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Bi-stiffness property of motion
structures transformed into
square cells
rspa.royalsocietypublishing.org
Hiro Tanaka
Department of Mechanical Engineering, The University of Tokyo,
Bunkyo-ku, Tokyo 113-8656, Japan
Research
Cite this article: Tanaka H. 2013 Bi-stiffness
property of motion structures transformed
into square cells. Proc R Soc A 469: 20130063.
http://dx.doi.org/10.1098/rspa.2013.0063
Received: 31 January 2013
Accepted: 23 April 2013
Subject Areas:
mechanical engineering, structural
engineering
Keywords:
motion structure, eightfold rotational
symmetry, square cells, bi-stiffness property
Author for correspondence:
Hiro Tanaka
e-mail: [email protected]
Cellular solids with internal microstructures enable
the reduction in some environmental loads because
of their lightweight bodies, and deliver unique elastic,
electromagnetic and thermal properties. In particular,
their large deformability without topological change
is one of their most interesting solid properties. In
this study, we propose a bar-and-joint framework
assembled with a basic unit of motion structure,
which has eightfold rotational symmetry (MS-8). The
MS-8 is made of tetragons, arranged in a concentric
fashion, which are transformed into either one of two
different aligned patterns of square cells according
to the coordinated rotations of the inside squares.
Square cells are extremely anisotropic, which is why
the stiffness of the MS-8 changes dramatically in
the transformation process. Thus, the MS-8 exhibits
bi-stiffness according to the two different motions.
Taking advantage of the bi-stiffness property, the
possibilities of deformation behaviours for repetitive
structures of MS-8s are discussed.
1. Introduction
Cellular solids with low relative densities have been
produced for mechanical design in response to the
need for lightweight bodies that are adequately stiff
and strong with mechanical properties that range
widely from high specific stiffness to shock absorption,
electromagnetic shielding/transmission and thermal
conduction/insulation [1,2]. Exceptional chemical properties such as catalysis, separation and gas storage can
also be provided using inorganic and organic cellular
materials: for example, metal-organic frameworks [3] and
porous coordination polymers [4].
Recently, these structural materials have become
increasingly important for developments in the latest
fields of engineering such as the design of electrode
2013 The Author(s) Published by the Royal Society. All rights reserved.
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materials for lithium-ion batteries [5,6], deformation-induced pattern transformations in
optical materials for photonic crystals [7,8] and in periodic elastomeric cellular solids [9,10].
In these applications, the large deformability that all cellular structures have in common is an
essential element, because their mechanical properties are accompanied by considerable changes
in their morphologies.
The deformation of a cellular solid depends strongly on the geometry of its internal
microstructure, and the structural characteristics contribute directly to the linear elastic modulus
and geometrical nonlinearity such as buckling. In particular, the in-plane deformation behaviour
of honeycombs has been studied extensively because it is simple to understand spatially. The
deformations of honeycombs are approximately classified into two major groups: stretchingdominated behaviour or bending-dominated behaviour [11,12]. For instance, the honeycombs
packed by hexagonal cells, which are typically observed in nature [13], belong to the group
of bending-dominated structures and have some useful and interesting mechanical properties:
for example, high capacity for energy absorption [14,15], and multiple buckling shapes and
post-buckling behaviour [16,17]. Conversely, triangular cells, i.e. truss constructions, display
stretching-dominated behaviours and their high specific stiffness is one of the outstanding
mechanical properties for architecture. Square cells have intermediate characteristics such as
(i) they are readily deformed because they are bending-dominated under shear loading and
(ii) they are stiff unless and until they are buckled or collapsed because they are stretchingdominated under compressive/tensile loading. The square honeycombs with a large surface area
and high stiffness have been commonly used for multi-functional applications such as catalytic
converters and heat exchangers [18], and buckling problems for periodic squares have been
discussed [19–21].
It is well known that some structural materials, as typified by re-entrant hexagonal
honeycombs, exhibit a negative Poisson’s ratio and expand laterally when stretched, or shrink
when compressed [22]. This is termed auxetic behaviour by Evans et al. [23]. A variety of structural
materials with negative Poisson’s ratios have been designed and developed [24–27] and it has
been reported that there are a few natural auxetic materials: for example, a cristobalite (SiO2 )
shows auxetic behaviour at certain temperatures [28,29]. Auxetic behaviour seems to be similar
to the mechanism of a machine, so structures with negative Poisson’s ratios can be modelled
by assemblies with rigid components connected pivotally to each other [30]. As seen in the
attempts to make a link between auxetic properties and general continuum mechanics [24,31],
the local rotational manner of the internal members is a key factor in solid deformation, and the
controllability of their connections enables the structure to perform with novel deformability: for
example, a near-zero Poisson’s ratio [32], and a Poisson’s ratio that switches from a positive to a
negative value [33].
