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19 JANUARY 2017 | VOL 541 | NATURE | 347
ARTICLE doi:10.1038/nature20824
Rational design of reconfigurable
prismatic architected materials
Johannes T. B. Overvelde1,2, James C. Weaver3, Chuck Hoberman3,4,5 & Katia Bertoldi1,6
In the search for materials with new properties, there have been great
advances in recent years aimed at the construction of architected
materials, whose behaviour is governed by structure, rather than
composition1–3. Through careful design of the material’s architecture,
new material properties have been demonstrated, including negative
index of refraction
4,5
, negative Poisson’s ratio
6
, high stiffness-to-weight
ratio7,8 and optical9 and mechanical10 cloaking. However, most of the
proposed architected materials (also known as metamaterials) have a
unique structure that cannot be reconfigured after fabrication, making
each metamaterial suitable only for a specific task and limiting its appli-
cability to well known and controlled environments.
The ancient art of origami provides an ideal platform for the design
of reconfigurable systems, since a myriad of shapes can be achieved
by actively folding thin sheets along pre-defined creases. While
most of the proposed origami-inspired metamaterials are based on
two- dimensional folding patterns, such as the miura-ori11–17, the
square twist18 and box-pleat tiling19, it has been shown that cellular
structures can be designed by stacking these folded layers13, or
assembling them in tubes20–23. Furthermore, taking inspiration from
snapology
24,25
—a modular origami technique—a highly reconfigurable
three- dimensional (3D) metamaterial assembled from extruded cubes
has been designed26. Although these examples showcase the potential
of origami-inspired designs to enable reconfigurable architected
materials, they do not fully exploit the range of achievable deformations
and cover only a small region of the available design space. As a result,
ample opportunities for the design of architected materials with tunable
responses remain.
Here, we introduce a robust strategy for the design of 3D recon-
figurable architected materials and show that a wealth of responses
can be achieved in periodic 3D assemblies of rigid plates connected
by elastic hinges. To build these structures, we use periodic space-
filling tessellations of convex polyhedra as templates, and extrude
arbitrary combinations of the polygon faces. In an effort to design
architected materials with specific properties, we systematically
explore the proposed designs by performing numerical simulations
and characterize the mobility (that is, number of degrees of freedom)
of the systems. We find that qualitatively different responses can
be achieved, including shear, uniform expansion along one or two
principal directions, and internal reconfigurations that do not alter
the macroscopic shape of the materials. Therefore, this research paves
the way for a new class of structures that can tune their shape and
function to adapt and even influence their surroundings, bringing
origami- inspired metamaterials closer to application.
Design strategy
To design 3D reconfigurable architected materials, we start by selecting
a space-filling and periodic assembly of convex polyhedra (Fig. 1).
We then perform two operations on the tessellation. (i) We separate
adjacent polyhedra while ensuring that the normals of the overlapping
faces remain aligned. This can be achieved by imposing that for each
overlapping face pair
−=pp ndd L2(1)
jj jj
,b ,a
where dpj denotes the displacements applied to the polyhedra to
separate the jth pair of faces, and the subscripts a and b indicate to
which polyhedron the two overlapping faces belong. Moreover, L
j
is
the distance between the jth pair of faces in the separated state, and nj
is the unit normal to the faces pointing outward from the polyhedron
indicated by the subscript a. (ii) We extrude the edges of the polyhedra
in the direction normal to their faces to form a connected thin-walled
structure (Fig. 1), which we refer to as a prismatic architected material
(Supplementary Video 1).
Importantly, for the periodic space-filling tessellations considered
here, it is sufficient to focus on a unit cell that consists of only a few
polyhedra and covers the entire assembly when translated by the three
lattice vectors
li
0
(i = 1, 2, 3). While equation (1) can be directly imposed
on all internal face pairs in the unit cell, for overlapping faces that are
periodically located (that is, lie on the external boundary of the unit
cell) the constraint needs to be updated as
−+−=
ppRR n
dd L2(2)
jj
jjjj
,b ,a
0
Advances in fabrication technologies are enabling the production of architected materials with unprecedented properties.
Most such materials are characterized by a fixed geometry, but in the design of some materials it is possible to incorporate
internal mechanisms capable of reconfiguring their spatial architecture, and in this way to enable tunable functionality.
Inspired by the structural diversity and foldability of the prismatic geometries that can be constructed using the snapology
origami technique, here we introduce a robust design strategy based on space-filling tessellations of polyhedra to create
three-dimensional reconfigurable materials comprising a periodic assembly of rigid plates and elastic hinges. Guided by
numerical analysis and physical prototypes, we systematically explore the mobility of the designed structures and identify
a wide range of qualitatively different deformations and internal rearrangements. Given that the underlying principles
are scale-independent, our strategy can be applied to the design of the next generation of reconfigurable structures and
materials, ranging from metre-scale transformable architectures to nanometre-scale tunable photonic systems.
1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA. 2AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands. 3Wyss Institute for
Biologically Inspired Engineering, Harvard University, Cambridge, Massachusetts 02138, USA. 4Hoberman Associates, New York, New York 10001, USA. 5Graduate School of Design, Harvard
University, Cambridge, Massachusetts 02138, USA. 6Kavli Institute, Harvard University, Cambridge, Massachusetts 02138, USA.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
348 | NATURE | VOL 541 | 19 JA NUARY 2017
ARTICLE
RESEARCH
where
α=∑=
Rl
j
i
ji i
1
3
,
and
α=∑=
Rl
jiji i
0
1
3
,
0
denote the distance between
the two periodically located faces in the expanded and initial
configuration, respectively, li being the lattice vectors of the expanded
unit cell and αj,i ∈ {− 1,0,1}. As shown by equations (1) and (2), for a
unit cell with F face pairs the expanded configuration is fully described
by F extrusion lengths Lj (j = 1,…, F) (Fig. 1). However, for most unit
cells the extrusion lengths cannot all be specified independently owing
to the constraints introduced by equations (1) and (2). As a result, each
unit cell is characterized by F
indep
≤ F independent extrusion lengths
as illustrated in Supplementary Fig. 6. For the sake of convenience we
chose the F
indep
independent extrusion lengths to be as close as possible
to an average extrusion length
L
, by solving
∑−
... =
LLmin()(3)
LL j
F
j
1
2
F1indep
while ensuring that the constraints imposed by equations (1) and (2)
are not violated.
Finally, we note that all periodic and space-filling assemblies of
convex polyhedra tested in this study were successfully extruded
following the proposed design strategy (that is, we always found
Findep ≥ 1). As an example, in Fig. 1 we show three prismatic
architected materials based on unit cells containing two triangular
and one hexagonal prism (Fig. 1a), an octahedron and cuboctahedron
(Fig. 1b), and four triangular prisms (Fig. 1c).
Characterizing reconfigurability
Although the aforementioned design strategy represents a robust and
efficient approach to construct prismatic architected materials, it does
not provide any indication of their reconfigurability. To determine
whether, and to what extent, the meso-structure of the designed
architected materials can be reshaped, we started by fabricating
centimetre-scale prototypes from cardboard and double-sided tape
(Fig. 2a–c), using a stepwise layering and laser-cutting technique
(see the ‘Methods’ subsection of Supplementary Information)26,27.
Focusing on the three architected materials shown in Fig. 1, we find
that the structure based on triangular prisms and the one based on a
combination of triangular and hexagonal prisms can be reconfigured by
bending the edges and without deforming the faces, and are respectively
characterized by one and two deformation modes (Fig. 2d, e and
Supplementary Video 2). In contrast, the material based on a combination
of octahedra and cuboctahedra is completely rigid (Fig. 2b and
Supplementary Video 2). Furthermore, our experiments reveal that
these architected materials have fewer degrees of freedom than their
constituent individual extruded polyhedra (Supplementary Fig. 7),
indicating that the additional constraints introduced by the connections
between the polyhedra effectively reduce their reconfigurability.
