# Properties of 3D Double-Arrow Negative Poisson’s Ratio Lattice Structure under Quasi-Static Compression and Low-Speed Impact

Authors: Fuzhen Sun; Meng Liu; Baoming Wang; Na Yang; Jianfeng Wang; Keqian Cai; Zizhi Liu
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

Structures with negative Poisson’s ratios have wide applications in automotive design, owing to their excellent mechanical properties and good energy absorption capability. This type of structure can be used to design lightweight components. We used a double-arrow negative Poisson ratio structure and embedded lock assembly process to design a 3D lattice structure with a negative Poisson ratio. Moreover, we selected different design parameters for a simulation analysis and experimental research. The performances of the lattice structures with different parameters were compared and analyzed under different test conditions, including out-of-plane compression, three-point bending, and low-speed impact. The findings reported here pave the way for the development of new 3D auxetic materials with significantly expanded design space and potential applications by reducing the difficulty of the manufacturing process.


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## Body

## 1. Introduction

Cellular solids, such as foams and honeycombs, are widely used in engineering applications [1, 2] because of their superior properties, including low mass and high energy absorption capacity [3]. Cellular solids can have high-energy absorption capacities, which are determined by the peak stress and its decay during the plastic-flow plateau stages. Cellular structures with a negative Poisson ratio are known to have superior mechanical properties and application prospects [4] because of their indentation resistance [5], light weight [6], impact resistance [7, 8], energy absorption [9, 10], and vibration reduction capability [11–13].Auxetic materials can be obtained with a 2D cell structure, as well as a 3D cell structure. A 2D auxetic structure has a simple form and is easy to obtain. To obtain new auxetic materials with desirable performances, many metastructures have been proposed to explain the mechanical mechanisms behind auxetic materials [14–18]. Among these, a reentrant structure was proposed by Lakes to describe the auxetic behavior of manufactured isotopic auxetic foam [19]. Reentry is one of the basic mechanisms that can lead to auxetic behavior. Reentrant, double-arrowed, and star structures are typical structures attributed to the reentrant mechanism [19–21]. The other two basic auxetic mechanisms are chiral and rotational [22–25]. In addition to these three basic auxetic mechanisms, researchers have proposed other novel auxetic structures, such as metamaterials with high stiffness [9] and materials with independently tailored elastic properties [26]. Moreover, it is quite popular to combine a negative Poisson ratio with other negative indices, such as negative stiffness, compressibility, stability, and thermal expansion [27, 28].Because 3D auxetic structures are complex and difficult to produce, there has been less relevant research on these than on 2D auxetic honeycombs. Andreassen et al. [29] designed a 3D double-headed structure based on an arrow-type honeycomb, with a model obtained via 3D printing used for experiments. Lim [30] further deduced the equivalent elastic parameters of a 3D double-headed structure. Fu et al. [31] designed a 3D auxetic structure with a double concave hexagon and performed numerical simulations and experiments using the equivalent parameters of the structure. Ha et al. [32, 33] proposed a 3D chiral adjustable auxetic structure and studied the equivalent isotropic parameters through numerical simulations. However, the Poisson ratio of this structure could only reach −0.2. Chen and Fu [34] designed a new 3D auxetic lattice material with enhanced stiffness and deduced its elastic modulus and Poisson’s ratio as functions of the geometric parameters of the cell. Fu et al. [35] proposed a new design for 3D auxetic materials based on the rotation mechanism of chiral honeycombs. Furthermore, based on the beam theory and micromechanical method, the analytical formulas for both the elastic modulus and Poisson’s ratio were deduced for this category of 3D chiral materials, which were verified by numerical simulations, as well as experiments on the 3D printed model. Eyvazian et al. investigated the crush response and crashworthiness characteristics of round aluminum tubes with and without vertical corrugations [36]. The crashworthiness characteristics and collapse behaviors of these structures were analyzed, and an analytical relationship for the average force was established. Taghipoor et al. experimentally investigated the effect of holes on the crushing force of thin tubes with a square cross section under axial impact with a high strain rate [37]. Their crushing characteristics were extracted and compared with those of simple absorbers to evaluate the effects of different parameters. Consequently, it was found that in square thin-walled tubes with holes with a high strain rate, increasing the tube length does not affect the maximum force. Palomba et al. [38] performed a preliminary static analysis of aluminum honeycomb sandwich panels under both three-point and four-point bending conditions, evaluated the impact resistance parameters, and found slightly better performance under four-point bending conditions. The combination of static testing and computed tomography analysis led to an observation of the phenomena involved in the static bending response of aluminum honeycomb sandwich structures, which depended mainly on the flexure of the cell walls. Wang et al. [39] proposed an octagon auxetic reentrant honeycomb structure, quasistatic compression experimental test, and finite element method are conducted obtaining the effects of NPR of the proposed structure, and the results show the new structure has a superior energy absorption capacity than the traditional honeycomb structure. Wu et al. [40] combined four spherical reticulated shell structures with thin-walled tubes and reticulated tubes and compared the energy absorption capacity under quasistatic compression and impact status. The results showed that these composite structures can meet the requirements of light weight, high energy absorption, and low impact effect. Zhang et al. [41, 42] proposed a vertical strut and hexagon combined structure and investigated the quasistatic compression and dynamic crushing behaviors of it through simulation and theoretical analysis; the results indicate that the VSHCS has a better energy-absorption capacity than other reentrant structures. Researchers have done plenty of experiments compared with theoretical analysis and simulations on 2D auxetic structures, but there is less relevant research on 3D auxetic structures; it is necessary to study the mechanical properties for better application.In this study, a 3D lattice structure was designed to realize a negative Poisson ratio effect based on a double-arrow negative Poisson ratio structure. To facilitate manufacturing, we designed cells that could be assembled using the snap-fitting method. The mechanical properties with different parameters were compared and analyzed under different test conditions, including out-of-plane pressure, three-point bending, and low-speed impact. The structure in this study had a lower mass and superior impact absorption performance compared with those of existing structures, which can further enrich the study of 3D auxiliary structures.


