Large deformation of an auxetic structure in tension: Experiments and finite element analysis

Accepted Manuscript Large deformation of an auxetic structure in tension: experiments and finite element analysis Jianjun Zhang, Guoxing Lu, Zhihua Wang, Dong Ruan, Amer Alomarah, Yvonne Durandet PII: DOI: Reference:

S0263-8223(17)31687-2 https://doi.org/10.1016/j.compstruct.2017.09.076 COST 8939

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

29 May 2017 5 September 2017 26 September 2017

Please cite this article as: Zhang, J., Lu, G., Wang, Z., Ruan, D., Alomarah, A., Durandet, Y., Large deformation of an auxetic structure in tension: experiments and finite element analysis, Composite Structures (2017), doi: https:// doi.org/10.1016/j.compstruct.2017.09.076

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.

Large deformation of an auxetic structure in tension: experiments and finite element analysis Jianjun Zhang1, 2, Guoxing Lu 1*, Zhihua Wang2, Dong Ruan 1, Amer Alomarah1 and Yvonne Durandet1, 1

Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, VIC 3122, Australia 2

Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China

Abstract: The present paper reports on the post-yield behaviors of an auxetic structure, honeycomb with representative re-entrant topology. Specimens were made of stainless steel and polymer, respectively. Quasi-static uniaxial tensile tests were conducted in the two principal directions, followed by simulations using the commercial code - ABAQUS 6.11-2. The deformation, tensile stress-strain curves and Poisson’s ratio were of interest. A good agreement was observed between the numerical simulations and the experimental results. Subsequently, the effect of cell wall thickness and initial cell angle was studied by means of finite element analysis. An analytical equation was also given for the yield stress of such materials under tension. Keywords: Re-entrant hexagonal honeycomb; Polymer and stainless steel; Image correlation; Poisson’s ratio; Finite element analysis 1. Introduction Over the past several decades, developments in structural engineering design and technology in aircraft industry as well as automotive, sports, and leisure sectors have demanded novel materials to meet higher engineering specifications [1]. Such materials are to possess a combination of high stiffness and strength with significant weight savings. Structural material with negative Poisson’s ratio was explored [1, 2], known as auxetic materials [3, 4]. Lakes [5] first discovered this negative Poisson’s ratio effect in polyurethane (PU) foam with reentrant structures and responded to a comment in Ref. [6] on this negative trait. The key to the *

Corresponding author: [email protected] Phone: +61-3-9214-8669

1

auxetic behavior was the negative value of Poisson’s ratio and this structural material was foreseen with a wide range of applications: better artificial bones, improved sound and shock absorbers and enhanced bulletproof vests; gaskets and seals were another evident area of applications [7]. Subsequently, researchers managed to purposely fabricate a wide range of synthetic auxetic materials covering all major classes of materials, such as metals [8, 9], polymers [10, 11], textile [12, 13], composites [14, 15] and ceramics [16]. A methodology to convert conventional foams into auxetic ones was first reported by Lakes [5] for polymer foams, which was followed by Choi and Lakes [8] and Friis et al. [9] to fabricate auxetic metallic foams. Fabrication processes such as multi-phase auxetic fabrication [17, 18] were then developed and modified by some other researchers [19]. Recently, soft lithography has become an advantageous technology to make structural materials on different scales [20], especially for metal cellular structures. The selective laser melting (SLM) or electron beam melting (EBM) techniques were employed to fabricate the auxetic materials [21, 22], and the direct laser writing method (DLW) was also used [23, 24]. One disadvantage of the re-entrant honeycomb is, however, the difficulty in manufacturing them on a commercial scale [1]. Other topologies including chiral shape [25, 26], star shape [27, 28], arrowhead shape honeycombs [29] and certain designed foams [30, 31] leading to an auxetic effect might eventually yield a commercially acceptable structure in terms of manufacturability and performance. Materials with negative Poisson’s ratio demonstrate a series of particular characteristics over conventional ones, such as enhanced toughness and shear or indentation resistance [32], along with improved sound and vibration absorption [33]. They have been also exploited for applications as fasteners [34], tougher composites [35], medicine [36], tissue engineering [37] and others[33]. Up to now, studies have concentrated mainly on the small deformation elastic properties of negative Poisson’s ratio materials, in terms of the elastic modulus and elastic Poisson’s ratio [19, 38-41]. Chan and Evans [10, 42] fabricated auxetic foams and characterized their mechanical behaviors under compression and tension, respectively, with small deformation. Koudelka et al. [43] printed a three dimensional re-entrant structure to test its stress variation under quasi-static compression. Unlike the unconverted foams, the re-entrant structured foams showed no significant plateau region up to a maximum compression at 50% strain [44], similar to the outcomes of Ref. [10]. Auxetic effect and compression behavior of 3D auxetic textile structure were investigated by Zhou et al. [45] and they found that the auxetic composite 2