Numerical algorithm
While the examples of Fig. 2a–e illustrate the potential of our strategy
to design reconfigurable architected materials, they also show that the
design of systems with specific behaviour is not straightforward. To
improve our understanding of the reconfigurability of the proposed
architected materials, we implemented a numerical algorithm that
predicts their mobility and corresponding deformation modes. In our
numerical analysis, each extruded unit cell is modelled as a set of rigid
faces connected by linear torsional springs, with periodic boundary
conditions applied to the vertices located on the boundaries. To
characterize the mobility of the structure we solved the following eigen-
problem
ω
=
−
~~
aaM K
1m
2
m
, in which
~
M
and
~
K
are respectively the mass
and stiffness matrices, which account for both the rigidity of the faces
and the periodic boundary conditions through master–slave
elimination. Moreover, ω is an eigenfrequency of the system and am is
the amplitude of the corresponding mode (see the ‘Mode analysis for
1
1
2
2
3366
4
4
5
5
Triangular prisms and hexagonal prisms
cTriangular prisms
bOctahedra and cuboctahedra
a
Select space-lling
tessellation
Identify unit cell
Expand unit cell
Identify face pairs
Prismatic architected material
Tessellate extruded unit cell
Extrude expanded
unit cell
l0
3
l0
2
l0
1
l0
1
l0
2
R0
5
R0
2
R0
1
R0
4dp1 = 0
R1
R
5
dp2
R2R4
dp3
l1
(i)
(ii)
F = 11
F
indep
= 1
F = 10
F
indep
= 4
n1L1
n5L5
l2
n2L2
n1L1
n3L3
n4L4
n2L2
n6L6
n5L5
n4L4
F = 9
Findep = 4
Figure 1 | Design strategy to construct 3D prismatic architected
materials. Space-filling and periodic assemblies of convex polyhedra
are used as templates to construct prismatic architected materials
(Supplementary Video 1). After selecting a space-filling tessellation,
we focus on a unit cell spanned by the three lattice vectors
li
0
(i = 1, 2, 3)
and identify all pairs of overlapping faces. We then separate the polyhedra
while ensuring that the normals of all face pairs remain aligned. Finally,
we extrude the edges of the polyhedra in the direction normal to their
faces to construct the extruded unit cell. Note that the architected material
can be constructed by tessellating the extruded unit cell along the three
new lattice vectors li. Using this approach, we designed three architected
materials that are based on space-filling tessellations comprising triangular
prisms and hexagonal prisms (a), octahedra and cuboctahedra (b) and
triangular prisms (c).
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
19 JANUARY 2017 | VOL 541 | NATUR E | 349
ARTICLE RESEARCH
3D prismatic architected materials with rigid faces’ subsection of
Supplementary Information).
Figure 2f, g shows the simulated eigenmodes for the two reconfig-
urable architected materials considered in Fig. 1a, c. Although the
simulations predict only the deformation for small rotations, the modes
are strikingly similar to the deformation observed in the experiments
(Fig. 2d, e). Solving the aforementioned eigenproblem therefore
provides a convenient approach to determine the mobility of the
structures and gives insight into their deformation without the need
for specific boundary conditions.
Designs based on uniform tessellations
To further explore the potential of prismatic architected materials, and
to establish relations between their reconfigurability and the initial
space-filling polyhedral assembly, we next focus on extruded materials
based on the 28 uniform tessellations of the 3D space, which comprise
regular polyhedra, semiregular polyhedra and semiregular prisms
28–30
.
Owing to their relative simplicity, these uniform templates provide a
convenient starting point to explore the design space.
Using the numerical algorithm, we first determined the number of
degrees of freedom, ndof, of the resulting 28 architected materials
(Supplementary Fig. 9). We find that the mobility of the unit cells is
affected by two parameters: the average connectivity of the unit cell,
=∑
=
zz
P
p
Pp
1
1
, and the average number of modes of the individual
polyhedra,
=∑
=
n
n
P
p
Pp
1
1
, where P is the number of polyhedra in the
unit cell and zp and np are the number of extruded faces and modes of
the pth polyhedron, respectively (Supplementary Fig. 8). The results
for the 28 architected materials reported in Fig. 3 show three key
features. First, higher values for
z
lead to rigid materials (that is, n
dof
= 0
for
z
> 8). Second, if all the constituent extruded polyhedra are rigid
(that is,
=n0
), the resulting architected material is rigid as well. Third,
only 13 of the 28 designs are reconfigurable (that is, ndof > 0).
Interestingly, we find that all of the 13 reconfigurable structures are
based on unit cells comprising only prisms, such that they recover the
relation previously demonstrated for extruded individual prisms,
=−n z 3
(ref. 31). Moreover, our results indicate that most of the
reconfigurable structures are characterized by fewer degrees of freedom
than the constituent individual polyhedra (that is,
<n n
dof
), with the
exception of the architected materials based on the cube (number 22)
and the triangular prism (number 11) for which
=n n
dof
.
Having determined the number of modes for the 28 architected
materials, we next characterize the macroscopic deformation associated
to each of them. More specifically, we determine the macroscopic
volumetric strain
δ
=∑ =ε
j
j
1
3
for each mode, where ε
j
are the macro-
scopic principal strains (see the ‘Mode analysis for 3D prismatic archi-
tected materials with rigid faces’ subsection of Supplementary
Information). Interestingly, we find that for the 13 reconfigurable
architectures all modes are characterized by δ = 0, which indicates pure
macroscopic shearing deformation, as also confirmed by visual
inspection of the modes (Supplementary Fig. 9).
To characterize the reconfigurability of prismatic architected
materials, so far we had assumed the faces to be completely rigid and
abc
de
fg
Rigid
Mode 1 Mode
2M
ode 1
Figure 2 | Deformation modes of 3D prismatic architected materials.
a–c, Prototypes of the 3D prismatic architected materials shown in
Fig. 1 were constructed using cardboard (rigid faces) and double-sided
tape (flexible hinges). d, The structure based on a combination of
triangular and hexagonal prisms can be reconfigured in two different
ways (that is, has two degrees of freedom). e, The structure based
on triangular prisms has a single deformation mode. Note that the
architected material based on the octahedra and cuboctahedra cannot
be reconfigured. f, g, Simulated modes of the reconfigurable architected
materials. The obtained deformation modes were linearly scaled to match
the experiments (scale bar in a, 10 cm).
0
1
2
3
123456 7
8910 11 12 13 14
15 16 17 18 19 20 21
23 24 25 26 27 28
22
7
6
5
4
3
2
1
00246810 12 14
z
n
ndof n=z – 3
Figure 3 | Number of degrees of freedom for architected materials based
on the 28 uniform tessellations of the 3D space. The mobility of the
structures is affected by the average connectivity,
z
, and the average
mobility,
n
. Overlapping data were separated for clarity; the small black
lines indicate the original position of the data in each cluster. The
prismatic architected materials and their deformation modes are shown in
Supplementary Fig. 9.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
350 | NATURE | VOL 541 | 19 JA NUARY 2017
ARTICLE
RESEARCH
the hinges to act as linear torsional springs. However, fabrication will
always result in deformable faces, raising the question of whether
prismatic architected materials can be reconfigured when their faces
are deformable. To explore this direction, we updated our numerical
algorithm by introducing a set of springs to account for the deformabil-
ity of the faces
12,13,21
(see the ‘Stiffness of 3D prismatic architected mate-
rials with deformable faces’ subsection of Supplementary Information).
