Novel auxetic structures with enhanced mechanical properties

Accepted Manuscript Novel auxetic structures with enhanced mechanical properties Xiang Li, Qingsong Wang, Zhenyu Yang, Zixing Lu

PII: DOI: Reference:

S2352-4316(18)30212-8 https://doi.org/10.1016/j.eml.2019.01.002 EML 426

To appear in:

Extreme Mechanics Letters

Received date : 2 October 2018 Revised date : 8 December 2018 Accepted date : 3 January 2019 Please cite this article as: X. Li, Q. Wang, Z.Y. Yang et al., Novel auxetic structures with enhanced mechanical properties, Extreme Mechanics Letters (2019), https://doi.org/10.1016/j.eml.2019.01.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Novel Auxetic Structures with Enhanced Mechanical Properties

Xiang Li, Qingsong Wang, Zhenyu Yang* and Zixing Lu* Institute of Solid Mechanics, Beihang University (BUAA), Beijing 100083, China E-mail: [email protected] (Z.Y. Yang), [email protected] (Z.X. Lu) Abstract Normally, enhancement of mechanical properties of auxetic cellular materials is at the expense of its negative Poisson's ratio. Here, two novel auxetic cellular structures are proposed, with the Young's modulus and yield strength of the auxetic structures significantly enhanced and without sacrifice of their negative Poisson's ratio in the corresponding perpendicular direction. Analytical solutions and finite element simulations are carried out to predict the mechanical properties of these new auxetic materials, and the results are validated by the experiments. In addition, the Young's moduli and strengths of the novel structures can be designed independently with the corresponding Poisson's ratios. Keywords Auxetic Honeycomb, Yield strength, Young's modulus, Enhanced mechanical properties 1. Introduction In nature, some materials are auxetic, including human tendons[1], nuclei in embryonic stem cells

, -Cristobalite (SiO2) and HT-AlPO4 etc[3, 4]. In addition,

[2]

some nanomaterials also show negative behaviors, for instance, defected graphene, single-layer black phosphorus and single-wall carbon nanotubes with certain

defects[5-7]. Negative Poisson's ratios provide materials with many unique mechanical properties, for instance, excellent shear stiffness, indentation resistance, high fracture toughness, high energy dissipation and acoustic absorption abilities[8, 9]. This is believed to be the motiviation for people to design and manufacture man-made auxetic materials with desired performance, which has board applications in the fields of aerospace[10], biomedical engineering[11,

12]

, sensors[13], and new functional

structures[8]. The first man-made auxetic material was created by Lakes[14]. After this, polymers, an-isotropic fibrous composites and cellular structures were also designed and fabricated to be auxetic[15, 16]. Recently, this metamaterials with negative Poisson's ratios has attracted more and more attention from researchers. The 3D printing technology gave new ideas to the design of the mesostructures of materials, leading to the booming of materials with negative Poisson's ratio[17-19]. By using digital micro-mirror device projection printing and dip-in direct-laser-writing optical lithography, complex auxetic microstructures could be made[20,

21]

. 2D metallic sheet with mutually orthogonal

elliptical voids was experimentally proved to have negative Poisson’s ratio[22]. Babaee et al. also manufactured a series of cellular structures with using soft materials by 3D printing technology[23]. In addition, metamaterials with negative Poisson’s ratio are also investigated by origami[24], which showed bistable and self-locking properties[25]. In the most scenarios, due to the highly-porous micro-structure of auxetic materials, the stiffness of the auxetic materials are relatively low[26]. Recently, researchers were focused on the enhancement of the auxetic materials’ mechanical

