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1
Article published in 1
International Journal of Mechanical Sciences, Vol. 223 (2022) 107286 2
https://doi.org/10.1016/j.ijmecsci.2022.107286 3
Design and mechanical characteristics of auxetic metamaterial with 4
tunable stiffness 5
Xian Cheng a, Yi Zhang a, Xin Ren a, *, Dong Han a, Wei Jiang a, Xue Gang Zhang a, Hui Chen Luo a, Yi Min Xie b
6
a Centre for Innovative Structures, College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China 7
b Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne 3001, Australia 8
* Corresponding author. 9
Email: xin.ren@njtech.edu.cn 10
Abstract 11
Auxetic materials are a class of mechanical metamaterials. They exhibit lateral expansion 12
under tension, and lateral contraction under compression. Such metamaterials have attracted 13
increasing attention due to their unusual mechanical behaviour and various potential applications. 14
However, the stiffness of auxetic structures is much weaker than that of solid ones due to their 15
porous structure. Therefore, many researchers try to enhance the stiffness of auxetic cellular 16
structures to broaden their potential applications. In this study, re-entrant unit cells with different 17
variable stiffness factors (VSF) were designed to achieve the tunability of stiffness from the aspect 18
of tuning the densification strain. Experimental and numerical analyses were carried out to verify 19
2
the accuracy between the designed and the actual VSF. It is found that the compaction points of 1
the proposed re-entrant structures could be tuned quantitatively using the defined VSF, which 2
provides a new method for the optimal design of negative Poisson’s ratio unit cell. 3
Key words: Negative Poisson’s ratio, Auxetic, Mechanical metamaterial, Densification strain, 4
Variable stiffness factor 5
1. Introduction 6
Materials with negative Poisson’s ratio [1, 2] are called "auxetic" materials by Evans et al. [3] 7
Such materials contract (expand) under compression (tension) [4], exhibiting counter-intuitive 8
deformation characteristics. Compared with traditional engineering materials, auxetic materials 9
have excellent shear resistance [5], indentation resistance [6], synclastic behavior [7], fracture 10
resistance [8] and energy absorption performance [9]. Because of these desirable properties, 11
auxetics possess many potential applications in civil engineering [10, 11], protective engineering 12
[12, 13], medical treatment [14], intelligent materials [15], and so on. To be more specific, they 13
could be used in automobile anti-collision devices [16, 17], esophageal stents [18], nails [19], and 14
shock absorbers [20]. When an auxetic structure is compressed, the internal periodic cellular 15
structure gradually compacts. Therefore, the stress-strain curve of the auxetic structure has a 16
longer stress plateau stage than that of the traditional material, and the stiffness of the structure 17
increases again after compaction. 18
There are many kinds of auxetic materials, at present, Ren et al. [2] have reviewed several 19
auxetic material structures, including concave structure [21, 22], chiral structure [23-25], rotating 20
polygon structure [22, 26], perforated plate structure [27, 28] and other structures [29]. The re-21
3
entrant structures are a typical kind of auxetic material, including the re-entrant honeycomb model 1
[30, 31], double arrow model [32-34] and star model [35]. Among these auxetic structures, the re-2
entrant structure [36] is particularly concerned in consideration of the desirable auxetic 3
performance [37] and manufacturing convenience. 4
As the stiffness of the re-entrant structure is low generally, many researchers have tried to 5
enhance their overall stiffness in different ways while maintaining the auxetic effect. Li et al. [38] 6
tried to replace the side of re-entrant honeycomb with a rigid rectangle and found that the effective 7
Young’s modulus of rigid rectangle structure and negative Poisson’s ratio effect are better than 8
that of conventional re-entrant honeycomb. The inclined ribs were added to the traditional re-9
entrant recesses by Baran et al. [39] in 2020, which improved the in-plane stiffness of the re-10
entrant unit cells without significantly losing the auxetic effect. In the same year, the vertical part 11
of the traditional re-entrant structure was replaced by the diamond structure to improve the 12
stiffness and strength of the structure by Logakannan et al. [40]. The traditional stretch lattice has 13
the characteristics of negative Poisson’s ratio but weak stiffness, a cell with enhanced stiffness in 14
which the dilatant lattice is perpendicular to the re-entrant direction is proposed by Chen et al. 15
[41]. This work is applied to the application of auxetic materials that not only need negative 16
Poisson’s ratio but also need high stiffness, e.g., energy absorption devices. It is worth noting that, 17
by coating nickel layer on the surfaces using electroless plating method, Zheng et al. [42] found 18
that the stiffness of auxetic metamaterials were enhanced for several orders without sacrificing 19
