Analysis of the mechanical properties of double arrowhead auxetic metamaterials under tension

Abstract

In order to study the deformation behavior of double arrowhead (DAH) auxetic metamaterials under different forces with different directions as well as the effects of the geometry parameters on the Poisson's ratio, this paper assembled an advanced researching method by combining the industrial design of SolidWorks software, the finite element analysis of Abaqus software and three-dimensional printed technology. Results show that the DAH structure expanded in the direction perpendicular to the force first and contracted with the strain increasing when it was pulled by the uniaxial force, no matter which direction the force is applied. Besides, the auxetic effect of the DAH structure under the tensile force in the X direction is longer than that under the tensile force in the Y direction. It is more resistant to compression when the DAH structure is subjected to transverse tensile forces than when it is subjected to longitudinal tensile forces. The angle of the V-shaped short truss has a significant impact on the negative Poisson's ratio of the DAH structure, while the angle of the V-shaped long truss angle has little influence on its auxetic performance. Specifically, the smaller the angles, the better the auxetic effect.

Keywords

negative Poisson's ratio auxetic double arrowhead finite element simulation tensile properties

Metamaterials are macroscopic composites of periodic or non-periodic structures, whose effective properties arise from both cellular architectures and their constituents.¹ Auxetic metamaterials are materials whose unique mechanical features originate from their underlying architecture, rather than from their chemical composition.² Due to their extraordinarily properties, auxetic metamaterials have attracted great attention in the scientific communities over the past few years.^3,4 Besides, auxetic metamaterials have many engineering advantages, such as increased indentation resistance, shear resistance, synclastic curvature, superior energy absorption capability and acoustic properties.^5,6 Therefore, auxetic materials have broad applications in many areas, including the civil and aeronautical engineering,⁷ vehicle crashworthiness,⁸ defensive equipment,^9,10 smart sensors,^11–13 filter cleaning,⁵ biomechanics^14,15 and protective seat pads.¹⁶

Recently, three-dimensional (3D) printing technology has gained significant attention for inventing auxetic metamaterials, since it offers unparalleled flexibility in achieving controlled composition, geometric shape, function and arbitrary complex geometry with low cost compared with traditional manufacturing methods.^17–19 Among many mainstream 3D printing technologies, Fused Deposition Modeling (FDM) technology is a relatively advantageous one due to its relatively low costs, high utilization rate of raw materials and its association with desktop 3D printers.^2,20 FDM is an additive manufacturing process belonging to the material extrusion family. In FDM, an object is built by selectively depositing melted material in a pre-determined path layer-by-layer. The materials used are thermoplastic polymers and protrudes in a filament form.²¹

Finite element analysis (FEA) is a modern method for structural mechanics analysis. It was firstly used as an effective numerical analysis method in the field of continuous mechanics analysis of static and dynamic characteristics of aircraft structures in the 1950s, and then it was widely used to solve continuity problems, such as heat conduction, electromagnetic fields and fluid mechanics. Now, it has been used extensively to predict the stress distributions within auxetic cellular metamaterials upon any kind of force.^22–26 So far, finite element (FE) design combined with 3D printing has become the expected method and concept of modern scientific research workers.