performance. 3D auxetic cellular structure designed by Cabras et al. [27] could be used in the large-scale structural systems, for instances, civil and aeronautical engineering. Auxetic composites were manufactured and showed high tensile strength and excellent energy absorption capabilities[28-30]. Moreover, auxetic structures were also modified to have high Young's modulus and compressive strength[31-33]. However, enhanced mechanical properties generally lead to the decrease of the negative Poisson's ratio[27, 30-32]. It is of great significance to propose novel auxetic structures with enhanced mechanical properties and non-reduced negative Poisson's ratio. In this paper, based on the classical auxetic structures, two novel auxetic structures are proposed to combine the auxetic behaviors and the high mechanical performance together. The manufacturing of these novel materials is achieved by means of 3D printing technology. The experiment tests, analytical solutions and finite element simulations show that the Young's modulus and yield strength of the novel auxetic structures can be highly enhanced on one principle direction and keeping the negative Poisson's ratio in the plane almost unchanged. 2. Analytical results of the auxetic structures 2.1. Topology of the novel auxetic structures The novel auxetic structures presented here are 2D metamaterial and consist of beam components made of same material. Based on the classical re-entrant honeycomb (RH) (Fig. 1a) and double arrowhead honeycomb (DAH) (Fig. 1e), the novel structures are named as augmented re-entrant honeycomb (ARH) (Fig. 1b) and augmented double arrowed honeycomb (ADAH) (Fig 1f), respectively. Fig. 1 shows a

top view of the novel structures which are fabricated by a Stereo-lithigraphy apparatus (SLA) printer. A Cartesian coordinate system with in-plane x-y coordinates is built. The pictures of the specimen with the novel auxetic structures are shown in Figs. 1d and 1h, respectively. Details of the specimens can be found in the Methods Section and Supplementary material.

RH

DAH

h

2

l2





t

l

l1

1

t

t/2

(a )

ARH  l

t

(e)

ADAH l2  2

h

t

*

t/2

( b)

t* 2

l1

(c)

(d )

t*  1

(f )

 t* 2

(g )

(h )

Fig. 1. Unit cell and manufactured auxetic honeycombs (a) unit cell of the RH where, h and l are horizontal and inclined cell walls,  is the angle between the horizontal and inclined cell walls and t is the thickness of the cell walls (b) unit cell of the ARH where, t* is the thickness of the augmented cell walls (c) RH with h=11 mm, l=8 mm, t=1.5 mm and =70º(d) ARH with h=11 mm, l=8 mm, t= t*=1 mm and =70º(e) unit cell of the DAH where, l1 and l2 are the length of the two inclined cell walls, 1 and 2 are the angle between the two inclined cell walls and direction y and t is the thickness of the incline cell walls (f) unit cell of the ADAH where, t* is the thickness of the augmented cell walls (g) DAH with l1=10 mm, l2=20 mm, t=1.5 mm and =50º and (h) ADAH with l1=10 mm, l2=20 mm, t= t*=1 mm and =50º. (The 2D

honeycombs are made of photosensitive resin using 3D printing and the red points are used for the DICM) 2.2. Mechanical properties of the novel auxetic structures According to the classic RH structure, Masters et al. gave the effective Young's modulus as a function of the geometrical parameters [34]: 𝐸 𝑡3 𝑠𝑖𝑛𝜃

(1)

𝐸 RH = ℎ s ( −𝑐𝑜𝑠𝜃)𝑙3 𝑐𝑜𝑠2 𝜃 𝑙

where Es denotes the Young's modulus of the matrix and the other parameters are defined in Fig. 1a. With a cell wall added in the vertical direction in the unit cell, the Young's modulus of the ARH can be obtained as 𝐸 ARH = 𝐸 Ad + 𝐸 RH

(2)

where, EAd is the effective Young's modulus of the adding cell walls and is depended on the thickness of the adding cell walls t*. Therefore, the Eq. (2) can be rewritten as 𝐸 ARH =

𝐸𝑠 𝑡 ∗ ℎ ( −cos𝜃)𝑙 𝑙

+

𝐸𝑠 𝑡 3 sin𝜃 ℎ ( −cos𝜃)𝑙3 cos2 𝜃 𝑙

(3)

The macroscopic Poisson’s ratio of the RH is[34]: RH 𝜈yx =−

sin2 𝜃

(4)

