One-way transmission in topological mechanical metamaterials based on self-locking

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One-way transmission in topological mechanical metamaterials based on self-locking Xiao-Fei Guo Formal analyisis , Li Ma PII: DOI: Reference:

S0020-7403(19)32591-3 https://doi.org/10.1016/j.ijmecsci.2020.105555 MS 105555

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International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

16 July 2019 21 January 2020 23 February 2020

Please cite this article as: Xiao-Fei Guo Formal analyisis , Li Ma , One-way transmission in topological mechanical metamaterials based on self-locking, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105555

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Highlights 

A novel topological mechanical metamaterial based on self-locking mechanism is proposed.



The proposed topological mechanical metamaterial exhibits significant one-way displacement transmission property.



Design guide of displacement reduction factor is proposed to achieve more significant directional response.

1

One-way transmission in topological mechanical metamaterials based on self-locking Xiao-Fei Guo, Li Ma* Center for Composite Materials, Harbin Institute of Technology, Harbin 150080, PR China

Abstract Directional response is of great use both in daily life and in technological areas, for example, in a semiconductor diode or the directional propagation of waves. We realize a mechanical directional response, i.e., one-way displacement transmission, using a novel topological mechanical metamaterial. The one-way transmission is predicted by theoretical analysis and verified by simulations and experiments. The difference in the directional response when the structure is loaded at opposite sides can reach many dozen-fold. A design guide for the displacement reduction factor is proposed to achieve a more significant one-way transmission property. The proposed topological mechanical metamaterial provides a new concept in achieving a directional response and could be applied as signal transport and isolation, mechanical diodes, mechanical logic gates and so on.

Keywords: One-way transmission; Directional response; Topological mechanical metamaterials 1. Introduction Mechanical metamaterials are materials with engineered structures that exhibit unprecedented mechanical properties that are difficult to achieve in bulk materials. Over several decades, various attractive and unusual properties of mechanical metamaterials have been proposed, including negative Poisson’s ratio[1-8], pentamode[9-11], multi-stable[12-16], programmable[17-20], tension-twist[21-23], negative thermal expansion[24-27], reconfigurable architected[28-30], acoustic black holes[31] and so on. The unusual properties make mechanical metamaterials have many attractive potential applications, such as invisibility cloaks[9], crash mitigation devices[14], soft robots[32, 33], soliton splitters and diodes[34], piezoelectric phononic crystal nanobeams[35],

*

Corresponding author, Tel.:+86 451 86402739; fax: +86 451 86402739. E-mail address: [email protected] (Li Ma) 2

multi-resonator mechanical metamaterials[36] and so on. The directional response[37-43] of metamaterials has been a focus of attention in recent years. These metamaterials break transmission symmetry and realize signal propagation along specific directions or paths. Due to their intrinsically polarized property, topological metamaterials have become a novel class in the realization of a directional response. Topological metamaterials are inspired by topological insulators[44-46], which are insulating in the bulk but have protected conductive states on the surface, even in the presence of impurities. In the past few years, topological metamaterials have been applied to many fields, such as magnetic systems[47], acoustics[48-50], photonics[51, 52] and piezoelectric materials[53]. In 2014, Kane and Lubensky[54] applied the topological idea to classical mechanical systems by establishing a connection between a mechanical dynamical matrix and quantum electronic Hamiltonians, and these authors first presented the concept of topological mechanical metamaterials. These authors proposed that topological mechanical metamaterials are mechanically polarized, which provides a novel way to realize a directional response in static mechanical systems. Moreover, similar to the topological insulators, the directional response of topological mechanical metamaterials is robust and topologically protected and is not affected by imperfection or external perturbation. However, only a few papers have been devoted to the studies of the static mechanical directional responses of topological mechanical metamaterials. Rocklin[55] proposed that if a force is exerted at a point in the bulk of a topological Maxwell lattice, stress or strain is induced only on one side (the polarized side) of the force, which is an entirely directional bulk response in static systems. Coulais[56] presented a fishbone structure and a 2D topological mechanical metamaterial that can break reciprocity and show different output displacements under excitation at different sides. Bilal[57] presented a truss-like periodic lattice, which can obtain an elastic polar response and realize a different surface stiffness at two of its opposing faces. Guo[58] proposed a 2D topological honeycomb structure and 3D topological lattice whose opposite surfaces have significantly different indentation hardness.

