
the cellular metamaterials which are the space-filling of the tubes using the elements
17
or unit cells. In addition, the last column of the Fig.2 (a, b, c) corresponds the
18
self-locking stage of the cellular metamaterials. Using this strategy, the next
19
generation of cellular metamaterials are constructed (see Fig.2(d)). Finally, the
20
folding motion of the Model c-4 and the other closed cellular metamaterials will be
21
shown in the Supplementary Movie 1 and Supplementary Movie 2 respectively.
22
The folding motion of the unclosed cellular metamaterials, which follow the strategy
23
(i)-(iii) will be given in Supplementary Movie 3-5 respectively.
24
3
25 26
Figure 3 | The experimental setup. This experimental equipment is used to measure
27
the Poisson’s ratio.
28
4
29
30
31
32 5
33
Figure 4 | Poisson’s ratio changes of Three-layers CCO unit cells The geometric
34
parameters are
35
dividing line between the positive and negative Poisson’s ratio. The contour plot of
36
the Poisson’s ratio represents the function of initial angle and the folding ratio in
37
Fig.4(b). The changes of Poisson's ratio with length-width ratio and folding ratio are
38
shown in the Fig.3 (d), (f) and (h). The folding ratio is defined as (90° −
39
addition, the initial folding ratio is 1%. The line of the Poisson's ratio is very close
40
(especially the part which corresponds to the experiment) in Fig. 4(a, c, e and f).
41
Therefore, the error bars overlap together are difficult to distinguish. To reflect the
42
error more clearly, we only give the error of one group of experimental data.
ℎ = 20
and
+
= 60
43
6
. The white dashed line is the
)/90°. In
44
45
46 47 48
Figure 5 | The effect of angle different on Poisson’s ratio of the four-layers unit cells.
Fig. 5 shows the Poisson’s ratio of the four-layers CCO unit cells in the same
7
49
length parameters ( ,
, ℎ ) with different angles (∆α = 5° and
∆α = 15° ).
50
51 52
Figure 6 | Poisson’s ratio of closed tube
53
and three layers are p = 1 and q = 2 respectively in Fig 6. (a)-(b). And the height
54
of every layer is 15mm. Three basic orthographic views are given in the
55
Supplementary Fig. 12 and 13, which correspond to Fig. 6 (a, b) respectively. Note
56
that the characteristic length correspond to the size of the model as is shown in
57
Supplementary Fig (1,2). In addition, the initial folding ratio is 1%.
The number of the tube with two layers
58
8
59
60 61
Figure 7 | The in-plane stiffness of closed tubes and cellular metamaterials
62
/
represents
the in-plane dimensionless stiffness. The write dash indicates the
63
transform of the stiffness between the positive and negative. The initial folding angle
64
is
65
S1-S3. The concrete length of the closed tube and the cellular metamaterials are given
66
in the based elements. (see the Supplementary Fig. 1).
=0° . The other initial folding angle
and ! could be obtained by equation
67
9
Conflict of interest: The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted Zhihua Wang and Xuefeng Shu
S0266-3538(19)31342-9
DOI:
https://doi.org/10.1016/j.compscitech.2020.108015
Reference:
CSTE 108015
To appear in:
Composites Science and Technology
Received Date: 14 May 2019 Revised Date:
25 December 2019
Accepted Date: 17 January 2020
Please cite this article as: He YL, Zhang PW, You Z, Li ZQ, Wang ZH, Shu XF, Programming mechanical metamaterials using origami tessellations, Composites Science and Technology (2020), doi: https://doi.org/10.1016/j.compscitech.2020.108015. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Author statement Manuscript title: Programming mechanical metamaterials using origami tessellations I have made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; And I have drafted the work or revised it critically for important intellectual content; And I have approved the final version to be published; And I agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. All persons who have made substantial contributions to the work reported in the manuscript, including those who provided editing and writing assistance but who are not authors, are named in the Acknowledgments section of the manuscript and have given their written permission to be named. If the manuscript does not include Acknowledgments, it is because the authors have not received substantial contributions from nonauthors. Zhihua Wang and Xuefeng Shu