## 2. Material and Methods

### 2.1. Physical Design

A double-arrow negative Poisson ratio structure has a 3D structure, and its topology optimization rules can be controlled, which makes its negative Poisson’s ratio characteristics more obvious, as shown in Figure1. Based on this structure and the interlocking assembly technique, the unit cells of a new structure and the overall structure are shown in Figure 2. Compared with the machining process of additive materials, the interlocking assembly technique is easier and cheaper to use when machining large parts, while ensuring the overall strength and stiffness of the structure. It is also more conducive to the promotion of engineering practice.Figure 1
3D double-arrow negative Poisson’s ratio structure.Figure 2
Schematic of negative Poisson’s ratio structure. (a) Designed negative Poisson’s ratio structure and (b) unit cell structure and (c) equivalent mechanical model diagram.
(a)(b)(c)Figure2(b) shows an enlarged view of the unit cell, and Figure 2(c) shows the equivalent mechanical model diagram. The structure was geometrically symmetric, and the long and short oblique bars were the main components of the unit cell structure. Here, T is the width of the bars, and l1 and l2 are the lengths of the long and short oblique bars, respectively. θ is the angle between a long oblique bar and the axis of symmetry, φ is the angle between a short oblique bar and the axis of symmetry, N is the length of the crossbar in a unit cell, and the horizontal length of a unit cell connected to the other side is N/2. Under the action of external force F, the deflection of the bar can be expressed by the following equation:(1)d=Fyl312E,where E is the modulus of elasticity of the matrix material and I is the secondary moment of inertia of the bar. The external forces along the vertical direction can be expressed using the following equations:(2)Fy1=sbN+l2Sinφ,(3)Fy2=sbN+l1Sinθ,where S is the stress caused by the external forces. Deflection d1 on the long oblique bar and deflection d2 on the short oblique bar can be expressed as follows:(4)d1=sN+l2sinϕl13sinθET3,(5)d2=sN+l2sinθl23sinφET3.Because the structure is compressed in the vertical and horizontal directions, the strain of the unit cell along both directions can be expressed as follows:(6)εy=d1sinθ+d2sinϕl1cosθ−l2cosϕ=sl12sin2θN+l1sinθsinϕ+sinθET3sinϕcosθ−sinθcosϕ,(7)εx=−d2cosϕ+d1cosθN+l1sinθ=−sl13sinθsin2θcosϕ−sin2ϕcosθET3sin2ϕ.According to the definition of the equivalent performance of a negative Poisson ratio structure, the equivalent Poisson ratio and equivalent elastic modulus of the unit cell structure can be obtained as follows:(8)v¯=εxεy=−l1sin2θcosϕ−sin2ϕcosθsinϕcosθ−sinθcosϕsinθsin2ϕN+l1sinθsinϕ+sinθ,(9)E¯=sεy=ET3sinϕcosθ−sinθcosϕl12sin2θN+l1sinθsinϕ+sinθ.A negative Poisson ratio structure has variable stiffnesses. The stiffness of the structure will become progressively larger as external forces are applied, and the resistance to deformation will become progressively stronger. The mechanical properties of structures with different parameters can be predicted using an equivalent mechanical model, which provides a theoretical basis for subsequent tests and analyses.The relative density of a structure refers to the ratio of the density of the negative Poisson ratio structure to that of the matrix material. The relative density is an important index for describing the performance of a negative Poisson ratio structure. Because a negative Poisson ratio structure has a honeycomb structure, the relative density is between 0 and 1, according to