behaved more like a damping material with a lower range of compression stress. Further, Zhang et al. [46] simulated the in-plane dynamic crushing behaviors of auxetic honeycombs and found that the plateau stress at the proximal end and the energy absorption could be improved by increasing the negative cell angle, the relative density, the impact velocity, and the matrix material strength. An effective numerical model was developed to simulate auxetic composite sandwich panels subjected to blast loading [47] and demonstrated that this proposed auxetic panel was a promising protective structure against blast loadings. In addition, Alderson et al. [48] simulated the responses of re-entrant hexagonal and re-entrant trichiral honeycombs subject to out-of-plane bending and found that they underwent synclastic (dome-shape) curvature deformation. All the previous studies have been concerned with small deformation behavior, mainly under compression. Research on the large deformation post-yielding behaviors of auxetic structures under tension is limited, but in this case, their mechanical behaviors such as deformation and Poisson’s ratio may be of much difference. In this paper, a simple but typical auxetic structure, re-entrant honeycomb, is employed, and its large deformation tensile behavior of is experimentally and numerically investigated. Polymer and stainless steel specimens were made by 3D printing. The deformation modes, stresses and Poisson’s ratios of the re-entrant honeycombs are studied. 2. Experiments 2.1 Specimens The re-entrant honeycombs were manufactured by using 3D printing technology with stainless steel and polymer, as shown in Fig.1.
 is the original length of the inclined cell walls and ℎ that of the vertical cell walls.  is the initial angle between the inclined and vertical cell walls. T and t are the specimen thickness and cell wall thickness, respectively. Five cells of each sample were chosen randomly for measurement of their dimensions by a digital caliper, which represents 12.5% of the total cell number in the stainless steel sample, or 14.3% for the polymer sample. Average dimensions of the printed samples are given in Table 1. From Fig. 1(a), it has H0=2(h0-l0cosθ0) and L0=2l0sinθ0. For a repeatable cell in a re-entrant specimen, the thickness of the vertical walls on both sides is t / 2, the relative density (the ratio

3

between the volume occupied by the solid cell walls and the nominal volume of a reputable cell, H0L0) of the specimen can thus be calculated as

ρ0 =

( 2l0 + h0 ) t ρ∗ = ρ s 2l0 sin θ0 ( h0 − l0 cos θ0 )

(1) where ρ* and ρs are the densities of the re-entrant honeycomb and the solid cell wall material, respectively. Consequently, for stainless steel auxetic structure, the relative density is 21%, while it is 8.59% and 9.91%, respectively, for polymer-X1 and polymer-X2. The slight difference in the relative density of polymer-X1 and polymer-X2 is due to a small difference in the value of angle  (as shown in Table 1). The stainless steel samples (Figs. 1b and c) were 3D printed by using the selective laser melting machine ProX DMP 200 from 3D Systems, which has a maximum laser power of 300 W. Reentrant stainless steel structures were built layer-by-layer using a laser beam of 1.07 µm wavelength to melt the 17-4 PH powder with a particle size of 20 - 32 µm. The laser power and laser focal plane position were selected as 240 W and 0.6 mm, respectively. The travelling speed of laser beam was set at 2500 mm/sec. The total number of layers for each sample was 180 with an approximate thickness of 40 µm for each layer. After printing the re-entrant stainless steel samples, wire cutting was employed to cut off the samples from the printing substrate, using CNC Wire cutting machine (model Alpha OC, Fanuc Robocut). Liquid epoxy resin containing reactive diluents was used to fabricate the re-entrant polymer samples. Plastic/photopolymer auxetic samples (Figs. 1d and e) were created after hardening epoxy resin using stereo lithography (SL) process. In the ProJet 6000 HD 3D printer, SL process uses visible or ultraviolet (UV) laser and scanning mechanism to selectively solidify liquid photopolymer in order to form a layer of the sample. AUTOCAD 3D model was sliced to 40 layers and then the UV laser scanned according to the in-plane profile of the model, which solidified the liquid resin. The resin was hardened, one slice at a time, to a prescribed depth of 0.125 mm. The build platform moved down 0.125 mm after each layer was polymerized. The total thickness of 40 layers was T=5 mm. The dimensional accuracy was within 0.025-0.05 mm per 25.4 mm.

4

In both the manufacturing methods, in order to eliminate the effect of building direction on the mechanical properties, all the samples were built in the same orientation. The out-of-plane direction aligned with the building direction (i.e. the direction of the laser). In order to capture the local displacement fields, Vic-2D image correlation system was employed, which required adequate and distinguishable stochastic spots on the specimens. Therefore, all the samples were first coated with a layer of white paint-Dura Max (Dulux) and then left for at least two hours till the paint had dried. Afterwards, the samples were sprayed with Spray Easy (British Paints) to have black dots on the surface.

