Experimental and computational investigations of novel 3D printed square tubular lattice metamaterials with negative Poisson’s ratio

Abstract

Novel 3D printed square auxetic tubular lattice (SATL) structures were designed, fabricated and investigated. Their mechanical properties were examined by the finite element method and experiments. The height and wall thickness show different effects on the mechanical properties of SATL structures. Compared with the circular auxetic tubular (CATL) structures, the SATL structure has a lower peak force under axial load. Under lateral load, the SATL structure has higher stiffness and specific energy absorption. Moreover, the auxetic effect of the proposed SATL structure is also obvious under lateral load. Then, numerical investigations of several improved SATL structures were carried out, the results show that the improved square auxetic tubular lattice (ISATL) structures have stronger energy absorption capacity under axial and lateral loads. Due to their unique structural design and excellent mechanical properties, the SATL structures and ISATL structures have great potential for applications in civil engineering, vehicle crashworthiness and protective infrastructure.

Full text

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Article published in
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Additive Manufacturing,
Vol. 55 (2022) 102789
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https://doi.org/10.1016/j.addma.2022.102789
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Experimental and computational investigations of novel 3D printed square
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tubular lattice metamaterials with negative Poisson’s ratio
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Dong Han a, Xin Ren a, *, Chen Luo a, Yi Zhang a, Xiang Yu Zhang a, Xue Gang Zhang a, Wei Jiang a,
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Jian Hao a, Yi Min Xie b
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a Centre for Innovative Structures, College of Civil Engineering, Nanjing Tech University, Nanjing, Jiangsu,
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211816, PR China
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b Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne,
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3001, Australia
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* Corresponding author: Email address: xin.ren@njtech.edu.cn (X. Ren).
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Abstract
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Novel 3D printed square auxetic tubular lattice (SATL) structures were designed, fabricated and
15
investigated. Their mechanical properties were examined by the finite element method and experiments. The
16
height and wall thickness show different effects on the mechanical properties of SATL structures. Compared
17
with the circular auxetic tubular (CATL) structures, the SATL structure has a lower peak force under axial
18
load. Under lateral load, the SATL structure has higher stiffness and specific energy absorption. Moreover, the
19
auxetic effect of the proposed SATL structure is also obvious under lateral load. Then, numerical investigations
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2
of several improved SATL structures were carried out, the results show that the improved square auxetic
1
tubular lattice (ISATL) structures have stronger energy absorption capacity under axial and lateral loads. Due
2
to their unique structural design and excellent mechanical properties, the SATL structures and ISATL
3
structures have great potential for applications in civil engineering, vehicle crashworthiness and protective
4
infrastructure.
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Keywords: Auxetic; Tubular structure; Metamaterial; 3D printing; Energy absorption.
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1. Introduction
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Mechanical metamaterials are man-made materials with special properties [1], which can be divided into
8
negative Poisson’s ratio metamaterials [2], metamaterials with zero Poisson’s ratio [3] and negative stiffness
9
metamaterials [4] and so forth according to their specific properties. As part of mechanical metamaterials,
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negative Poisson’s ratio metamaterials have been extensively studied due to their excellent mechanical
11
properties [5-7], which was named as ‘auxetics’ by Evan et al. [8, 9]. Auxetics have many desirable mechanical
12
properties [10-14], such as indentation resistance [15-17], shear resistance [18], fracture resistance [19, 20]
13
and energy absorption capacity [21-24]. According to the geometrical micro-structure, auxetics can be divided
14
into the following categories: foam [25-27], re-entrant honeycomb [28-30], chiral and anti-chiral structures
15
[31-33], etc. It can be applied in different fields according to its mechanical properties and geometric
16
characteristics, such as cushion pads [34], nails [35], intelligent filters [36, 37], and intelligent sensors [38,
17
39]. In addition, auxetics also have a wide range of seismic applications. Zhang et al. [40] designed a novel
18
perforated auxetic core buckling restraint support to explore the hysteretic performance of auxetic
19
metamaterials under cyclic loading. Huang et al. [41] proposed a novel 2D seismic metamaterial that combined
20
auxetic foam and steels for seismic attenuation.
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With the increasing application of auxetics, more and more scholars have joined the researchers of
1
auxetics. Grima et al. [42] firstly claimed that two-dimensional materials containing diamond or star
2
perforations exhibit auxetic behavior when stretched and compressed. Bertoldi et al. [43] investigated a
3
perforated plate structure with an auxetic effect due to elastic instability. Chen et al. [44] designed a novel oval
4