ℎ ( −cos 𝜃) cos 𝜃 𝑙

In fact, when stretched on the vertical direction, the adding cell walls do not contribute to the deformation of the inclined cell walls. Consequently, the effective Poisson’s ratio of the ARH is equal to the RH, i. e. ARH RH 𝜈yx = 𝜈yx

Poisson’s ratio of the DAH was obtained by previous researchers

(5) [35]

. However,

their solutions do not take the cell wall thickness into consideration. In fact, the negative Poisson’s ratio of the DAH has an obvious decrease when the cell wall

thickness increase. Here, we give the Poisson’s ratio of the DAH with considering the cell wall thickness: DAH 𝜈yx =−

𝛿𝑥 (𝑙1 𝑐𝑜𝑠 𝜑1 −𝑙2 𝑐𝑜𝑠 𝜑2 ) 𝛿𝑙1 𝑠𝑖𝑛 𝜑1

(6)

In addition, the effective Young's modulus of the DAH is obtained as 𝐸 DAH = 𝐸𝑠

𝑓1 −𝑓2 𝑙1 𝑠𝑖𝑛 𝜑1

×

(𝑙1 𝑐𝑜𝑠 𝜑1 −𝑙2 𝑐𝑜𝑠 𝜑2 ) 𝛿

(7)

Where, 𝑓1 = (𝛿𝑥 𝑠𝑖𝑛 𝜑1 𝑐𝑜𝑠 𝜑1 + 𝛿𝑦 𝑐𝑜𝑠 2 𝜑1 )𝑡/𝑙1

(8)

𝑓2 = (𝛿𝑥 𝑠𝑖𝑛 𝜑1 𝑐𝑜𝑠 𝜑1 − 𝛿𝑦 𝑠𝑖𝑛2 𝜑1 )𝑡 3 /𝑙13

(9)

The detail deduction of Eqs. (6) and (7) can be found in the Supplementary material. The effective Young's modulus of the ADAH EADAH can be calculated by adding the effective Young's modulus of the adding cell walls EAd to the DAH one: 𝐸 ADAH = 𝑙

𝐸s 𝑡 ∗

1 sin𝜑1

+ 𝐸 DAH

(10)

The effective Poisson’s ratio of the ADAH equals to that of the DAH: ADAH DAH 𝜈yx = 𝜈yx

(11)

The yield stress of the DAH is expressed as[35]: DAH 𝜎ys =

𝜎ys sin𝜑2 cos𝜑1 sin𝜑 2(sin𝜑1 +sin𝜑2 )2

(𝜌𝑟DAH )2

(12)

Where, 𝜌𝑟DAH is the relative density of the DAH. The yield strain of the matrix we used in the 3D printing is about 2%. During the tension of the novel auxetic structures, the main deformations of the adding cell walls and original auxetic cell walls are stretching and bending, respectively. Generally, cell walls could have much larger bending deformation than the stretching one. Therefore, before the yield of the adding cell walls, the deformation of the original part of the novel structures could be regarded as linear. Therefore, the yield strengths of the augmented structures can be

computed as 𝑡

ARH 𝜎ys = 𝜎ys ℎ−𝑙cos𝜃 + 𝐸 RH 𝜀ys ADAH 𝜎ys = 𝜎ys 𝑙

𝑡

1 𝑠𝑖𝑛𝜑1

+ 𝐸 DAH 𝜀ys

(13) (14)

where, ys=ys/Es is the yield strain of the matrix. 3. Results and Discussion To evaluate the mechanical properties of the new auxetic structures, uniaxial tension tests are carried out using an electronic universal testing machine (Instron5565) with a loading rate of 1 mm/min. Digital image correlation method (DICM) is adopted to calculate the Poisson’s ratios of the specimen. Four dots are marked at the corner of the center unit cell of the specimen to calculate the effective Poisson’s ratios with DICM (the four markers are shown in Figs.1c, 1d, 1g and 1h). Displacements of the four dots are recorded by the camera, then the dispacements are adopted to calculate the local strains. The effective Poisson’s ratio of a sample is defined as the negative slope of the local strains in two principle directions. To insure the deformation is linear, the strain scope selected to calculate the effective Poisson’s ratio is small than 0.5%. Therefore, the effective Poisson’s ratio can be regarded as an averaged value over a small range of strain. The detail calculation of the effective Poisson’s ratio could be found in the Supplementary material. In addition, Finite element model (FEM) simulations are also carried out to calculate the mechanical properties of the auxetic honeycombs under uniaxial tension. More details on the determination of the mechanical properties of the auxetic structures can be found in the Experimental Section and Supplementary material.