(a)

(d)

(b)

(c)

(e)

(f)

Fig. 1 Photographs of samples: (a) sketch of a re-entrant unit cell; (b) auxetic stainless steel sample to be loaded in the X2 direction (5×8 cells); (c) auxetic stainless steel sample to be loaded in the X1 direction (5×8 cells); (d) auxetic polymer sample to be loaded in the X2 direction (5×7 cells); (e) auxetic polymer sample to be loaded in the X1 direction (5×7 cells); (f) details of the fixtures holding one side of the sample.

5

Table 1 Material and dimensions of the samples (Note: Letters X1 and X2 are the loading directions)

Name

Polymer- X1

Polymer- X2 Stainless steel- X1 Stainless steel- X2

Material

h0 (mm)

l0 (mm)

16.8

8.4

16.8

17-4 PH

17-4 PH

VisiJet SL Flex VisiJet SL Flex

θ0

Number

t (mm)

T (mm)

45

0.33

5

5×7

8.4

40

0.33

5

5×7

16.8

8.4

30

0.5

7.2

5×8

16.8

8.4

30

0.5

7.2

5×8

(degree)

of Cells

2.2 Properties of the parent materials In order to obtain the properties of cell wall material, standard samples to the ASTM standards E8/E8M−15a (for stainless steel) and D638-14 (for polymer) were also printed using identical values of operating parameters to those for the auxetic samples. For both stainless steel and polymer samples, their gauge length is 25.0 mm. The average dimensions from several measurements are 6.34 mm in width and 5.28 mm in thickness for the stainless steel samples. For the polymer samples, the average width and thickness are 5.99mm and 3.35 mm, respectively. Engineering stress-strain curves of the different samples were obtained and are given in Fig. 2. Meanwhile values of the density of the stainless steel and polymer were measured as 7550 kg/m3 and 1164 kg/m3, respectively, by calculating the average ratio of mass of the sample in the gauge length to the corresponding volume. Three nominally identical samples were tested and the stress-strain curves of both the materials were repeatable except for the value of the fracture strain for the polymer. The difference in fracture strain might be due to the manufacturing technique. Since fracture strain will not be used in the subsequent analysis in this study, the exact reason for the difference was not explored further. 6

The mechanical behavior of the cell wall material may be slightly different from that of the standard samples. Therefore, some cell wall pieces were cut from the auxetic samples and then subjected to micro- and nano- indentation tests. From the indentation tests it was found that the hardness of the auxetic samples was approximately 20% lower than that of the standard tensile test coupons. This may be because material properties are sensitive to printing process such as the multiple passes, which affects the porosity, residual stresses etc. Therefore, in the finite element simulation a factor of 80% was applied to the flow stress in the stress-strain curves measured from the tensile tests. 2.3 Experimental setup Uniaxial tensile tests were conducted by using Zwick Roell machine (Zwick/Z010) with a load cell of 1 kN. Polymer- X1 and Stainless steel- X1 were tested in the X1 direction, while PolymerX2 and Stainless steel- X2 in the X2 direction. In each test, two sides of the specimen were pinjoined, allowing the edge to rotate freely, and the other two free. Small fixtures were fabricated from 17-4 PH stainless steel to allow the lateral movement in the X2 direction (Fig. 1f). Meanwhile, grease was applied on the contact surfaces to reduce friction between the specimen and the fixture during the tests. One edge of the sample was attached to a fixed position while the other moved with the cross-head at 3 mm/min. Each test was stopped when fracture occurred.

(a)

(b)

Fig. 2 Engineering stress-strain curves of the parent materials: (a) stainless steel; (b) polymer.

3. Finite element analysis Finite element (FE) models with the same dimensions as those in the tests were set up and corresponding numerical simulations were performed by using ABAQUS 6.11-2. In the 7

simulations, three-node quadratic Timoshenko beam element (B22) was used to mesh the cell walls of re-entrant honeycombs with the element size 0.24 mm (35 elements in each inclined wall), which was shown to produce converged FE results. The stress-strain relationships obtained from ASTM standard tests were employed in the FE models as the material behavior. In regard of loads and confinements, a uniform displacement was applied as loading to the corresponding nodes of the models whilst the supporting nodes were constrained in the loading direction, but free in the other perpendicular direction. Failure (fracture) of materials was not considered in simulations. 4. Experimental and Finite Element Analysis Results 4.1 Deformation pattern For each test, a load-displacement curve was obtained directly from the machine. The load can be converted to the nominal engineering stress by taking its ratio to the nominal cross-sectional area of the specimen, and the displacement divided by the initial length of the specimen gives the engineering strain. As expected, when the polymer auxetic sample was stretched in the X2 direction, lateral expansion occurred, as shown in Figs. 3a and b. This expansion indicates a negative Poisson’s ratio, until the initially inclined cell walls became horizontal (Fig. 3c). Afterwards, cells shrank in the X1 direction as they were further stretched in the X2 direction (Fig. 3d), which indicated that the sample then possessed a positive Poisson’s ratio during this stage, similar to conventional materials. When loaded in the X1 direction, as exhibited in Figs. 3(e)-(h), the polymer sample kept expanding in both the X1 and X2 directions until the initially inclined cell walls became vertical and finally broke. The characteristic of a negative Poisson’s ratio was present during the complete tensile process. Similarly, the deformation patterns of the stainless steel samples are shown in Figs. 4(a)-(d) (X2 direction loading) and Figs. 4(e)-(f) (X1 direction loading). A difference from the polymer sample is that the sample in Fig. 4(d) fractured at some cells before the cells started to contract. In addition, considerable frictional effect (Figs. 3c, 3h, 4b and 4g) was observed even though grease was applied to minimise friction in the tests. The overall specimen became barrel shaped and the cells deformed inhomogeneously, especially those within the bend of two columns in Fig. 3(h). Corresponding to the experimental observations, Figs. 5 and 6 show the FE deformation patterns (with free lateral movements, or no friction). The cells expanded homogeneously. 8