the auxetic effect. Although the structures proposed in the aforementioned publications have 20
improved the stiffness of auxetics, a systematic method to design the auxetic unit cells with 21
enhanced stiffness is rare reported. Compared with other methods, the advantage of the VSF 22
4
method is that it can quantitatively control the compacted point, so that the structure design is 1
more accurate. The disadvantage is that currently it can only be designed with a single material 2
and applied to two-dimensional structures. 3
Achieving the tunable mechanical properties of the structure by changing the configuration 4
of unit cells is a common approach. An adaptive auxetic composite tubular structure with tunable 5
stiffness and great stability was proposed by Zhang et al. [43], and the densification point of the 6
structure can be adjusted artificially. Ren et al. [44, 45] and Zhang et al. [46] proposed novel 7
auxetic tubular structures [47] and three-dimensional metamaterials. The mechanical properties 8
of proposed structures could be easily tuned by adjusting PSF (Pattern Scale Factor). Zhang et al. 9
[48] proposed a design concept to adjust the densification strain of auxetics for tuning the 10
compression stiffness indirectly. However, only the design of the auxetic oval unit cell was studied 11
by the VSF method and the applicability of this design method is limited. The re-entrant unit cell 12
is more suitable to design using the VSF method due to its high porosity, compressibility and 13
designability. 14
Researchers have carried out various studies to improve the compressive strength and other 15
superior mechanical properties of auxetic structures. Two types of foam-filled honeycomb 16
structures were introduced by Luo et al. [49] which could be used in protective engineering. The 17
foam-filled re-entrant honeycomb had higher stiffness and bearing capacity. Meena et al. [50] 18
introduced the star-shaped re-entrant unit cells between two consecutive S-shaped unit cells, it 19
shows better tensile results and higher compressive strength, stiffness and Young’s modulus [36]. 20
The auxetic effect of the proposed structure was achieved by sacrificing the structural stiffness. A 21
preliminary study was carried out by Chen et al. [41], aiming at predicting the effective 22
5
mechanical properties of concave structures and designing new cellular structures with enhanced 1
stiffness and negative Poisson’s ratio. Nevertheless, it does not possess the characteristics of 2
stiffness regulation. 3
In this work, the VSF method is extended to the re-entrant unit cells. A series of novel re-4
entrant unit cells are designed, manufactured, and tested. The designed accuracy is experimentally 5
and numerically estimated by comparing the real and designed VSF. The influences of the VSF 6
value, re-entrant angle and thickness on the real densification compacted point are examined using 7
the validated finite element models. 8
2. Design method of auxetic unit cell with a variable stiffness of re-entrant 9
In this study, the re-entrant honeycomb is introduced for variable stiffness optimization. Other 10
re-entrant structures could also be optimized in a similar way. This method mainly includes the 11
following three steps: Firstly, determining the size of a single re-entrant cell; Secondly, designing 12
the initial and variable stiffness areas; Thirdly, cutting out the variable stiffness unit cell for 13
meeting the required VSF value, then the variable stiffness structure could be generated by array 14
isometric transformation. 15
The deformation mode of re-entrant structure (VSF = 100%) compaction is the cellular 16
contact of ligament a1, a2, in the lateral and ribs d1, d2 in the upper and lower (Fig. 1(a)). While 17
the upper ligament b1 and lower ligaments b2 do not contact densely at the same time. The main 18
innovation of this design is to increase the variable stiffness area. In order to make the upper and 19
lower ligaments of the original re-entrant unit cell structure contact before the lateral ligaments 20
when compression, so convenient to quantitatively reach the preset dense point. 21
6
The first step is to determine the size of a single re-entrant unit cell, as shown in Fig. 1(a), 1
where d = 25.26 mm and a = 11.34 mm. The overall offset of 1.61 mm along the outer side is used 2
as the outer frame, and to ensure that 2a1 (a2) < d1 (d2) (to escape early contact between b1 and b2), 3
VSF = 100% (e.g., the initial cell without variable stiffness) is obtained (Fig. 1(a)). 4
In the second step, in order to tune the densification strain conveniently and accurately, only 5
the variate θ is changeable and the length a is equal to c (Fig. 1(b)). The re-entrant unit cell is 6
divided into the initial area (blue) and variable stiffness area (pink), as shown in Fig. 1(c). The 7
size of the unit cell is shown in Tab le 1 . 8
The third step is to build the whole variable stiffness structure through an array arrangement. 9
The variable stiffness models with VSF = 40%, 60% and 80% are designed respectively. And the 10
array arrangement in two orthogonal directions is carried out to form the auxetic structure with 11
variable stiffness, as shown in Fig. 1. 12