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Although the topological concept is an effective way to realize directional response, few studies have been conducted in this field. Little is known about the relationship between topological metamaterials and mechanical directional responses. Therefore, we propose a novel mechanical metamaterial based on a topological mechanism, which can realize one-way displacement transmission. The distribution of the zero mode of the topological mechanism is calculated to predict the directional response. This prediction is verified by theoretical analysis. Deformation of the proposed topological mechanical metamaterial under single-side loading is simulated by finite element method and verified by experiments. Moreover, a detailed parametric analysis is conducted by simulations to provide a design guide that realizes different degrees of the one-way transmission property. 2. Configuration design To obtain a one-way transmission property, it is necessary to identify a structure that breaks symmetrical deformation. A topological mechanism is an effective way to realize this design concept because of the intrinsic polarization. The topological mechanism adopted is deformed from Kagome structure[54], as shown in Fig. 1(a). The transformation rule is given by defining the position of three nodes r of representative elements[54], shown as Eqs. (1) and (2) and in Fig. 1 (b), where x p is the transformation coefficient.

x p  ( x1 , x2 , x3 ),

(1)

r  r0  3x p  x 1a 1 ,

(2)

0 0 0 where r is the vector of the nodes of the initial Kagome structure, and r1  a1 / 2 , r2  a3 / 2 , and

r30  0 . ai is the primitive vectors[59]. pi is the vector perpendicular to ai . The lengths of both ai and pi are 2l .

4

Fig. 1 (a) Illustration of the transformation from the initial Kagome structure (grey) to the corresponding topological mechanism (blue). (b) Transformation rule based on three nodes of a representative element. (c) Basic geometric parameters of a representative element. (d) Sketch of the proposed topological mechanism. Motion decreases from the right boundary to the left side (green arrows) and increases from the left boundary to the right side (red arrows).

The topological mechanism consists of triangular rigid parts connected by hinges. The triangular rigid part cannot be deformed and can only rotate around the hinge point. When a triangular rigid part at the boundary moves one unit, according to the constraint, triangular parts are rigid and cannot be deformed, we can deduce the movement of the next triangular part. If the next triangular part just moves less than one unit, due to the periodicity of the topological mechanism, similarly, the movement of the third triangular part should be less than the second triangular part. This reduction continues along the horizontal direction, as illustrated by green arrows at right side in Fig. 1(d), so that motions could occur freely at this boundary without stretching the triangular rigid parts. This is also called a zero mode. In contrast, when a triangular rigid part at the opposite boundary (the left boundary in Fig. 1(d)) moves one unit, the next triangular rigid part must move more than one unit to keep the rigid part undeformed. This amplification continues along the horizontal direction, shown as red arrows on left side in Fig. 1(d). The rightmost cell must move exponentially more than one unit provided that the number of cells is sufficient. However, the configuration cannot allow this motion to amplify infinitely; thus, the structure will incur self-locking. Moreover, the deformation of the elastomer structure does not completely conform to the rule of topological 5

mechanism. Due to the principle of minimum potential energy, the structure tends to stretch triangular blocks at the self-locking side to a greater degree instead of rotating around the links. Therefore, even if the number of cells is small, the boundary without a zero mode still shows a self-locking property. According to the conclusions above, to realize a one-way transmission property, a topological mechanism should be designed whose one side (the right boundary) has a zero mode while the other side (the left boundary) does not have a zero mode. The calculation method of zero mode count refers to the index theorem in [54]. A zero mode count per unit cell on surface has two contributions:

   T  L , where

(3)

 L is the local count, and  T is the topological count. The local count  L is only related to

the location of the selected boundary and it is independent of the transformation coefficient x p . According to [54], the local counts on the left and right boundary in a topological mechanism transformed from Kagome structure are

 Lleft   Lright  2.