3.1. Experiment results Stress-strain curves and local strains in the two principle directions are obtained by the standard experiments. The stress-strain curves and local strains are then used to obtain the mechanical properties, including Young's modulus, yield strength and Poisson's ratio of the auxetic structures. 10 9 8 7 6 5 4 3 2 1 0 (b) 0.00

ARH (=70) RH (=70)

y (MPa)

ARH (=60) RH (=60)

y (MPa)

10 9 8 7 6 5 4 3 2 1 0 (a) 0.00

0.01

0.02 y 0.03

0.04

0.05

0.01

0.02 y

0.03

0.04

8

6

7

ADAH (=30) DAH (=30)

6

y (MPa)

y (MPa)

5

5

4 3

ADAH (=50) DAH (=50)

4 3 2

2

1

1 0

(c) 0.00

0.02

y

0.04

0.06

0 (d) 0.00

0.02

0.04 y 0.06

0.08

0.10

Fig. 2. Stress-strain curves of the auxetic structures (a) RH and ARH with =60º, (b) RH and ARH with =60º, (c) DAH and ADAH with =30ºand (d) DAH and ADAH with =50º. 3.1.1. Stress-strain curves of the new auxetic structures The stress-strain curves of the auxetic structures obtained by experiments are illustrated in Fig. 2. Compared with the RHs and DAHs, both the Young’s modulus and strength of the ARHs and ADAHs in direction y are remarkably enhanced. One can find that both the fracture strains of the ARHs with =60ºand =70ºare about

3.5%. The fracture strains of the RHs are less than that of the ARHs, which could be attributed to the different in-plane thickness of the cell walls of the two structures. The thicknesses of the cell walls of the RHs with =60ºand =70ºare about 1.60 mm and 1.61 mm, respectively. Both the thicknesses of the cell walls of the ARHs with

=60ºand =70ºare about 1mm. As the fracture of RH structure is due to the bending of the inclined cell walls, thus large thickness of the cell walls reduces bending deformation of the RH structure. 0.08 0.10

ARH (=60) RH (=60)

0.07 0.06

0.08

ARH (=70) RH (=70)

x

0.05 0.06 x

0.04 0.03

0.04

0.02 0.02

0.01 0.00 (a) 0.00

0.01

0.02 y 0.03

0.04

0.06

0.00 0.05 (b) 0.00

0.01

0.02 y 0.03

0.04

0.05

0.08

0.10

0.012 0.010

ADAH (=30) DAH (=30)

0.008

x

x

0.04

ADAH (=50) DAH (=50)

0.006 0.004

0.02

0.002 0.00 (c) 0.00

0.02

y

0.04

0.06

0.000 (d) 0.00

0.02

0.04 y 0.06

Fig. 3. Local strains in the two principle directions of the auxetic structures (a) RH and ARH with =60º, (b) RH and ARH with =60º, (c) DAH and ADAH with =30º and (d) DAH and ADAH with =50º. The fracture strains of the DAHs with =30ºand =50ºare about 6.2% and 8.5%, respectively. However, the fracture strains of the ADAHs are less than 3%. The thicknessnes of the cell walls of the ADAHs with =30ºand =50ºare about 1mm.