We then deformed the extruded unit cells uniaxially and investigated
their macroscopic stiffness for different loading directions (identified
by the two angles γ and θ as shown in Fig. 4).
In Fig. 4 we report the normalized stiffness K/E as a function of
γ and θ for four prismatic architected materials characterized by
t/
L
= 0.01, where E is the Young’s modulus of the material and t is the
thickness of the faces. We find that the response of the architected
material based on template number 28, which was previously qualified
as rigid (that is, n
dof
= 0), is fairly isotropic because its stiffness does not
vary much as a function of the loading direction (that is,
3.1 × 10
−3
≤ K/E ≤ 4.0 × 10
−3
). In contrast, the stiffness of architected
materials for which ndof > 0 drops noticeably for specific directions
(that is, Kmin/Kmax = O(10−3)). Interestingly, these are the loading
directions for which the reconfiguring modes are activated, as indi-
cated by the deformed structures shown in Fig. 4. Therefore, these
results indicate that the deformation modes we found in the limit of
rigid faces still persist even when the faces are deformable. We used
the same stiffness for bending of the faces and bending of the hinges,
and from the results we can therefore conclude that the architecture
of these systems makes bending of the faces energetically costly
(because it is typically accompanied by stretching and shearing of the
faces). Finally, materials characterized by higher n
dof
are characterized
by more ‘soft’ deformation modes. As such, materials with ndof = 1
seem most promising for the design of reconfigurable architected
materials, since they can be reconfigured along a specific direction,
while still being able to carry loads in all other directions (Fig. 4 and
Supplementary Fig. 10).
Enhancing the reconfigurability
Although we have shown that by extruding the edges of expanded
assemblies of polyhedra we can construct reconfigurable architected
materials, our results indicate that the mobility of the resulting
structures is strongly reduced by their connectivity. Furthermore,
the modes of all reconfigurable designs show a qualitatively similar
shearing deformation. To overcome these limitations, we next introduce
an additional step in the design strategy and reduce the connectivity
of the materials by extruding some of the faces of the unit cell, while
making the remaining faces rigid.
As an example, in Fig. 5 we consider the architected material based
on a tessellation of truncated octahedra (number 28). When all faces
are extruded,
=z14
, leaving the structure rigid (that is, ndof = 0).
However, by making 8 of the 14 faces rigid instead of extruding them
(Fig. 5a and Supplementary Video 3) we can reduce the connectivity
to
=z6
and the resulting architected material is no longer rigid, because
n
dof
= 1. As shown in Fig. 5b and Supplementary Video 3, this response
was also confirmed experimentally. Finally, we note that by varying the
face pairs in the unit cell that are made rigid instead of extruded, a total
of 2
F
= 128 different architected materials can be designed using the
truncated octahedra as a template. However, only 82 combinations are
possible (as all the other cases will result in structures with discon-
nected parts) and of those designs only four are reconfigurable. Owing
to symmetries in the truncated octahedron, these four configurations
are identical to the one shown in Fig. 5.
Next, to determine the range of deformations that can be achieved
in the proposed structures, we apply the same brute force strategy
to the other 27 uniform space-filling tessellations depicted in
Fig. 3. For this study we considered a maximum of 216 designs per
tessellation, randomly selected from the 2F possibilities, so that for 11
of the tessellations (numbers 4, 5, 9, 10, 16, 17, 20, 21, 23, 25 and 27)
the results are not complete, but rather indicate a trend. We expanded
the number of possible designs by removing the polyhedra for which all
faces have been made rigid from the extruded unit cell, because those
would have resulted in rigid parts completely disconnected from the
architected materials.
Of the approximately 0.6 × 106 connected designs investigated here
(Supplementary Table 1), 90% are rigid (that is, n
dof
= 0) while the other
10% are reconfigurable (that is, ndof > 0). Supplementary Fig. 11a, b
shows that to achieve reconfigurability we still need
≤z8
, with the
exception of six designs based on number 5 for which
=z9
(see Supplementary Fig. 12). Moreover, fully extruded architected
materials characterized by
=n0
always remain rigid, independent of
the reduced number of connections. Finally, and perhaps more
importantly, we also find that using this design approach the mobility
of the architected materials can be greatly enhanced, as 0 ≤ n
dof
≤ 16
and for many of the structures
>n n
dof
(Supplementary Table 1).
Inspection of the modes also reveals that a variety of qualitatively
different types of deformation can be achieved. To characterize them
better, in Fig. 6 and Supplementary Fig. 11c–f we report the magnitude
Loading axis
min max
a dcb
min max min max min max
22 26 12 28
00Tπ
J
π
00Tπ
J
π
00Tπ
J
π
00Tπ
J
πK/E
5 × 10–3
y
z
x
0
T
J
K/E = 1.2
×
10–6 4.9
×
10–3 4.6
×
10–6 5.5
×
10–3 1.0
×
10–6 4.4
×
10–3 3.1
×
10–3 4.0
×
10–3
Figure 4 | Normalized stiffness K/E of prismatic architected materials.
a–d, The results for architected materials based on template numbers
22 (a), 26 (b), 12 (c) and 28 (d). To determine the stiffness in all loading
directions, the architected materials are rotated by angles γ and θ before
loading. In each contour plot we indicate the minimum and maximum
stiffness with white and black squares, respectively. We also show
the deformed architected materials for the minimum and maximum
stiffness direction. Note that the deformation is magnified to facilitate
visualization.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
19 JANUARY 2017 | VOL 541 | NATUR E | 351
ARTICLE RESEARCH
of the principal strains,
=∑
=
εε
ii
1
32
, versus the volumetric strain,
δ, for each deformation mode observed in the reconfigurable archi-
tected materials investigated here. Interestingly, we find that for many
modes
δ==ε0
. These modes do not alter the global shape of the
structure, but result only in internal rearrangements. The design
labelled a, shown in Fig. 6, is an example of a structure undergoing such
a local deformation. Here, most of the structure is rigid except for
one-dimensional tubes that can deform independently. In contrast, the
design labelled b is an example in which the whole internal structure
is deforming, while maintaining the same macroscopic shape
(Supplementary Video 4).
Besides these local modes, Fig. 6 also indicates that there are designs
capable of achieving types of macroscopic deformation that differ from
pure shear (for which δ = 0 and
>ε0
). For example, we find that
some of the structures are characterized by an effective vanishing strain
in two directions (labelled c in Fig. 6). The deformation of such
architected materials is characterized by ε
1
≠ 0 and ε
2
= ε
3
≈ 0, resulting
in
δ=ε
. Moreover, the results also reveal that there are a variety of
structures capable of uniform biaxial expansion (or contraction), for
which ε
2
= ε
3
≠ 0 and ε
1
= 0 and
δ=/ε2
. This deformation mode
is exemplified by the design labelled d shown in Fig. 6 (Supplementary
Video 4). Finally, we note that
δ=ε3
corresponds to uniform
expansion (or contraction) characterized by ε1 = ε2 = ε3, and defines a
boundary for possible combinations of δ and
ε
. In fact, none of the
designs considered here exhibits this type of deformation.