(a) ε=0

(e) ε=0

(b) ε=0.275

(c) ε=0.7

(f) ε=0.184

(g) ε=0.282

(d) ε=1.1

(h) ε=0.38

Fig. 3 Deformed polymer samples in the tests: (a)-(d) for loading in the X2 direction; (e)-(h) for loading in the X1 direction.

(a) ε=0

(b) ε=0.1

(e) ε=0

(f) ε=0.151

(c) ε=0.146

(g) ε=0.315

(d) ε=0.2

(h) ε=0.493

Fig. 4 Deformed stainless steel samples in the tests: (a)-(d) for loading in the X2 direction; (e)-(h) for loading in the X1 direction.

9

(a) ε=0

(e) ε=0

(b) ε=0.275

(c) ε=0.7

(f) ε=0.184

(d) ε=1.1

(g) ε=0.282

(h) ε=0.38

Fig. 5 Deformed polymer specimens from FE simulations: (a)-(d) for loading in the X2 direction; (e)-(h) for loading in the X1 direction.

(a) ε=0

(e) ε=0

(b) ε=0.1

(c) ε=0.146

(f) ε=0.151

(g) ε=0.315

(d) ε=0.2

(h) ε=0.493

Fig. 6 Deformed stainless steel specimens from FE simulations: (a)-(d) for loading in the X2 direction; (e)-(h) for loading in the X1 direction.

10

4.2 Tensile stress versus strain curves As mentioned before, the engineering stress-strain curves can be obtained from the loaddisplacement curves of the specimens and they are shown in Fig. 7, for re-entrant honeycombs made of polymer and stainless steel. Generally, the experimental results agree with the FE data with respect to the characteristics of curves and magnitudes of stress. In Fig. 7(c), in the test some cell walls of the stainless steel honeycomb fractured at approximately 0.15 of the engineering strain, resulting in a sudden decrease in the stress at a small strain. Furthermore, in terms of the shape of the curves for the loading in the X2 direction, there exists a plateau stage after an initial elastic stage, followed by a final stage with a rapid increase in stress, which is similar to cellular materials under compression [49-51]. It should be noted that, from Fig. 7(a), the test result is a slightly higher than that of the simulation. This may be explained by recalling the comparison between the deforming profiles in Fig. 3(c) and Fig. 5(c). The cells deform nonuniformly in the test resulting from the horizontal friction on the contacting surfaces at the top and bottom ends, while they deform uniformly in simulation. This shows that the slight transverse restraint leads to a small increase of stress.

(a)

(b)

11

(c)

(d)

Fig. 7 Engineering stress-strain curves: polymer sample loaded in the X2 direction (a) and in the X1 direction (b); and stainless steel sample loaded in the X2 direction (c); and in the X1 direction (d).

4.3 Poisson’s ratio Poisson’s ratio is an important parameter of a material, which generally represents the negative of the ratio of the lateral extension to the longitudinal extension (extension in the loading direction). Poisson’s ratio calculated from the engineering strain reflects the macroscopic and averaged value. To observe the progressive deformation of auxetic honeycombs in both the X1 and X2 directions, a series of images (frames) were captured by the Digital Image Correlation (DIC) System during the testing process. Consequently, the successive increment of the total extension between frames was captured and the Poisson’s ratio could then be calculated at each increment. From these images, local displacement field and hence engineering strain field) were calculated. Here, eight pairs of points were selected and monitored (four in the horizontal direction as marked in orange, and the other four in the vertical direction, in blue) (Fig. 8). Using the first frame as the reference, the vertical relative displacement between the corresponding points divided by the initial distance gave the engineering strain in the loading direction. Similarly, the lateral engineering strain was calculated. The lateral strains and longitudinal strains measured at the various locations are plotted against the cross-head displacement in Fig. 9(a) and their averaged value is shown in Fig. 9(b). As expected, the strain in the loading direction increased linearly with the cross-head displacement. However, strain in the other direction increased initially and reached a peak, then it decreased gradually. This demonstrates the expansion and then shrinking of the auxetic honeycombs in uniaxial tensile tests, as observed before. 12

Fig. 8 Points traced for determining the longitudinal (blue points) and lateral (orange points) engineering strain during the tests. Note that this figure is similar to Figs. 3(a)-(d).