the definition of the relative density. When the clearance between the unit cells of the structure is reduced and the cell walls are filled with matrix materials, the density of a negative Poisson ratio structure will approach the density of the matrix materials; that is, the value is close to 1. When the gaps between the unit cells of a negative Poisson ratio structure are large and there is no filling material between the structures, the relative density of a negative Poisson ratio structure will approach 0. There is no direct relationship between the size of the unit cell and its relative density. Instead, it mainly depends on the shape of the unit cell.The relationship between the relative density of the structure and the parameters of the unit cell structure is defined as follows:(10)ρR=Tbl1+2l2+2/sinθ3l2cosϕN+l1sinθ2.The relative density values of different structures can be obtained from a mechanical model describing the relative density and parameters of cellular structures, which provides a basis for subsequent simulations and analyses.LetH be the monocyte height in Figure 2(b), let the rod width be T, and let the combined angle be θ-φ. Refer to previous literature [41]; the selected parameters are presented in Table 1.Table 1
Structural parameters.
Unit cell heightBar widthAngleH (mm)T (mm)θ-φ (°)16140–55181.545–6020250–65The structural parameters of nine samples were determined from orthogonal test results and are listed in Table2.Table 2
Test scheme.
Sample numberUnit cell heightBar widthAngleH (mm)T (mm)θ-φ (°)116140–55218145–60320150–654161.545–605181.550–656201.540–55716250–65818240–55920245–60


### 2.2. Flat Compression Test

#### 2.2.1. Fabrication of Experimental Samples

A basic single-cell diagram for each group of samples was based on the single-cell structure shown in Figure2(b) and Table 2, as shown in Figure 3. Aluminum alloy 6061 was used as the structural core material, and its basic mechanical properties are listed in Table 3. If a combined row of four unit cells is regarded as a complete piece of material to be processed, it can be seen from Figure 2(a) that 24 pieces were required to construct a sample model for assembly. The assembled samples are shown in Figure 4.Figure 3
Unit cell plots of different samples.Table 3
Basic properties of aluminum alloy 6061.
Aluminum alloy 6061DensityPoisson’s ratioElasticity modulusYield strengthValue2.75 g/cm30.33068.9 GPa110 MPaFigure 4
Samples of core material after assembly: samples (a) 1–3, (b) 4–6, and (c) 7–9.
(a)(b)(c)


#### 2.2.2. Test Process

To obtain the strength and stiffness of this structure, we performed a flat compression test on each sample. Each sample was analyzed three times to ensure the repeatability of the measurements. A Tianchen WAW-10 microcomputer-controlled hydraulic universal test machine was used for this test. The flat compression test rate was 0.2 mm/min, which was continued after the test specimen reached a limit load (maximum load), and the test ended when the load dropped to 60% of the limit load. A comparison of each structure before and after the experiment is shown in Figure5.Figure 5
Comparison of samples before and after tests. (a) Sample 1 before and after test. (b) Sample 2 before and after test. (c) Sample 3 before and after test. (d) Sample 4 before and after test. (e) Sample 5 before and after test. (f) Sample 6 before and after test. (g) Sample 7 before and after test. (h) Sample 8 before and after test. (i) Sample 9 before and after test.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 2.3. Three-Point Bending Test