Discussion and conclusion
In this work we introduced a convenient and robust strategy for the
design of reconfigurable architected materials, and explored the design
space by considering structures based on the 28 uniform space-filling
tessellations of polyhedra. Our study uncovered architected materials
with a wide range of qualitatively different responses and degrees of
freedom, but many more designs are made possible by using different
assemblies of convex polyhedra as templates (including assemblies
based on Johnson solids and irregular polyhedra, and assemblies that
do not fill space), by considering different extrusion lengths, or by
removing faces (instead of making them rigid before the extrusion
step). Given these additional possibilities in the design of recon-
figurable architected materials, we have made our numerical algorithm,
implemented in Matlab, available for download as Supplementary
Information, to be used and expanded upon by the community. Finally,
we believe that, building on the results presented in this work, prismatic
Extrude
Make rigid
Adapt
unit cell
a
b
Recongure the architected material along the only degree of freedom
= 1
= 6
= 1
n
z
ndof
= 5
= 14
= 0
n
z
ndof
Figure 5 | Enhancing the reconfigurability of 3D prismatic architected
materials. a, To enhance the reconfigurability of the architected material
based on the space-filling assembly of truncated octahedra (number 28 in
Fig. 3), we extrude only six of its faces and make the remaining eight faces
rigid. Using this approach, the average connectivity is reduced from
=z14
to
=z6
and the resulting structure is no longer rigid, because ndof = 1.
b, Experimental validation of the numerical predictions (scale bar, 10cm).
b
a
d
c
d
a,b
c
0.5
Fig. 5
0.5
1234567
G
1
2
3
4
5
6
7
¬
¬¬¬
= 1
n
dof
1 = –2
3 = 0
1 = 0
2 = 0
3 = 0
1 = 0
2 = 0
3 = 2
1 = 2 = 3
00G
¬¬¬¬
Figure 6 | Deformation modes of 3D prismatic architected materials
with enhanced reconfigurability. Relation between the volumetric strain,
δ, and the magnitude of the principal strains,
ε
, for all the architected
materials characterized by ndof = 1. The colour of the markers refers to the
uniform tessellation that has been used as a template, as shown in Fig. 3.
Structures a–d and the one in Fig. 5 are indicated by grey circles on the
main panel. The solid and dashed lines and associated schematics and
conditions on ε1, ε2 and ε3 highlight how different choices of strains lead to
different types of deformation (see text). Structures labelled a and b
(based on tessellations 24 and 9, respectively) are characterized by
δ==ε0
and experience internal rearrangements that do not alter their
macroscopic shape. The structure labelled c (based on tessellation 16)
deforms only in one direction (that is, δ = 4.21,
=.ε476
), while the
structure labelled d (based on tessellation 14) experiences uniform biaxial
extension (or contraction) (that is, δ = 2.45,
=.ε173
). The grey shaded
region corresponds to combinations of strains that do not permit
deformation.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
352 | NATURE | VOL 541 | 19 JA NUARY 2017
ARTICLE
RESEARCH
architected materials with specific properties may be efficiently
identified by combining our numerical algorithm with stochastic
optimization algorithms such as genetic algorithms. Such optimization
algorithms could prove essential in the design of reconfigurable
architected materials capable of handling changing environments or
multiple tasks (that will probably lead to pareto optimal solutions).
To realize prismatic architected materials, in this study we used
cardboard for the rigid faces and double-sided tape for the hinges. This
fabrication process enables the realization of centimetre-scale
prototypes (for our models we used
L
= 35 mm) that closely match the
conceptual origami-inspired mechanisms, but real-world applications
depend on the ability to efficiently manufacture assemblies comprising
a large number of unit cells at different length scales using different
fabrication techniques. Taking advantage of recent developments in
multi-material additive manufacturing, we also built the proposed
architected materials using a stiff material (with Young’s modulus
E ≈ 1 GPa) for the faces and a soft material (E ≈ 1 MPa) for the hinges
(see the ‘Methods’ section of Supplementary Information).
Supplementary Video 5 shows 3D printed models for two designs
based on assemblies of truncated octahedra (for both models we used
L
= 6 mm). Although additional local deformation arises from the
finite size of the flexible hinges, the 3D printed structures exhibit the
same deformation modes predicted by our numerical analysis and
observed in the cardboard prototypes. As such, recent advances in
fabrication, including projection micro-stereolithography7, two-photon
lithography8,32,33 and ‘pop-up’ strategies34–40, open up exciting
opportunities for miniaturization of the proposed architectures. Our
strategy thus enables the design of a new class of reconfigurable systems
across a wide range of length scales.
Data availability The Matlab model used to determine the mobility and
deformation modes of the prismatic architected materials is provided
in Supplementary Information. Other models and datasets generated
during and/or analysed during the current study are available from the
corresponding author on request.
Received 24 May; accepted 22 November 2016.
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Supplementary Information is available in the online version of the paper.
Acknowledgements This work was supported by the Materials Research
Science and Engineering Center under NSF Award number DMR-1420570.
K.B. also acknowledges support from the National Science Foundation
(CMMI-1149456-CAREER). We thank M. Mixe and S. Shuham for assistance
in the fabrication of the cardboard prototypes, and R. Wood for the use of his
laboratory.
Author Contributions J.T.B.O., C.H. and K.B. proposed and designed the
research; J.T.B.O. performed the numerical calculations; J.T.B.O., C.H. and
J.C.W. designed and fabricated the models; J.T.B.O. performed the experiments;
J.T.B.O. and K.B. wrote the paper.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial
interests. Readers are welcome to comment on the online version of the paper.
Correspondence and requests for materials should be addressed to
K.B. (bertoldi@seas.harvard.edu).
Reviewer Information Nature thanks J. Paik, D. Pasini and the other anonymous
reviewer(s) for their contribution to the peer review of this work.
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
In the following, we first describe the fabrication approaches used to make cardboard prototypes and 3D printed
prototypes. We next describe the numerical algorithms we implemented in Matlab (i) to predict the number of
degrees of freedom and corresponding deformation modes of 3D prismatic architected materials with rigid faces; and
(ii) to characterize the elastic response of 3D prismatic architected materials with deformable faces. Moreover, this
document contains the supplemental figures and table.
METHODS
Fabrication of Cardboard Prototypes
Our cardboard prototypes were fabricated from two layers of cardboard with a thickness of 0.7 mm (13001-2506,
Blick) and one layer of double-sided tape with a thickness of 0.07 mm (23205-1009, Blick), using a stepwise layering and
laser cutting technique on a CO2 laser system (VLS 2.3, Universal Laser Systems). To fabricate each of the extruded
polygons that together form the architected material, we started by cutting one of the cardboard sheets, after which
we removed it from the laser system (steps 1-2 as shown in Supplementary Fig. 1). Using a different pattern, cutting
slits were introduced in the second cardboard sheet (steps 3-5). The double-sided tape was bonded to the second
cardboard sheet still in the laser system, to which the initially cut cardboard sheet was attached (steps 6-7). A third
cutting step was then performed to finalize the different components (step 8), which were completely separated from
the main sheet (step 9-10). The individual components were assembled to form the extruded architected material
using the tape that was exposed during the cutting process (steps 11-15).
Fabrication of 3D Printed Prototypes
We furthermore manufactured prismatic architected materials using multi-material additive manufacturing (Con-
nex500, Stratasys). We used a rigid material (VeroWhitePlus RGD835, Young’s modulus E ≈ 1 GPa) for the faces
and a softer (TangoPlus FLX930, E ≈ 1 MPa) for the hinges. The final structure was designed using a custom made
Matlab script, which was based on extruded polyhedra with an edge length of L0 = 6 mm and an average extrusion
length of L¯ = 6 mm. Both the faces and hinges were given a thickness of t = 1 mm. Moreover, the size of the faces
was reduced by 0.5 + 1.5|θ|/π mm on each side to account for the finite size of the rounded hinges, θ being the initial
angle of the hinges (Supplementary Fig. 2).
MODE ANALYSIS FOR 3D PRISMATIC ARCHITECTED MATERIALS WITH RIGID FACES
Here, we describe the algorithm that we implemented to predict the number of degrees of freedom and corresponding
deformation modes of 3D prismatic architected materials. Focusing on an extruded unit cell comprising 2F rigid faces
(F face pairs) connected by H hinges (i.e. torsional springs), we first determine the elastic and kinetic energy required
to deform the structure. Then, we describe the constraints that we impose to ensure that the faces remain rigid and
the unit cell deforms in a periodically repeated manner (i.e. we model the response of an infinitely large structure
without considering boundary effects), after which we describe the eigenfrequency problem that we solve to find
the characteristic deformation modes of the architected materials. Finally, we discuss the assumptions made in the
numerical model.