Analogous to auxetic mechanics, some expandable movements have been realized by
assembling-resistant bodies connected by movable joints [34–36]. Frameworks that provide
such unique transformations have been introduced as motion structures by You [37]. The most
common types of motion structures are fabricated by the in-plane connection of rigid bars
with scissor-hinge elements, and the combinations in which they are assembled have been
extensively investigated in the past [38]. The subtypes of a basic scissor-hinge structure have
some deployable mechanisms because of the coordinated relative rotations of the scissor
elements [39–41].
In a brief review from conventional cellular solids to unique deformable solids such as
auxetic materials and motion structures, the simple modelling approach within the framework
of the component configuration and connectivity promotes an understanding of the overall
mechanical properties of structural materials. In this respect, bar-and-joint frameworks are valid
for representing the concise deformations of these materials by mathematical or numerical
analysis: for example, the rigidity or flexibility of the frameworks modelled on repetitive cells [42],
zeolites [43,44], auxetic materials [45] and the loop structures connected by revolute joints [46].
The aim of this study is to explore structural materials with bi-stiffness, using the abovementioned modelling approach. The mechanical property of passively switching stiffness under
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3
ek + 1
ek + 3
ek
m=0
O
ek + 7
ek + 4
e+5
ek + 6
Figure 1. The proposed motion structure with eightfold rotational symmetry (MS-8). The adjacent bars painted in the same
colour, blue or red, are rigidly connected at the normal pivot joints (smaller circles) or scissor-type joints (larger circles). The
configurations of the scissor-type joints are determined by x k (m, n) of equation (2.4). For example, the position of a joint with
m = 0 is located at the origin O, and all the positions of joints with m = 1 are located on the broken line shown here.
different conditions of compressive loading (or tensile loading) is called a bi-stiffness property in
this paper. The development of a cellular structure with bi-stiffness enables us to create a new
concept of structural material design as a multi-functional system.
In §2, we develop the basic unit of a motion structure, which is capable of transforming
square cells, from the point of view of building a periodic bar-and-joint framework with a bistiffness property. Simple kinematic consideration will show that the proposed unit has two
different motions, and after these motions, it transforms into two patterns of square cells that are
tilted by 45◦ in relation to each other. When taking account of cell-to-cell contact, the numerical
structural analyses will clarify that this kinematic mechanism produces bi-stiffness because of the
anisotropy of square cells. Taking advantage of the deformation characteristic with the bi-stiffness
property, we then construct some repetitive structures with the proposed units and discuss their
curious deformation behaviours.
2. Geometrical configuration and transformation of a proposed
motion structure
Figure 1 shows that the motion structure has a geometrical configuration with eightfold rotational
symmetry. In this paper, the proposed structure is called MS-8 for convenience. All the rigid beam
segments (bars), that are parts of the cell walls, are of equal length and the adjacent bars painted
in the same colour, blue or red, are rigidly connected using two types of joints: a normal pivot
joint (smaller circles in figure 1) and a scissor-type joint (larger circles in figure 1). The MS-8 is
constructed from rigid square cells that are arranged in a concentric manner and behave like
a kinetic transformation from the rotating squares with a single degree of freedom [41]. In this
section, we formulate the MS-8 and explain the geometrical characteristics and transformation.
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We define reference vectors as unit directional vectors with eightfold rotational symmetry,
which are expressed by
(2.1)
where k ∈ Z and (·) indicates an inner product that is defined by a · b = aT b (a, b ∈ R2 ).
The reference vectors then satisfy the following three relationships:
ek · ek±2 = 0,
ek · ek±4 = −1,
ek · ek±8 = 1.
(2.2)
From equations (2.1) and (2.2), the subscript k represents a circular number in that k ≡ k mod 8.
We also define the other reference vectors as the median of the two original reference vectors, thus,
e∗k+1/2 =
ek + ek+1
.
|ek + ek+1 |
(2.3)
In equation (2.3), we add the superscript of ∗ to make it easily recognizable. It is apparent that the
reference vectors e∗k+1/2 themselves have the same closed relationship to equations (2.1) and (2.2).