Smardzewski et al.²⁷ conducted a thorough analysis of 3D printed models and numerical models of different spring structures having concave cell walls, subjected to uniaxial compression. By changing the elasticity modulus for the material and geometry of the internal structure, the non-linear characteristics of rigidity and negative Poisson's ratios were established for designed springs. Wang et al.²⁸ used dual-material 3D printing to create a dual-material auxetic metamaterial. After comparisons between FEA and mechanical testing, they found that the Poisson's ratio values significantly influenced the mechanical properties of the structures. Zhang and Yang²⁴ combined FEA models and 3D printed samples to perform a parametric analysis of the effects of the Poisson's ratio (cell angle) and the relative density (cell thickness) of different scales of the honeycombs on the bearing capacity and dynamic performance of the auxetic material. Fu et al.²⁹ designed a novel 3D auxetic structure based on two-dimensional (2D) re-entrant honeycombs. Through compressing the corresponding 3D printed samples and simulating the auxetic behavior of the proposed structure by FEA, the conclusion is that the 3D structure can achieve a negative Poisson's ratio in two principal orthogonal coordinate directions when it was compressed along the other principal axis. Wang et al.³⁰ proposed an interlocking assembly method that can be used to manufacture a 3D auxetic cellular structure by combining FEA and 3D printing technology. Frenzel et al.³¹ designed a kind of 3D chiral mechanical metamaterial and presented this material showing twists per axial strain as large as 2 °/% by both FEA and 3D laser microprinting technology. With the help of some FEA and 3D printing technology, Bertoldi³² proved that mechanical instabilities in architected cellular materials can be harnessed to design auxetic materials, to control the propagation of elastic waves, and to realize reusable energy-absorbing materials. Bertoldi's study highlights a new strategy to design tunable systems across a wide range of length scales. Li and Wang³³ fabricated a new class of sandwich composite structures with 3D printed core materials and carbon fiber-reinforced polymer face sheets. By combining the 3D printing technique, FEA, and three-point bending tests, they found that architected core structures can be utilized to tailor the bending properties as well as failure mechanisms. Jiang et al.¹¹ incorporated auxetic metamaterials into stretchable strain sensors to enhance the sensitivity of sensors via the reduced synergistic effect of the structural Poisson's ratio and strain concentration. Then they verified this viewpoint by means of both experiments and numerical simulations. Zhang et al.³⁴ calculated stresses and Poisson's ratios of the re-entrant honeycombs for different materials properties by means of FE simulations and experiments.

In recent years, double-arrow research has become a hot topic. Brighenti et al.³⁵ presented a thorough discussion on the linear and geometrically non-linear deformability of the 2D double arrowhead (DAH) plate. They not only identified certain key geometric parameters that affect the deformability of the plate, but also offered some analytical expressions for calculating the Poisson's ratio, as a function of the applied strain. Different approaches, including FEA and experimental tests on 3D printed specimens, were carried out to verify the theoretical findings. Chen et al.³⁶ designed three types of novel lattices with negative Poisson's ratio to study their properties. FEA and quasi-static experiments showed that by embedding some ribs into classic re-entrant cell, three types of lattice structures were proposed to improve mechanical properties and these novel lattices significantly increased the Young's modulus along the loading direction. Nady et al.³⁷ developed mechanical and numerical homogenization models and evaluated the effective non-linear mechanical response of periodic networks prone to auxetic behavior. Gao et al.^38–40 conducted a series of studies on 3D structures based on DAH and their applications in the field of energy absorption and shock absorption. Similarly, Wang et al.^41–45 have also been focusing on 3D double-arrow structures and their performance for many years and have achieved a great deal of research results. Lim⁴⁶ proposed a 3D geometrical model (an intersecting DAH microstructure) by extending the DAH structure. Through further study of the models, he found that geometry, in terms of either the relative linkage lengths or the half angles, can be changed to substantially control the Poisson s ratio. In 2019, Lim⁴⁷ made thermal expansion analysis on rectangular cells with triangular array and triangular cells with rectangular array composite microstructures. Results reveal that the microstructures lead to positive and negative thermal expansivities under the constraint condition.

Although researchers have already focused on the geometry analysis of the 3D DAH structures, physical properties of 2D hyperelastic DAH metamaterials have not been under consideration. In particular, the theoretical, numerical and experimental analysis of the mechanical properties of the 2D DAH structure pad has not been investigated thoroughly. In addition, in previous studies, the theoretical prediction of the mechanical properties did not consider the effect of the tensile direction.

In this paper, the influence of the force direction and the effects of the geometry parameters on the Poisson's ratio were investigated. Three approaches, namely theoretical, numerical and experimental analysis, were adopted to study the mechanical properties of DAH auxetic metamaterials. A theoretical model is established, and the influence of the force direction and structural parameters on the tensile performance of DAH structures is analyzed firstly. Two types of DAH structures were designed by using SolidWorks 2015, then FEA non-linear simulation was conducted on the models to identify their deformation profile and also to assess their auxetic potential. Tensile tests were performed on the corresponding 3D printed samples to verify the effectiveness of the theoretical results from the FEA simulations. In view of the deformation mechanism of the auxetic structure in different directions, its geometric structure is studied and explained.

Theoretical analysis

The influence of the direction of force on the auxetic property

The basic cell of the DAH and its deformation under tensile action is shown in Figure 1. The long truss and short truss are the cell walls with high and low slopes, respectively.

Figure 1.

Stretching of the DAH cell structure and its main parameters, where s is the length of the long truss, t is the length of the short truss, α is the angle of the V-shaped long truss, β is the angle of the V-shaped short truss, h₁ is the original longitudinal length and h₂ is the deformed longitudinal length.