Energy
Elastic Energy
Assuming that each hinge acts as a linear torsional spring of stiffness Kh, that no energy is required to maintain
the hinges in their initial configuration and that the faces are rigid, the total elastic energy of the unit cell, Eelastic,
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Supplementary Fig. 1: Fabrication of cardboard prototypes. Steps 1-10 show the cutting proces to fabricate flat three layer
composites of cardboard and double-sided tape, and steps 11-16 depict the assembly process using the fabricated fabricated
pieces.
is given by
Eelastic =Ehinge =
H
X
i=1
1
2Kh
idθ2
i=1
2dθTKhdθ,(1)
where dθidenotes the change in angle for the i-th hinge, dθ= [dθ1, dθ2, . . . , dθH]T, and Kh= diag(Kh
1, Kh
2, ...., Kh
H).
Note that dθican be expressed in terms of the displacement of the Vvertices (corner points of the faces) as
dθi=
V
X
v=1 ∂θi
∂x1,v
dx1,v +∂θi
∂x2,v
dx2,v +∂θi
∂x3,v
dx3,v!,(2)
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Supplementary Fig. 2: Fabrication of 3D printed prototypes. (a) Multi-material design based on a polyhedron template. Here,
L0depicts the edge size, θthe angle between faces, and tthe thickness of the faces. (b) 3D printed prototype using stiff
materials for the faces, and soft material for the hinges. (scale bar 6 mm)
where dx1,v,dx2,v and dx3,v denote the displacement components of the v-th vertex with initial coordinates x1,v,x2,v
and x3,v. Substitution of Eq. (2) into Eq. (1) yields
Ehinge =1
2uTJT
hKhJhu,(3)
in which Jhis the Jacobian matrix with entries
Jh[i, 3(v−1)+j]=∂θi
∂xj,v
, j = 1,2,3 and v= 1, ..., V, (4)
and u= [dx1,1, dx2,1, dx3,1, . . . , dx1,V , dx2,V , dx3,V ]T.
Since the angle of the i-th hinge, which connects two faces with unit normals naand nband rotates around the
axis pointing in the direction ah, is given by
θi= tan−1ah·(na×nb)
na·nb,(5)
it follows that
∂θi
∂xj,v
=
∂y
∂xj,v z−y∂z
∂xj,v
y2+z2,(6)
in which we have used y=ah·(na×nb) and z=na·nb. The derivatives can then be found according to
∂y
∂xj,v
=∂ah
∂xj,v
·(na×nb) + ah· ∂na
∂xj,v
×nb+na×∂nb
∂xj,v !,(7)
and
∂z
∂xj,v
=∂na
∂xj,v
·nb+na·∂nb
∂xj,v
(8)
Finally, since the unit normal to any of the faces, n, can be calculated as
n=v1×v2
q|v1|2|v2|2−(v1·v2)2,(9)
where v1and v2are two non-parallel vectors lying on the face (Supplementary Fig. 3), the derivatives of nin Eqs.
(7) and (8) are given by
∂n
∂xj,v
=∂v1
∂xj,v ×v2+v1×∂v2
∂xj,v c−(v1×v2)∂c
∂xj,v
c2,(10)
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Supplementary Fig. 3: Schematic of the extruded unit cell.
in which
c=q|v1|2|v2|2−(v1·v2)2,(11)
and
∂c
∂xj,v
=1
c
∂v1
∂xj,v
·v1!|v2|2+|v1|2 ∂v2
∂xj,v
·v2!−(v1·v2) ∂v1
∂xj,v
·v2+v1·∂v2
∂xj,v !
.(12)
Note that ∂n/∂xj,v = 0 for vertices not belonging to the face with normal n.
Kinetic Energy
Next, we determine the kinetic energy, Ekinetic , associated with the displacements of the vertices of the unit cell
Ekinetic =1
2
V
X
v=1
Mv∂x1,v
∂t +∂x2,v
∂t +∂x3,v
∂t 2
=1
2˙uTM ˙u,(13)
where ˙u =∂u/∂t,Mvis the mass assigned to the v-th vertex and Mis the 3V×3Vdiagonal mass matrix
diag(M1, M1, M1, . . . , MV, MV, MV). Note that each face to which the v-th vertex belongs contributes a mass M/N
to the vertex, where Mis the mass of the face (which we take equal to the area by assuming a unit thickness and
density), and Nis the number of vertices of the face.
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Constraints
Rigidity of the Faces
To ensure that all the faces are rigid and do not deform, we triangulate them (Supplementary Fig. 3) and impose
that the length of each edge of the triangulation remains constant,
(xa−xb)·(xa−xb) = L2,(14)
where xaand xbare the two vertices connected by the edge, which has initial length L. We then linearize Eq. (14)
to obtain an expressions for each constraint that depends explicitly on the displacements of the two vertices
(xa−xb)·(ua−ub)=0,(15)
in which ua= [dx1,a, dx2,a, dx3,a] and ub= [dx1,b, dx2,b, dx3,b].
Furthermore, we also ensure that all the faces remain flat (i.e. each face can undergo rigid body translation and
rotations, but cannot bend). To this end, we impose that all vertices of each face remain on the same plane spanned
by the two vectors w1and w2(Supplementary Fig. 3) [41],
wi·(w1×w2) = 0,for i= 3, ..., Vf−1,(16)
in which Vfis the number of vertices of the face. Note that this constraint is automatically satisfied for faces that
only connect three vertices. We again linearize the constraints to obtain
u1·(wi×w2) + u2·(w2×wi) + u3·(wi×w1) + ui·(w1×w2) = 0,for i= 3, ..., Vf−1.(17)
Finally, we note that the constraints of Eqs. (15) and (17) are only valid for small displacements, since the constraints
are linearized around the initial coordinates of the vertices.
Periodic Boundary Conditions
For the infinitely large periodic prismatic architected materials considered here, it is sufficient to focus on a unit
cell that consists of a few extruded polyhedra and covers the entire assembly when translated by the three lattice
vectors li(i= 1,2,3). To ensure that the extruded unit cell deforms in a periodically repeated manner we constrain
the deformation of each periodically located vertex pair on its boundary as
ub−ua=
3
X
i=1
αidli,(18)
where uaand ubare the displacements of the two periodically located vertices, dlidenotes the deformation of the
lattice vectors, and
xb−xa=
3
X
i=1
αili,(19)
with αi∈ {−1,0,1}. In our implementation we treat dlias additional degrees of freedom, which we include in Eqs. (3)
and (13) as
Eelastic =1
2u
dlT
Jh0TKhJh0u
dl,(20)
and
Ekinetic =1
2"˙
u
d˙
l#TM 0
0 0"˙
u
d˙
l#,(21)
in which dl= [dlT
1, dlT
2, dlT
3]Tand d˙
l=∂dl/dt.
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Master-slave Elimination
To enforce the constraints given by Eqs. (15), (17) and (18), we adopt the master-slave elimination method [42].