Using the reference vectors ek , we denote the position of the scissor-type joints forming the
MS-8 as follows:
xk (m, n) = {m(ek + ek−1 ) + n(−ek−1 + ek+1 )},
n ≤ m,
(2.4)
where n, m ∈ {0, N} and indicates the cell length. From now on, we assume = 1 for convenience.
Based on equation (2.4), the position of a joint with m = 0 is located at the origin O, and all the
positions of joints with m = 1 are located on the broken line illustrated in figure 1. Hence, the
joints with m ≤ 1 constitute the minimum structural unit of the MS-8. It is apparent from figure 1
that geometrically the proposed structure has D8 invariance with eightfold rotational symmetry
and eight axes of symmetry [47]. In equation (2.4), the transformation of k → k + 1 indicates a
one-eighth rotation of the overall structure. The eight rotational symmetries also confirm that
equation (2.4) satisfies xk (m, m) = xk+1 (m, 0). The same configuration of MS-8 with m ≤ 1 is actually
observed in a part of a two-dimensional octagonal quasi-lattice consisting of squares and 45◦
rhombi [48].
Comparing equation (2.4) with figure 1, it is found that the connecting bars between the two
adjacent scissor-type joints via a pivot joint, correspond to a set of two reference vectors. Figure 2
clearly illustrates the constitutive vectors of xk in equation (2.4). The combination of (ek + ek−1 ) or
(ek + ek+1 ) forms a single rhombic element, the acute angle of which equals arccos(ek · ek+1 ) = π/8.
Conversely, the combination of (−ek−1 + ek+1 ), that is (ek+3 + ek+1 ), forms a single square element.
As shown in figure 2, the scissor-type joints are categorized into three types of connections that
are expressed by sets of reference vectors as follows:
⎧
⎪
⎪
⎨J0 = {ek , ek+1 , ek+2 , ek+3 , ek+4 , ek+5 , ek+6 , ek+7 }, if m = n = 0,
k
J (m, n) = J1 = {ek , ek+1 , ek+3 , ek+4 , ek+6 , ek+7 }, if m ≥ 1, n = 0,
⎪
⎪
⎩J = {e , e , e , e , e , e }, otherwise.
2
k
k+1
k+3
k+4
k+5
(2.5)
k+7
Here, J0 corresponds to the joint with eight bars (eight-bar joint) located at the centre of the
structure. Meanwhile, J1 and J2 correspond to six-bar joints. We note that the joints classified
as J2 appear from m = 2.
Based on the connectivity of the MS-8, all the pairs of bars corresponding to ek−1 and ek are
connected pivotally, and all the pairs of bars corresponding to ek−1 and ek+1 are connected rigidly.
In this situation, the MS-8 has an overconstrained mechanism and it can transform kinematically
because of the coordinated relative rotations of the central eight-bar and six-bar joints [41]. Thus,
all the even-numbered bars at any of the joints rotate coordinately and all the odd-numbered
..................................................
1
ek := ek ∈ R | ek · ek = 1, ek · ek+1 = ,
2
2
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5
ek + 1
ek + 4
ek + 5
ek + 2
ek + 1
ek + 1
ek + 3
ek
ek + 4
J0
ek + 7
ek + 5
ek
ek + 7
ek + 3 J1
ek
ek – 1
ek + 4
O
ek + 6
m=1
ek + 1
ek
ek + 7
ek + 6
m=2
Figure 2. A set of three reference vectors, ek−1 , ek and ek+1 , forms one-eighth of the region of the MS-8. The black arrows
illustrated here correspond to the reference vectors described by equation (2.4) and also represent the connecting bars. The
three types of scissor-type joints coloured grey are represented by J0 , J1 and J2 in equation (2.5).
bars rotate coordinately in the opposite rotational direction. Using a rotation matrix R(φ) ∈ SO(2),
that is,
cos φ − sin φ
R(φ) =
,
(2.6)
sin φ
cos φ
we obtain the common rotated vectors of any bars at all the joints as follows:
êk = R((−1)k φ)ek ,
(2.7)
where the superscript of •ˆ indicates the movement of the vector rotated by φ. Therefore, such a
coordinated motion still complies with equation (2.4) during the overall structural transformation.