When stretched in the Y direction, the DAH cell will first expand from cell A to cell A′. The length of trusses s, t does not change, while angles α, β become larger. Longitudinal length h changes from h₁ to h₂. If it keeps getting stretched in the Y direction, cell A′ will turn to cell A″. The angle β will become a convex angle from a concave angle and the DAH cell shows a positive Poisson's ratio property. That is to say, the length of the trusses stays the same and the angle changes throughout the Y stretch. Since there is no stretching of the trusses themselves, a small force can deform the structure in this way. When is stretched in the X direction, the DAH cell will also expand from cell A to cell A′. If it keeps getting stretched in the X direction, short truss t will be stretched. It takes a greater force to complete the tensile deformation of the structure, which means that the DAH has a greater bearing capacity when it is stretched in the X direction than in the Y direction. In this process, the DAH structure will show the negative Poisson's ratio effect for a long time before it shows the positive Poisson's ratio effect. So, it can be concluded that regardless of the force applied in any direction, the DAH structure will eventually behave as a positive Poisson's ratio effect. However, the auxetic effect will last longer and the force needed to meet the deformation will be larger when the DAH structure is subject to the tensile force in the X direction.

Effect of cell angle on the Poisson's ratio

As the DAH structure changes from cell A to cell A′, the angles α, β become large, while trusses s and t remain the same. Longitudinal length h changes from h₁ to h₂. The transversal strain ε_x and tensile strain ε_y can be calculated from the following equations

ɛ_{x} = t - t sin \frac{β}{2}

(1)

ɛ_{y} = h_{2} - h_{1} = t cos \frac{β}{2}

(2)

The Poisson's ratio value is shown in the following equations.

v = - \frac{ɛ_{x}}{ɛ_{y}} = - \frac{t - t sin \frac{β}{2}}{t cos \frac{β}{2}} = \frac{sin \frac{β}{2} - 1}{cos \frac{β}{2}}

(3)

0 < α < β < 180^{\circ}

(4)

cos \frac{β}{2} > 0, sin \frac{β}{2} - 1 < 0

(5)

\frac{π}{4} < \frac{β}{2} + \frac{π}{4} < \frac{3 π}{4}

(6)

sin (\frac{β}{2} + \frac{π}{4}) > \frac{\sqrt{2}}{2}

(7)

cos \frac{β}{2} + sin \frac{β}{2} > 1

(8)

- 1 < v < 0

(9)

According to its structure characteristics, the value range of angles α, β of the DAH is limited (as shown in Equation (4)). It is easy to get Equations (5) and (6) according to Equation (4). Equation (7) is calculated from Equation (6). Equation (8) is calculated from Equation (7). Equations (5) and (8) are substituted into Equation (3) to get Equation (9). From Equation (9), it can be seen that when the DAH structure changes from cell A to cell A′, it exhibits a negative Poisson's ratio performance and its Poisson's ratio is in the range −1<v<0.

According to Equation (3), angle β has a significant impact on the negative Poisson's ratio of the DAH structure. Specifically, as angle β increases, the Poisson's ratio value of the DAH structure increases. It can be seen from Figure 1 that the change of angles α, β will affect the change of longitudinal length h. Assuming that the length of the whole DAH structure is constant in the Y direction and the X direction, then the smaller the longitudinal length h is, the more DAH cells in the Y direction there are, and the more obvious the auxetic effect of the DAH structure. However, if the size of angle α is close to that of angle β, both the long truss and short truss of the DAH structure tend to become horizontal when the DAH is stretched in the X direction. Thus, the auxetic effect of the DAH structure is not obvious. In other words, when the angle β and the length of the truss t remain the same, the Poisson's ratio value of the DAH structure decreases with the angle α increasing within a certain range.

Experimental verification

Design of geometrical models with auxetic structures by SolidWorks

Two auxetic pore parts with the same geometric data, namely double arrowhead-X (DAHX) and double arrowhead-Y (DAHY), were built by using SolidWorks software. The detailed geometries of the parts are shown in Figure 2. The dimensions of plates investigated in this work for tensile loading are 100 × 100 × 2 mm³ (3.9370 × 3.9370 × 0.0787 inch³). The geometrical parameters of each cell are listed in Table 1. Edges of 20 mm long were set on both sides of the models to ensure that the models could be evenly clamped by the chuck of the extender in the third part of the actual test, and the stress on the stretched part at both ends of the model was balanced. To facilitate the stretching of the DAH sketch (Figure 2(a)) into a 3D drawing (Figures 2(c) and (d)), a distance n is set between the two triangles.