We start by rewriting all the constraints in matrix form as
Au
dl=0,(22)
where each row of Arepresents one constraint. Next, we rewrite Ain its reduced row echelon form, Arref. The
dependent constraints correspond to rows of all zeros in Arref and are therefore automatically satisfied. Moreover, all
the columns of Arref with a single entry correspond to the slave degree of freedom, ds, while the remaining degrees of
freedom are referred to as the master degree of freedom, dm. We then rewrite Eq. (22) as
I Brrefds
dm=0,(23)
where Iis the identity matrix and Brref comprise the columns of Arref that correspond to the master degrees of
freedom. It follows from Eq. (23) that
ds=−Brrefdm,(24)
so that
ds
dm=−Brref
Idm.(25)
Finally, since the vectors [dT
s,dT
m]Tand [uT, dlT]Tcontain exactly the same degrees of freedom arranged in a different
order, we rearrange the rows of the matrix [Brref,I]Tin Eq. (25) to obtain
u
dl=Tdm,(26)
where Tis a transformation matrix.
Using Eq. (26), the elastic and kinetic energies from Eqs. (20) and (21) can be rewritten as
Eelastic =dT
mTTJh0TKhJh0Tdm,(27)
and
Ekinetic =˙
dT
mTTM 0
0 0T˙
dm.(28)
Mode Analysis
The equations of motion for the extruded unit cell are derived using Lagrange’s equations
∂
∂t ∂Epotential
∂˙
u−∂Epotential
∂u=0,(29)
where Epotential =Eelastic −Ekinetic. Substitution of Eqs. (27) and (28) into Eq. (29) yields
TTM 0
0 0T¨
dm−TTJh0TKhJh0Tdm=0,(30)
in which we assumed that T,M,Jhand Khdo not depend on the displacement and do not change in time. Next,
we assume the solution to have the form
dm=amsin(ωt +β),(31)
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and substitute into Eq. (29) to obtain the eigenproblem
˜
M−1˜
Kam=ω2am,(32)
in which
˜
M= TTM 0
0 0T!,(33)
˜
K=TTJh0TKhJh0T.(34)
Moreover, ωis an eigenfrequency of the system and amis the corresponding mode. Finally, the displacements of all
the vertices associated to each mode are obtained from Eq. (26) as
u
dl=Tamsin(ωt +β).(35)
Characterizing the deformation modes
To characterize the macroscopic deformation associated to each prismatic material, we determine the macroscopic
infinitesimal strain tensor for each of the computed modes as
¯
e=1
2H+HT,(36)
where His the macroscopic displacement gradient, which can be determined from the infinitesimal deformation of
the three lattice vectors, dli, by solving the following set of equations
dli=Hli,for i= 1,2,3.(37)
Note that we normalized dliby the maximum change in angle between connected faces. To characterize the type of
macroscopic deformation associated to each mode we introduce the volumetric strain
δ=
3
X
j=1
j,(38)
where jare the principal strains, which can be determined by solving ¯
e−jIvj=0,vjbeing the principal
directions.
Discussion
While our numerical analysis proved essential in the exploration of the design space for prismatic architected
materials, it is important to note that the algorithm is only valid for small rotations. However, we found that the
numerical results still provide valuable insights into the large deformations typically experienced by the structures,
as demonstrated by the excellent agreement with the experiments. Moreover, in our model we assumed that the
deformation of the architected material can be fully captured by an extruded unit cell to which periodic boundary
conditions are applied, thus neglecting boundary effects. Although in our experiments we observed additional modes
that arise from the reduced connectivity of the unit cells near the boundaries (Supplementary Fig. 4), we do not
expect them to significantly influence the behavior of the bulk material as they are confined to the outer surfaces.
These assumptions significantly reduced computation time and also removed the need for applying specific boundary
conditions, allowing us to model and compare many different systems.
STIFFNESS OF 3D PRISMATIC ARCHITECTED MATERIALS WITH DEFORMABLE FACES
While in the analysis used to characterize the reconfigurability of the structures we assumed the faces to be rigid,
we now account also for their deformability by introducing a set of springs. More specifically, for each rectangular
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Supplementary Fig. 4: Deformation modes of a finite-size prototype of the architected material based on the space-filling
assembly of hexagonal prisms (#26). Besides the two bulk modes predicted by our numerical simulations, we also observe 6
boundary modes (of which 2 are shown) that arise from the reduced connectivity of the unit cells near the boundaries (scale
bar 5 cm).
face we used four linear springs placed along the perimeter to capture its stretching, two linear springs placed along
the diagonal to capture its shearing, and a linear torsional spring placed along an arbitrary diagonal to capture its
bending [43–45] (Supplementary Fig. 5). Therefore, the elastic energy required to deform an extruded unit cell is
given by
Eelastic =Ehinge +Estretch
face +Eshear
face +Ebend
face ,(39)
where Ehinge is the elastic energy as defined in in Eq. (1), and Estretch
face ,Eshear
face and Ebend
face denote the contribution
to the elastic energy of the unit cell due to stretching, shearing and bending of the faces. Focusing on unit cells
comprising 2Ffaces, the energy required to stretch the faces can be determined from the extension of their edges as
Estretch
face =
8F
X
i=1
1
2Kst
i(dest
i)2=1
2deT
stKst dest,(40)
where Kst
iand dest
idenote the stiffness and change in length of the i-th edge, dest = [dest
1, dest
2, . . . , dest
8F]T, and
Kst = diag(Kst
1, Kst
2, ...., Kst
8F). Following the same approach used for Ehinge (Eqs. (2)-(4)), we rewrite Estretch
face in
terms of the displacement of the Vvertices. We first note that
dest
i=
V
X
v=1 ∂est
i
∂x1,v
dx1,v +∂est
i
∂x2,v
dx2,v +∂est
i
∂x3,v
dx3,v!.(41)
Next, we substitute Eq. (41) into Eq. (40) and obtain
Estretch
face =1
2uTJT
stKst Jstu,(42)
in which Jst is the compatibility matrix with entries
Jst [i, 3(v−1)+j]=∂est
i
∂xj,v
, j = 1,2,3 and i= 1, ..., 8F. (43)
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Supplementary Fig. 5: Schematic of the extruded unit cell indicating elements used to model the deformation of the faces.
Since shearing of faces is also modeled using springs (placed along the diagonals of the face), following the same
procedure used for Estretch
face (Eqs. (40)-(43)), Eshear
face can be determined as
Eshear
face =1
2uTJT
shKsh Jshu,(44)
where Ksh = diag(Ksh
1, Ksh
2, ...., Ksh
4F) (Ksh
idenoting the stiffness of the i-th diagonal spring introduced to capture
shearing) and
Jsh [i, 3(v−1)+j]=∂esh
i
∂xj,v
, j = 1,2,3 and i= 1, ..., 4F. (45)
Finally, the energy associated to bending of the faces, Ebend
face , can be determined following the procedure used to
determine Ehinge (Eq. (1)-(12)), yielding
Ebend
face =1
2uTJT
bKbJbu.(46)
where Kb= diag(Kb
1, Kb
2, ...., Kb
2F) (Kb
idenoting the stiffness of the i-th torsional spring placed on the diagonal to
capture shearing) and
Jb [i, 3(v−1)+j]=∂ψi
∂xj,v
, j = 1,2,3 and i= 1, ..., 2F, (47)
ψbeing the angle between the two triangulated faces separated by the diagonal on which the torsional spring is placed
(Supplementary Fig. 5).