Namely
(2.8)
x̂k (m, n, φ) = m(êk + êk−1 ) + n(−êk−1 + êk+1 ), n ≤ m.
When φ = +π/8, the pairs of transformed vectors reach the identical vectors from equation (2.7)
so that ê2p = ê2p+1 = e∗2p+1/2 , (p = 0, . . . , 3). Substituting them into equation (2.8) gives
π
= m(e∗2p+1/2 + e∗2p−3/2 ) + n(−e∗2p−3/2 + e∗2p+1/2 )
x̂2p m, n, +
8
= (m + n)e∗2p+1/2 + (m − n)e∗2p−3/2 .
e∗2p−3/2
(2.9)
· e∗2p+1/2
= 0, equation (2.9) represents the position of the square cells constructed
Because
using the two orthonormal bases e∗2p−3/2 and e∗2p+1/2 . On the other hand, we can obtain the
following expression using ê2p−1 = ê2p = e∗2p−1/2 as φ = −π/8:
π
= m(e∗2p−1/2 + e∗2p−1/2 ) + n(−e∗2p−1/2 + e∗2p+3/2 )
x̂2p m, n, −
8
(2.10)
= (2m − n)e∗2p−1/2 + ne∗2p+3/2 .
We note that e∗2p−1/2 · e∗2p+3/2 = 0 and e∗2p−3/2 · e∗2p−1/2 = 1/ 2. Therefore, equation (2.10) also
represents the position of the different square cells that are tilted at 45◦ with respect to the square
cells described by equation (2.9). In this paper, the former transformation to the square cells of
equation (2.9) is called motion I and the latter of equation (2.10) is called motion II. Figure 3a,b
..................................................
J2
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ek + 3
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6
e3
e0
e4
e7
x1
O
e5
(a)
e6
(b)
e*+5/2
e*+3/2
e*+7/2
e*+1/2
e*–1/2
e*–7/2
e*–5/2
e*–3/2
Figure 3. Two types of transformations of the MS-8 (m ≤ 1): (a) motion I; (b) motion II. The red bars exhibit the positive
rotations, and the blue bars exhibit the negative rotations for motion I and vice versa for motion II. In addition, the purple
bars illustrated by bold lines represent complete overlaps between the red and blue bars.
shows motions I and II of the MS-8 as m ≤ 1, where we set k = 0 and an anticlockwise direction
is defined to be positive. It is clear that the structure can transform into two types of square cells
according to the above formulation. This kinematic system nominally exhibits D4 invariance [49].
Motions I and II belong to the same kinematic path of movement in opposite rotational directions
to each other.
3. Elastic property of the MS-8
For real structures, a beam segment has finite stiffness, and a joint exhibits some rotational
resistance against the bar-and-joint framework. We here investigate the large deformability of
the smallest unit of the MS-8 subjected to a uniaxial compression load to understand the integrity
of the structural framework that depends on joint flexibility.
(a) Flexibly jointed modelling with large deformation
Based on the total Lagrangian formulation for our flexibly jointed modelling [21,33], we
characterize the deformable beam segments (cell walls), each of which is represented by 10 beam
elements with finite translation and rotation displacements under infinitesimal strain. To solve
the equilibrium path of the transformation, we also adopt the arc-length method that enables us
to overcome local minimum and maximum points under load control [50]. As shown in figure 4,
the material parameters are given as follows: all the beam segments of equal length have both
axial stiffness EA and bending stiffness EI. All the flexible joints typed by J0 and J1 have two
types of rotational springs with a rotational rigidity r1 or r2 between first/second-neighbour bars.
The relative characteristic of the flexible joint is given by the joint flexibility, which is defined
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e1
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e2
x2
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W0
W0
x2
r2
P
10 beam elements
EA, EI
A
(a)
r1
a
O
(b)
b
r1
r2
r2
W0
W0
Figure 4. The analytical model for numerical computing. (Online version in colour.)
by μi = ri /EI (i = 1, 2). We set μ2 as a fixed parameter which is adequately rigid. However, μ1
is a controllable parameter. Hence, the connected beam segments around a joint rotate rigidly
as μ1 → ∞ and they rotate relatively as scissor-like elements as μ1 → 0. We note that the MS-8
with μ1 ≈ 0 behaves as a motion structure with rigid bars. On the other hand, the cell walls are
no longer rigid, with increasing μ1 , and the MS-8 then behaves as a beam-like structure with
bending deflections. Figure 4 also illustrates the load and displacement constraints for a numerical
analysis. In the analytical model, the compression load W0 acts on the point located at P and the
line length along segment AP is described by the parameter ỹ = y/. The concentrated load W0 is
normalized as W̃0 = W0 2 /EI.