13
Fig. 1. Illustrations of design method: (a) The initial area combined with (b) the variable stiffness 14
area obtained (c) the new cell, longitudinally stretched D distance (d). The four re-entrant 15
7
structures (e) with VSF = 40%, 60%, 80% and 100% obtained by periodic arrangement (c). 1
2
Table 1. Size of the unit cell 3
Name L H a1(a2) d1(d2) c t D
Length (mm)
31.33 17.86 11.34 25.26 11.34 1.61 20
4
= 1
(1) 5
(：Design proportion ：Variable stiffness angle ：Re-entrant angle) 6
The range of variable stiffness ratio factor (VSF) value of inner auxetic cell satisfies: 7
Under the circumstances of
＞
2 8
The value of 1 and can be estimated as ： 9
0 (2) 10
01 (3) 11
In the cases of < 2 12
The value of 1 and can be estimated as ：
13
θθ
d
2c
arctan 1≤≤
(4) 14
1
η
θ
2d
c
arctan
≤≤
(5) 15
This work only considers the case of 2a1 (a2) < d1 (d2), in which the left and right nodes inside 16
(Fig. 1(a) the yellow one) the cell could not contact during compaction. What’s more, the local 17
buckling caused by misalignment of internal angle during compaction can be avoided, and the 18
areas with variable stiffness could contact and reach densification point in prediction. 19
8
3. Experiment 1
In the experimental section, four parts are included. Firstly, the design and fabrication of the 2
specimens were carried out. Secondly, the mechanical properties were tested with the same 3
stainless steel material. Thirdly, the uniaxial compression experiment of the variable stiffness re-4
entrant structure was carried out. Finally, the experimental results were analyzed and discussed. 5
3.1 Design and fabrication of specimens 6
Two specimens with VSF = 40% and 100% were manufactured by laser cutting technology. 7
The length × height × thickness was designed as 114 mm×73 mm×20 mm, as shown in Fig. 2. 8
The 304 stainless steel was adopted as the base material due to its superb ductility, section 9
shrinkage and impact value. 10
3.2 Mechanical properties of stainless steel materials 11
Five stainless steel dog bone samples were designed according to standards [51] and 12
manufactured using the same batch of stainless steel with the above compression samples. A 13
universal tester (WAW-600-G) was used for the uniaxial tensile test with a loading speed of 3 14
mm/min. The properties of stainless steel are given in Tabl e 2 . 15
Table 2. Material properties of stainless-steel. 16
Type T (mm) E（GPa）
σ
y (MPa)
σ
u (MPa)
ε
u
Stainless-steel 1 190 360 863 0.158
Note: Where the T is thickness, and the E is Young’s modulus. 17
9
3.3 Uniaxial compression experiment of re-entrant structure with variable stiffness 1
A camera is used to record the transverse and longitudinal deformation of the structure to 2
calculate Poisson’s ratio by using the image processing method. The calculation method of 3
Poisson’s ratio is discussed in the next section in detail. The reaction force of the auxetic 4
metamaterial before densification is smaller than the densification stage. Therefore, a testing 5
machine with a loading range of 600 kN was adopted to ensure the accuracy of experimental data. 6
The compression process is stopped manually when the force is close to 600 kN. The variable 7
stiffness characteristics of the designed structure were studied by quasi-static compression test 8
with a strain rate of 10-3 s-1. The compression process goes on until the specimens were completely 9
compacted. 10
The lateral and uniaxial deformation is recorded by a camera. As can be seen in Fig. 2, 16 11
white points are marked in the re-entrant node for calculating the Poisson’s ratio using the image 12
processing method. The computing equations of the equivalent Poisson’s ratio are shown in 13
formulas (3 and 4): 14
=
=
= ()()
()()
(23, 2 3) (3) 15
=
(23, 2 3) (4) 16
Where the horizontal and vertical distances of the interspaced points of the undeformed 17
structure are represented by Dx and Dy respectively, and the white points in Fig. 2 are P11, P12... 18
Pmn. X and Y are the real-time coordinates of these points. Based on the image processing method, 19
the Poisson’s ratio is calculated by the position changes of the 16 connecting ribs on the surface 20
(Fig. 2). 21
10
1
Fig. 2. The specimens fabricated by laser cutting of VSF = 40% (a) and 100% (b), their length L 2
is 114 mm, height H is 73 mm and thickness d is 20 mm. 16 white points P11–P44 have been 3
marked in the figure, which is convenient for calculating the structural Poisson’s ratio later. 4
5
Fig. 3. First row (a) gives the deformational processes of the specimen with VSF = 40% until the 6
structure is compacted. Second row (b) gives the deformational processes of the specimen with 7
VSF=100% until the structure is compacted. (Scale bar: 10 mm) 8
3.4 Analysis and discussion of experiments 9
The experimental deformation process of two metal samples with VSF = 40% and VSF = 10
100% is shown in Fig. 3. The phenomenon of compressive shrinkage can be clearly found in both 11
11
VSF = 40% and VSF = 100% metallic structures in the red box (Fig. 3), which indicates the 1
Poisson’s ratio of the two specimens is negative. Compared with the specimens with VSF = 40%, 2