(4)

Let the zero mode count on the left boundary be zero, that is

 left   Lleft  Tleft  0.

(5)

 Tleft  2.

(6)

 T  G  RT / 2 ,

(7)

We obtain

The topological count

 T is

where G is the reciprocal lattice vector, which is normal to the specific surface. RT is the topological polarization, which depends on the transformation coefficient x p and remains invariant in the whole structure.

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G left 

4 aˆ1 , a

RT 

G right  

4 aˆ1 , a

1 3  a p sgn  x p  , 2 p 1

(8)

(9)

where aˆ1  a1 / a . From Eqs. (6-9), we obtain

x1  0,

x2  0,

x3  0,

(10)

that is,

x p  (, , ).

(11)

From Eqs. (4) and (7-9), we obtain the zero mode count on the right surface.

 right   Lright  Tright  4.

(12)

According to Eq. (11), it can be concluded that we only need to ensure the value of the transformation coefficient x p , that is, x1 is negative, and x2 and x3 are positive. Thus a topological mechanical metamaterial whose right boundary has four zero modes and left boundary has no zero mode is obtained, which can realize a one-way transmission property. Representative elements of the topological mechanical metamaterial are shown in Fig. 1(c). To cause the structure to be more simply visualized, three simple basic geometrical parameters, s , a and b are adopted to express the structure configuration instead of the transformation coefficient x p in the following sections, whose substitutional relations are shown as Eqs. (13-15).

s  2 3x1l ,

(13)

1 l a  3( x1  x2  x3 + ) , 2 s

(14)

l b   2( x3  x2 )+1 . s

(15)

3. Theoretical analysis

7

To verify the feasibility of the design, a theoretical analysis of the proposed topological mechanism is conducted. We assume that the system consists of stiff triangular parts connected with ideal hinges. As the topological mechanism is symmetric along the horizontal central axis, half of a symmetric structure is adopted for analysis (as shown in Fig. 2(a)). We choose three rows of cells (Fig. 1(d)), which is the minimum number of rows that can ensure the mechanism has at least one degree of freedom. There are 6(3N  2) degrees of freedom and 18N 16 geometrical constraints, where N is the number of columns. Thus the topological mechanism has four degrees of freedom. Therefore, four initial displacements must be given to ensure the topological mechanism has only one mode of motion, as shown in the green arrow in Fig. 2(a) (this figure only shows the two initial displacements xi and yi of the half structure).

Fig. 2. (a) Nonlinear behaviour of the topological mechanism. Undeformed configuration is shown as a grey solid line; deformed configuration is shown as a blue dashed line. (b) Illustration of linear motion of a representative element simply supported at top right corner. (c) Diagram of displacement of each node in the topological mechanism under linear motion.

All the parameters shown in Fig. 1(c) are given as follows. 8

h  (1  a) s,

(16)

l1  s 1  b 2 ,

(17)

l2  s (1  a ) 2  b 2 ,

(18)

1  arctan

1 a 1  arctan , b b

(19)

b . 1 a

(20)

 2  arctan

The nonlinear motion of the topological mechanism is shown in Fig. 2(a). The displacement of each node of the topological mechanism caused by the initial displacements xi and yi is calculated by the following equations.

l2 (sin   sin  )  yi  2h,

(21)

l2 (cos   cos  )  xi ,

(22)

xi  xi  bs  l1 cos(  1 ),

(23)

yi  l1 sin(  1 )  yi  s,

(24)

(h  yi)2  ( xi  bs  Ui 1 )2  l22 ,

(25)

  arctan

h  yi  2 , xi  bs  U i 1

(26)

xi 1  Ui 1  as cos  ,

(27)

yi 1  as(sin  1).