Meanwhile, the thicknesses of the cell walls of the DAHs with =30ºand =50ºare about 1.45 mm and 1.52 mm, respectively. The fracture strains of the DAHs with

=30ºand =50ºare much larger than that of the RHs with =60ºand =70º. One reason is that the difference between the topology of the two structures, and another reasons is that the thickness of the cell walls of the DAH structure are less than that of the RHs. 3.1.2. Local strains in the two principle directions of the the auxetic structures Local strains in the two principle directions of the the auxetic structures are shown in Fig. 3. All the structures have positive strain in direction x under tension in direction y. Consequently, the Poisson’s ratios 𝜈𝑦𝑥 of the four auxetic structures are negative. The Poisson’s ratios of the auxetic structures depend on the slope of its local strain curves. Apparently, the slope of the novel auxetic structures is larger than that of the classical ones. Therefore, the novel auxetic structures possess smaller Poisson’s ratio than the original ones. It can be find that the local strains of the RHs with =60º and =70º in direction x are larger than that in direction y (Figs. 3a and 3b). Consequently the Poisson’s ratios of the RHs are smaller than -1. However, the local strains of the DAHs with =30ºand =50ºin direction x are larger than that in direction y (Figs. 3c and 3d), which means the Poisson’s ratios of the DAHs are larger than -1. Both the main deformation modes of the RH and DAH are the bending of the inclined cell walls. For the RHs with =60ºand =70º, most of the deformation of the inclined cell walls contribute to the local strain in direction x. However, for DAHs with =30ºand =50º, most of the deformation of the inclined cell walls contribute to

the local strain in direction y. This is the reason why the fracture strains in direction y of the DAHs are much larger than that of the RHs.

Fig. 4. Comparison between the displacements of the ARH obtained by the DICM and the FEM in (a) direction x and (b) direction y, with a tensile strain y=2.2%. 3.1.3. Comparison between the FEM and DICM Comparisons between in-plane displacement distributions in the ARH obtained by the FEM and DICM are shown in Fig. 4. When the honeycombs is subjected to a tensile loading in direction y, the displacement in direction y is positive, and the maximum and minimum value of displacement of the selected area are the top and bottom boundary of the area, respectively. During the tension in direction y, the honeycomb expands in direction x, consequently the displacement in direction x of the left and right side of the honeycomb is negative and positive, respectively. An

excellent agreement could be found between the DICM and the FEM. The movie of the DICM results of the ARH could be found in the Supplementary material. 3.1.4. Deformation process of the novel auxetic structures The deformation process of the 3D printed models during the tensile test are shown in Fig. 5. Two red dashed lines parallel to the loading direction are used to understand the deformation of the novel auxetic structures. With the help of the red dashed lines the expand of the ARH with =70º is relatively easy to observed. However, the expand of the ADAH with =30ºis small. In fact, the Poisson’s ratio of the ARH with =70ºis -2 and the Poisson’s ratio of ADAH with =30ºis -1. This could explain the different expand degree of the two structures. For both of the structures, an brittle fracture can be observed in the experiments, and both the augmented cell walls and original cell walls are falied. It is believed that the augmented cell walls failed first and then the sudden stress redistribution caused the failure of the re-entrant cell walls.

Fig. 5. Deformation process of the new auxetic structures subjected to tensile loading (a) ARH with =70º and (b) ADAH with =30º. 3.2. Enhancement of the Young’s modulus and yield strength

Young’s moduli, strengths and Poisson’s ratios of the novel auxetic structures are obtained as functions of the angles, as shown in Fig. 6. Mechanical properties of the RHs and DAHs with similar relative density obtained by the experiment method are also given in Fig. 6, which shows that the analytical prediction and FEM simulation agree with the experimental results quite well. Compared with the classical auxetic structures, the Young’s moduli are significantly enhanced. For instance, the average experimental normalized Young’s modulus of the ADAH with l1=10 mm, l2=20 mm, t= t*=1 mm and =30ºis 0.147, which is about 9.8 times than that of DAH with the same relative density. Similarly, the strengths of the novel structures also show remarkable increase. The average experimental normalized strength of the ARH with h=11 mm, l=8 mm, t=1.5 mm and =60ºis 0.156, which is about 4.4 times than that of RH. Cellular structures can be distinguished as stretching dominated structure and bending dominated structure[36]. Stretching dominated honeycombs, for instance triangle and Kagome[37, 38], can be used as lightweight structures as they have higher stiffness and strength than bending dominated ones[39, 40]. However, the RH and DAH are bending dominated structures. The analytical solutions show that the Young’s moduli of RH and DAH scale as [34, 35] and the strengths of the RH and DAH scale as [35, 41] which limits the structural applications of the auxetic structures. However, the novel structures we proposed here overcome the limitation. From Eqs. (3) and (8), the Young’s moduli of ARH and ADAH scales as , i. e. the novel structures could be considered

as

stretching dominated

ones.