1
Programming mechanical metamaterials using origami
2
tessellations
3
Y.L. He1,2, P.W. Zhang3, Z. You4, Z.Q. Li1,2, Z.H. Wang1,2,∗ X.F. Shu1,2,**
4
1
Institute of Applied Mechanics, College of Mechanical and Vehicle Engineering, Taiyuan University
5 6
of Technology, Taiyuan 030024, China. 2
Shanxi Key Laboratory of Material Strength and Structural Impact, Taiyuan University of Technology,
7
Taiyuan 030024, China
8
3
9
College of management & engineering, Shanxi University of Finance and Economics, Taiyuan 030006, China
10
4
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
11
Abstract
12 13
Origami structures, whose mechanical properties can be tuned by programing
14
the crease network, have been sought in application including DNA origami
15
nanorobots and deployable space structures. However, the existing researches mainly
16
placed on the cylindrical origami tubes or the periodic cellular metamaterials, which
17
are based on the stacking of individual 3D origami units. Here, we present a novel
18
mechanical metamaterial, which can be constructed via using the curved-crease
19
origami (CCO), with tunable Poisson’s ratio and stiffness. We further analytically and
20
experimentally demonstrate that the Poisson's ratio of these structures exhibits some
∗
Corresponding authors. Tel.: +86 351 6018560 E-mail address: [email protected], [email protected]
1
21
new properties by altering the length to width ratio of the structures. In addition, we
22
verify the tunable stiffness of the CCO-based metamaterials by changing the patterns
23
of these structures. Our approach can be used to design and construct the next
24
generation of the CCO-based metamaterials with tunable mechanical properties for
25
engineering applications.
26 27
Keywords: Origami structures, Tunable Poisson’s ratio, Tunable stiffness.
28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 2
43 44
Origami, an ancient handcraft art, has stimulated curiosity and research
45
interest of scientists from different disciplines and fields, not only because origami
46
patterns can be found in the nature1-4, but also because it shows some exotic
47
properties that exist large potential for engineering application5-11. Meanwhile,
48
origami-inspired metamaterials provide the possibility for scientific communities to
49
manipulate the physical properties by reprogramming the crease and tessellating the
50
structures into an ideal pattern. These properties have been verified, including tunable
51
Poisson’s ratio and stiffness12-17, tunable chirality18 and thermal expansion19,
52
programmable collapse20 and curvature21, self-foldability22,23 and self-locking24-26,
53
high stiffness-to-weight27, multi-stability28 and multi-bit memory29. One good case is
54
that the multi-stable stacked-origami30 can be used to control wave propagation31.For
55
example, the sound absorption and noise reduction can be achieved by the design of
56
the origami structures. And origami structure also can be used to manipulate the
57
electromagnetic response18 by transforming the geometrical configurations. Previous
58
studies reported that reentrant origami structure28 exhibited tunable negative Poisson’s
59
ratio and structural bistability simultaneously. What’s more, the negative stiffness and
60
ultra-stiffness origami cellular solids20 have been exploited to program collapsible
61
mechanical metamaterials. Furthermore, the lattice structures32 of such metamaterial
62
have been manufactured by 3D-Printer or Schematic Drawing of the Pre-Folding
63
Technique33,34. In addition, a set of basic theories of the origami structure were
64
reported. For example, the geometric folding algorithms35 that can be used to 3
65
manufacture any 3D structure with the least creases. The mechanical characterization
66
of creased sheets36 and the elastic theory of the folding paper37 provide a platform for
67
designing and analyzing the origami structures. However, the theories were mainly
68
used on the reconfigurable architected materials38 or the periodic origami
69
structures24,25,39,40 which were constructed in the ordered unit-cells. The mechanical
70
behavior of CCO-based metamaterials, which are constructed in different strategies,
71
has rarely been investigated.