(28)

Due to the system of Eqs. (16-28) being implicit, we solved it numerically using MATLAB and obtained U i . Detailed discussions are shown in Section 5. To obtain an explicit result, the condition is degenerated into a linear deformation. Under a linear small motion, the displacement of each node of the representative element, simply supported at the top right corner as shown in Fig. 2(b), is expressed as follows. 9

uA  l1d , xA  u As i n A , yA  u Ac o s

A

,

uB  l2d , xB  u B s i n B , yB  u B cos B .

(29) (30)

By solving Eqs. (29) and (30), the relationship between the displacements of two nodes in the representative element are obtained.

yB  yA  bxA ,

(31)

xB  (1  a)xA.

(32)

Using Eqs. (31) and (32), we deduce the displacements of the nodes in Fig. 2(c), shown as follows.

xi  Ui  x1, xi  xi  x2 ,

yi  0,

(33)

yi  yi  bx2 ,

(34)

xi  xi  (1  a)x2 ,

xi1  xi  x3 ,

yi  yi  bx2 , yi1  yi  bx3 ,

Ui1  xi  (1  a)x3 , Vi 1  yi  bx3.

(35) (36) (37)

To solve Eqs. (33-37), three displacement constraints are given. First, as the mechanism is symmetric along the horizontal central axis, the vertical displacements of the central nodes are zero, that is

Vi 1  0.

(38)

Second, the displacement constraint of the top cell caused by the simple support shown in Fig. 2(b) is

bxi  (1  a) yi.

(39)

Third, the topological mechanism is assumed to be infinite the in horizontal direction.

U i 1 xi 1  . Ui xi Substituting Eqs. (38-40) into Eqs. (33-37), we obtain equations as follows. 10

(40)

x1  Ui , x2  x3 

(41)

Ui . 1 a

(42)

Substituting xi into Eqs. (33-37), we obtain the displacement of each point in Fig. 2(c), as follows.

xi  2Ui ,

xi  (1 

yi  0,

a )U i , 1 a

xi  2Ui ,

xi 1 

yi  

yi  

2a Ui , 1 a

U i 1 

b Ui , 1 a

(43)

(44)

b Ui , 1 a

(45)

yi 1  0,

(46)

a Ui . 1 a

(47)

Thus,

1 U i 1 a   . k Ui 1  a

(48)

According to Eq. (48), U i decreases exponentially from the right side to the left side along the horizontal direction under a small linear motion. Therefore, motion can occur at the right side, which conforms to the design goal. We define k as the displacement reduction factor, which only relates to parameter a and increases with a decrease in a . In contrast, U i obviously increases exponentially from the left side to the right side along the horizontal direction. This increase cannot occur under actual conditions due to the limitation of the configuration, which causes the left side boundary to be self-locking. The conclusions above conform to the design aim based on the distribution of zero modes. The validity of realizing a one-way transmission property using the proposed design is confirmed. More importantly, the displacement reduction factor k is a key parameter to measure the 11

degree of the one-way transmission property. The one-way transmission property is more significant when the displacement reduction factor k is larger. We discuss this in detail in Section 5.2. 4. Methods To apply the proposed design in a real situation, it is necessary to develop the proposed topological mechanism to a mechanical metamaterial. A tiny link is used to replace the hinge point, and polyurethane elastomer is adopted as the material, which ensures that the triangular block can rotate around the tiny link, similar to the motion of the proposed topological mechanism. Numerical simulations and experiments are conducted to verify that the deformation of the topological mechanical metamaterial conforms to the theoretical result analysed above and can realize a one-way transmission property. 4.1 Finite element model The commercial finite element package Abaqus/Standard is used in the simulations. The model is illustrated in Fig. 3. The structure is 194 mm length, 80 mm wide and 4.75 mm thick, and the geometric parameters are s  5 mm , a  1 , b  3 , corresponding to a displacement reduction factor of k =2 . The thickness of the connection between the triangular parts is 1 mm. The number of rows of the structure is three, and the number of columns is eleven. A hexahedral 8-node linear reduced-integration element (C3D8R) is used in the models. The number of finite elements is 47800, which ensures the accuracy of the simulation results. The material is polyurethane, which is assumed to be an isotropic hyperelastic that adopts the Yeoh model. The mechanical properties of the polyurethane are obtained by standard test ASTM D412-16, shown in Fig. 4. Poisson’s ratio is 0.49. The top and bottom rectangular boundaries are fixed with all displacements being zero. The displacement load is applied on the second cell (Fig. 6) to avoid boundary effect, until the force increased to 4 N.