Figure 2.

Auxetic structures and their dimensions (in mm) for tensile loading: (a) double arrowhead-Y (DAHY) sketch before arc processing of sharp corners; (b) DAHY sketch after arc processing of sharp corners; (c) double arrowhead-X (DAHX) model; (d) DAHY model.

Table 1.

Dimensions (in mm) of the DAH structure

Arcs		Angles		Length		Width	Height
r₁ (mm)	r₂ (mm)	α (°)	β (°)	s (mm)	t (mm)	l (mm)	h (mm)	n (mm)
0.5	1.50	60	120	20	11.547	20	12.100	0.2

Three-dimensional printing

The samples for mechanical testing were built in the FDM system from Raise3D N2 (3D system, China) using thermoplastic polyurethane (TPU; Polyflex™) with super elastic properties. In the process, samples were produced with a layer thickness of 0.1 mm, a fill rate of 100% and the total depth of each sample was 2.0 mm, where the whole production process of each sample took approximately 6 h.

Tensile testing

To capture the mechanical response of the 3D printed constituent materials, the uniaxial tensile tests of plate samples are performed using a fabric strength machine (YG026MB, Fangyuan Co., Ltd, China). All the experiments are conducted with an axial constant speed of 5 mm/min. The sample was mounted in a vertical position, as shown in Figure 3. It was gradually stretched while the tensile force, axial (vertical) deformation and lateral deformation were measured. The specified elongation is 100 mm (strain 100%). A high-definition camera with 16 megapixels (wide angle) + 20 megapixels (telephoto) was set to record the transverse change of the samples.

Figure 3.

Schematic system of the tensile test.

Although only two types of the DAH auxetic metamaterial were available at the modeling stage and testing stages, six different samples of each structure were tested, in order to better prove the agreement between simulation and experiment.

Numerical verification

The numerical models of DAH structures are models of the DAHY structure and the DAHX structure, which were mentioned in the Design of geometrical models with auxetic structures by SolidWorks section. FEA was utilized to simulate the tensile loading conditions for both the DAHY structure and the DAHX structure by ABAQUS software.

Material properties for FEA

TPU has attracted a great deal of interest since it is a soft shape-memory polymer that exhibits unique thermomechanical storage and strong capability of shape recovery upon the cycle of heating–deformation–fixing–recovery.⁴⁸ Besides, compared with other hyperelastic printing materials, polyurethane does not easily to block the nozzle of 3D printing machines. So, Polyflex™ offered by 3D system was selected as the raw material in the test.

In FE simulation, various parameters of raw materials need to be input. The objective of the testing described herein is to define and to satisfy the input requirements of mathematical material models that exist in structural, non-linear FEA software. Hyperelastic material is a kind of complex non-linear material with various mathematical models of strain potential energy. So, there are no clearly defined experiments by national or international standards organizations.⁴⁹ The performance test of hyperelastic materials is according to the test method specially provided by Axel Products, Inc. As shown in Figure 4, uniaxial tensile tests, biaxial tensile tests, plane shear experiments and volume compression tests were implemented using YG026MB (fabric strength machine, Fangyuan Co., Ltd, China), KSM-BX5450ST (Kato Tech Co., Ltd, Japan) and YG026MB and HD026G (Nantong Hongda Experimental Instrument Co., Ltd, China) respectively. All the patterns in Figure 2 were prepared by 3D printing with a 100% filling rate.

Figure 4.

Testing elastomers for hyperelastic material models in finite element analysis: (a) uniaxial tensile test; (b) biaxial tensile test; (c) plane shear experiment; (d) volume compression test.