It

has been

shown that the

stretching-dominated 2D structures generally have minimum 6 nodes connectivity[36]. Obviously the novel structure ADAH proposed here is one of those materials.

0.15

0.15

0.10

Ey/Es

0.20

ys

0.20

ARH (FEM) ARH (Experiment) ARH (Analytical) RH (Experiment)

0.10

0.05 0.00 40

(a)

0.05

50

60



70

80

90

0.00 20

0.15

y/ys

0.15

yys

0.20

ARH (FEM) ARH (Experiment) ARH (Analytical) RH (Experiment)

0.10

0.05 0.00 40

(c)

0.05

50

60



70

80

90

ARH (Analytical) ARH (FEM) ARH (Experiment) RH (Experiment)

50

ADAH (FEM) ADAH (Experiment) ADAH (Analytical) DAH (Experiment)

0.00 20

30

(d)



40

50

ADAH (Analytical) ADAH (FEM) ADAH (Experiment) DAH (Experiment)

yx

-1.5

yx

-2.5

40

-2.0

-3.5 -3.0



30

(b)

0.20

0.10

ADAH (FEM) ADAH (Experiment) ADAH (Analytical) DAH (Experiment)

-1.0

-2.0 -1.5

-0.5

-1.0

(e)

40

50

60



70

80

90

(f)

0.0

30



40

50

Fig. 6. Comparison between the normalized Young’s modulus, strength and Poisson’s ratio of the novel honeycombs and the classical ones (a) normalized Young’s modulus of RH and ARH, (b) normalized Young’s modulus of DAH and ADAH, (c) normalized strength of RH and ARH (d) normalized strength of DAH and ADAH (e) Poisson’s ratio of RH and ARH and (f) Poisson’s ratio of DAH and ADAH.

3.3. Non-reduction of the Poisson’s ratio With the increase of the Young’s modulus and strength, the Poisson’s ratio does not reduce at all. From the experimental results we can find that the novel structures have higher negative Poisson’s ratios than that of the classical ones with similar relative density, because the thickness of the cell walls of the classical structures is larger than that of the novel ones. For example, the average experimental Poisson’s ratio of the ARH with h=11 mm, l=8 mm, t=1.5 mm and =60ºis about -1.6, while that of RH is about -1.4. In fact, we can also prove that by Eqs. (5) and (10). The Poisson’s ratios of the ARH and ADAH are mostly dependent on the geometries of the original cell walls of the original part of the structures (solid black line in Figs. 1b and 1f). However, the Young’s modulus and strength of the novel structures mainly depend on the augmented part of the novel structures (dash blue line in Figs. 1b and 1f). Consequently, the Young’s modulus and strength can be independent with the Poisson’s ratio. This is very important and useful for the mechanical design of the auxetic structures. It is possible to design a structure with certain Poisson’s ratio firstly and then get the expected Young’s modulus and strength by adding the augmented part. Effective length of the cell walls should be considered in the calculation of the mechanical properties of the novel structures[43, 44], especially for the ADAH, to have a good prediction of the cellular materials with relative high density. The significance of the work presented in this paper goes beyond the results obtained here. For instance, one can even get a ‘J’ shape stress-strain curve by adding

a horseshoe structure or curved ligaments in the RH (see the Supplementary material) [42, 19]