The MS-8 with m ≤ 1 for μ1 = 10−2 and μ2 = 102 gives rise to two types of transformations
similar to motions I and II under concentrated loading as shown in figure 5a–f. When the MS-8
is loaded by W̃0 at ỹ = 0.8, it transforms into the square cells tilted at a 45◦ angle as in motion I
(figure 5a–c). The MS-8 transforms into square cells parallel to the loading direction as in motion
II when the load is applied at the vertex of the square surface, thus, ỹ = 1.0 (figure 5d–f ). Each
transformation is generated by the coordinated rotations of rigid squares under low loading
as W̃0 < 0.1. It is noted that W̃0 slightly decreases during the transformation of motion I from
figure 5a to b, because the internal moment corresponding to the acting load increases as a result
of the rotation of the squares such as in the principle of leverage.
As μ1 is small, the boundary of ỹ ≈ 0.8284 makes a decision on whether motion I or II occurs.
Based on a simple geometrical consideration, the equilibrium point is derived from equation (A 1)
in appendix A. Therefore, the loading routine is a key factor in selecting either motion I or II.
If W̃0 is to the right (ỹ < 0.8284) motion I occurs and if to the left (ỹ > 0.8284), motion II is the
consequence.
Figures 6 and 7 show the deformation of the MS-8 with increasing μ1 . For μ1 = 1 and μ2 = 102 ,
the MS-8 exhibits similar behaviour to motion I or II for the loading points at ỹ = 0.8 or ỹ = 1.0
although these deformations are imperfect motions accompanied by small deflections of the beam
segments (figure 6a,b). Furthermore, as μ1 is over 102 , the two deformation shapes can no longer
be distinguished as a part of motion I or II (figure 7a,b).
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x1
P
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+
7
y
y
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(c)
(d)
(e)
(f)
Figure 5. Snapshots of motion I and II of the MS-8 (m ≤ 1): (a) (μ1 , ỹ, W̃0 ) = (0.01, 0.8, 0.0634), (b) (0.01, 0.8, 0.0632),
(c) (0.01, 0.8, 0.0707), (d) (0.01, 1.0, 0.0486), (e) (0.01, 1.0, 0.0565) and (f ) (0.01, 1.0, 0.0633). (Online version in colour.)
(a)
(b)
Figure 6. Snapshots of the deformations of a flexibly jointed structure (m ≤ 1): (a) (μ1 , ỹ, W̃0 ) = (1, 0.8, 2.0472),
(b) (1, 1.0, 3.9940). (Online version in colour.)
(a)
(b)
Figure 7. Snapshots of the deformations of a rigidly jointed structure (m ≤ 1): (a) (μ1 , ỹ, W̃0 ) = (100, 0.8, 5.3584),
(b) (100, 1.0, 6.1801). (Online version in colour.)
8
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(b)
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(a)
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(a)
9
~
y = 1.0
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v
v
W0
W0
(b)
2W0
~
y=0
P
A
A
O
O
2W0
W0
W0
Figure 8. Analysis modelling with nonlinear springs for evaluating the post-motion behaviours: (a) motion I, (b) motion II. The
rigidities of the inserted nonlinear springs are adjusted to increase when the different cell walls approach each other. (Online
version in colour.)
Table 1. Individual material parameters for numerical modelling with nonlinear springs.
EA
5.0 × 104
EI
5.0 × 103
10
μ1
0.01
μ2
100
k1
1.0 × 10−3
k2
5.0 × 10−12
..........................................................................................................................................................................................................
(b) Stiffness changes involved in the transformations
It is well known that the cellular aggregate with squares is extremely anisotropic, compared
with triangular and hexagonal cells [2]. Taking the anisotropy of square cells into account,
we now simulate further transformations following motions I and II to evaluate the change
in stiffness after cell-to-cell contacts. To avoid the overlapping of beam segments (cell walls)
during the motions, we adopt the improved analytical model that corresponds to each motion, by
implementing the nonlinear springs as shown in figure 8a,b. The model for motion I corresponds
to figure 8a and for motion II to figure 8b. The equation for a single nonlinear spring is given by
f = k1 u + k2 u15 ,
(3.1)
where f and u are the force and displacement of the spring. An additional model can represent
the pseudo-contact interaction between two close cells by adjusting the two spring stiffness
parameters of k1 and k2 .