the lateral contraction of the specimen with VSF = 100% is not obvious until the nominal strain 3
reaches 0.338. The experiment shows that the deformation of structure with VSF = 100% is 4
asymmetric during the compaction. However, as for VSF = 40% one, the auxetic effect is obvious 5
and symmetric throughout the whole process. The reason may be that when the VSF value is small, 6
the variable stiffness area is larger. When deformed, it can be easier to contact in advance, which 7
makes the deformation more stable. On the contrary, when the variable stiffness area is small, the 8
deformation is easy to lose stability. The experimental samples of VSF = 40% and VSF = 100% 9
showed that the deformation modes of the re-entrant structures with variable stiffness are 10
consistent with the initial re-entrant ones, maintaining the original compression shrinkage 11
characteristics. Besides, the two designed structures with variable stiffness have finally reached a 12
complete dense state. 13
4. Finite element analysis 14
The numerical section mainly includes two parts. The validity of the finite element was firstly 15
verified. Then, through three different aspects, VSFs, different re-entrant angles θ and different 16
thicknesses t, to analyze the influence of their changes on the real VSF, and then optimize the 17
structure. 18
4.1 The setup of the finite element model 19
To verify the consistency of the experimental and parametric analysis, a finite element 20
12
analysis was performed by using the Abaqus/Explicit solver. The geometric model of finite 1
element analysis was consistent with the parameters of experimental specimens. The upper and 2
lower were two discrete rigid plates, and the specimens are in the middle of them. The upper part 3
was fixed, and the lower part was loaded with static compression, as shown in Fig. 4. The red 4
points in Fig. 4 were marked by P11, P12... Pmn for calculating the Poisson’s ratio and illustrated 5
in section 3.3. The large geometrical deformation and complex contact nonlinearity of the whole 6
structure were taken into account in simulation [52]. The models were established using solid 7
element (C3D8R) with a global size of 0.6 mm. The narrowest ligaments were divided into at 8
least three layers to satisfy the most basic tension-compression and flexural effects. The general 9
contact was adopted for all finite element models. For the interaction module in the software, the 10
normal behaviour of hard contact and tangential behaviour of the penalty function with the friction 11
ratio of 0.15 was chosen in general contact. The verification of mesh density and loading rate had 12
been conducted. Plastic elasticity properties are adopted in finite element simulation, where the 13
elastic modulus is 190 GPa and the Poisson’s ratio is set as 0.3 (Table 2). 14
15
Fig. 4. The schematic diagram of the finite element model. The top and bottom are rigid plates, 16
13
and the specimens are placed in the middle. The upper rigid plate is fixed, the lower one is applied 1
with static load, and the displacement is carried out until the specimen is compacted. 2
3
Fig. 5. First row (a) gives the deformation process of the finite element model with VSF = 40% 4
until the structure is compacted. Second row (b) gives the deformation process of the finite 5
element model with VSF = 100% until the structure is compacted. 6
4.2 Finite element model verification 7
The deformation process of the specimen with VSF = 40% and VSF = 100% is shown in Fig. 8
5, which is basically the same as the experimental deformation process shown in Fig. 3. By 9
comparing the simulated deformation process and stress-strain curve with the experiment, it can 10
be found that the experiment and simulation curves basically coincide, indicating the finite 11
element model is verified effectively. The finite element model is further verified by comparing 12
the stress-strain curves, as shown in Fig. 6. Although the compacted point of the simulated results 13
is higher than that of the experimental results, the overall trends of the two curves are similar. 14
After careful examination of the deformation, some ribs of the structure compact first because the 15
deformation process was unstable when the VSF value is low. Due to the lack of variable stiffness 16
area in the model of VSF=100%, the joints are easily deformed under compression, so the overall 17
14
deformation is unstable. When VSF=40%, the rib will contact the variable stiffness area (Fig. 1(c)) 1
first, and the joint is more stable, which makes the overall deformation stable. Since the surface 2
of the laser-cut component is not completely flat, it can be speculated that the interior is not 3
completely homogeneous. Compared with the inhomogeneous material, the finite element is a 4
homogeneous material, which makes the difference between them. As shown in Fig. 6, the 5
stiffness increases sharply when the two specimens are compacted. Failure criteria are not defined 6
in finite element simulation. Therefore, the slight difference between the two curves might be 7
attributed to the imperfection of the specimens during the process of fabrication. 8
In order to explore whether the compacted point of the structure could be tuned, the 9