(a)

(b)

Fig. 9 Total engineering strain in the two directions against the cross-head displacement (a); and the averaged value (b).

For plastic deformation, Poisson’s ratio was calculated as the ratio of strain increments in the two directions[52, 53], instead of using the total engineering strain. It represents the instantaneous incremental deformation features of the re-entrant honeycombs. Values of the Poisson’s ratio such calculated from the tests and finite element analysis are plotted in Fig. 10. In all the plots, the Poisson’s ratio changes with the total strain in a non-linear way, with both negative and positive values. For auxetic polymer samples loaded in the X2 direction, Poisson’s ratio from both the test and simulation increases gradually from about -2.5 to about 2.0, reaching zero at an approximate strain of 0.6. This indicates that when a polymer re-entrant honeycomb is subjected to tension in the longitudinal direction, it expands firstly and then contracts along the lateral direction. This agrees with the observed deformation patterns during the tensile tests (Fig. 3). It 13

also shows that as the deformation progresses the re-entrant honeycomb behaves from an auxetic material to a conventional one with positive Poisson’s ratio. Under the load in the X1 direction (Fig. 10b), the Poison’s ratio was always negative and it decreased to as low as -2.8 at a longitudinal strain of 0.38. However, around this point the lowest values from the test and simulation are very different. The reason can be found by referring to Fig. 3(h). The two columns of cells did not move freely due to friction on both the loading and fixed edges, resulting in reduced lateral expansion. Similarly, for the stainless steel sample in Fig. 10(c), the Poisson’s ratio was negative firstly, about -4.0, and then it increased gradually. The magnitude from the test becomes lower than that from the simulation after a longitudinal engineering strain of 0.1, due to the lateral constraint from friction and fracture of some cell walls in the test. For the stainless steel sample loaded in the X1 direction (Fig 10d), the trend is the same as that in Fig. 10(b).

(a)

(b)

(c)

(d)

Fig. 10 Poisson’s ratio from the tests and finite element simulations: polymer samples loaded in the X2 (a) and X1 (b) directions; stainless steel samples loaded in the X2 (c) and X1 (d) directions 14

5. Discussions 5.1 Yield stress It would be interesting to give some theoretical consideration on the yield stress of a sample. For reasonably thick cell walls, this global yielding is governed by the plastic collapse of a cell. Consider a single cell under tensile force F2 in the X2 direction (Fig. 11). When plastic collapse occurs, plastic hinges form at all the corners. Assuming the bending moment at the hinges reaches the fully plastic bending moment  = 1⁄4    , where σys is the yield stress of the parent material of cell walls, from the equilibrium of cell wall AB, the force F2 is  = 2 
  . The corresponding nominal stress is hence the ratio of F2 and the cross-sectional area: 2

1 σ2 1  t  =   σ ys 2  l0  sin 2 θ 0

(2) Similarly, the tensile yield stress in the X1-direction, σ1, is given as 2

σ1 1  t  1 =   σ ys 2  l0  cos θ  h0 − cos θ  0 0 l 

0



(3) Analytical predictions from Eqs. (2) and (3) are compared with the experimental results for polymer and stainless steel samples loaded in both the X1 and X2 directions in Fig. 12. As in the finite element simulations, the yield stress of the parent materials is taken as 80% of that from the coupon tests. The comparison indicates that Eqs. (2) and (3) could predict the initial yield stress of stainless steel samples for both the directions (Fig. 12b), but it is not the case for the polymer samples. This is because, compared with conventional manufacturing processes, the stereo lithography (SL) process method lacks dimensional accuracy due to parameters involving layer thickness, hatch over-cure and hatch spacing [54].

15

F2

σ2 A

M A

X2 B X1

F2 θ0 2 F2 2

X2

σ2 X1 (a)

F2

(b)

B

M

Fig. 11 A single re-entrant cell loaded in the X2 direction (a); and the free body diagram of the cell wall AB (b).

(a)

(b)

Fig. 12 Tensile stress and theoretical yield stress of polymer sample (a); and stainless steel sample (b), loaded in both the X1 and X2 directions.