perforated plate structure, which greatly improved the specific energy absorption of the oval perforated plate
5
structure. Linforth et al. [45] designed a novel oval perforated plate with a quadrilateral in the center of the
6
oval removing material to increase specific energy absorption. Although auxetics could be used in many fields,
7
the disadvantages of low early stiffness are often exposed. Therefore, some scholars adopt some methods to
8
enhance the stiffness of auxetics. Zhang et al. [46] proposed a novel re-entrant unit cell, which improves the
9
stiffness and stability of re-entrant structures. Zhang et al. [47] designed a novel auxetic unit cell based on an
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elliptic unit cell, which can enhance the stiffness of auxetic metamaterials by adjusting the geometric
11
parameters. Luo et al. [48] filled foams into re-entrant structures to enhance the stability and stiffness of re-
12
entrant structures. Zhang et al. [49] combined chiral structures with foam materials to improve the energy
13
absorption capacity and stability of chiral structures.
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As a branch of auxetic metamaterials, auxetic tubular lattice structure have become a hot research topic.
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Auxetic tubular lattices were mainly proposed and investigated for the application in medical devices [50] in
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early publications. However, these medical devices just utilized the auxetic behavior under tension and its
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potential application under compression was ignored. Ren et al. [51] proposed the Pattern Scale Factor method
18
to design 3D auxetic tubular lattice structures, enabling the tubular lattice structures to exhibit auxetic behavior
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both in tensile and compression. Subsequently, Ren et al. [35] fabricated and investigated the first auxetic nail,
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which is easy to push in and difficult to pull out. Zhang et al. [52] designed an auxetic tubular lattice structure
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with auxetic effect in both wall thickness and radial direction. Many kinds of auxetic tubular lattice structures
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4
show excellent mechanical properties. However, to the best knowledge of authors, the most studies on auxetic
1
tubular structures mainly focus on the mechanical properties of circular auxetic tubular lattice structures, the
2
research work of square auxetic tubular lattice structure has not been reported yet. The square section has a
3
larger section moment of inertia and a larger section modulus than the circular one, it is known that the energy
4
absorption capacity of the square tubular lattice is better than that of the circular tubular lattice under different
5
loads. Therefore, square section tubular lattice is widely used because of its unique geometric shape and
6
excellent mechanical properties in engineering practice [53]. Therefore, it is of significance to design and
7
explore the mechanical properties of the square auxetic tubular lattice structures, which could also expand the
8
family of auxetic tubular lattice metamaterials.
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The unique geometric design of auxetics makes fabrication relatively difficult, so 3D printing is adopted
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for the fabrication of specimens. Zhang et al. [54] fabricated and investigated a circular auxetic tubular lattice
11
with enhanced stiffness by 3D printing technology. Delcuse et al. [55] explored the effect of 3D printing
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process parameters on the mechanical properties of auxetics. Johnston et al. [56] studied the mechanical
13
properties of auxetics for dual-material printing. Albertini et al. [57] prepared and studied auxetic
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metamaterials by 3D printing composite technology. Jiang et al. [58] studied the mechanical properties of 3D-
15
printed auxetic lattice structure under bending. Novak et al. [59] investigated the mechanical properties of 3D
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printed chiral structures. In addition, Carneiro et al. [60] manufactured and studied printing 3D auxetic
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honeycomb structures. The auxetics fabricated by 3D printing showed excellent mechanical properties,
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however, 3D-printed square auxetic tubular lattice (SATL) with metal is rarely studied. Therefore, it is of great
19
significance to study the mechanical properties of 3D-printed metal SATL structures.
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In this work, a stainless steel SATL structure was fabricated using 3D printing technology and its
21
mechanical properties were studied by the finite element method and experiments. Secondly, based on a
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5
verified numerical model, the effects of height and wall thickness on the mechanical properties of SATL were
1
studied by the finite element method. Then, the effects of different cross section forms on the mechanical
2
properties of auxetic tubular lattice structures were studied numerically. Finally, several improved auxetic
3
tubular lattice structure was proposed and studied.
4
2. Design methods and geometric modelling
5
2.1. Square auxetic tubular lattice structure modeling method
6
The specific generation steps of SATL structure are shown in Fig. 1. In this paper, the dimensions of the
7
unit cell are a = 12 mm, b = 4 mm, L = 17.8 mm. First, the 2D plane is obtained by unit cell array method.
8
Then, the 3D shell is obtained by method of coordinate transformation. In the end, the 3D SATL structure is
9
obtained by the offset method.
10
11