. The 2D structures proposed in this paper can also be possiblely extended to 3D

metamaterials (see the Supplementary material). In addition, the superior mechanical properties can be used in other classical auxetic structures, for example star shape or miss ribs structures. 4. Conclusion In this paper, two novel auxetic cellular structures are proposed, which have negative Poisson’s ratio, high Young's modulus and yield strength simultaneously. The proposed structures can be regarded as stretching dominated structures, at least in one principle direction. The Young’s modulus and strength of the novel structures can be designed independent with its Poisson’s ratio. The results of the study provide a new idea for designing new auxetic metamaterials in the future. 5. Experimental Section Materials. A photosensitive resin is used to 3D print the experimental specimen. The Young's modulus, strength and Poisson’s ratio of the manufactured photosensitive resin matrix are about 2.5 GPa, 55 MPa and 0.3 respectively. These material properties were obtained by testing multiple dog-bone shaped samples in a tensile loading machine and DICM. Detailed stress-strain curves of the manufactured photosensitive resin can be found in the Supplementary material. Fabrication using 3D printing. 3D printing technology is used to fabricate the dog-bone shaped photosensitive resin, ARH, ADAH, RH and DAH systems. The 3D printed specimens were manufactured using a 3D stereolithography laser printer (with

a print resolution of ~50 𝜇m). Before printing, the photosensitive resin is liquid. During the printing an UV light is used to curing the liquid resin to solid with a speed of 2.5 m/s. Following printing, the specimens were cured under UV light with a power of 30 W for 5 mins. All the specimens are printed parallel to the printing surface to eliminate the anisotropy of the bulk materials due to the print direction. For all the systems, 2 unit cells in direction x and 9 unit cells in direction y are contained in each sample. The thickness of the third direction is about 3.2 mm. ARHs with angles =60ºand =70ºare manufactured. Similarly, ADAHs with angles =30ºand

=50ºare manufactured. In addition, RHs and DAHs with similar relative destinies are manufactured. The in-plane thicknesses of the cell walls of the ARHs and ADAHs are about 1 mm. To have a more uniform tensile displacement distribution, a rectangular rigid beam section with 25 mm width is added in the ends of the systems. The as-fabricated specimens are kept at room temperature for at least 7 days to allow for the saturation of the curing. See the Supplementary material for more detail of the specimens. Mechanical testing. Uniaxial tension tests are carried out using an electronic universal testing machine (Instron5565) with a 5 kN load cell and a displacement rate of 1 mm/min. The load-displacement curves measured by the uniaxial tensile tests are then transferred into stress-strain curves based on the measured dimensions of the specimens. A digital camera is used to take photographs of the deformed configurations. Then the photographs are used in the DICM to calculate the local strains of the auxetic honeycombs. Four points are selected to calculate the local

strains of the auxetic honeycombs (see Figs. 1c, 1d, 1g and 1h). The local strains are then used to obtain the Poisson’s ratio of the systems. See the supplementary material for more detail of the mechanical testing. Numerical simulations. Commercial software ABAQUS/ Explicit is used to perform the numerical analyses. The matrix is assumed as elastic perfect plastic with a Young’s modulus of 2.5 GPa and a yield strength about 55 MPa for all the FEM simulations. The Poisson’s ratio of the matrix is 0.3. The stress-strain curve of the matrix is shown in the Supplementary material. The dimension of the FEM is the same with its corresponding experimental ones. The mechanical properties of the auxetic structures are investigated by giving the structures a thickness in the out-of-plane direction. The movement of the samples in the out-of-plane direction is constrained. And 2 elements are meshed in the model on the out-of-plane direction. The models are developed using C3D8R quadrilateral 8 nodes linear brick with reduced integration. We meshed 4 elements in the thickness of the cell wall to guarantee good convergence of results. For ARH with h=11 mm, l=8 mm, t=1.5 mm and =70º, the total number of elements is 72224. Acknowledgments The authors thank the support from the National Natural Science Foundation of China (Grant Nos. 11672013, 11672014 and 11472025), Aviation Science Foundation of China (28163701002) and the Fundamental Research Funds for the Central Universities (Grant No. YWF-17-BJ-Y-30). References

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