5.2 Effect of geometrical parameters Combining Eqs. (1) and (2), the tensile yield stress in the X2-direction, σ2 , can be re-written as

16

 h0  − cos θ0  2  ∗ ρ  l σ2  = 2   0 2 σ ys  ρs   h0  2+  l0  

2

(4) It indicates that, in addition to the cell dimensions l0 and h0, the yield stress is also related to both the relative density (ρ0) and initial angle of cells (θ0). Influence of the relative density on the tensile stress was studied by using the FE models in Section 3. This was achieved by changing the cell wall thickness, t, in Eq. (1). Four values of the cell wall thickness were considered: t=0.15 mm (ρ0=4.5%), t=0.20 mm (ρ0=6.0%), t=0.25 mm (ρ0=7.5%), and t=0.30 mm (ρ0=9.0%). FE results are plotted in Fig. 13. As indicated in Eq. (4), a higher value of relative density leads to a higher value of the tensile stress plateau, as well as the initial slope of the stress-strain curves. The initial angle of cell, θ0, also affects the tensile stress. FE models with θ0=30o, 35o, 40o, 45o were established and the cell dimensions, l0 and h0, were fixed at the same values as those in the tests. The relative density of all these FE models was kept the same at 7.5% by changing the value of thickness t. Therefore, a larger value of the initial angle corresponds to a thicker cell wall. Fig. 14 presents effect of the cell angle on the tensile stress in the X2-loading direction. A larger initial angle leads to a higher tensile stress plateau and the elastic slopes of stress-strain curves, but it decreases the value of the maximum strain when the stress increases rapidly, similar to the locking strain in cellular materials. For a single cell in Fig. 1(a), the initial height of this cell is 2(h0-l0cosθ0) and the maximum extension is 2l0(1+cosθ0), the maximum critical strain, εmax, can be expressed as

ε max =

1 + cos θ 0 h0

l0

− cos θ 0

(5) It shows that, for a special case of h0=2l0, the corresponding maximum strain is related only to the initial angle of cells: a larger one reduces the critical strain of the final stage of curves.

17

Fig. 13 Effect of cell wall thickness (t) on engineering stress-strain curves of the re-entrant honeycombs loaded in the X2 direction (θ0=40o).

Fig. 14 Effect of the cell angle on the engineering stress-strain curves of re-entrant honeycomb s loaded in the X2 direction (ρ0=7.5%)

5.3 Effect of friction

This section explores effect of possible lateral constraint due to friction. One loading case shown in Fig. 1(d) was considered. Details of the fixture used in the experiment were reproduced in the FE model and different values of the friction coefficient, µ, were defined for the contacting surfaces (General Contact). In the simulation, three-node quadratic Timoshenko beam element (B32) was used for meshing. Other definitions such as element size, loading and boundary conditions were the same with those in Section 3. The stress-strain relationship of polymer in the section 2.2 was employed in the FE models. Fig. 15 shows the deformed samples at ε=0.7 (corresponding to the displacement 72.5 mm). The deformation type at µ=0.3 is similar to that in Fig. 3(b). Meanwhile, for µ= 0.8, no lateral 18

movement of cells at both the proximal and distal ends was observed, as if the lateral movement was fully constrained. Fig. 16 illustrates the force-displacement curves for different values of friction coefficient. A large value of friction coefficient brings forward the final stage. However, when the friction is small (μ≤0.3) the curves are almost the same with exception of slight fluctuations. For the friction coefficient lager than 0.3, the curves are all similar to that of the fully constrained case (µ= 0.8).

(a)

(b)

(c)

(d)

Fig. 15 Deformed (ε=0.7) patterns for different values of friction coefficient of the polymer samples: (a) =0.1; (b) =0.3; (c) =0.5; (d) =0.8. The original specimen is also shown in dotted lines.

Fig. 16 Force-displacement curves with different values of friction coefficient.

19

6. Conclusions

As typical auxetic structures, several stainless steel and polymer re-entrant honeycomb samples were printed using the 3D printing technology. Quasi-static tensile tests were then performed in both the X1 and X2 directions, followed by the finite element simulation. The deformation, tensile stress and Poisson’s ratio of re-entrant honeycombs were studied in the tests and simulations. A good agreement between the experimental and FE results has been obtained. It has been observed that when loaded in the X2 direction the re-entrant honeycombs had a large strain of 1.3 and the value of Poisson’s ratio changed during the deformation. The honeycombs initially exhibited an auxetic feature with negative Poisson’s ratio almost as low as -4. Beyond a certain point of deformation, they behaved as conventional honeycombs with a positive value of Poisson’s ratio. In the X1-direction, the auxetic samples kept expanding until the inclined cell walls became horizontal. The Poisson’s ratio was negative for the whole process. In terms of the tensile stress, all the stress-strain curves had a plateau stage, similar to compression of conventional cellular materials. An analytical formula based on the plastic collapse of the cell walls has been given, which could lead to a reasonable estimate of the yield stress for stainless steel honeycombs, but not so for the polymer samples. A higher relative density led to a higher stress plateau and the elastic slopes of stress-strain curves, as expected. For a given density, a larger initial cell angle also enhanced the plateau stress, but reduced the maximum strain in the final stage of the stress-strain curves. Acknowledgements