The most significant requirement for the simple tension experiments is that in order to achieve a state of pure tensile strain, the specimen should be much longer in the direction of stretching than in the width and thickness dimensions. The objective is to create an experiment where there is no lateral constraint to specimen thinning. The experimental model is a “dog-bone” shape specimen. The size of the test area is 170 × 5 × 2 mm³, as shown in Figure 4(a). After stretching 10 samples of the same size, their average values were taken as the test results. There are two kinds of biaxial tensile styles: one is a square flake with a thickness of 1 mm and a side length of 70 mm, and the other is a cube with a side length of 70 mm and a thickness of 2 mm. Three cubes of each type were prepared for biaxial tensile tests. In order to eliminate the precision error, it is necessary to firstly calculate the difference between the average values of the X and Y direction values after the stretching of the two styles, and then calculate the data. The test is shown in Figure 4(b). The pure shear experiment used for analysis is not what most of us would expect. The experiment appears at first glance to be nothing more than a very wide tensile test. However, because the material is nearly incompressible, a state of pure shear exists in the specimen at a 45 ° angle to the stretching direction. The most significant aspect of the specimen is that it is much shorter in the direction of stretching than in the width. The objective is to create an experiment where the specimen is perfectly constrained in the lateral direction such that all specimen thinning occurs in the thickness direction. The sample size is 140 × 40 × 2 mm³, of which 140 ×10 × 2 mm³ on both sides is the clamping part of the fixture and 10 × 10 × 2 mm³ in the center is the test area, as shown in Figure 4(c). The compression experiment, as shown in Figure 4(d), is also a popular test for elastomers. The experiment is used instead of volumetric compression.

The tested data are shown in Figure 5, where Figure 5(a) shows uniaxial, biaxial and plane shear curves, while Figure 5(b) shows a simple volume compression test. Since the latter is too large for the first three, the material is set as incompressible.

Figure 5.

Performance curves of hyperelastic material: (a) tensile property curve; (b) volumetric compressibility test.

Most models share common test data for input requirements. In general, stress and strain data sets developed by stretching the elastomer in several modes of deformation are required and “fitted” to sufficiently define the variables in the material models. A typical set of three stress–strain curves appropriate for input into fitting routines are shown in Figure 6.

Figure 6.

Material evaluation curves of hyperelastic material: (a) uniaxial material evaluation; (b) biaxial material evaluation; (c) material evaluation.

Due to TPU being a hyperelastic material, the uniaxial test, biaxial test and planar test should be done and the data utilized as the property parameters in ABAQUS. Then, through material evaluation, the Van der Waals constitutive equation model fits the material. The property of Polyflex™ used for the structures is shown Table 2.

Table 2.

Material properties

Materials	Density (kg/m³)	Poisson's ratio
TPU	1200	0.5

TPU: thermoplastic polyurethane.

Finite element analysis

After determining the properties of the material, the mechanical process of DAH structures can be simulated. Firstly, the model constructed by SolidWorks was imported to ABAQUS. The bottom surface of the model was fixed for all degrees of freedom and a displacement control boundary condition was applied on a plate, which was placed on the top of the model. The interaction property for contact was applied as tangential behavior and frictionless. Two constraints, namely a rigid body and tie, were set between the rigid plate and model. Then, a static general step was created and the top surfaces of the rigid plate were set as a constant speed of 0.833 mm/s in the Y direction. The time period for the step is 120 s. Since the TPU material in this paper is considered an incompressible material, DAH models were meshed by using hybrid (mixed formulation) elements⁵⁰ and the approximate global size was 1. The element family 3D stress is chosen for the stress analysis. In order to obtain a higher computational accuracy at a lower computational cost, the DAH model should be meshed into hex-dominated types.⁵¹ Finally, a job was run to submit it. After simulation, ABAQUS would show the results data. ABAQUS would show the theoretical results and deformation under the tensile load for TPU.

Results and discussion

Comparisons between experimental and numerical deformation photos

It obviously shows that the samples in the actual test (Figure 7, 8) have the same deformation with the models in simulation (Figure 9, 10). Besides, experimental results and numerical simulation show a satisfactory agreement with the theoretical analysis. Moreover, the transverse length of the structure is greater after stretching than before and explains why the structure can show auxetic behaviour at this time. Results also revealed thatDAH structure expanded in the transverse direction firstly and contract then with the strain increasing no matter X direction (Figure 7, 9) or Y direction (Figure 8, 10) the force is applied. DAHX structure (Figure 7, 9) has a more obvious auxetic effect than DAHY structure (Figure 8, 10).The mean mises stress of DAHX structures(Figure 9) is bigger than that of DAHY structures (Figure 10)which means DAHX has a bigger bear capacity than DAHY. Most of the stress is distributed on the short trusses and there is almost no stress distribution on the long trusses when DAH structure is stretched. The above phenomenon validates the theories discussion in The influence of the direction of force on the auxetic property section.