deformation process of the finite element is observed. According to the results in the finite element 10
analysis, the stress is mainly distributed at the cell junction before densification. With the 11
development of densification, the whole unit cell gradually enters the plastic stage, and then, the 12
stage of stiffness enhancement appears. The corresponding strain-Poisson’s ratio curve is 13
outputted, and it is found that the simulated densification strain is close to the test results, with an 14
acceptable error of 0.005. The error is caused by the tilt of the hydraulic press head and the non-15
uniformity of the components. 16
15
1
Fig. 6. The comparison of experimental and numerical results, the nominal stress-nominal strain 2
curves of VSF = 40% (a) and VSF = 100% (b), Poisson’s ratio-nominal strain curves of VSF = 3
40% (c) and VSF = 100% (d). 4
In this section, data results from different VSFs experiments and simulations are compared. 5
The nominal strain and nominal stress of the cellular materials could be 6
written as (5 and 6): 7
=
(5) 8
16
=
(6) 1
where Δy was the compressive distance, and F0 was the external force of the indenter. L0 2
and S0 denoted the initial height and initial cross sectional area of the honeycomb, respectively. 3
4.3 Parameter analysis of different VSF value 4
In order to further confirm the accuracy of the four kinds of variable stiffness models with 5
VSF = 40%, 60%, 80% and 100%, involved models are all established. The representative volume 6
element is shown in Fig. 1 (e). Finite element analysis is conducted respectively according to the 7
above simulation method. 8
9
Fig. 7. The comparison of numerical results of four different VSFs, nominal stress-nominal strain 10
curves (a) for four models with VSF = 40%, 60%, 80%, 100%; Poisson’s ratio-nominal strain 11
curves (b) for four models with VSF = 40%, 60%, 80%, 100%. (Where the star is the strain when 12
each structure is compacted) 13
The finding of the compacted point of the structure is the position of abrupt slope change in 14
the stress-strain curve or the stress-Poisson’s ratio curve. The calculation method of real VSF is 15
17
based on the compacted strain when VSF = 100% (u0) and the compacted strain obtained by other 1
different VSFs (ui) values are divided by it (ui/u0). If this is consistent with our design proportion, 2
then the compaction of this structure is tunable. Fig. 7 shows the nominal stress-nominal strain 3
and Poisson’s ratio nominal strain curves of four models with VSF = 40%, 60%, 80% and 100%. 4
The compacted points are marked with asterisks and corresponding densification cells are given 5
in it. These curves show the expected phenomenon of adjustable compression points. The real 6
VSF obtained by the stress-strain curve is compared with that obtained by the strain-Poisson’s 7
ratio curve. As shown in Fig. 8, it is easy to see that the real proportion obtained by simulation is 8
basically similar to the proportion originally designed. It is found that the real VSF obtained by 9
strain-Poisson’s ratio curves becomes smaller than that of stress-strain curves. The Poisson’s ratio 10
is taken from the node displacements, the stress transfer is basically on the longitudinal rib, and 11
the node is also on the rib, so the Poisson’s ratio has a great influence on it. The reason may be 12
that the expansion effect increases after compaction, while the Poisson’s ratio is smaller and the 13
compaction occurs later. 14
The ideal state is the real VSF exactly corresponding to the designed proportion when the 15
structure reached the densified strain. From the comparison of real VSF (Fig. 8), the design 16
proportion is basically consistent with the real proportion. The design proportion could be used to 17
replace the real proportion, reaching the standard of adjustable compaction points. 18
18
1
Fig. 8. Bar chart comparison of the real VSF achieved by nominal stress-nominal strain curves 2
and Poisson’s ratio-nominal strain curves. 3
4.4 Parameter analysis of different re-entrant angles 4
Four models with angles of 45°, 50°, 55° and 60° are set up to investigate the effect of the re-5
entrant angle θ on structural compaction and deformation patterns. Fig. 9 shows the stress-strain 6
curves corresponding to the real VSF at different re-entrant angles. With the increase of the re-7
entrant angle θ, both the initial peak stress and the plateau stress increase, which indicated that the 8
larger the envelope area would obtain, the more energy could be absorbed by the structure. As the 9
re-entrant angle θ increases, the amount of material increases. Thus, the bearing capacity of the 10
overall structure improved. 11
19
1
Fig. 9. The comparison of numerical results of four different re-entrant angles θ, nominal stress-2
nominal strain curves for four models with VSF = 40% (a), 60% (b), 80% (c), 100% (d). (Where 3
the star is the strain when each structure is compacted) 4
When VSF = 40%, four sets of models with different re-entrant angles θ of 45°, 50°, 55°and 5
60° were generated, respectively. In order to further study the influence of θ value, a quasi-static 6