72
Here, we provide a strategy for the design of the 3D cellular metamaterials
73
based on the curved-crease origami (CCO) pattern. In order to provide reference for
74
the design of future origami-based cellular structures which based on the stitching and
75
cutting of the cellular metamaterials, we systematically explored the mechanical
76
response of the CCO-based cellular metamaterials. In addition, previous 3D
77
origami-based structures such as the cellular origami structure24,25 and origami-based
78
cylindrical structures27-29, which are often made up of individual tessellating repeat
79
unit cells along out-of-plane direction. Therefore, the mechanical behavior of these
80
structures is uniform throughout. Different to these structures, the CCO-based unit
81
cells have the rigid folding stage and the plastic deformation stage when the number
82
of the layer is more than three and the divide line of these two stages is the
83
self-locking. Therefore, the displacement-force curve of the structures has
84
multi-platforms as shown in the supplementary Figure 15, which may be an excellent
85
energy absorption equipment. In this paper, we only pay attention to the mechanical
86
properties of the origami structures during the folded motion. Thus, we firstly gave 4
87
the generation expression of the Poisson’s ratio and the stiffness in several
88
representative unit cells, closed tubes and the cellular metamaterials and verified that
89
our design strategy is an effective way to program the Poisson’s ratio and stiffness.
90
And we can judge change trend of the force-displacement curve from the contour map
91
of the stiffness during the folding motion. Therefore, this research paves the way for
92
the next generational cellular metamaterials which will bring the origami-inspired
93
metamaterials closer toward the engineering application.
94
Design strategy
95
To build the 3D origami-inspired programming cellular metamaterials, we
96
started by making up the CCO unit cells that were made up of two unsymmetrical
97
elements, whose geometrical patterns can be described by the length parameters
98 99
( ,
, ℎ ) and the original angles
(see Fig.1). We then proposed three methods
on the stacking sequences of the unit cells along the out-of-plane direction (see
100
Fig.2(a)). (i) We linked the two-layers unit cells with another CCO unit cells while
101
ensuring that the motions were compatible between them. The adjacent faces and the
102
holes between different unit cells remain aligned along the out-of-plane direction.
103 104
That is, the geometrical parameters ( ,
, ℎ ) are identical between the connected
faces. (ii) We connect two kinds of unit cells to form a closed tube (see Model a-2).
105
The patterns of these two kinds of the unit cells are identical when the flat rectangular
106
grids are excluded. For example, the tubes are connected by the four-layers and
107
five-layers unit cells. Moreover, the tubes are composed of two-layers and
108
three-layers unit cells also is a special example of strategy (i). To avoid confusion, we 5
≥ 2. (iii) We introduce the two-layers unit cells into
109
set the strategy (ii) meet with
110
the structure based on strategy (ii) (see Model a-3). Complementing these strategies,
111
we don’t introduce other types of unit cells in Model a-3, because these structures are
112
regarded as the derivatives of the above models.
113
What is more, there are two methods on the design of the tube along the
114
in-plane direction. Based on these two methods, the closed tubes and unclosed tubes
115
are constructed. For one thing, the tube is completely closed as shown in Fig.2(a). To
116
ensure that the elements can be connected together to form a closed tube, the curve
117
creases are compatible for the two overlapping faces (see Supplementary Fig.1(a,b)).
118
In addition, there is a special closed tube (see Model c-4) whose crease patterns are
119
shown in the Supplementary Fig.3, and its three basic orthographic views are shown
120
in the Supplementary Fig.13. What’s more, the structure is deployable along the
121
in-plane direction (see the Supplementary Movie 1). And the crease patterns need
122
match the combination method of the tube as Supplementary Fig. (3,4) shown. For
123
another thing, the tube is unclosed as shown in the Fig.2(b). Moreover, there are the
124
holes in the unclosed metamaterials which is beneficial to reduce the accumulation of
125
moisture. These unclosed metamaterials will be investigated in detail in the next work.
126
Two elements or unit cells are attached by the sheet with the same shape of the two
127
overlapping planes. These closed tubes and unclosed tubes have the rigid folding
128
stage and the plastic deformation stage, which is an improvement of the previous
129
closed-loop tubes27-29. Considering the space-filling, the patterns of the tube and the
130
space-filling elements should be compatible. Finally, the tubes could be expanded or 6
131
filled by the elements to form the cellular metamaterials which can be actively altered
132
into numerous patterns by changing the assembly and the size of elements.
133
Furthermore, the space-filling assemblies of cellular metamaterials could form the
134
next generation cellular metamaterials as shown in Fig.2(d), providing a strategy for
135
the design of deployable next generation cellular metamaterials.