The authors wish to thank Dr Andrew Ang for his help with the nano-indentation tests. They acknowledge, with thanks, the financial support from the Australian Research Council (Grant No. DP160102612), National Natural Science Foundation of China (Grant Nos. 51578361 and 11572214), Shanxi Scholarship Council of China (2013-046) and the State Scholarship Fund of China Scholarship Council (CSC). References [1] Alderson A, Alderson KL. Auxetic materials. Proceedings of the Institution of Mechanical Engineers Part G-Journal of Aerospace Engineering. 2007;221:565-75. 20

[2] Bhullar SK. Three decades of auxetic polymers: a review. e-Polymers. 2015;15:205-15. [3] Evans KE, Nkansah MA, Hutchinson IJ, Rogers SC. Molecular network design. Nature. 1991;353:124. [4] Evans KE. Auxetic polymers - a new range of materials. Endeavour. 1991;15:170-4. [5] Lakes R. Foam structures with a negative Poisson's ratio. Science. 1987;235:1038-40. [6] Burns S. Negative Poisson's ratio materials. Science. 1987;238:551. [7] Cherfas J. Stretching the point. Science. 1990;247:630. [8] Choi J, Lakes R. Non-linear properties of metallic cellular materials with a negative Poisson's ratio. Journal of Materials Science. 1992;27:5375-81. [9] Friis E, Lakes R, Park J. Negative Poisson's ratio polymeric and metallic foams. Journal of Materials Science. 1988;23:4406-14. [10] Chan N, Evans K. The mechanical properties of conventional and auxetic foams. Part I: compression and tension. Journal of Cellular Plastics. 1999;35:130-65. [11] Scarpa F, Pastorino P, Garelli A, Patsias S, Ruzzene M. Auxetic compliant flexible PU foams: static and dynamic properties. Physica Status Solidi B-Basic Solid State Physics. 2005;242:681-94. [12] Wang ZY, Hu H. Auxetic materials and their potential applications in textiles. Textile Research Journal. 2014;84:1600-11. [13] Reda H, Rahali Y, Ganghoffer JF, Lakiss H. Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models. Composite Structures. 2016;141:328-45. [14] Zhang G, Ghita O, Evans KE. The fabrication and mechanical properties of a novel 3-component auxetic structure for composites. Composites Science and Technology. 2015;117:257-67. [15] Miller W, Ren Z, Smith C, Evans K. A negative Poisson’s ratio carbon fibre composite using a negative Poisson’s ratio yarn reinforcement. Composites Science and Technology. 2012;72:761-6. [16] Huang X, Blackburn S. Developing a new processing route to manufacture honeycomb ceramics with negative Poisson's ratio. Key Engineering Materials. 2001;206:201-4. [17] Grima JN, Attard D, Gatt R, Cassar RN. A novel process for the manufacture of auxetic foams and for their re‐conversion to conventional form. Advanced Engineering Materials. 2009;11:533-5. [18] Bianchi M, Frontoni S, Scarpa F, Smith C. Density change during the manufacturing process of PU– PE open cell auxetic foams. Physica Status Solidi B-Basic Solid State Physics. 2011;248:30-8. [19] Critchley R, Corni I, Wharton JA, Walsh FC, Wood RJ, Stokes KR. A review of the manufacture, mechanical properties and potential applications of auxetic foams. Physica Status Solidi B-Basic Solid State Physics. 2013;250:1963-82. [20] Xu B, Arias F, Brittain ST, Zhao X-M, Grzybowski B, Torquato S, et al. Making negative Poisson’s ratio microstructures by soft lithography. Advanced Materials. 1999;11:1186-9.