6
Fig.1. Square auxetic tubular lattice structure modeling method.
1
2.2. Geometry modelling of SATL
2
The specific geometric dimensions are shown in the Fig. 2. The length of SATL is H, the inner side length
3
is L, and the wall thickness is T. To maintain the integrity of the SATL, the number of unit cells on one side
4
of the SATL needs to be set to an integer.
5
6
Fig.2. Geometry parameter of SATL structure.
7

7
3. Experimental tests
1
3.1. Fabrication of the test specimens
2
3
Fig.3. 3D printed stainless-steel specimens: (a) dumbbell shaped specimens; (b) front view of SATL
4
structure.
5
All specimens were fabricated by ISLM280 printer (Zrapid Technologies Ltd., China), which is a high-
6
resolution 3D printer with a resolution of 0.1 mm. The rated power and rated voltage of the 3D printer are 6
7
KW and 220 V, respectively. As shown in Fig. 3, this paper adopts SLM laser powder sintering technology to
8
3D printing SATL and its corresponding dumbbell shaped specimens. The matrix material is 316L stainless
9
steel powder. The sintering temperature of stainless steel is 650 ℃, which is the specified working temperature,
10
if the temperature exceeds this, the performance will decline very badly.
11
3.2. Uniaxial tensile test of 3D printing specimens
12
In order to obtain material properties, 3D printing technology was used to manufacture three same
13
dumbbell stainless-steel specimens for parallel tests in Fig. 4. The specific geometric dimensions are shown
14
in Fig. 4 (b), and the thickness of the dumbbell stainless-steel specimen was set to 1 mm. The setting of the
15

8
dumbbell stainless-steel specimens conforms to the requirements of international standards [61]. As shown in
1
Fig. 4 (a), a universal testing machine was used to conduct uniaxial tensile tests on dumbbell specimens with
2
a loading speed of 2 mm/min. Fig. 5 shows a real stress-strain curve fitted with the three dumbbell stainless-
3
steel specimens. The specific material parameters are shown in Table. 1.
4
5
Fig. 4. Uniaxial tensile test: (a) illustration of the tensile experiment; (b) standard size of 3D-printed
6
stainless-steel specimen; (c) 3D printing dog-bone.
7
8
Fig. 5. The stress-strain curve of the test specimen.
9
Table 1.
10

9
Property parameters of stainless-steel specimen.
1
T (mm)
ρ (g/cm3)
E (MPa)
ν
σy (MPa)
εy
1.00
7.93
190,000
0.29
466
0.02
Note: T, ρ, E, ν, are the thickness, density, Young’s modulus, and Poisson’s ratio, respectively. σy is the yield
2
stress, εy is the yield strain.
3
3.3. Uniaxial compression test of the specimens
4
A universal testing machine (KEXIN WDW-100) with the measurement range of 100 kN was used to
5
conduct a uniaxial compression test on SATL structure, as shown in Fig. 6. The loading speed of the testing
6
machine was 2 mm/min. In the whole process, a high-definition camera was used to take photos to record the
7
deformation modes the of the SATL structure. The fill-in light was turned on for light supplement in the course
8
of the experiment. The data acquisition system was used to collect and store time, displacement and load data.
9
10
Fig. 6. Quasi-static uniaxial compression test.
11
3.4. Calculation of Poisson’s ratio
12
In the uniaxial compression test, the SATL structure was marked as shown in Fig. 7, and the deformation
13
modes of the SATL structure were recorded by high-definition cameras. The Poisson’s ratio can be calculated
14