Figure 7.

Photographs of sample DAHX at different stages of deformation: (a) ε_x = 0; (b) ε_x = 0.25; (c) ε_x = 0.50; (d) ε_x = 0.75; (e) ε_x = 1.

Figure 8.

Photographs of sample DAHY at different stages of deformation: (a) ε_y = 0; (b) ε_y = 0.25; (c) ε_y = 0.50; (d) ε_y = 0.75; (e) ε_y = 1.

Figure 9.

Deformations of FEA models DAHX at different tensile strains when stretched in the course direction: (a) ε_x = 0; (b) ε_x = 0.25; (c) ε_x = 0.50; (d) ε_x = 0.75; (e) ε_x = 1.

Figure 10.

Deformations of FEA models DAHY at different tensile strains when stretched in the course direction: (a) ε_y = 0; (b) ε_y = 0.25; (c) ε_y = 0.50; (d) ε_y = 0.75; (e) ε_y = 1.

From the above conclusions, it can be concluded that the DAH structure has a more obvious auxetic effect when it was stretched in the X direction than in the Y direction. The DAHX structure has a greater bearing capacity than the DAHY structure. Most of the stress is distributed on the short trusses t and there is almost no stress distribution on the long trusses s when the DAH structure is stretched.

Comparisons of tensile parameters between numerical and experimental results

Comparisons of tensile curves between theoretical and experimental results are shown in Figure 11. Corresponding key parameters, including the tensile energy and the maximum tensile force, are featured from the four curves in Figure 11 and are listed in Table 3.

Figure 11.

Comparison of simulation result curves and the actual test result. DAHX: double arrowhead-X; DAHY: double arrowhead-Y; FEA: finite element analysis.

Table 3.

Physical property parameters of different structures obtained from tests and finite element analysis (FEA) results

Structures	Results	Tensile energy (J)	Maximum tensile force (N)
DAHX	Test results	22.184	342.513
DAHX	FEA results	19.816	329.704
DAHY	Test results	9.687	196.221
DAHY	FEA results	7.096	174.560

DAHX: double arrowhead-X; DAHY: double arrowhead-Y.

It can be concluded from Figure 11 that the curves of the FEA results have a consistent trend of change with the curves of measurement. The tensile force curves of the FEA at different displacements are a little higher than those of experiment, which is due to the fact that the parameters of TPU in FE simulation is not perfect. As stated by Kurt Miller AP, Inc., there are no clearly defined experiments by national or international standards organizations as of now.⁴⁹ So, the data of the property test of TPU is close to the real data, but not completely consistent, which results in some difference between the force curves of the FEA and tensile test.

In addition, when stretched to the same displacement, DAHY needs a smaller force than DAHX, which means that the DAHX structure is stronger than the DAHY structure. Moreover, compared with DAHY, the DAHX structure has a more obvious auxetic effect. These are decided by the structure and the properties of the samples, which was discussed in the The influence of the direction of force on the auxetic property and Comparisons between experimental and numerical deformation photos sections.

It is shown in Table 3 that relative errors of tensile energy and maximum tensile force between test results and FEA results are 2.368 and 12.809 for DAHX and 2.591 and 21.661 for DAHY, respectively. These errors are negligible, since some differences are only caused by experimental measurements as well as the difference between TPU parameters for FE simulation and that for the real test.

Comparisons of Poisson's ratio between numerical and experimental results

The Poisson's ratio values of the models in the FEA are obtained from corresponding data, such as the longitudinal displacement and transverse displacements of the model. In the experiment phrase, the longitudinal distances and the longitudinal force value corresponding to different time points can be obtained by the machine report when the sample is subjected to the longitudinal stretching. Thus, the longitudinal strain ε_y can be calculated from Equation (11). The transverse strain of the sample can be obtained by subsequent experimental processing and experimental data calculation. Firstly, a tensile process video of the sample described in the Tensile testing section was shot via Software Video Snapshots and one picture was grabbed every 10 seconds. The first 120 pictures were selected for the next process. Secondly, the lateral changes of the sample in the 120 pictures was measured using the screen ruler Nano Measurer. At this stage, at least 10 pairs of different spots were selected and the average value was calculated to measure the lateral deformation in each picture. Then, transversal strain ε_x could be calculated from Equation (10). Finally, Poisson's ratio ν could be calculated from Equation (12)