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Fig. 3 Finite element model of the topological mechanical metamaterial

Fig. 4 Stress-strain curve of the standard test (ASTM D412-16)

4.2 Experiments The samples are made of polyurethane 8400, produced by casting the three-component mixed liquor into a mould. The mould is made of photosensitive resin and is produced by 3D printing using stereo lithography apparatus (SLA) technology (UnionTech Lite600). The geometric parameters of the samples are the same as those in the finite element analysis in Section 4.1. The boundary of the sample is fixed by a U-shaped fixture, which is fixed on the uniaxial testing device (Instron 3344), as shown in Fig. 6. Three samples are tested on each side. The speed of the test is 0.5 mm/min. Compared to other unit cells, the cell at boundary is lack of constraint on one side, therefore the motion of boundary cells is more flexible. As shown in Fig. 5, when the rightest boundary cell is loaded, deformation basically occur in this boundary cell, which almost does not transmit to the next cell by the joint node, especially in loading direction. Unfortunately, this deformation is not caused by topological effect, only due to the lack of constraint at boundary. The special boundary deformation will be confused with the deformation caused by topological displacement transmission. Therefore, to avoid boundary 13

effect, displacement load is applied on the second cell by a specific puller till force up to 4N (Fig. 6). The puller is produced by 3D printing and made of resin, whose stiffness is much higher than the material of the samples (polyurethane 8400). Meanwhile, we collected displacement data using a high-resolution camera (Sony ILCE-6000). GOM Correlate (2D digital image correlation (DIC) commercial software) is employed to acquire the displacements of specific nodes within an accuracy of 0.05 mm. A black background plate is placed behind the sample to increase contrast.

Fig. 5. Illustration of boundary effect. Due to the lack of constraint at boundary, the motion of boundary cell is very flexible. The motion applied at the central node hardly transmits to the next cell by the joint node.

5. Results and discussion 5.1 One-way response The deformations of the topological mechanical metamaterial in the experiments are shown in Fig. 6. It could be found from the figure that the one-way transmission property of the proposed topological mechanical metamaterial in this paper is remarkable (see supplementary Movie 1). With a force continually loaded at the left side by a puller, as shown in Fig. 6(a), the whole structure does not incur great deformation, just as if the structure locks. In contrast, when the force is loaded at the right side, the triangular blocks rotate around tiny links, which leads to the whole structure significantly deforming, shown in Fig. 6(b). Thus, the proposed topological mechanical metamaterial is movable at the right side, while it locks at the left side. For the convenience of further analysis, numerical simulations were conducted and compared with the experimental results. Fig. 6(c) shows the comparative deformation chart of the topological mechanical metamaterial between the experiments and simulations with an input force of 4 N. The simulation results show the 14

displacement contour of each point in the horizontal direction. It can be concluded from Fig. 6(c) that the experimental result (upper half) and numerical result (lower half) are in great agreement.

Fig. 6 Deformations of the topological mechanical metamaterial loaded at a single side. Load is applied on the second cell to avoid boundary effect. (a) Structure incurs self-locking when it is loaded at the left side. (b) The structure is deformed markedly when it is loaded at the right side due to the rotation of the triangular parts around the links. (c) Comparative deformation chart of the topological mechanical metamaterial between experiments (upper half) and simulations (lower half) when the input force is 4 N.

