## Abstract

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 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: stretching-dominated 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 stretching-dominated 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 (SiO_{2}) 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 above-mentioned modelling approach. The mechanical property of passively switching stiffness under 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 bi-stiffness 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.

We define reference vectors as unit directional vectors with eightfold rotational symmetry, which are expressed by
2.1where and (⋅) indicates an inner product that is defined by ** a**⋅

**=**

*b*

*a*^{T}

**. The reference vectors then satisfy the following three relationships: 2.2From equations (2.1) and (2.2), the subscript**

*b**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, 2.3In equation (2.3), we add the superscript of * to make it easily recognizable. It is apparent that the reference vectors themselves have the same closed relationship to equations (2.1) and (2.2).

Using the reference vectors *e*_{k}, we denote the position of the scissor-type joints forming the MS-8 as follows:
2.4where and ℓ indicates the cell length. From now on, we assume ℓ=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 *D*_{8} invariance with eightfold rotational symmetry and eight axes of symmetry [47]. In equation (2.4), the transformation of indicates a one-eighth rotation of the overall structure. The eight rotational symmetries also confirm that equation (2.4) satisfies *x*^{k}(*m*,*m*)=*x*^{k+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 *x*^{k} in equation (2.4). The combination of (*e*_{k}+*e*_{k−1}) or (*e*_{k}+*e*_{k+1}) forms a single rhombic element, the acute angle of which equals . Conversely, the combination of (−*e*_{k−1}+*e*_{k+1}), that is (*e*_{k+3}+*e*_{k+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:
2.5Here, *J*_{0} corresponds to the joint with eight bars (eight-bar joint) located at the centre of the structure. Meanwhile, *J*_{1} and *J*_{2} correspond to six-bar joints. We note that the joints classified as *J*_{2} appear from *m*=2.

Based on the connectivity of the MS-8, all the pairs of bars corresponding to *e*_{k−1} and *e*_{k} are connected pivotally, and all the pairs of bars corresponding to *e*_{k−1} and *e*_{k+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 bars rotate coordinately in the opposite rotational direction. Using a rotation matrix *R*(*ϕ*)∈*SO*(2), that is,
2.6we obtain the common rotated vectors of any bars at all the joints as follows:
2.7where the superscript of 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
2.8When *ϕ*=+*π*/8, the pairs of transformed vectors reach the identical vectors from equation (2.7) so that , (*p*=0,…,3). Substituting them into equation (2.8) gives
2.9Because , equation (2.9) represents the position of the square cells constructed using the two orthonormal bases and . On the other hand, we can obtain the following expression using as *ϕ*=−*π*/8:
2.10We note that and . 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 3*a,b* 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 *D*_{4} 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 ℓ have both axial stiffness *EA* and bending stiffness *EI*. All the flexible joints typed by *J*_{0} and *J*_{1} have two types of rotational springs with a rotational rigidity *r*_{1} or *r*_{2} between first/second-neighbour bars. The relative characteristic of the flexible joint is given by the joint flexibility, which is defined by *μ*_{i}=*r*_{i}ℓ/*EI* (*i*=1,2). We set *μ*_{2} as a fixed parameter which is adequately rigid. However, *μ*_{1} is a controllable parameter. Hence, the connected beam segments around a joint rotate rigidly as and they rotate relatively as scissor-like elements as . We note that the MS-8 with *μ*_{1}≈0 behaves as a motion structure with rigid bars. On the other hand, the cell walls are no longer rigid, with increasing *μ*_{1}, 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 *W*_{0} acts on the point located at P and the line length along segment is described by the parameter . The concentrated load *W*_{0} is normalized as .

The MS-8 with *m*≤1 for *μ*_{1}=10^{−2} and *μ*_{2}=10^{2} gives rise to two types of transformations similar to motions I and II under concentrated loading as shown in figure 5*a*–*f*. When the MS-8 is loaded by at , it transforms into the square cells tilted at a 45^{°} angle as in motion I (figure 5*a*–*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, (figure 5*d*–*f*). Each transformation is generated by the coordinated rotations of rigid squares under low loading as . It is noted that slightly decreases during the transformation of motion I from figure 5*a* 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 *μ*_{1} is small, the boundary of 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 is to the right () motion I occurs and if to the left (), motion II is the consequence.