21

[21] Li S, Hassanin H, Attallah MM, Adkins NJ, Essa K. The development of TiNi-based negative Poisson's ratio structure using selective laser melting. Acta Materialia. 2016;105:75-83. [22] Yang L, Harrysson O, West H, Cormier D. Modeling of uniaxial compression in a 3D periodic reentrant lattice structure. Journal of Materials Science. 2013;48:1413-22. [23] Jang D, Meza LR, Greer F, Greer JR. Fabrication and deformation of three-dimensional hollow ceramic nanostructures. Nature Materials. 2013;12:893-8. [24] Walia S, Shah CM, Gutruf P, Nili H, Chowdhury DR, Withayachumnankul W, et al. Flexible metasurfaces and metamaterials: a review of materials and fabrication processes at micro-and nano-scales. Applied Physics Reviews. 2015;2:011303. [25] Sigmund O, Torquato S, Aksay IA. On the design of 1–3 piezocomposites using topology optimization. Journal of Materials Research. 1998;13:1038-48. [26] Liu X, Hu G. Elastic metamaterials making use of chirality: A review. Strojniški vestnik-Journal of Mechanical Engineering. 2016;62:403-18. [27] Grima JN, Gatt R, Alderson A, Evans K. On the potential of connected stars as auxetic systems. Molecular Simulation. 2005;31:925-35. [28] Theocaris P, Stavroulakis G, Panagiotopoulos P. Negative Poisson's ratios in composites with starshaped inclusions: a numerical homogenization approach. Archive of Applied Mechanics. 1997;67:27486. [29] Larsen UD, Signund O, Bouwsta S. Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio. Journal of Microelectromechanical Systems. 1997;6:99-106. [30] Babaee S, Shim J, Weaver JC, Chen ER, Patel N, Bertoldi K. 3D Soft metamaterials with negative Poisson's ratio. Advanced Materials. 2013;25:5044-9. [31] Alderson A, Alderson KL, McDonald SA, Mottershead B, Nazare S, Withers PJ, et al. Piezomorphic materials. Macromolecular Materials and Engineering. 2013;298:318-27. [32] Greaves GN, Greer A, Lakes R, Rouxel T. Poisson's ratio and modern materials. Nature Materials. 2011;10:823-37. [33] Jiang JW, Park HS. Negative poisson’s ratio in single-layer black phosphorus. Nature Communications. 2014;5. [34] Choi J, Lakes R. Design of a fastener based on negative Poisson's ratio foam. Cellular Polymers. 1991;10:205-12. [35] Sun Y, Pugno N. Hierarchical fibers with a negative Poisson’s ratio for tougher composites. Materials. 2013;6:699-712. [36] Scarpa F. Auxetic materials for bioprostheses. IEEE Signal Processing Magazine. 2008;25:125-6.

22

[37] Park YJ, Kim JK. The effect of negative Poisson’s ratio polyurethane scaffolds for articular cartilage tissue engineering applications. Advances in Materials Science and Engineering. 2013;2013:1-5. [38] Yang W, Li ZM, Shi W, Xie BH, Yang MB. Review on auxetic materials. Journal of Materials Science. 2004;39:3269-79. [39] Alderson A, Evans K. Microstructural modelling of auxetic microporous polymers. Journal of Materials Science. 1995;30:3319-32. [40] Prall D, Lakes R. Properties of a chiral honeycomb with a Poisson's ratio of -1. International Journal of Mechanical Sciences. 1997;39:305-14. [41] Hou X, Hu H, Silberschmidt V. A novel concept to develop composite structures with isotropic negative Poisson’s ratio: effects of random inclusions. Composites Science and Technology. 2012;72:1848-54. [42] Chan N, Evans K. The mechanical properties of conventional and auxetic foams. Part II: shear. Journal of cellular plastics. 1999;35:166-83. [43] Embala A, Folije N. Corrosion behavior and the weak-magnetic-field effect of aluminum packaging paper. Materiali in Tehnologije. 2016;50:165-73. [44] Allen T, Shepherd J, Hewage T, Senior T, Foster L, Alderson A. Low‐kinetic energy impact response of auxetic and conventional open‐cell polyurethane foams. Physica Status Solidi B-Basic Solid State Physics. 2015;252:1631-9. [45] Zhou L, Jiang L, Hu H. Auxetic composites made of 3D textile structure and polyurethane foam. Physica Status Solidi B-Basic Solid State Physics. 2016;253:1233. [46] Zhang XC, Ding HM, An LQ, Wang XL. Numerical investigation on dynamic crushing behavior of auxetic honeycombs with various cell-wall angles. Advances in Mechanical Engineering. 2015;7:1-12. [47] Imbalzano G, Tran P, Ngo TD, Lee PV. A numerical study of auxetic composite panels under blast loadings. Composite Structures. 2016;135:339-52. [48] Alderson A, Alderson K, Chirima G, Ravirala N, Zied K. The in-plane linear elastic constants and out-of-plane bending of 3-coordinated ligament and cylinder-ligament honeycombs. Composites Science and Technology. 2010;70:1034-41. [49] Gibson LJ, Ashby MF. Cellular solids: structure and properties: Cambridge University Press, 1999. [50] Li ZQ, Zhang JJ, Fan JH, Wang ZH, Zhao LM. On crushing response of the three-dimensional closed-cell foam based on Voronoi model. Mechanics of Materials. 2014;68:85-94. [51] Ruan D, Lu G, Wang B, Yu T. In-plane dynamic crushing of honeycombs-a finite element study. International Journal of Impact Engineering. 2003;28:161-82. [52] Wang DA, Pan J. A non-quadratic yield function for polymeric foams. International Journal of Plasticity. 2006;22:434-58. 23

[53] Deshpande V, Fleck N. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids. 2000;48:1253-83. [54] Park WS, Lee SH, Cho HS, Leu MC. Evaluating the contributions of process parameters in SLA using artificial neural network. Intelligent Manufacturing Systems. 1997;IMS'97:189-94.

24