The experimental data is shown as charts in Fig. 7-9. Figure 7 shows the experimental output force-displacement curves in comparison with numerical results. It can be concluded that the dispersion of the experiment results is small (the grey area is the error bar). Moreover, the experimental results and numerical results match well. From the data, we can see that under the same input force, the output displacement when the structure is loaded at the right side is much 15

larger than the displacement when it is loaded at left side. This result conforms to the structure being self-locking at the left side and movable at the right side. Figure 8 shows the displacement of each node of the topological mechanical metamaterial when it is loaded at a single side. The nodes at the middle of the structure are selected for output displacements. The labelling of nodes is from small to large from the loading end, as shown in Fig. 1(d) (the loading end is to the right, for example). The label of each node is opposite if it is loaded at the left side. The data for the left-most and right-most nodes are not extracted because of the boundary effect. It can be concluded from Fig. 8(a) and (c) that , during the whole loading process, regardless of the side at which the structure is loaded, displacements of the selected nodes always decrease along the direction away from the loading end, i.e., Ui 1  Ui . The coloured areas represent the error bars of the experiments. Combined with Fig. 8(b), it is shown that if the structure is loaded at the right side, the displacements of the nodes decrease relatively softly away from the right side. When the structure is loaded at the left side, unlike loading at right side, displacements already start to decline quickly during transmission from the loading end (Node 2) to the next cell (Node 3), which embodies the self-locking property in design.

Fig. 7. Force-displacement curves of the topological mechanical metamaterial when it is loaded at single side. (a) Loaded at the left side. (b) Loaded at right side. Under the same external load, compared with right side, the structure incurs only a tiny deformation when it is loaded at the left side due to self-locking.

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Fig. 8. Displacement of each node of the topological mechanical metamaterial when it is loaded at a single side. (a) Displacement-force curve of each node when loaded at the right side. (b) Displacement of each node when the force is 4 N loaded at the right side. (c) Displacement-force curve of each node when loaded at the left side. (d) Displacement of each node when the force is 4 N loaded at left side. Due to self-locking, the displacement of each node when the structure is loaded at the left side is much smaller than when the structure is loaded at the right side. Moreover, displacements start to decline quickly from the loading end (Node 2) to the next unit (Node 3) when the structure is loaded at the left side.

Comparing Fig. 8(b) with Fig. 8(d), it is shown that the displacement of each node when it is loaded at the left side is much smaller than when it is loaded at the right side. This result demonstrates the one-way transmission property of the topological mechanical metamaterial. To U iright side right side evaluate this one-way transmission property, the ratio Ri  left side is defined, where Ui is Ui left side the displacement of the i-th node when the structure is loaded at the right side. Similarly, Ui is

the displacement of the i-th node when the structure is loaded at the left side. Ri reflects the degree of unidirectionality at the position of the i-th node. The larger the value of Ri is, the stronger the

17

unidirectional transmission property is. The results are shown in Fig. 9. From Fig. 9(a), it is shown that the topological mechanical metamaterial has a one-way transmission property during the whole loading procedure, i.e., Ri  1 during the whole procedure. The solid lines are the simulation results. Dashed lines are the average of experimental results, and the colour shadow areas are the error bars of the experiments. Due to the limitation of the accuracy of test facilities, all U i extracted by DIC left side method fluctuate to some extent. Because of the self-locking structure, Ui is very small,

especially when

i is large. Thus, Ri fluctuates significantly even if the measured values of Uileft side

fluctuate slightly, especially the last few nodes (Fig. 9(a)), but the average values of the experimental results basically agree with the simulation results. Furthermore, the dispersion of experimental results of the first few nodes is very small, and experimental results agree well with the simulation results, which proves the simulation results are credible. Figure 9(b) shows Ri of each node when the input force is 4 N. It is shown that the ratios of all the nodes are much larger than one (