Figures 6 and 7 show the deformation of the MS-8 with increasing *μ*_{1}. For *μ*_{1}=1 and *μ*_{2}=10^{2}, the MS-8 exhibits similar behaviour to motion I or II for the loading points at or although these deformations are imperfect motions accompanied by small deflections of the beam segments (figure 6*a,b*). Furthermore, as *μ*_{1} is over 10^{2}, the two deformation shapes can no longer be distinguished as a part of motion I or II (figure 7*a,b*).

### (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 8*a,b*. The model for motion I corresponds to figure 8*a* and for motion II to figure 8*b*. The equation for a single nonlinear spring is given by
3.1where *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 *k*_{1} and *k*_{2}.

In this simulation, we use the material parameters listed in table 1. Similar to common finite-element simulation, there is no scale factor in our modelling. Thus, the relative value such as *EI*/*EA* or *μ*_{1} 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 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 and *μ*_{1}=0.01. The other curve indicated by the solid line is the equilibrium path of motion II for and *μ*_{1}=0.01, which is divided into two paths branched at the bifurcation point. We also draw the curve of rigidly jointed behaviour as and *μ*_{1}=100 to compare the behaviours of the two motions.

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 *x*_{2}-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 9*b*). 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 *x*_{2}-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 and *E**_{dia}/*E*_{s}=0.25*r*^{3}, where , or *E**_{dia} indicates the Young modulus of a cell structure, applied in the *x*_{1}, *x*_{2} or diagonal directions in the *x*_{1}–*x*_{2} plane, and *E*_{s} indicates the Young modulus of the fully dense cell wall material. In addition, *r*=*ρ**/*ρ*_{s} is the relative density determined by 2*t*/ℓ 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 *E*_{1}/*E*_{dia}=2/*r*^{2} 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.

## 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 10*a*). When the compressive stress *σ*_{2} parallel to *x*_{2} 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 10*b*. Therefore, from appendix A, the movement of the periodical assembly under *σ*_{2} is dominated by Motion II for any boundary conditions. The behaviour of the assembly under *σ*_{1} parallel to *x*_{1} is similar to that under *σ*_{2}. On the other hand, when the assembly is subjected to a stress *σ*_{dia} in the diagonal direction, the two types of forces act on the basic unit as shown in figure 10*c*: one is the same as the vertical force illustrated in figure 10*b*, 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 (A2) and (A3) in appendix A, we can calculate the total moment of the rotating square under *σ*_{dia} as follows:
4.1The 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 *σ*_{dia} is dominated by motion I. We demonstrate the compressive analysis as illustrated in figure 11*a* to confirm the diagonal characteristic of the assembly, where *μ*_{1}=0.001 and the other parameters are the same as those used in table 1. Figure 11*b,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 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.

### (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 12*a*); the other where local compressive forces act on the re-entrant part around the centre vertex of the upper-middle unit (figure 12*b*). Here, *k*_{1}=5.0, *k*_{2}=0, *μ*_{1}=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 13*a*). 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 13*b*,*c*). Subsequently, the deformation becomes concentrated at the upper cells of the upper-middle unit, and the distortions of other cells are released (figure 13*d*,*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 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.

## 5. Conclusions

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 scissor-type 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.

## Acknowledgements

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, *W*_{V} or *W*_{H}, at the square edge. In addition, we set the normal reaction forces *R*_{O} at an origin O, and the two normal reaction forces *R*_{A} and *R*_{B} at both sliders with free-rotation.

When *W*_{V} acts on the point D, the square is equilibrated because the line of action of *W*_{V} passes through the common point C of the intersection of forces *R*_{O}, *R*_{A} and *R*_{B}. The distance of AD is then derived from
A1In addition, the moment of the rotating square AO′BF as *W*_{V} or *W*_{H} acts on the vertex (the point *F*) of the square is respectively calculated by
A2
A3where an anticlockwise direction is defined to be positive.

- Received January 31, 2013.
- Accepted April 23, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.