compression simulation was carried out, and the analysis results are shown in Fig. 10. It can be 7
seen that when the re-entrant angle θ increases, the compacted strain gradually decreases, which 8
is very unfavorable for determining the real VSF later. Herein, the real VSF is defined by the 9
compacted strain of different VSFs divided by the compacted strain of VSF = 100%. 10
20
1
Fig. 10. The comparison of numerical results of four different angles θ at VSF=40%, nominal 2
stress-nominal strain curves (a) for four models with VSF = 40%; Poisson’s ratio-nominal strain 3
curves (b) for four models with VSF = 40%. (Where the star is the strain when each structure is 4
compacted) 5
6
Fig. 11. Bar chart comparison of real VSF and designed proportions of four angles. 7
The bar chart of the real VSF and design proportion at different re-entrant angles is shown in 8
Fig. 11. With the increase of the angle, the real proportion of the structure gradually become low. 9
However, the maximum deviation is 4.8%, which is within the allowable error range. The possible 10
reason is the material of the variable stiffness area and the compacted strain gradually decreases 11
21
as the re-entrant angle increases. 1
4.5 Parameter analysis of different t value 2
By comparing the finite element models, it can be found that the stressed part of the structure 3
under compression is mostly on the lateral ribs, and the stress spreads from the ribs to the variable 4
stiffness areas until compaction. This shows that thickness is an important parameter for 5
parametric studies. In the case of VSF = 40%, five groups of models with different element wall 6
thicknesses of t = 1.2 mm, 1.4 mm, 1.6 mm, 1.8 mm and 2.0 mm, respectively. Combined with 7
the article of Masters and Evans [53], t/L < 0.2 can be seen that the local hinging dominates. The 8
simulation found that when the model with t = 0.5 mm was compacted, only the bending 9
deformation occurred. But we clearly saw that when the structure with t = 0.1 mm, not only 10
bending deformation, but also higher-order buckling deformation occurs before compaction, as 11
shown in Fig. 12. It can be seen that the ligament bends in a higher-order flexion manner at t < 12
0.1 mm, not just bending. In reality, structures usually do not sacrifice too much stiffness to ensure 13
the Poisson's ratio effect, so t=0.5 mm can be considered the critical point. 14
15
Fig. 12. Higher-order buckling before densification at t = 0.1 mm. 16
In order to further study the influence of thickness t on the structure, quasi-static compression 17
was simulated, and the analysis results are shown in Fig. 13. 18
22
1
Fig. 13. The comparison of numerical results of five different thicknesses t at VSF = 40%, nominal 2
stress-nominal strain curves (a) for five models with VSF = 40%; Poisson’s ratio-nominal strain 3
curves (b) for five models with VSF = 40%. 4
5
Fig. 14. The curve of critical (compacted) strain (a) as a function of t at VSF = 40% and real VSF 6
(b) obtained by different thicknesses t. 7
The results show that the stress plateau gradually increases with the increase of t (Fig. 13), 8
which contributes to the improvement of energy absorption performance. The relationship 9
between compaction strain and t is basically linear, as shown in Fig. 14(a). Comparing the five 10
23
groups of finite element results again, the compression area of the structure is basically 1
concentrated on the lateral ribs. The compacted strain is largely influenced by the thickness of the 2
lateral ribs, which is related to the deformation in the later stage. Similarly, it is worth noting that 3
t has little influence on Poisson’s ratio of these models before compression. It can be seen from 4
Fig. 14(b), the real VSF increases with the increase of the thickness in a certain thickness range. 5
The controllability of the designed structure with t = 2.0 mm has the highest real VSF 38%, which 6
is the closest one to the design proportion. On the one hand, by adjusting t to a certain extent, not 7
only the original auxetic behavior of the structure could be maintained, but also the plateau stress 8
could also be tuned. On the other hand, in order to effectively modify the accuracy of the designed 9
VSF percentage, the compacted strain points can be adjusted to a small extent by modifying the t 10
value. 11
Conclusion 12
In this study, a series of metal auxetic metamaterials are proposed using the variable stiffness 13
factor (VSF) method. Two 2D metal samples were designed, manufactured and tested to verify 14
the effectiveness of uniaxial compression. In order to verify the accuracy of the VSF method under 15
quasi-static conditions, a finite element model was established for verification. A parametric 16
analysis is carried out for different VSF values, re-entrant angles, and cell wall thickness. Based 17
on the obtained results, the following conclusions can be drawn: 18
The compacted strain of the auxetic metamaterials designed by the VSF method is highly 19
tunable. Several parameters that have a substantial effect on the real VSF, including designed 20
VSFs, different re-entrant angles θ and different thicknesses t. 21
24