) and the

largest Ri can have a value of up to 60, which indicates the remarkable one-way transmission property of the topological mechanical metamaterial. It is worth noting that a structure already has a one-way transmission property when Ri  1 ( Ri  1 in this paper), while the value of Ri only reflects the degree of the one-way transmission property. In addition, the ratio Ri first increases and then declines. This result is caused by the boundary effect. To verify that the decline in Ri for the last few nodes is indeed caused by the boundary effect, models with different lengths are simulated, and the results are shown in Fig. 10. The numbers of the nodes of unit cells in the horizontal direction are set to 11, 16, 21 and 26. Figure 10(a) and (b) shows the displacement of each node when the input force is 4 N. It is shown that the data of the structures with different lengths nearly completely coincide. In other words, an increase in the length of the structure does not influence the displacement of each node in that part of the original length. Moreover, it still has a one-way transmission property even if the dimension of structure is very long. Figure 10(c) shows the value 18

of Ri for each node when the input force is 4 N. The data of the first few nodes nearly completely coincide, and Ri always starts to decline from around the fourth node from the end, which suggests that the decline of Ri is caused by boundary effect. Moreover, with the increase in the length of structure, Ri has a limit value when the displacement reduction factor k of the topological mechanical metamaterial remains unchanged. As shown in Fig. 10(c), when k  2 , the limit value of

Ri is approximately 83.

Fig. 9. One-way transmission property of the topological mechanical metamaterial. (a) Ri for each node with an increase in force. (b) Ri for each node when the input force is 4 N. Ri is the ratio of displacements of each node when the structure is loaded at opposite sides and reflects the degree of directional difference of the displacement transmission. In the whole loading process, Ri  1 , which indicates that the topological mechanical metamaterial has a significant one-way transmission property.

19

Fig. 10. Influence of the length of the structure on the one-way transmission property of the topological mechanical metamaterial. The number of nodes reflects the number of unit cells in the horizontal direction. (a) Displacement of each node when the structure is loaded at the left side. (b) Displacement of each node when the structure is loaded at the right side. (c) Ri for each node always decreases for the last few nodes, which proves that the decrease is due to the boundary effect. All the results above are evaluated when the input force=4 N.

5.2 Parametric studies According to Section 3, it is concluded that U i decreases exponentially with factor k from the right side to the left side under a small linear deformation as shown in Fig. 11(a). Thus, the displacement reduction factor k is the key factor that affects the degree of the one-way transmission property of the topological mechanical metamaterial. To investigate the influence of the displacement reduction factor on the directional response of the topological mechanical metamaterial in detail, parametric studies are conducted by numerical analysis and verified by comparing these results with the theoretical results. Since the displacement reduction factor k  1  a and is independent of other geometric parameters, s  5 and

1 only relates to parameter a

b  3 remain the same in all

cases. Seven levels of the displacement reduction factor are adopted, namely, 2, 2.25, 2.5, 2.75, 3, 20

3.25, and 3.5, corresponding to a  1 ,

4 2 4 1 4 2 , , , , , and , respectively. The results are 5 3 7 2 9 5

shown in Fig. 11(b)-(d). Figure 11(b) and (c) show the displacement of each node of the topological mechanical metamaterial with different displacement reduction factors when the input force is 4 N. It can be concluded that with an increase in k , the displacement of each node decreases slightly. In addition, the theoretical results match well with the numerical results. Figure 11(d) shows the influence of the displacement reduction factor on the one-way transmission property. It is obvious that the value of Ri for each node increases with the increase in k . In other words, the one-way transmission property of the topological mechanical metamaterial is stronger when k is larger. The decline in Ri for the last few nodes is also caused by the boundary effect, as was discussed in Section 5.1.

21

Fig. 11. Influence of the displacement reduction factor

k

on the one-way transmission of the topological

mechanical metamaterial. (a) Displacement of each node decreases exponentially with factor

k

under a small

linear deformation. (b) Displacement of each node under a large nonlinear deformation when the displacement reduction factor is different and when loaded at the left side. (c) Displacement of each node under a large nonlinear deformation when the displacement reduction factor is different and when loaded at right side. (d) Ri increases significantly with the increase in the displacement reduction factor, which reflects that the displacement reduction factor

k is

an essential index to evaluate the degree of the one-way transmission property of the topological

mechanical metamaterial and shows a positive correlation between them. All the results above are evaluated when the input force=4 N.