Spring Stiffnesses
We assume that the faces are made from a material with Young’s modulus Eand Poisson’s ratio ν= 1/3, and their
thickness tis chosen so that t/¯
L= 0.01 (¯
Lbeing the average extrusion length). For such systems the stiffnesses of
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the springs introduced in our model can be determined as [45]
Kst =Et
2L2
L2−νL2
⊥
1−ν2,(48)
Ksh =Et
2LLHLW
ν(L2
H+L2
W)3/2
1−ν2,(49)
Kb=Cb
Et3
12(1 −ν2)L
t1/3
(50)
in which Lis the length of the edge on which the spring is placed, L⊥is the length of the edge perpendicular to that
on which the spring is placed, LWand LHare the width and the height of the face, and Cb= 0.441. Moreover, while
in our experiments we used hinges that can bend more easily than the faces, here we consider the extreme case for
which the bendability of the hinges is similar to that of the faces and use
Kh=Cb
ELt3
24(1 −ν2)1
t1/3
.(51)
Periodic boundary conditions
Next, we apply periodic boundary conditions to the unit cell and express them in terms of the macroscopic dis-
placement gradient H. To ensure that the extruded unit cell deforms in a periodically repeated manner under applied
loading we update Eq. (18) as
ub−ua=
3
X
i=1
αiHli.(52)
Furthermore, we also introduce three fictitious nodes, (v1,v2,v3), to conveniently apply Hto the unit cell [46]. The
displacement components of three fictitious nodes are assigned to be the components of H. Virtual work is then used
to determine the macroscopic stress tensor as
sij =1
V0
rvi
j(53)
where V0is the initial volume occupied by the extruded unit cell and rviis the “reaction force” corresponding to the
assigned “displacement components” of the fictitious nodes vi.
The periodic boundary conditions specified by Eq. (52) are then enforced using the master-slave elimination method.
Following the procedure detailed in Eqs. (22)-(28) for Ehinge , we obtain
Estretch
face =dT
mTTJst 0TKst Jst 0Tdm,(54)
Eshear
face =dT
mTTJsh 0TKsh Jsh 0Tdm,(55)
Ebend
face =dT
mTTJb0TKbJb0Tdm,(56)
where
u
h=Tdm,(57)
and h= [H11, H 12, H 13, H21 , . . . , H 33]T.
Deformation Under Uniaxial Loading
Finally, assuming that the deformation is applied quasi-statically, the equilibrium equations for the extruded unit
cell can be obtained as
∂Epotential
∂u= 0,(58)
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where Epotential =Eelastic −W,W=hTrbeing the external work [46], for which we have defined r=
(rv1)T,(rv2)T,(rv3)TT. It follows from Eqs. (54)-(58) that
TTJh0TKhJh0+Jst 0TKst Jst 0+Jsh 0TKsh Jsh 0+Jb0TKbJb0Tdm=TT0
r.
(59)
Having determined the equilibrium equations for an extruded unit cell, we apply a uniaxial loading to the system
by imposing H11 6= 0 and Hij = 0 for i6=j, while leaving H22 and H33 unset (i.e. allowing the structure to freely
expand in the lateral directions, while constraining macroscopic shear deformations). Note that we also constrained
rigid body translations by fixing the displacement of a single vertex of the unit cell. To determine the response of the
architected material along all directions, we rotate the unit cell about two axis according to
x0=RzRy0x,(60)
in which
Rz=
cos γ−sin γ0
sin γcos γ0
0 0 1
,(61)
Ry0= cos θI+ sin θ
0−y0
3y0
2
y0
30−y0
1
−y0
2y0
10
+ (1 −cos θ)
(y0
1)2y0
1y0
2y0
1y0
3
y0
2y0
1(y0
2)2y0
2y0
3
y0
3y0
1y0
3y0
2(y0
3)2
(62)
with y0=Rzey(see schematic in Supplementary Fig. 10). For each direction, we can determine the stiffness according
to
K=s11/(H11 ).(63)
[41] Lalibert´e, T. & Gosselin, C. Construction, mobility analysis and synthesis of polyhedra with articulated faces. J. Mecha-
nisms Robotics 6, 011007 (2013).
[42] Cook, R. Concepts and applications of finite element analysis (Wiley, 2001).
[43] Wei, Z. Y., Guo, Z. V., Dudte, L., Liang, H. Y. & Mahadevan, L. Geometric mechanics of periodic pleated origami. Phys.
Rev. Lett. 110, 215501 (2013).
[44] Schenk, M. & Guest, S. D. Geometry of miura-folded metamaterials. Proc. Natl. Acad. Sci. U.S.A. 110, 3276-3281 (2013).
[45] Filipov, E. T., Tachi, T. & Paulino, G. H. Origami tubes assembled into stiff, yet reconfigurable structures and metama-
terials. Proc. Natl. Acad. Sci. U.S.A. 112, 12321-12326 (2015).
[46] Danielsson, M., Parks, D. & Boyce, M. Three-dimensional micromechanical modeling of voided polymeric materials. J.
Mech. Phys. Solids 50, 351-379 (2002).
[47] Gru¨nbaum, B. Uniform tilings of 3-space. Geombinatorics 4, 49-56 (1994).
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SUPPLEMENTAL FIGURES
Supplementary Fig. 6: For a unit cell with Fface pairs the expanded configuration is fully determined by choosing Findep ≤F
extrusion lengths. As an example, here we consider four prismatic architected materials based on the space-filling assembly of
hexagonal prisms (for which F= 4 and Findep = 2) and show the effect of the two independent extrusions lengths L1/L0and
L2/L0on the final extruded shape, in which L0denotes the length of the edges of the polyhedra. All four designs considered
here have the same degrees of freedom (ndof = 2).
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Supplementary Fig. 7: Reconfigurability of individual extruded polyhedra. The extruded units based on a (a) tetrahedron, (c)
octahedron, (d) truncated tetrahedron and (f) truncated cube are rigid, while those based on the (b) cube, (e) cuboctahedron,
(g) truncated octahedron, (h) rhombicuboctahedron, (i) truncated cuboctahedron, and (j-m) prisms are reconfigurable. For
reference, we also denoted the polyhedra on which the unit cells are based by their Sch¨afli symbols. Note that only a selected
number of deformation modes is shown, as it is not straightforward to determine all of them experimentally. For all the
prototypes the edges are 35 mm.
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Supplementary Fig. 8: Numerically determined modes of individual extruded polyhedra. The extruded geometries based on
the (a) tetrahedron, (c) octahedron, (d) truncated tetrahedron and (f ) truncated cube are rigid, while those based on the
(b) cube, (e) cuboctahedron, (g) truncated octahedron, (h) rhombicuboctahedron, (i) truncated cuboctahedron, and (j-m)
prisms are reconfigurable. Importantly, using our numerical algorithm we can easily identify the degrees of freedom, n, and the
deformation modes for the extruded units. Note that modes characterized by the same eigenvalue ω2are identical, so that we
only show one of these modes. For reference, we also denoted the polyhedra on which the unit cells are based by their Sch¨afli
symbols.
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Supplementary Fig. 9: Reconfigurability of architected materials based on on the 28 uniform tessellations of the 3D space,
which comprise regular polyhedra, semiregular polyhedra and semiregular prisms. The tessellations used as a template are
indicated with the notation introduced in [47]. Specifically, the individual polyhedra are indicated by their Sch¨afli symbol,
and the superscript shows the number of polyhedra of the given kind that meet at each vertex. Moreover, we also provide the
number of each kind of polyhedra in the unit cell. (#1-2) The architected materials based on (3.3.3)8.(3.3.3.3)6(tetrahedra
and octahedra) are rigid. Note that #2 differs from #1 as it comprises reflected layers of tetrahedra and octahedra. (#3-4)
The assemblies based on (3.3.3)8.(3.3.3.3)3.(3.4.4)6(tetrahedra, octahedra and triangular prisms) are rigid. Note that #4
differs from #3 as it comprises reflected layers of tetrahedra, octahedra and triangular prisms. (#5) (3.3.3)4.(3.4.4.4)3.(4.4.4)
(tetrahedra, rhombicuboctahedra and cubes) is rigid.