10
by the method of image digitization. The equations for calculating Poisson’s ratio of the SATL structure are
1
shown in formulas (1) and (2).
2
    

 
      (1)
3



      (2)
4
5
Fig. 7. Monitoring points and calculation of Poisson's ratio in two directions: (a) top view; (b) front view.
6
4. Finite element modelling
7
The finite element model was established by commercial software Abaqus in Fig. 8. In the compression
8
simulation, Abaqus/Explicit solver was used to analyze the deformation and force of the model. The upper
9
and lower rigid bodies were respectively used to simulate the base and pressure plate in the test, and reference
10
points were set on the two rigid bodies. The upper reference point was used to apply vertical loads, and the
11
reference point on the bottom rigid body limits all degrees of freedom. The whole process adopted general
12
contact, where the tangential friction coefficient was set to 0.15 and the normal behavior was set to hard
13
contact. After the convergence analysis, the mesh size of 0.4 mm was chosen. Finally, the cell type C3D8R
14
was selected. In the process of model calculation, the horizontal and vertical displacements of 7 points are
15
detected to calculate the Poisson ratio of the SATL structure.
16

11
1
Fig. 8. Finite element model of the SATL structure.
2
5. Experimental results and finite element verification
3
5.1. The curve of the load-displacement and Poisson’s ratio-displacement.
4
The results of the finite element method (FEM) and experiment (EXP) show a good agreement, as shown
5
in Fig. 9. However, 3D printing metal SATL structure has insufficient accuracy and certain defects. Therefore,
6
some certain differences exist between the results of FEM and EXP, especially in the initial elastic stage, the
7
slope of the finite element is high. In addition, the defects in the test lead to some uncertain fractures, resulting
8
in a temporary load drop and a certain difference in the peak load. For Poisson’s ratio-displacement curve, the
9
manual image processing method is used in the test, so some errors may exist. On the whole, the load-
10
displacement curve and Poisson’s ratio-displacement curves of EXP are relatively consistent with the result
11
of FEM, so FEM can represent the basic law of study about SATL structure.
12
The SATL structure has obvious auxetic behavior before reaching the densification point where two sides
13

12
of the short axis of the ellipse contact, and the load begins to increase. Then, with the generation of folds, the
1
load begins to decline and each wave peak produces a fold.
2
3
Fig.9. Comparison of load-displacement and Poisson’s ratio-displacement under uniaxial compression
4
between FEM and EXP: (a) load-displacement curves; (b) Poisson’s ratio curves.
5
5.2. The comparison of deformation modal between the EXP and FEM
6
The deformation modes of FEM are in good agreement with the result in the test, as shown in Fig. 10.
7
As the specimen used in the test is processed by 3D printing technology and has some defects, so the
8
deformation of EXP is also different from the result of FEM.
9
The first stage of the energy absorption process of SATL structure is also the contraction stage with
10
auxetic effect in the deformation process, as shown in Fig. 10 (a). At this stage, the auxetic elliptic unit cell
11
rotates and the short axes of the elliptic unit cells contact each other to produce the first stiffness increase point
12
finally. In the first stage, the load fluctuation is small and the energy absorption of the SATL structure is
13
relatively stable, which also improves the disadvantage of the traditional square tube with a large peak load.
14
It is the second stage of energy absorption of SATL structure, as shown in Fig. 10 (b), namely the folding
15
deformation stage. Two folds are generated in Fig. 10 (b), the folds are first generated in the upper and lower
16

13
parts of the tube due to the great friction between the tube and the pressure plate and the base plate. Each fold
1
leads to a wave peak in the load-displacement curve. Folding deformation is a good energy-absorbing
2
deformation form of a tubular structure, which also makes the material utilization rate of the tubular structure
3
higher.
4
5
6
Fig.10.The comparison of deformation modal between EXP and FEM: (a) deformation stage with auxetic
7