136
Geometry of unit cells and tube
137
In order to calculate the geometrical relationship of the CCO origami unit cells
138
during its folding motion, we take out a half model of the CCO unit cells, which
139
corresponds to an element that has been folded along the crease. The lengths of
140
L and H correspond to the displacement as shown in Fig.1(b). The dihedral angle and
141
the angle between lines were given in Fig. 1(b). And the corresponding geometrical
142
relationship in the 3D unit cells was given in the Supplementary Fig. (8, 9). The
143
mathematical expression of the geometrical relationship was described in the
144
Supplemental Information 2 in detail (That is, geometry of the elements and unit cells
145
based
146 147
on
CCO
∈ 0, ⁄2 ,
150
,
supplemental
∈ 0,
tan
materials).
Noted
=!
. #$
,
where the
%
the
angle
=
=
was given as follow (1) at the self-locking state
in the unit cells. Therefore, we can obtain the variation range of arccos (!
that
could be obtained by the given geometry.
In addition, we can obtain the expression
151 152
>
the
Moreover, the relationship between the angle
148 149
in
, ,
⁄!
%)
≤
%
≤
%
%
is the natural dihedral angles in the undeformed state. 7
as follow (2)
153 154
The Poisson’s ratio of the unit cells and closed tubes
155 156 157 158 159 160
In order to investigate the “Poisson’s ratio” of the CCO cells in different directions, we define them according to the previous study28. It was given as follow
+,- = −
(/-/-)
(/,/,)
, +,1 = −
(/1 ⁄1) (/, ⁄, )
+,2 = −
(/2/2)
(/,/,)
(3)
Here, we only give the general expression. The specific expression of
Poisson’s ratio +,- , +,1
+,2 have been given in the Supplemental information
3.2.
161
To verify the analytical solutions, we manufacture three groups prototypes of the
162
CCO unit cells using the sheets (See the Supplementary Information 3.1 for details).
163
Accordingly, we measure the dimensions of the length
164
some height intervals of H, as we gradually compress the CCO unit cells by the
165
experiment set up along the out-of-plane direction. As the Fig. 3 shown, an acrylic
166
plate with the equal size grids is placed on the top surface of prototypes and the plate
167
is fixed at a certain height. The other two acrylic plates are fixed on the bottom, which
168
are used to fix the scissor-type lifting platform. This allows the upper plate of the
169
scissor-type lifting platform to transitional motion in vertical direction. The position
170
of the upper plate of the scissor-type lifting can be controlled by the knob. The height
171
of the prototype could be measured by digital display caliper. The level gauge is used
172
to ensure that the upper acrylic plate is parallel to the top plate of the lifting platform.
173
In addition, the camera is fixed above the upper acrylic plates to capture the digital
174
image of the prototypes’ cross-sectional area. Based on the pictures from camera, we 8
, ,L,H and the angle 2ξ in
175 176
obtain the length parameters ( ,
, 4) and the angle 2 in different height (5) by
the Image processing software. We compare the Poisson’s ratio that is measured with
177
the analytical solutions which is calculated from the Eq. (3). They are in good
178
agreement between theoretical and experimental results.