ɛ_{x} = - \frac{X - X_{0}}{X_{0}}

(10)

ɛ_{y} = - \frac{Y - Y_{0}}{Y_{0}}

(11)

where X₀, X, Y₀ and Y are the distances between two dots in the transverse direction and tensile direction in the initial state and deformed state, respectively. X is the average of 10 pairs of different data. ε_x and ε_y are the tensile strain and the transverse strain

ν = - \frac{ɛ_{x}}{ɛ_{y}}

(12)

For the FEA, 10 pairs of nodes located in the boundary part of the model structure in the transverse directions are selected to determine the size changes in the lateral direction of the simulated model. The average data of the 10 distances is the last distance data. One reference point, as shown in Figures 9 and 10, located in the central part of the sample edge in the transverse directions, is selected to determine the vertical changes of the simulated sample. The original distances of each pair of nodes are 100 mm, which is equal to the same value used in the experiment. As the original position and the displacement of the nodes can be directly obtained from ABAQUS, both the tensile strain and transverse strain can be calculated. When the tensile strain and transverse strain were known, the Poisson's ratio could finally be calculated. The Poisson's ratio–tensile force curves from the FEA and experiment in both the course and wale directions are shown in Figure 12.

Figure 12.

Poisson's ratio of auxetic structures obtained from tests and finite element analysis (FEA) results. DAHX: double arrowhead-X; DAHY: double arrowhead-Y.

As shown in Figure 12, the curves from the FEA are consistent with that from the experiment. Some subtle differences between the FEA results and experiment results still exist. This phenomenon mainly results from the simplification of the FE models, material properties and experimental error. After all, in the experiment phrase, the lateral displacement data of the model was recorded by the tester, which made the experimental data have some artificial subjective influence and some errors. The data before 50 N in the actual test curve are not very stable, which is due to the fact that the force at this time is small and both the transverse and longitudinal deformation of the material are not obvious. Besides, Figure 12 shows that the overall tendency of these curves is increasing and all of the Poisson's ratios increased with the increase of tensile force, until infinitely approaching a certain positive value, which means that when the DAH structure is stretched by forces in the X or Y direction, it first exhibits negative Poisson's ratio performance and then positive Poisson's ratio effect. Moreover, as shown in Figure 12, the Poisson's ratio of the DAHX structure is always lower than that of the DAHY structure, which means that the DAH shows a more obvious auxetic property when it is subjected to the tensile force in the X direction.

Effect of cell angle on the Poisson's ratio

In order to further study the effect of the cell angles α and β on the Poisson's ratio, 13 DAHX models with different cell angles were designed by SolidWorks. The schematic diagram of these structures is shown in Figure 2, while the specific parameters are listed in Table 4.

Table 4.

Specifications (in mm) of the DAH structure

β	α	s	l	t	h	n	r ₁	r ₂
(°)	(°)	(mm)	(mm)	(mm)	(mm)	(mm)	(mm)	(mm)
90	60	20	20	14.142	7.321	0.2	0.5	1.5
105	60	20	20	12.605	9.647	0.2	0.5	1.5
120	60	20	20	11.547	12.100	0.2	0.5	1.5
135	60	20	20	10.824	13.178	0.2	0.5	1.5
150	60	20	20	10.353	14.641	0.2	0.5	1.5
120	50	23.662	20	11.547	15.872	0.2	0.5	1.5
120	70	17.434	20	11.547	8.708	0.2	0.5	1.5
120	80	15.557	20	11.547	6.344	0.2	0.5	1.5
120	90	14.142	20	11.547	4.426	0.2	0.5	1.5

These structures were simulated by FEA software ABAQUS. Most of the simulation parameters were the same as in the Finite element analysis section. The result is shown in Figure 13.

Figure 13.

Effect of the cell angle on the engineering displacement–Poisson's ratio curves of double arrowhead-X (DAHX) structures: (a) Poisson's ratios of DAHX structures with different cell angle β; (b) Poisson's ratios of DAHX structures with different cell angle α.