14
behavior; (b) folding deformation stage.
1
6. Parametric study of the auxetic lattice structure
2
The validity of FEM has been verified, and parametric analysis could be carried out by FEM below.
3
Firstly, the influence of height (H) and thickness (L) on the mechanical properties of the auxetic lattice
4
structures is studied. Secondly, the effects of cross section form on the mechanical properties of auxetic lattice
5
structures under axial load are studied. Then, the mechanical properties of the SATL and CATL under lateral
6
load are studied. Finally, two improved square auxetic tubular lattice (ISATL) structures are proposed and
7
examined.
8
6.1. The effect of the H
9
Quasi-static compression of SATL structures which have different heights were carried out by FEM. In
10
order to control the uniqueness of the variables, the inner side length of the SATL structures is 35.6 mm and
11
the wall thickness is 2 mm. To preserve the symmetry and integrity of the SATL structures, the vertical unit
12
cells of the tube are set as 3, 4 and 5 respectively. The specific geometric dimensions of the tube are shown in
13
Table 2.
14
Table 2
15
Geometric parameters of three SATL structures.
16
H (mm)
T (mm)
L (mm)
N1
N2
H/L
53.4
2
35.6
2
3
1.5
71.2
2
35.6
2
4
2.0
89.0
2
35.6
2
5
2.5
Note: H, T, L, are the height, wall thickness, and inner side length of SATL structures, respectively. N1 is the
17
number of unit cells in the width direction. N2 is the number of unit cells in the direction of height.
18

15
The total energy absorption (EA) and specific energy absorption (SEA) are important indexes [42] of
1
structural energy absorption and can be expressed as follows:
2
1) Energy absorption
3
 

 (3)
4
Where F is the compression load, d is the loading displacement.
5
2) Specific energy absorption
6
  
 (4)
7
Where m is the mass of the SATL structure.
8
The height of SATL structures has little influence on the load, as shown in Fig. 11 (a). The higher height
9
of SATL, the higher the peak load compressed to the same strain. Due to the lower height of SATL with a
10
height of 53.4 mm, its compression displacement is small and the number of folds is also small. The
11
appearance of each fold could cause a wave peak. Due to the higher height of SATL structures at 71.2 mm and
12
89.0 mm and the large compression displacement, two folds appear before the nominal strain is equal to 0.35.
13
In the stage with auxetic effect, the auxetic effect of SATL is basically not affected by the height of the
14
tube, as shown in Fig. 11 (b). Some differences are related to the position of the points used to calculate the
15
Poisson’s ratio of the SATL structures. As shown in Fig. 11 (c), EA and SEA enhance as the height of SATL
16
increases. The peak load of SATL which has the height of 89.0 mm is higher, resulting in higher EA and SEA.
17
18
Fig.11. The mechanical properties of SATL structures which have different heights: (a) load-nominal strain
19

16
curves; (b) Poisson’s ratio curves; (c) EA and SEA.
1
6.2. The effect of the T
2
Quasi-static compression of SATL structures which have different wall thickness are carried out by FEM.
3
In order to control the uniqueness of the variables, the inner side length of the AST is 35.6 mm and the height
4
is 71.2 mm. The specific geometric dimensions of the SATL structures are shown in Table 3.
5
Table 3
6
Geometric parameters of SATL structures.
7
H (mm)
T (mm)
L (mm)
N1
N2
L/T
71.2
1.0
35.6
2
4
35.6
71.2
1.5
35.6
2
4
23.7
71.2
2.0
35.6
2
4
17.8
71.2
2.5
35.6
2
4
14.2
Note: H, T, L, are the height, wall thickness, and inner side length of SATL structures, respectively. N1 is the
8
number of unit cells in the width direction. N2 is the number of unit cells in the direction of height.
9
The initial platform load of the SATL structures becomes progressively higher as the wall thickness
10
increases, as shown in Fig. 12 (a). The contact between the two sides of the elliptic short axis forms the
11
densification point of SATL structures, then the load quickly increases and the first peak load appears. The
12
value of peak load gradually increases with the increase of wall thickness. When the thickness of SATL
13
structure reaches 2.5 mm, the structure is difficult to form uniform fold deformation and the load fluctuation
14
is large. Two folds are generated when the wall thickness of the SATL structure is in the range from 1 mm to
15
2.0 mm, the deformation is relatively uniform and the load fluctuation is relatively small. When the wall
16
thickness is 1 mm, the SATL structures has a lower load resulting in lower energy absorption. However, its
17
long platform area is a desirable advantage in protection engineering.
18
The auxetic effect of the SATL with a wall thickness of 1 mm is weaker than that of the SATL with a wall
19