179
The result of the Poisson’s ratio of three -layers unit cells was given in Fig.3,
180
which has different initial angles and length-width ratio in different directions. The
181
other results of Poisson’s ratio of CCO unit cells have been given in the
182
Supplementary Fig.10 such as the angle difference between
183 184
and
in the
four-layers unit cells. Here, Fig. 4(a) shows the Poisson’s ratio +,2 of the CCO unit cells in the cases of α=55°, α=65°, α=75° and length-width ratio is
:
= 2 ∶ 1. The
185
corresponding analytical contour plot, which is a function of the continuous α and the
186
is shown in Fig. 4(b). The Fig.4 (c, d)
187 188 189 190 191 192 193 194 195 196
folding ratio defined as (90° −
% )⁄90°,
indicates the change of Poisson’s ratio in different length-width ratio, while the initial
angle α = 65° remains unchanged. We find that the CCO unit cells emerge the positive in-plane Poisson’s ratio +,2 at initial angle α = 65°, when the length-width ratio of
:
and :
= 1 ∶ 3 . Previous research reported that the zigzag-based origami14-16
is above approximately 3:1 or below approximately 1:3. Meanwhile,
the Poisson’s ratio +,2 are identical when the length-width ratio
:
=3∶1
produced the positive in-plane Poisson’s ratio at initial angle above α = 75°, but the
fact that positive Poisson’s ratio produces at a smaller angle by changing the
length-width ratio have not been reported. In addition, the Poisson’s ratio +,- of the
CCO unit cells become negative when the initial angle α is above approximate 9
197 198 199 200 201 202
75° (see the Supplementary Fig.11(a)). Therefore, the conventional 2D Miura-ori also has multiple sign flips of the Poisson’s ratio just like the reentrant origami-based
metamaterial28. Figure 3 (e, f) indicate that the Poisson’s ratio +,- of the CCO unit cells changes from positive to negative, when the length-width ratio
:
is below
approximately 2:3 at the initial angle α = 65°. Besides, the Poisson’s ratio +,1 of the CCO unit cells has a transient transition between the positive and negative, when :
is below approximately 1:2 at the initial angle α = 65°
203
the length-width ratio
204
as Fig. 4 (g, h) shown. These unique features of the CCO unit cells have not been
205 206 207 208 209 210
reported in the previous research. By solving d = 0 and d4 = 0, we could obtain the analytical expression of the transition between positive and negative for the Poisson’s ratio +,2 , +,- . We put the concrete expression of d = 0 as follow:
2
#$ 2
%
+ ∑%N
$C
$C
DEF GH K IFJ% LH MFJ%K GH
⋅
DEF LH PQ% RH
IDEF K RH MDEF K LH
=0 (4)
It is plotted with a white dashed curve in Fig.4 (f).
211
In addition, Fig.5 shows the effect of the angle difference on the Poisson’s
212
ratios of the four-layers CCO unit cells with different initial angles. Meanwhile, the
213
effect of the different initial angle and length to width ratio on the Poisson’s ratio of
214 215 216 217 218
the four-layers CCO unit cells, whose angle difference is S = 10∘ , is given in the Supplementary Fig.10. Compare the Fig.5 (a) and (c) with the Fig. 5 (b) and (d), we
can find that the transform of the Poisson’s ratio +,- , +,2 between negative and positive disappear when the different angle is S = 15∘ . This is because that the self-locking emerge at a smaller folding ratio as the angle different increases. That is 10
219 220
consistent with the expression tan −
=!
∙ #$
,
=
=
and ∆α =
. Compare with the Supplementary Fig.10 (a) and (b), the positive Poisson’s
221
ratio and the negative Poisson’s ratio will produce at a smaller initial angle in the Fig
222
5. (c) and (a). They indicate the transform between the negative Poisson’s ratio and
223
positive Poisson’s ratio at smaller initial angle as the angle difference decreases.
224 225
To further explore the properties of the cellular metamaterial, we investigate
226
the Poisson’s ratio of the Model c-4 and the closed tube which is the basic framework
227
of the cellular metamaterials. The theoretical expression of Poisson’s ratio is given in
228
the Supplementary Information 3.3.
229 230 231 232 233 234
Fig.6 (a, b) indicate that the experimental and the analytical results are
excellent agreement. Fig.6(a) shows the Poisson’s ratio +,2 is monotonically increasing in the case of α=55°, α=65° and α=75°. The Poisson’s ratio +WX of the Model c-4 (see Fig.6 (b)) is monotonically increasing in the fold motion, while the
Poisson’s ratio +WY undergoes the change from positive to negative, which is similar to the previous research39-41. Fig.6 (c) indicates that the combination of the two-layers
235
and three-layers CCO unit cells has the same trend with the single three-layers CCO
236
unit cells, but the size of Poisson’s ratio is different.