As shown in Figure 13, angle α has little influence on the auxetic performance of DAH structures, while angle α has a greater influence. Figure 13(a) shows that DAHX structures have the most obvious auxetic properties during the initial stretching. With the continuation of the stretching, the Poisson's ratios gradually increase and the auxetic effects gradually decrease until the structures present a positive Poisson's ratio effect, like traditional materials. Moreover, if the number of DAHX structure elements perpendicular to the stretching direction does not change much, then the DAHX structure with small angle β has an obvious auxetic effect. The DAHX structure with angle β of 150 ° has a large Poisson's ratio value at the beginning of the stretch because it has fewer units along the direction perpendicular to the force. Figure 13(b) shows that as angle α increases, the initial auxetic effect of DAHX structures become more obvious. DAHX structures with angle α of 80 °, 90 ° are exceptions because the size of angle α is close to that of angle β in these structures. When the DAH is stretched in the X direction, both the long truss and short truss of the DAH structure tend to become horizontal, and thus the auxetic effect of the DAH structure is not obvious. Besides, when the sketches are extruded into 3D structures, the long trusses overlapped short trusses heavily because of interference in DAH structures with angle α of 80 °, 90 °, especially structures with angle α of 90 °. Although the Poisson's ratio curve of the DAHX structure with angle α of 90 ° decreases at the displacement of 60–100 mm, it will go up if it keeps being stretched. Figure 13 verifies the theoretical analysis in the Effect of cell angle on the Poisson's ratio section.

Conclusions

The influence of the direction of force on the auxetic property and the effect of the cell angle on the Poisson's ratio of the DAH structure has been discussed in this paper. Parametric 3D models of auxetic polymeric structures were manufactured through 3D printing technology using TPU material. Then their tension properties and Poisson's ratios were tested in the X direction and Y direction. Meanwhile, FE simulations were carried out to evaluate these different structural auxetic models. The numerical results in ABAQUS were adopted to compare with the experimental results and make a validation of the experiments. Some conclusions can be drawn from this study as follows.

The FE simulation could well predict the tensile deformation behaviors of the auxetic samples. The simulated auxetic structures deformations at different tensile force show good accordance with those obtained from experiments. The simulated Poisson's ratio values of the auxetic samples agree well with the experimental Poisson's ratio values. The FE simulation method in this paper can be used in future research.

The DAH structure expanded in the direction perpendicular to the force first and contracted with the strain increasing when it was pulled by the uniaxial force, no matter which direction the force is applied. Besides, the auxetic effect of the DAH structure under the tensile force in the X direction is longer than that under the tensile force in the Y direction. The reason for the auxetic behavior of the DAH structure is that the short truss has a tendency to change from inclined to horizontal. In this change, the structure expands perpendicular to the direction of force. If it keeps getting stretched in the Y direction, the short truss changes from horizontal to convex, and the DAH structure shows positive Poisson's ratio performance immediately. If it keeps getting stretched in the X direction, short truss t of the DAH structure will be stretched. The Poisson's ratio does not immediately go up.

The DAH structure is more resistant to compression when subjected to transverse tensile forces than when subjected to longitudinal tensile forces. When the DAH structure is stretched along the Y direction, the short truss first becomes horizontal, then convex, until it becomes a straight line in the direction of the force, and it does not take much force to complete this process. However, when the DAH is stretched in the X direction, the short truss will be stretched along the horizontal direction after it becomes horizontal. It takes a great deal of force for the DAH structure to move in the X direction.

Angle β has a significant impact on the negative Poisson's ratio of the DAH structure, while angle α has little influence on its auxetic performance. Specifically, the smaller the angles β and α, the better the auxetic effect. Computationally, the Poisson's ratio of the DAH structure is a trigonometric function with β $(ν = \frac{sin \frac{β}{2} - 1}{cos \frac{β}{2}})$ . The magnitudes of angles α, β affect the magnitude of longitudinal length h. The smaller the longitudinal length h is, the more DAH cells in the Y direction there are, and the more obvious the auxetic effect of the DAH structure. However, if the size of angle α is close to that of angle β, both the long truss and short truss of the DAH structure tend to become horizontal when the DAH is stretched in the X direction. Besides, when the sketches are extruded into 3D structures, the long trusses overlapped short trusses heavily because of interference in these DAH structures. Thus, the auxetic effect of the DAH structure with too large an angle α is not obvious.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was jointly supported by the Jiangxi Provincial Bureau for Quality and Technical Supervision (GZJKY201807), the Jiangxi Provincial Administration for Market Regulation (GSJK201909) and the Fundamental Research Funds for the Central Universities [CUSF-DH-D-2018036].

ORCID iDs

Longxin Gu

Zhaoqun Du

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