Investigating different VSF values by comparing real VSF and designed proportion found 1
that the controllability of the structural compacted point is basically realized. Re-entrant angle θ 2
is also an important structural parameter. With the increase of the angle θ, the real VSF of the 3
structure gradually becomes low. The stress plateau and auxetic effect of the re-entrant structure 4
can be improved through quantitatively adjusting the compacted strain by changing the θ value. 5
By adjusting the thickness of the unit cell t in the range of t/L<0.2 and 1.2 mm < t < 2.0 mm, the 6
real VSF increases as the thickness t increases. The results show that the accuracy of the real VSF 7
percentage can be effectively modified with fine-tuning of the thickness t, while the original 8
negative Poisson’s ratio effect of the structure can be maintained. 9
The results provide new ideas for the design of novel metal metamaterials with superior 10
mechanical properties in the future. However, this design method also has some limitations, for 11
example, asymmetric deformation in the experiment has a great influence on the stress-strain 12
curve, especially in the compacted stage. It can be seen that asymmetric deformation is inevitable 13
when VSF is high. In addition, it is also worth further research to design three-dimensional 14
metamaterials based on unit cells with variable stiffness, such as variable stiffness tubular 15
structures. These auxetic metamaterials with different geometrical configurations could expand 16
potential applications of auxetics in civil, medical and protective engineering. 17
Acknowledgments 18
This work was supported by the National Natural Science Foundation of China (grant 19
numbers 51978330, 51808286, 51778283); Qing Lan Project of Jiangsu Province; Natural 20
Science Foundation of Jiangsu Province (grant number BK20180710). 21
25
1
References 2
[1] Lakes R. Foam structures with a negative poisson’s ratio. Science 1987; 235:1038-40. 3
[2] Ren X, Das R, Tran P, Ngo TD, Xie YM. Auxetic metamaterials and structures: a review. Smart Mater Struct 4
2018;27:23001. 5
[3] Evans K, Nkansah M, Hutchinson I, Rogers S. Molecular network design. Nature 1991;12:124. 6
[4] Liu J, Yao X, Wang Z, Ye J, Fu J. A flexible porous chiral auxetic tracheal stent with ciliated epithelium. 7
Acta Biomater 2021;124:153-165. 8
[5] Yang W, Li ZM, Shi W, Xie BH, Yang MB. Review on auxetic materials. J Mater Sci 2004;39:3269-79. 9
[6] Li T, Liu F, Wang L. Enhancing indentation and impact resistance in auxetic composite materials. Compos 10
Part B Eng 2020;198:108229. 11
[7] Zeeshan M, Ali M, Riaz R, Anjum AS，Ahmad S. Optimizing the auxetic geometry parameters in few Yarns 12
based auxetic woven fabrics for enhanced mechanical properties using grey relational analysis. J Nat Fibers 13
2021:1-12. 14
[8] Hu JS, Wang BL, Li JE, Wang KF. Thermal shock resistance behavior of auxetic ceramic honeycombs with 15
a central crack or an edge crack. Ceram Int 2020;46:11835-11845. 16
[9] Ingrole A, Hao A, Liang R. Design and modeling of auxetic and hybrid honeycomb structures for in-plane 17
property enhancement. Mater Design 2017;117:72-83. 18
[10] Zhang Y, Ren X, Zhang XY, Huang TT, Sun L, Xie YM. A novel buckling-restrained brace with auxetic 19
perforated core: Experimental and numerical studies. Eng Struct 2021;249:113223. 20
[11] Huang TT, Ren X, Zeng Y, Zhang Y, Luo C, Zhang X Y, Xie YM. Based on auxetic foam: A novel type of 21
26
seismic metamaterial for Lamb waves. Eng Struct 2021;246:112976. 1
[12] Bodaghi M, Serjouei A, Zolfagharian A, Fotouhi M, Rahman H, Durand D. Reversible energy absorbing 2
meta-sandwiches by FDM 4D printing. Int J Mech Sci 2020;173:105451. 3
[13] Wei Y, Yang Q, Tao R. SMP-based chiral auxetic mechanical metamaterial with tunable bandgap function. 4
Int J Mech Sci 2021;195:106267. 5
[14] Kelkar PU, Kim HS, Cho K, Kwak JY, Kang C, Song H. Cellular Auxetic Structures for Mechanical 6
Metamaterials: A Review. Sensors-Basel 2020;20:3132. 7
[15] Guo M, Yang H, Ma L. Design and characterization of 3D AuxHex lattice structures. Int J Mech Sci 8
2020;181:105700. 9
[16] Jiang H, Ren Y, Jin Q, Zhu G, Hu Y, Cheng F. Crashworthiness of novel concentric auxetic reentrant 10
honeycomb with negative Poisson's ratio biologically inspired by coconut palm. Thin Wall Struct 11
2020;154:106911. 12
[17] Chen G, Cheng Y, Zhang P, Cai S, Liu J. Blast resistance of metallic double arrowhead honeycomb 13
sandwich panels with different core configurations under the paper tube-guided air blast loading. Int J 14
Mech Sci 2021;201:106457. 15
[18] Ali MN, Rehman IU. Auxetic polyurethane stents and stent-grafts for the palliative treatment of squamous 16
cell carcinomas of the proximal and mid oesophagus: A novel fabrication route. J Manuf Syst 2015;37:375-17
395. 18
[19] Ren X, Shen J, Tran P, Ngo TD, Xie YM. Auxetic nail: Design and experimental study. Compos Struct 19
2018;184:288-298. 20
[20] Hajmohammad MH, Kolahchi R, Zarei MS, Nouri AH. Dynamic response of auxetic honeycomb plates 21
integrated with agglomerated CNT-reinforced face sheets subjected to blast load based on visco-sinusoidal 22
27
theory. Int J Mech Sci 2019;153-154:391-401. 1