237
The stiffness of the metamaterials
238
Now, we investigate the stiffness of the closed tubes and the cellular structures
239
to verify the programming stiffness of these CCO-based metamaterials. Base on the
240
previous study14-16, the structure is regarded as the rigid plane attached together along 11
241 242 243 244
the crease lines which are viewed as a torsional spring. The ZG is the torsional spring constant of the per unit length along the crease line. Therefore, the potential energy of the closed tubes and cellular structures can be expressed as: [
245
246 247 248 249 250 251 252
253 254
∏ =U +Ω
Ω %l = − ∫
θ
θ0
f l%
d l% dθ dθ '
'
(5)
Where \ is the elastic energy which stored in the elastic hinges. The ]^_ is
the work of the external force `^_ along the a_ . The a_ is the diagonal of the cross-section along in-plane direction in the close-loop tube, which corresponds the displacement b c of the unit cells.
Based on the first-order variation, the external force `^_ at equilibrium can be
obtained as following equation:
f%l = f D1F1 =
dU dθ dl% dθ
(6)
Taking a derivative of above equations, we obtaioned K %l = K D1F1 =
df %l df l% dθ = d l% d l% d θ
(7)
255
Here, we only give the general expression of the in-plane stretching stiffness.
256
The concrete expression of the dimensionless stiffness of the closed tube and the
257
cellular metamaterials is given in the Supplementary Information 4.
258
The stiffness becomes negative when the initial angle is up to one point as
259
shown in Fig.7 (a, b). Compared the contour line of the Fig. 7 (a, b), the closed tube
260
will generate the negative stiffness at a smaller initial angle α than the cellular 12
261
structure. The dimensionless stretching stiffness of cellular structure is higher than the
262
closed tube as shown in Fig 7. (c). Here, the magnitude of stiffness depends on the
263 264
absolute value. The trend of d^_⁄ZG is a monotonic decreasing during stretching motion. But the magnitude of stiffness is not monotonic decrease. Moreover, it
265
decreases sharply firstly and then slowly changes. That is, there is less change on the
266
negative stiffness. Meanwhile, the point, where the cellular stiffness and the tubes
267
stiffness cross, may be not very important in Fig.7. It is only caused by the different
268
change rate of the cellular stiffness and the tube stiffness at different folding ratio. In
269
addition, we can determine the change trend of the force displacement curve from the
270 271 272 273
stiffness contour plot d^_⁄ZG . For example, the closed tube is in monotonically decreasing state which corresponds to α ≤ 45° (see Fig.7(a)). However, the tube produces the negative stiffness when the initial angle α is above 45°. Moreover,
these stiffness count plots provide a quantitative tool for designing the stiffness of
274
structures based on the engineering requirements.
275
Discussion and conclusion
276
In this work, we introduce an effective strategy for the design of programming
277
mechanical metamaterial with tunable Poisson’s ratio and stiffness by the space-filling
278
of the CCO unit cells, which paves a way for the design of the next generation cellular
279
metamaterials based on the space-filling tessellations. Our study indicates that the
280
Poisson’s ratio could be changed by designing the patterns of the closed tube which
281
are the basic framework of the cellular metamaterials. We theoretically and
282
experimentally prove the transform of Poisson’s ratio from negative to positive at a 13
283
smaller initial angle (
284
negative and positive. Simultaneously, the changes of Poisson’s ratio between
285
negative and positive disappear when the angle difference of the four-layers CCO unit
286
cells reaches about 15 degrees (see Supplementary Fig.13)In addition, the stiffness of
287
the cellular metamaterials is tunable by using different assemblies of the CCO unit
288
cells as mentioned in the article or changing the space-filling tessellations of the CCO
289
elements in the rigid body motion stage. And the mechanical responses of these
290
strategies on the elastic-plastic deformation stage will be given in the next work. The
291
stiffness count plots provide a quantitative tool for designing the structures based on
292
the engineering requirements. What’s more, our strategy provides an effective
293
approach for assembling the elements into a cellular structure. The elements are
294
adjoined together by face, which is beneficial to ensure the integrity of the structure
295
and the coordination of deformation. Simultaneously, the pattern of the tubes is
296
programming, and we have given three kinds of strategies in this study. Furthermore,
297
many next generation cellular metamaterials can be manufactured via using different
298
assemblies of the cellular structures.