In this simulation, we use the material parameters listed in table 1. Similar to common finiteelement simulation, there is no scale factor in our modelling. Thus, the relative value such
as EI/EA or μ1 is a significant parameter with respect to the qualitative characteristics of the
structure. This is why the units of each parameter are not considered here.
Numerical modelling with nonlinear springs results in the load–displacement curves shown
in figure 9, where ṽ is a dimensionless vertical displacement at the loading point P. The curve
indicated by the dotted line is the equilibrium path of motion I for ỹ = 0 and μ1 = 0.01. The other
curve indicated by the solid line is the equilibrium path of motion II for ỹ = 1.0 and μ1 = 0.01,
which is divided into two paths branched at the bifurcation point. We also draw the curve of
rigidly jointed behaviour as ỹ = 1.0 and μ1 = 100 to compare the behaviours of the two motions.
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10
(e)
(d)
1
~
W0= W0
2
EI
rigidly jointed behaviour
5
(c)
secondary path
motion II
bifurcation point
(b)
motion I
0
0.25
~v = v
(a)
~
0.50 (v = 0.7)
Figure 9. Dimensionless load–displacement curves. The insets of (a), (c)–(e) show the deformation shapes of the MS-8
(m ≤ 1) at the arrowed point on each equilibrium path. The inset of (b) shows the deformation shape for ṽ = 0.7.
In the primary path (solid line) of motion II, it is observed that the tangent stiffness increases
monotonically and rapidly during the transformation into square cells. However, it has potential
to give rise to buckling as shown by the secondary path (dashed line). The inset c of figure 9
shows the post-buckling shape on the secondary path. It is found that the four pairs of coupled
beam segments bend along with the rhombic transformation of the surrounding squares. The
reflection symmetry of the vertical axis is broken with x2 -axial asymmetry. The buckling is called
symmetric bifurcation under loading [51]. After buckling, the applied load gradually decreases,
which specifically implies a negative tangent. The curve of motion I (dotted line) shows that the
structure maintains a low resistance for a while after the transformation of motion I because of the
shear deformation of the four square cells located at the centre (figure 9b). Such a low resistance
might be continued up to a densification behaviour, which is the result of limited deformation
with multiple cell-to-cell contacts.
The first buckling load is lower than the maximum load of the rigidly jointed structure,
although it is adequately higher than the plateau load during motion I. However, it is possible for
the MS-8 to trace the primary path without bifurcation if the reflection symmetry of the structure
holds under deformation. For example, applying displacement control instead of load control
for compression is the usual procedure for preventing the structure from the first bifurcation
breaking the x2 -axis of symmetry. In this situation, the structure has the potential to reach the
axial-stiffness of a beam segment. A square cell, in general, shows that E∗1 /Es = E∗2 /Es = 0.5r and
E∗dia /Es = 0.25r3 , where E∗1 , E∗2 or E∗dia indicates the Young modulus of a cell structure, applied
in the x1 , x2 or diagonal directions in the x1 –x2 plane, and Es indicates the Young modulus of
the fully dense cell wall material. In addition, r = ρ ∗ /ρs is the relative density determined by
2t/ for square cells with a thickness t [52]. Based on the in-plane elastic property of square
cells, it is expected that the stiffness achieved by motion II is E1 /Edia = 2/r2 times higher than
the stiffness achieved by motion I. This means that the basic unit of the MS-8 has a bi-stiffness
property under uniaxial loading and its structural capacity overcomes the strength of the rigidly
jointed structure.
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primary path
10
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:y~= 1.0, µ1 = 0.01
:y~= 1.0, µ1 = 0.01
:y~= 1.0, µ1 = 100
:y~= 0, µ = 0.01
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s2
(a)
(b)
W
W
11
sdia
sdia
s1
s1
(c)
W
W
W
x2
W
W
sdia
sdia
x1
W
s2
W
W
Figure 10. (a) Periodic assembly with the proposed structural units and the two directional types of uniaxial stress, σ1,2 or σdia .
The internal forces transferred on the each structural unit under (b) σ2 and (c) σdia .
4. Discussion on the possible deformations of repetitive structures
For material modelling, we now provide some cellular aggregations assembled with the basic
units of the MS-8 to discuss their possible deformation behaviours resulting from the bi-stiffness
property of the MS-8.