17
thickness of 1.5-2.0 mm, as shown in Fig.12 (b). It is clear that the critical point where the auxetic effect
1
enhances with the increase of wall thickness is between 1-1.5 mm. When the wall thickness exceeds 1.5 mm,
2
the auxetic effect of SATL structures does not enhance with the increase of wall thickness.
3
Before the densification point, the critical point where the SEA of SATL structures increases with the
4
increase of wall thickness is between 1.5-2.0 mm, as shown in Fig. 12 (c). When the wall thickness exceeds 2
5
mm, the SEA of SATL structure does not enhance with the increase of wall thickness. After the densification
6
point, a larger peak value is generated due to the larger mass of the SATL with a 2 mm wall thickness, and its
7
SEA is much larger than the SATL within a 2 mm wall thickness. Therefore, SATL structures of different wall
8
thicknesses could be selected for various situations of energy absorption.
9
10
Fig.12. The mechanical properties of SATL structures which have different wall thickness: (a) load-
11
displacement curves; (b) Poisson’s ratio curves; (c) EA and SEA.
12
6.3. The effect of cross section
13
In order to control the quality of SATL and CATL basically the same, the number of longitudinal unit
14
cells of the two tubes is 4, the number of transverse unit cells is 8, so that the internal girth of the sections of
15
two structures should be kept the same. Axial and lateral uniaxial compression of the two structures are carried
16
out and their mechanical properties are compared.
17

18
6.3.1 The effect of cross section on mechanical properties under axial load
1
Both the SATL and CATL show significant auxetic effects at the stage with auxetic effect, as shown in
2
Fig. 13 (a). At this stage, little difference exists in the deformation of the two structures. At about 12 mm, the
3
contact between the two sides of the short axis of the elliptic unit cells reaches the densification point. As
4
shown in Fig. 13 (b), both SATL and CATL enter the folding deformation stage of deformation process. It can
5
be clearly seen that the integrity of SATL is poorer than that of CATL due to the presence of connection
6
transition areas where one surface of SALT is connected to another surface. This results in more uniform
7
deformation of CATL and more irregular deformation of SATL in the folding deformation stage.
8
9

19
1
Fig.13. The comparison of deformation modal between the SATL and CATL: (a) stage with auxetic
2
behavior; (b) stage with folding deformation.
3
Within the loading displacement of the 20 mm, the load-displacement curves of SATL and CATL are
4
basically consistent, as shown in Fig. 14. However, as the deformation of SATL is not as uniform as that of
5
CATL and the material utilization rate is lower, the load of SATL is lower than that of CATL after 20 mm. As
6
a result, EA and SEA of SATL are lower than CATL. It can be seen from Fig. 14 (b) that the auxetic effect of
7
ACT is stronger than that of SATL in the deformation stage with auxetic effect. The reason is that the auxetic
8
effect is weakened by link transition areas of SATL.
9
10
Fig.14. The mechanical properties of SATL and CATL: (a) load-displacement curves; (b) Poisson’s ratio
11
displacement curves; (c) EA and SEA.
12