) and the transient transition of Poisson’s ratio between the
299 300 301 302 303 304 14
305 306
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400
Acknowledgements We thank Dr Y. Chen and J.Y. Ma at the Tianjin University for
401
technical support. This work was supported by the National Nature Science
402
Foundation of china (Grant Nos. 11772217, 11572214, 11802198 and 11772216), the
403
“1331 project” fund, Key Innovation Teams of Shanxi Province, the Top Young
404
Academic Leader of Shanxi and Opening foundation for state key laboratory for
405
Strength and vibration of Mechanical structures.
406
Author contributions Y.L. He, P.W. Zhang, Z.H. Wang and X.F. Shu proposed and
407
designed the research; Y.L. He, P.W. Zhang and Z. You design and fabricate the
408
models; Y.L. He and Z.Q. Li performed the experiment; Y.L. He, Z.H. Wang and X.F.
409
Shu prepared for manuscript.
410
Competing financial interests: The authors declare no competing financial interests.
411
19
412 413
Figure 1| The elements and unit cells of the CCO
414
the CCO unit cells on the sheet (see Fig. 1(a)), we can obtain two half model of the
415
CCO unit cells. The Fig.1(b) is one of the two half model of two-layers or
416
three-layers unit cells. In addition, the CCO unit cells (see Fig.1(c)) are consist of two
417
half model which are obtained from the crease pattern of the CCO unit cells.
20
Based on the crease pattern of
418
21
419
Figure 2 | Design strategies to construct 3D cellular metamaterials
420
Space-filling and tessellation of the unit cells are used to construct the cellular
421
metamaterials. The closed tubes and the unclosed tubes are constructed basing on the
422
strategy (i)-(iii) (see Model (a-1,2,3) and Model (b-1,2⋯6)). We then manufacture
423
the cellular metamaterials which are the space-filling of the tubes using the elements
424
or unit cells. In addition, the last column of the Fig.2 (a, b, c) corresponds the
425
self-locking stage of the cellular metamaterials. Using this strategy, the next
426
generation of cellular metamaterials are constructed (see Fig.2(d)). Finally, the
427
folding motion of the Model c-4 and the other closed cellular metamaterials will be
428
shown in the Supplementary Movie 1 and Supplementary Movie 2 respectively.
429
The folding motion of the unclosed cellular metamaterials, which follow the strategy
430
(i)-(iii) will be given in Supplementary Movie 3-5 respectively.
431
22
432 433
Figure 3 | The experimental setup. This experimental equipment is used to measure
434
the Poisson’s ratio.
435
23
436 24
437
Figure 4 | Poisson’s ratio changes of Three-layers CCO unit cells The geometric
438
parameters are
439
dividing line between the positive and negative Poisson’s ratio. The contour plot of
440
the Poisson’s ratio represents the function of initial angle and the folding ratio in
441
Fig.4(b). The changes of Poisson's ratio with length-width ratio and folding ratio are
442
In
443
ℎ = 20gg and
+
= 60gg. The white dashed line is the
shown in the Fig.3 (d), (f) and (h). The folding ratio is defined as (90° −
% )/90°.
addition, the initial folding ratio is 1%. The line of the Poisson's ratio is very close
444
(especially the part which corresponds to the experiment) in Fig. 4(a, c, e and f).
445
Therefore, the error bars overlap together are difficult to distinguish. To reflect the
446
error more clearly, we only give the error of one group of experimental data.
447
25
448 449 450 451
Figure 5 | The effect of angle different on Poisson’s ratio of the four-layers unit cells.
Fig. 5 shows the Poisson’s ratio of the four-layers CCO unit cells in the same
length parameters ( ,
, ℎ ) with different angles (∆α = 5° and 26
∆α = 15° ).
452 453 454 455
Figure 6 | Poisson’s ratio of closed tube
The number of the tube with two layers
and three layers are p = 1 and q = 2 respectively in Fig 6. (a)-(b). And the height
of every layer is 15mm. Three basic orthographic views are given in the
456
Supplementary Fig. 12 and 13, which correspond to Fig. 6 (a, b) respectively. Note
457
that the characteristic length correspond to the size of the model as is shown in
458
Supplementary Fig (1,2). In addition, the initial folding ratio is 1%.