[21] Zhang J, Lu G, Ruan D, Wang Z. Tensile behavior of an auxetic structure: Analytical modeling and finite 2
element analysis. Int J Mech Sci 2018;136:143-154. 3
[22] Yang H, Ma L. Design and characterization of axisymmetric auxetic metamaterials. Compos Struct 4
2020;249:112560. 5
[23] Attard D, Farrugia PS, Gatt R, Grima JN. Starchirals–A novel class of auxetic hierarchal structures. Int J 6
Mech Sci 2020;179:105631. 7
[24] Lu X, Tan VBC, Tay TE. Auxeticity of monoclinic tetrachiral honeycombs. Compos Struct 8
2020;241:112067. 9
[25] Ma C, Lei H, Liang J, Wu W, Wang T, Fang D. Macroscopic mechanical response of chiral-type cylindrical 10
metastructures under axial compression loading. Mater Design 2018;158:198-212. 11
[26] Mizzi L, Salvati E, Spaggiari A, Tan J, Korsunsky AM. Highly stretchable two-dimensional auxetic 12
metamaterial sheets fabricated via direct-laser cutting. Int J Mech Sci 2020;167:105242. 13
[27] Usta F, Türkmen HS, Scarpa F. Low-velocity impact resistance of composite sandwich panels with various 14
types of auxetic and non-auxetic core structures. Thin Wall Struct 2021;163:107738. 15
[28] Yao J, Sun R, Scarpa F, Remillat C, Gao Y, Su Y. Two-dimensional graded metamaterials with auxetic 16
rectangular perforations. Compos Struct 2021;261:113313. 17
[29] Wu W, Liu P, Kang Z. A novel mechanical metamaterial with simultaneous stretching- and compression-18
expanding property. Mater Design 2021;208:109930. 19
[30] Wang Z. Recent advances in novel metallic honeycomb structure. Compos Part B Eng 2019;166:731-741. 20
[31] Qi C, Jiang F, Yang S, Remennikov A. Multi-scale characterization of novel re-entrant circular auxetic 21
honeycombs under quasi-static crushing. Thin Wall Struct 2021;169:108314. 22
28
[32] Ren X, Das R, Tran P, Ngo TD, Xie YM. Auxetic metamaterials and structures: a review. Smart Mater 1
Struct 2018; 27: 23001. 2
[33] Wang H, Lu Z, Yang Z, Li X. In-plane dynamic crushing behaviors of a novel auxetic honeycomb with 3
two plateau stress regions. Int J Mech Sci 2019;151:746-759. 4
[34] Gao Q, Liao W, Wang L. An analytical model of cylindrical double-arrowed honeycomb with negative 5
Poisson's ratio. Int J Mech Sci 2020;173:105400. 6
[35] Wei L, Zhao X, Yu Q, Zhu G. A novel star auxetic honeycomb with enhanced in-plane crushing strength. 7
Thin Wall Struct 2020;149:106623. 8
[36] Li X, Wang Q, Yang Z, Lu Z. Novel auxetic structures with enhanced mechanical properties. Extreme 9
Mech Lett 2019;27:59-65. 10
[37] Zhang J, Lu G, You Z. Large deformation and energy absorption of additively manufactured auxetic 11
materials and structures: A review. Compos Part B Eng 2020;201:108340. 12
[38] Li D, Ma J, Dong L, Lakes RS. Stiff square structure with a negative Poisson's ratio. Mater Lett 13
2017;188:149-151. 14
[39] Baran T, Öztürk M. In-plane elasticity of a strengthened re-entrant honeycomb cell. Eur J Mech A Solid 15
2020;83:104037. 16
[40] Logakannan KP, Ramachandran V, Rengaswamy J, Gao Z, Ruan D. Quasi-static and dynamic compression 17
behaviors of a novel auxetic structure. Compos Struct 2020;254:112853. 18
[41] Chen Z, Wu X, Xie YM, Wang Z, Zhou S. Re-entrant auxetic lattices with enhanced stiffness: A numerical 19
study. Int J Mech Sci 2020;178:105619. 20
[42] Zheng X, Guo X, Watanabe I. A mathematically defined 3D auxetic metamaterial with tunable mechanical 21
and conduction properties. Materials & Design. 2021 Jan 15;198:109313. 22
29
[43] Zhang XY, Ren X, Wang XY, Zhang Y, Xie YM. A novel combined auxetic tubular structure with 1
enhanced tunable stiffness. Compos Part B Eng 2021;226:109303. 2
[44] Ren X, Shen J, Ghaedizadeh A, Tian H, Xie YM. Experiments and parametric studies on 3D metallic 3
auxetic metamaterials with tuneable mechanical properties. Smart Mater Struct 2015;24:95016. 4
[45] Ren X, Shen J, Ghaedizadeh A, Tian H, Xie YM. A simple auxetic tubular structure with tuneable 5
mechanical properties. Smart Mater Struct 2016;25:65012. 6
[46] Zhang XY, Wang XY, Ren X, et al. A novel type of tubular structure with auxeticity both in radial 7
direction and wall thickness. Thin Wall Struct 2021;163:107758. 8
[47] Luo C, Han CZ, Zhang XY, Zhang XG, Ren X, Xie YM. Design, manufacturing and applications of 9
auxetic tubular structures: A review. Thin Wall Struct 2021;163:107682. 10
[48] Zhang Y, Sun L, Ren X, Zhang XY, Tao Z, Xie YM. Design and analysis of an auxetic metamaterial with 11
tuneable stiffness. Compos Struct 2021:114997. 12
[49] Luo HC, Ren X, Zhang Y, et al. Mechanical properties of foam-filled hexagonal and re-entrant 13
honeycombs under uniaxial compression. Compos Struct 2022;280:114922. 14
[50] Meena K, Singamneni S. Novel hybrid auxetic structures for improved in- plane mechanical properties 15
via additive manufacturing. Mech Mater 2021;158:103890. 16
[51] ASTM, E8/E8M-16a, Standard test methods for tension testing of metallic materials, ASTM, PA, USA, 17
2016. 18
[52] Zhang XY, Ren X, Wang XY, Zhang Y, Xie YM. A novel combined auxetic tubular structure with 19
enhanced tunable stiffness. Compos Part B Eng 2021. 20
[53] Masters, I. G., and K. E. Evans. "Models for the elastic deformation of honeycombs." Composite 21
structures 35.4 (1996): 403-422. 22