20
6.3.2The effect of cross section on mechanical properties under lateral load
1
The mechanical properties of SATL and CATL under lateral load were also studied. Fig. 15 shows the
2
deformation process of the first 8 mm displacement. The results show that the surface perpendicular to the
3
loading plate of SATL also has auxetic effect under lateral loading. The SATL’s material with auxetic effect
4
is concentrated, eventually the two sides of the short axis of the ellipse contact and the load remains at a large
5
level. CATL has no auxetic effect and has smaller load.
6
7
Fig.15. The comparison of deformation modal between the SATL and CATL.
8
The load of SATL is much higher than that of CATL due to the support of the two surfaces perpendicular
9
to the loading plate of SATL, as shown in Fig. 16. At the same time, the EA and SEA of SATL are larger than
10
that of CATL. Therefore, SATL retains the auxetic effect while greatly improving the existing auxetic lattice
11
structure’s shortcomings of low lateral stiffness and poor energy absorption capacity under lateral compression.
12
13

21
Fig.16. The mechanical properties of SATL and CATL: (a) load-displacement curves; (b) EA and SEA.
1
6.4 The study of improved SATL structures
2
On the basis of the original square auxetic tubular lattice (SATL), improved square auxetic tubular lattice-
3
1 (ISATL-1) and improved square auxetic tubular lattice-2 (ISATL-2) are proposed by adding internal supports,
4
as shown in Fig. 17. Control the thickness of the internal support as well as the wall thickness of the external
5
tubular structure, and calculate wall thicknesses of the three structures by the principle of the same quality as
6
possible. The external tubular structure and the internal support are modeled as a whole. The specific
7
geometric parameters of the three structures are shown in Table 4. Axial and lateral quasi-static compression
8
of three kinds of structures are carried out by FEM, and the mechanical properties of three kinds of structures
9
are compared.
10
11
Fig.17. The three square auxetic lattice tubular structures: (a) original square auxetic tubular lattice; (b)
12
improved square auxetic tubular lattice-1; (c) improved square auxetic tubular lattice-2.
13
Table 4
14
Geometric parameters of three square auxetic lattice tubular structures.
15
Type
H (mm)
T (mm)
L (mm)
N1
N2
SATL
71.2
2.00
35.6
2
4

22
ISATL-1
71.2
1.60
35.6
2
4
ISATL-2
71.2
1.34
35.6
2
4
Note: H, T, L, are the height, wall thickness, and inner side length of square auxetic lattice tubular structure,
1
respectively. N1 is the number of unit cells in the width direction. N2 is the number of unit cells in the direction
2
of height.
3
6.4.1 The study of square auxetic tubular lattice structures under axial load
4
In the stage which has auxetic behavior, the load of SATL is slightly higher than that of the other two
5
structures, as shown in Fig. 18. However, the peak loads of ISATL-1 and ISATL-2 were lower than those of
6
SATL. During the folding deformation stage, the structures are more stable due to synergies between the
7
external tubes and internal supports of ISATL-1 and ISATL-2. It is clear from Fig. 18 (a) that the load of
8
ISATL-1 and ISATL-2 is greater than that of SATL in folding deformation stage. As can be seen from Fig. 18
9
(c), EA and SEA of ISATL-1 and ISATL-2 are also higher than those of SATL, and ISATL-2 has the most
10
outstanding energy absorption capacity.
11
The auxetic effect of ISATL-1 and ISATL-2 does not decrease significantly compared with SATL, as
12
shown in Fig. 18 (b). At the later loading stage, the deformation of the external tubular structure and the
13
internal support is more stable because of the cooperative deformation.
14
To sum up, when the auxetic effect of ISATL-1 and ISATL-2 does not decrease significantly, the
15
fluctuation of load-displacement curves is reduced and EA and SEA are improved. The advantage of ISATL-
16
2 is even more obvious.
17