459
27
460 461 462 463
Figure 7 | The in-plane stiffness of closed tubes and cellular metamaterials
Z^ /ZG represents the in-plane dimensionless stiffness. The write dash indicates the transform of the stiffness between the positive and negative. The initial folding angle =0° . The other initial folding angle j
464
is
465
S1-S3. The concrete length of the closed tube and the cellular metamaterials are given
466
in the based elements. (see the Supplementary Fig. 1).
467
28
and
could be obtained by equation
1
2 3
Figure 1| The elements and unit cells of the CCO
4
the CCO unit cells on the sheet (see Fig. 1(a)), we can obtain two half model of the
5
CCO unit cells. The Fig.1(b) is one of the two half model of two-layers or
6
three-layers unit cells. In addition, the CCO unit cells (see Fig.1(c)) are consist of two
7
half model which are obtained from the crease pattern of the CCO unit cells.
1
Based on the crease pattern of
8
9
10
2
11 12
Figure 2 | Design strategies to construct 3D cellular metamaterials
13
Space-filling and tessellation of the unit cells are used to construct the cellular
14
metamaterials. The closed tubes and the unclosed tubes are constructed basing on the
15
strategy (i)-(iii) (see Model (a-1,2,3) and Model (b-1,2⋯6)). We then manufacture
16
the cellular metamaterials which are the space-filling of the tubes using the elements
17
or unit cells. In addition, the last column of the Fig.2 (a, b, c) corresponds the
18
self-locking stage of the cellular metamaterials. Using this strategy, the next
19
generation of cellular metamaterials are constructed (see Fig.2(d)). Finally, the
20
folding motion of the Model c-4 and the other closed cellular metamaterials will be
21
shown in the Supplementary Movie 1 and Supplementary Movie 2 respectively.
22
The folding motion of the unclosed cellular metamaterials, which follow the strategy
23
(i)-(iii) will be given in Supplementary Movie 3-5 respectively.
24
3
25 26
Figure 3 | The experimental setup. This experimental equipment is used to measure
27
the Poisson’s ratio.
28
4
29
30
31
32 5
33
Figure 4 | Poisson’s ratio changes of Three-layers CCO unit cells The geometric
34
parameters are
35
dividing line between the positive and negative Poisson’s ratio. The contour plot of
36
the Poisson’s ratio represents the function of initial angle and the folding ratio in
37
Fig.4(b). The changes of Poisson's ratio with length-width ratio and folding ratio are
38
shown in the Fig.3 (d), (f) and (h). The folding ratio is defined as (90° −
39
addition, the initial folding ratio is 1%. The line of the Poisson's ratio is very close
40
(especially the part which corresponds to the experiment) in Fig. 4(a, c, e and f).
41
Therefore, the error bars overlap together are difficult to distinguish. To reflect the
42
error more clearly, we only give the error of one group of experimental data.
ℎ = 20
and
+
= 60
43
6
. The white dashed line is the
)/90°. In
44
45
46 47 48
Figure 5 | The effect of angle different on Poisson’s ratio of the four-layers unit cells.
Fig. 5 shows the Poisson’s ratio of the four-layers CCO unit cells in the same
7
49
length parameters ( ,
, ℎ ) with different angles (∆α = 5° and
∆α = 15° ).
50
51 52
Figure 6 | Poisson’s ratio of closed tube
53
and three layers are p = 1 and q = 2 respectively in Fig 6. (a)-(b). And the height
54
of every layer is 15mm. Three basic orthographic views are given in the
55
Supplementary Fig. 12 and 13, which correspond to Fig. 6 (a, b) respectively. Note
56
that the characteristic length correspond to the size of the model as is shown in
57
Supplementary Fig (1,2). In addition, the initial folding ratio is 1%.
The number of the tube with two layers
58
8
59
60 61
Figure 7 | The in-plane stiffness of closed tubes and cellular metamaterials
62
/
represents
the in-plane dimensionless stiffness. The write dash indicates the
63
transform of the stiffness between the positive and negative. The initial folding angle
64
is
65
S1-S3. The concrete length of the closed tube and the cellular metamaterials are given
66
in the based elements. (see the Supplementary Fig. 1).
=0° . The other initial folding angle
and ! could be obtained by equation
67
9
Conflict of interest: The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted Zhihua Wang and Xuefeng Shu