5.3 Discussion To further evaluate the one-way displacement transmission property of the proposed topological mechanical metamaterial, we compare it with other structures that can achieve static 22

directional responses and draw the plot shown in Fig. 12. It can be seen from Fig. 12 that the directional displacement transmission ratio is Rrot  1.5 of the topological rotating tetragon structure[56] and Rfish  9 of the fishbone structure[56]. For the proposed topological mechanical metamaterial based on Kagome, if considering the boundary effect, the directional displacement transmission ratio is RK  30 when the displacement reduction factor k  2 . If ignoring the boundary effect, the directional displacement transmission ratio can reach RK  83 . Obviously, the degree of static directional displacement transmission is much higher than other existing structures. It is worth noting that RK could be much higher if we allow the displacement reduction factor k to be larger by adjusting the geometrical parameters of the structure. Moreover, the directional response of the fishbone structure is due to its mechanical instability, and therefore, the directional response mainly occurs after the snapping of the structure. In contrast, the directional response of the proposed topological mechanical metamaterial based on Kagome structure continuously exists during the whole loading process. The directional response of the topological rotating tetragon structure relies on exposed boundaries to a certain extent, which leads to the number of unit cells being extremely limited. For the proposed topological mechanical metamaterial based on Kagome structure, the degree of the one-way transmission property remains almost constant, regardless of how many unit cells the structure has. This size-independent feature makes it possible to use the proposed topological mechanical metamaterials in practical application. Furtherly, it’s an interesting work to find out whether other kinds of structure could be topologized to realize directional response, such as honeycomb lattice, tetrahedron lattice and so on, which can enrich the existing configurations.

23

Fig. 12. Comparative plot of directional response property between the proposed topological mechanical metamaterial based on Kagome structure and other structures. Vertical axis is the directional displacement transmission ratio R , which reflects the degree of directional response. Horizontal axis is the number of unit cells in the loading direction. The degree of directional response of the proposed topological mechanical metamaterial based on Kagome structure is much higher than those of the topological rotating tetragon structure and the fishbone structure[56] and is not affected by the size of the structure.

6. Conclusion A novel mechanical metamaterial based on a topological mechanism is proposed, which can realize a one-way transmission property. According to a topological transformation rule, the configuration of the topological mechanism is designed from a transformed Kagome structure. The reason for the directional response is the self-locking of structure, which is predicted by the distribution of zero modes and verified by theoretical analysis. According to an index theorem, it is shown that motions could occur at the boundary with a zero mode, while the boundary without a zero mode is self-locking. The adopted configuration of the topological mechanism is further developed into a topological mechanical metamaterial by using a tiny link to replace the ideal hinge. The proposed topological mechanical metamaterial still retains its one-way transmission property. This observation is verified by numerical simulations and experiments and conforms to theoretical analysis. The difference in the directional response when the topological mechanical metamaterial 24

is loaded at opposite sides can reach many dozen-fold. According to the analysis results, a displacement reduction factor is defined to predict and evaluate the degree of the one-way transmission property of the topological mechanical metamaterial. By discussing the effect of the displacement reduction factor k , a design guide is proposed in which a structure with larger k shows more significant one-way transmission property. By comparing the proposed structure with other existing papers, it can be concluded that the topological mechanical metamaterial based on the Kagome structure extremely enhances the degree of the one-way transmission property. Moreover, the one-way transmission property is not affected by the size of the structure, which makes practical application possible. The one-way transmission property of the proposed topological mechanical metamaterial has broad prospects in many fields, including mechanical diodes, mechanical logic gates, signal transport or isolation and so on. Moreover, the proposed topological mechanical metamaterial provides a new concept in achieving directional response.

Author Contributions: Xiao-Fei Guo: Formal analyisis, Investigation, Methodology, Validation, Writing - original draft. Li Ma: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Writing - review & editing.

Acknowledgements The present work is supported by National Natural Science Foundation of China under Grant No. 11672085. Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References 25

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Graphical Abstract

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