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Supplementary Fig. 9: (continued). (#6) (3.3.3)2.(3.6.6)6(tetrahedra and truncated octahedra) is rigid. (#7)
(3.3.3.3)2.(3.4.3.4)4(octahedra and cuboctahedra) is rigid. (#8) (3.3.3.3).(3.8.8)4(octaheda and truncated octahedra) is
rigid. (#9) (3.4.3.4).(3.4.4.4)2.(4.4.4)2(cuboctahedra, rhombicuboctahedra and cubes) is rigid. (#10) (3.4.3.4).(3.6.6)2.(4.6.6)2
(cuboctahedra, truncated tetrahedra and truncated octahedra) is rigid.
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Supplementary Fig. 9: (continued). (#11-12) (3.4.4)12 (triangular prisms) are reconfigurable with ndof = 2 and ndof = 1,
respectively. Note that #12 differs from #11 as it comprises reflected layers of triangular prisms. (#13-15) (3.4.4)6.(4.4.4)4
(triangular prisms and cubes) are reconfigurable with ndof = 2, ndof = 2 and ndof = 1, respectively. Note that #14 differs from
#13 as the polyhedra are differently arranged in-plane, and #15 differs from #13 as it comprises reflected layers of triangular
prisms and cubes.
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Supplementary Fig. 9: (continued). (#16) (3.4.4)2.(4.4.4)4.(4.4.6)2(triangular prisms, cubes and hexagonal prisms) is reconfig-
urable with ndof = 2. (#17) (3.4.4)8.(4.4.6)2(triangular prisms and hexagonal prisms) is reconfigurable with ndof = 2. (#18)
(3.4.4)4.(4.4.6)4(triangular prisms and hexagonal prisms) is reconfigurable with ndof = 2. (#19) (3.4.4)2.(4.4.12)4(triangular
prisms and dodecagonal prisms) is reconfigurable with ndof = 2. (#20) (3.4.4.4).(3.8.8).(4.4.4).(4.4.8)2(rhombicuboctahedra,
truncated cubes, cubes and octagonal prisms) is rigid.
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Supplementary Fig. 9: (continued). (#21) (3.6.6).(3.8.8).(4.6.8)2(truncated tetrahedra, truncated cubes and truncated cuboc-
tahedra) is rigid. (#22) (4.4.4)8(cubes) is reconfigurable with ndof = 3. (#23) (4.4.4)2.(4.4.6)2.(4.4.12)2(cubes, hexagonal
prisms and dodecagonal prisms) is reconfigurable with ndof = 2. (#24) (4.4.4)2.(4.4.8)4(cubes and hexagonal prisms) is
reconfigurable with ndof = 2. (#25) (4.4.4).(4.6.6).(4.6.8)2(cubes, truncated octahedra and truncated cuboctahedra) is rigid.
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Supplementary Fig. 9: (continued). (#26) (4.4.6)6(hexagonal prisms) is reconfigurable with ndof = 2. (#27) (4.4.8)2.(4.6.8)2
(octagonal prisms and truncated cuboctahedra) is rigid. (#28) (4.6.6)4(truncated octahedra) is rigid.
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Supplementary Fig. 10: Normalized stiffness K/E of the 28 architected materials based on the uniform space-filling polyhedra
assemblies. To determine the stiffness in all loading directions, the architected materials are rotated by angles γand θprior to
loading.
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Supplementary Fig. 11: (a-b) Number of degrees of freedom, ndof, for the altered architected materials based on the 28 uniform
space-filling tessellations. Each point represents a design in which some of the faces of the unit cell are made rigid, instead of
extruded. (c-f) Relation between the volumetric strain, δ, and the magnitude of the principal strains, ||||, for all the architected
materials characterized by ndof = 2, 3, 4 and >5, respectively (Note that the results for ndof = 1 are shown in Fig. 6). The
color of the markers refers to the uniform tessellation that has been used as a template, as shown in Fig. 3.
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Supplementary Fig. 12: One of the six reconfigurable prismatic architected materials characterized by ¯z= 9 that we found
using the numerical algorithm. This specific architected material is based on #5, for which 18 faces of the polyhedra are
extruded and the remaining 18 faces are made rigid. Note that the faces of both tetrahedra are made fully rigid, and therefore
are not taken into account in the numerical analysis. The resulting architected material has ndof = 2.
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SUPPLEMENTAL TABLES
Supplementary Table 1: To enhance the reconfigurability of the proposed architected materials, we reduce their connectivity
by selectively extruding faces of the unit cell, while making the remaining faces rigid. In this table we summarize the results
obtained for the extruded structures based on the 28 uniform space-filling tessellations depicted in Fig. 3 and Supplementary
Fig. 9. For this study we considered a maximum of 216 designs per tessellation, randomly selected from the 2Fpossibilities
(where Fis the number of face pairs), so that for 11 of the tessellations (#4-5, #9-10, #16-17, #20-21, #23, #25, and #27)
the results are not complete, but rather indicate a trend. When determining the degrees of freedom, ndof, we only consider
the designs that do not contain any disconnected parts. However, we expanded the number of possible designs by removing
the polyhedra for which all faces have been made rigid from the extruded unit cell, as those would have resulted in rigid parts
completely disconnected from the architected materials.
unit cell #designs connected percentage of connected with ndof
ndof = 0 1 2 3 4 5 6 7 >7, < 17
#1 2824.7% 100.0% 0 0 0 0 0 0 0 0
#2 213/ 216 24.8% 100.0% 0 0 0 0 0 0 0 0
#3 213 16.7% 62.5% 34.0% 3.5% 0 0 0 0 0 0
#4 213/ 226 15.3% 62.1% 33.6% 4.3% 0 0 0 0 0 0
#5 213/ 220 91.6% 96.6% 3.2% 0.230% 0 0 0 0 0 0
#6 212 45.2% 100.0% 0 0 0 0 0 0 0 0
#7 211 66.1% 100.0% 0 0 0 0 0 0 0 0
#8 211 66.9% 100.0% 0 0 0 0 0 0 0 0
#9 213/ 229 94.4% 99.8% 0.249% 0 0 0 0 0 0 0
#10 213/ 222 96.4% 100.0% 0 0 0 0 0 0 0 0
#11 2512.9% 0 0 100.0% 0 0 0 0 0 0
#12 210 9.9% 0 81.2% 17.8% 0.990% 0 0 0 0 0
#13 2811.4% 0 0 75.9% 24.1% 0 0 0 0 0
#14 213/ 216 13.1% 43.7% 34.6% 19.6% 2.0% 0.070% 0 0 0 0
#15 213/ 216 3.1% 0 36.1% 43.8% 16.9% 3.0% 0.194% 0 0 0
#16 213/ 218 28.9% 36.5% 31.4% 18.7% 8.6% 3.8% 0.909% 0.048% 0 0
#17 213/ 224 11.4% 61.7% 31.1% 7.0% 0.175% 0 0 0 0 0
#18 2926.8% 27.7% 66.4% 5.8% 0 0 0 0 0 0
#19 212 52.9% 46.3% 7.3% 9.6% 7.0% 5.5% 12.5% 0 11.9% 0
#20 213/ 244 99.0% 100.0% 0.002% 0 0 0 0 0 0 0
#21 213/ 228 99.5% 100.0% 0 0 0 0 0 0 0 0
#22 2328.6% 0 0 0 100.0% 0 0 0 0 0
#23 213/ 224 60.1% 28.6% 20.0% 12.7% 9.8% 7.3% 5.2% 4.4% 3.3% 8.6%
#24 2844.3% 18.6% 42.5% 9.7% 15.0% 14.2% 0 0 0 0
#25 213/ 229 93.4% 100.0% 0 0 0 0 0 0 0 0
#26 2433.3% 0 0 40.0% 60.0% 0 0 0 0 0
#27 213/ 228 97.2% 100.0% 0 0 0 0 0 0 0 0
#28 2762.2% 94.9% 5.1% 0 0 0 0 0 0 0
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