(a) Periodic assembly with the minimum units of the MS-8
We first consider the periodical assembly in which the MS-8s are pivotally connected at their
vertices on the boundaries (figure 10a). When the compressive stress σ2 parallel to x2 is applied to
the periodical assembly, the internal forces act on the boundary of each unit via the four vertices
of the top and bottom squares as shown in figure 10b. Therefore, from appendix A, the movement
of the periodical assembly under σ2 is dominated by Motion II for any boundary conditions. The
behaviour of the assembly under σ1 parallel to x1 is similar to that under σ2 . On the other hand,
when the assembly is subjected to a stress σdia in the diagonal direction, the two types of forces
act on the basic unit as shown in figure 10c: one is the same as the vertical force illustrated in
figure 10b, and the other is the vertical force acting on the vertex of a lateral square; its moment
is equivalent to the moment against the horizontal force acting on the vertex of a top/bottom
square. Based on equations (A 2) and (A 3) in appendix A, we can calculate the total moment of
the rotating square under σdia as follows:
W(cos 18 π − 2 sin 18 π ) − W sin 18 π < 0
(4.1)
The first term on the left-hand side indicates the moment against the vertical force and the
second term that against the horizontal force. Hence, equation (4.1) shows that the movement
of the periodical assembly under σdia is dominated by motion I. We demonstrate the compressive
analysis as illustrated in figure 11a to confirm the diagonal characteristic of the assembly, where
μ1 = 0.001 and the other parameters are the same as those used in table 1. Figure 11b,c shows that
the finite repetitive assembly behaves as motion I although the external loads and reaction forces
act vertically on the vertices of the squares. As a result, motion I is the priority movement of the
assembly subjected to diagonal compression for any boundary conditions. The eventual stiffness
..................................................
W
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W
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(a)
(b)
12
(c)
W0
W0
W0
W0
W0
W0
(b)
(b)
W0 W0 W0
v
v
Figure 12. Compression problems of the repetitive assembly connected with linear springs between structural units: (a,b) two
types of compressive load conditions.
properties achieved by both motions under compressive stress are similar to the anisotropy
characteristics of square cells. The motions of the structure under tensile stress are inverted,
because all the component bars rotate in the opposite directions relative to that under compressive
stress. This gives the polar opposite stiffness from square cells.
..................................................
(a)
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Figure 11. (a) Compressive analysis of the MS-8s assembled in a diagonal fashion. (b,c) The transformation shapes of the
repetitive assembly.
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0.6
13
(b)
EI
~
W0= W0
2
(a)
0.2
0
(d)
–0.2
(b)
0
0.5
~
v= v
(c)
1.0
1.2
(a)
(d)
(e)
Figure 13. Dimensionless load–displacement curves of the compression analyses: the dot and solid lines correspond to
the paths for the load conditions shown in figure 12a,b. The insets of (a)–(e) show the deformation snapshots on the two
equilibrium paths.
(b) Repetitive structure of MS-8s with inserting springs
We next consider another case of a structural assembly as shown in figure 12, in which linear
springs are inserted between two adjacent structural units. For this modelling, we conduct two
types of compression analyses: one where equal interval compressive forces act on all the vertices
of the squares at the upper side of the structure (figure 12a); the other where local compressive
forces act on the re-entrant part around the centre vertex of the upper-middle unit (figure 12b).
Here, k1 = 5.0, k2 = 0, μ1 = 0.001 and the other parameters are the same as in table 1. It is noted
that we do not take into account the cell-to-cell contacts in the modelling.
Figure 13 shows the dimensionless load–displacement curves and the deformation snapshots
for both compressive load conditions. The curve indicated by the dotted line is the path of
the former load condition and all the units uniformly behave as motion II (figure 13a). It is
apparent that the repetitive structure exhibits the auxetic behaviour with ν = −1 and the enhanced
stiffness under motion II is similar to the prediction of some mechanical properties of auxetic
materials [25]. The other curve indicated by the solid line is the path of the latter load condition.
The snapshots on the solid line show that the upper-middle structural unit initially behaves
as motion I and then the movement of motion I spreads into the outside units (figure 13b,c).
Subsequently, the deformation becomes concentrated at the upper cells of the upper-middle unit,
and the distortions of other cells are released (figure 13d,e).