23
1
Fig.18. The mechanical properties of square auxetic tubular lattice structures under axial load: (a) load-
2
displacement curves; (b) Poisson’s ratio curves; (c) EA and SEA.
3
6.4.2 The study of square auxetic tubular lattice structures under lateral load
4
Both ISATL-1 and ISATL-2 have higher loads than SATL during lateral compression, as shown in Fig.
5
19 (a). The surface perpendicular to the loading direction of the three kinds of structures shows obvious auxetic
6
effect. After reaching the densification point, the stiffness of the three structures has been improved obviously.
7
Because ISATL-1 has the largest amount of vertical support material, it also has the highest peak load. It is
8
clear from Fig. 19 (b) that EA and SEA of ISATL-1 and ISATL-2 have been significantly improved. EA and
9
SEA of ISATL-2 also performed best, that thanks to lattice design for greater overall stability. By the way,
10
although the load of SATL is low, its load fluctuation is lower.
11
In conclusion, different square auxetic tubular lattice structures can be designed for different protection
12
scenarios. Among them, EA and SEA of the ISATL-2 are the largest, while the load fluctuation of the SATL is
13
the smallest.
14

24
1
Fig.19. The mechanical properties of square auxetic tubular lattice structures: (a) load-displacement curves;
2
(b) EA and SEA.
3
7. Conclusion
4
In this work, novel square auxetic tubular lattice (SATL) metamaterials have been designed, fabricated
5
and examined. Firstly, the mechanical properties and deformation modes of the proposed SATL are analyzed
6
by experiment (EXP) and finite element method (FEM). Secondly, the influence of geometric parameters of
7
the SATL on mechanical properties is explored. Then, the mechanical properties of the SATL and the circular
8
tubular lattice (CATL) under axial and lateral load compression are studied. Finally, the mechanical properties
9
of the two improved SATL are compared with those of the original SATL by FEM. From the results of the
10
above study, the following conclusions can be drawn:
11
1) With the increase of the height of the SATL, the auxetic effect is almost unchanged, but the peak force,
12
energy absorption (EA) and specific energy absorption (SEA) increase gradually when tubular lattices are
13
loaded to the same strain.
14
2) With the increase of the wall thickness of the SATL, the load increases gradually. When the wall thickness
15
is within a certain range, the auxetic effect increases gradually; when the wall thickness is beyond this
16

25
range, the auxetic effect does not increase.
1
3) Before the densification point, when the wall thickness of the SATL is within a certain range, SEA
2
gradually increases with the increase of the wall thickness, and when the wall thickness is beyond a
3
certain range, SEA does not increase with the increase of the wall thickness. After the densification point,
4
SEA enhances with increasing wall thickness.
5
4) Under axial compression, the auxetic effect of the SATL has a slight difference from that of the CATL and
6
the difference of early load is moderate. In the later stage, the deformation of the CATL is more uniform,
7
so the load is larger, and EA and SEA are also larger. However, the load fluctuation of the SATL is small.
8
5) Under lateral loading, SATL still shows a significant auxetic effect. Moreover, the indicators of lateral
9
stiffness, EA and SEA of the SATL are much higher than that of the CATL. In addition, two improved
10
SATL designs, ISATL-1 and ISATL-2, not only retain a large auxetic effect but also improve the overall
11
stability of the tubular lattice.
12
6) Under the axial load, the EA and SEA of ISATL-1 and ISATL-2 are improved to some extent compared
13
with SATL and the fluctuation of the load is also reduced, among these three, ISATL-2 has the best
14
mechanical properties. Under lateral load, the lateral stiffness of ISATL-1 and ISATL-2 increases
15
compared with SATL, and the peak load enhances more obviously. Both ISATL-1 and ISATL-2 have the
16
enhanced EA and SEA, but the indicators of ISATL-2 are better.
17
The novel square auxetic tubular lattice (SATL) metamaterials have desirable mechanical properties,
18
which could be applied in a wide range of scenarios, e.g., energy absorption and protection devices. However,
19
this paper has only explored the mechanical properties and energy absorption under the condition of quasi-
20
static compression. Investigating the impact resistance of the SATL metamaterials at middle and high speed
21
is necessary. The SATL and ISATL structures developed in the study have great potential for applications in
22

26
civil engineering, vehicle crashworthiness and protective infrastructure and other related fields in the future.
1
2
Acknowledgments
3
This work was supported by the National Natural Science Foundation of China (grant numbers 51978330,
4
51808286, 51778283); Qing Lan Project of Jiangsu Province; Natural Science Foundation of Jiangsu Province
5
(grant number BK20180710).
6
7
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