The above-mentioned deformation behaviour is explained as follows. The linear spring
connecting two structural units transfers the central force to the centre re-entrant vertex on each
side of a structural unit, and the adjusted spring constant enables each unit of the assembly
to have the state dominated equally by motions I and II, which means its movement depends
strongly on the boundary condition. Therefore, the external compressive load around the centre
..................................................
(e)
(c)
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0.4
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With the aim of creating a structural material with a bi-stiffness property, we have proposed a new
model based on a MS-8. The structural framework of the MS-8 is connected by pivot and scissortype joints, and their particular connections enable it to have the kinematic movement of rotating
squares (motions I or II). After the motions, the MS-8 is able to transform into two patterns of
square cells that are tilted at 45◦ with respect to each other. The compression analyses of the MS-8
with finite bending stiffness of its segments and rotational flexibility of its joints showed that its
deformation depends strongly on the joint flexibility and loading point. We further simulated the
post-motion behaviours to take account of the cell-to-cell contacts. The numerical results revealed
that the MS-8 exhibited the bi-stiffness property in response to compressive loading, that is, the
tangent stiffness differed substantially between the transformations of motions I and II because
of the strong anisotropy of square cells. In particular, the eventual strength after motion II could
overcome the strength of the beam-like framework with frozen joints.
We next considered the periodical assembly in which the smallest units of MS-8s were
pivotally connected with vertex-sharing and explained that its stiffness properties achieved by
both motions were similar to the anisotropic characteristics of square cells because it behaved
as motion II under vertical compressive stress and as motion I under diagonal compressive
stress although the stiffness properties under tensile stress were reversed. We also conducted
compression analyses of the repetitive assembly in which each linear spring is inserted at each
connection between two units. The simulations showed that the proposed assembly exhibited
two different deformations according to the types of compressive load conditions. These results
demonstrated that the proposed structure potentially has different deformation characteristics
according to the type of compression, such as broad pushing or local indenting.
Through this work, we have developed a structural framework equipped with a bi-stiffness
property that can be selected by a loading procedure. Based on the insights obtained, a variety of
novel structures with these unique mechanical properties could be created by conceptualizing the
subtypes of MS-8s and extending these to viscoelastic/dynamics problems and three-dimensional
frameworks. The control of the geometry and connectivity of microstructures as a manipulating
machine represents a critical problem for realizing such an advanced material design.
H.T. thanks Prof. H. Gao and his research group members Dr H. Kesari, Dr H. Yuan, Mr X. Yang and others for
the fruitful discussions at Brown University. This work was supported by the Japan Society for the Promotion
of Science for Young Scientists (grant nos. 23760086 and 25709001) and the JSPS Institutional Program for
Young Researcher Overseas Visits.
Appendix A. Solutions of a rotating square
Here, we solve the problem of the rotating square loaded by either a vertical or horizontal force.
Figure 14 illustrates the free-body diagram for the first of the eight structural elements. We apply
a vertical or horizontal concentrated load, WV or WH , at the square edge. In addition, we set the
normal reaction forces RO at an origin O, and the two normal reaction forces RA and RB at both
sliders with free-rotation.
When WV acts on the point D, the square is equilibrated because the line of action of WV passes
through the common point C of the intersection of forces RO , RA and RB . The distance of AD is
then derived from
(A 1)
|AD| = |AC| sec 18 π = 2 tan 18 π ≈ 0.8284 × .
..................................................
5. Conclusions
14
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re-entrant vertex of the upper-middle unit causes motion I movement of the assembly. However,
the springs are gradually elongated according to the movement of motion I, and the compressive
forces between the centre vertices of the two units become smaller. As a result, the assembly
becomes dominated by motion II because of the reaction forces at the ground except for the
distortion of the upper cells of the upper-middle unit under the compression loading.
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WV
G
E
..................................................
WH
F
rspa.royalsocietypublishing.org Proc R Soc A 469: 20130063
WH
15
WV
y
D
A
C
RA
RB
B
+
O'
1p
8
RO
O
Figure 14. The free-body diagram for the first of the eight structural elements of the MS-8 with m ≤ 1. (Online version in
colour.)
In addition, the moment of the rotating square AO BF as WV or WH acts on the vertex (the point
F) of the square is respectively calculated by
WV × |CE| = WV (cos 18 π − 2 sin 18 π ),
−WH × |EF| = −WH sin 18 π ,
(A 2)
(A 3)
where an anticlockwise direction is defined to be positive.
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