#### 2.3.1. Fabrication of Experimental Samples

When fabricating the samples, different core materials were required to fabricate different panels. Carbon fiber prepreg cloth was cut to match the sizes of the different core structures. Four layers of carbon cloth were placed so that the fibers had the same orientation. A small piece of felt was dipped in an appropriate amount of demolding agent, the surface of the steel plate was wiped clean, and the cut carbon cloth was spread on the steel plate. After its placement, the cloth was covered with a vacuum bag, which was glued to the steel plate with sealant, and the plate was placed in a curing furnace. After curing, the carbon fiber panel was bonded to the core material of the corresponding size. The complete structure is illustrated in Figure6.Figure 6
Completed samples. (a) Samples 1–3. (b) Samples 4–6. (c) Samples 7–9.
(a)(b)(c)


#### 2.3.2. Test Process

A Tianchen WAW-300 microcomputer-controlled hydraulic universal testing machine was used to test the samples at a rate of 5 mm/min. Each sample was analyzed three times to ensure the repeatability of the measurements. Clamps were placed on the lower surface of the test bed and leveled, and a carbon fiber-laminated plate sample was placed on the clamps. To ensure the consistency of the test and comparability of the results, the supporting points of the fixture were aligned with two outer pieces of the corresponding plate core material when determining the span because the samples had different sizes, as shown in Figure7. After determining the span for each sample, the upper pressure head was placed on the laminated plate. Figure 8 shows each structure before and after the experiment.Figure 7
Three-point bending diagram.Figure 8
Comparison of samples before and after experiments. (a) Sample 1 before and after experiment. (b) Sample 2 before and after experiment. (c) Sample 3 before and after experiment. (d) Sample 4 before and after experiment. (e) Sample 5 before and after experiment. (f) Sample 6 before and after experiment. (g) Sample 7 before and after experiment. (h) Sample 8 before and after experiment. (i) Sample 9 before and after experiment.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 2.4. Low-Velocity Impact Test

To test the impact resistance of the designed lattice structure with a negative Poisson ratio, an impact testbed was built, as shown in Figure9. The dimensions of the steel plate used in the impact test bed were 230 mm × 230 mm × 10 mm, and the falling height was 50 cm. After electromagnet adsorption, the steel plate moved freely after the power was cut off, and the lattice structure was impacted at a low speed. The impact resistance was demonstrated by measuring the compression deformation before and after impact. To facilitate the measurement of the compression deformation of the specimen before and after the impact, an aluminum plate was placed on the specimen during the measurement. The structure before and after the low-speed impact experiment is shown in Figure 10 (sample 8). The test was repeated three times.Figure 9
Impact testbed.Figure 10
Sample 8 (a) before and (b) after low-speed impact experiment.
(a)(b)


## 2.1. Physical Design

A double-arrow negative Poisson ratio structure has a 3D structure, and its topology optimization rules can be controlled, which makes its negative Poisson’s ratio characteristics more obvious, as shown in Figure1. Based on this structure and the interlocking assembly technique, the unit cells of a new structure and the overall structure are shown in Figure 2. Compared with the machining process of additive materials, the interlocking assembly technique is easier and cheaper to use when machining large parts, while ensuring the overall strength and stiffness of the structure. It is also more conducive to the promotion of engineering practice.Figure 1
3D double-arrow negative Poisson’s ratio structure.Figure 2
Schematic of negative Poisson’s ratio structure. (a) Designed negative Poisson’s ratio structure and (b) unit cell structure and (c) equivalent mechanical model diagram.
(a)(b)(c)Figure2(b) shows an enlarged view of the unit cell, and Figure 2(c) shows the equivalent mechanical model diagram. The structure was geometrically symmetric, and the long and short oblique bars were the main components of the unit cell structure. Here, T is the width of the bars, and l1 and l2 are the lengths of the long and short oblique bars, respectively. θ is the angle between a long oblique bar and the axis of symmetry, φ is the angle between a short oblique bar and the axis of symmetry, N is the length of the crossbar in a unit cell, and the horizontal length of a unit cell connected to the other side is N/2. Under the action of external force F, the deflection of the bar can be expressed by the following equation:(1)d=Fyl312E,where E is the modulus of elasticity of the matrix material and I is the secondary moment of inertia of the bar. The external forces along the vertical direction can be expressed using the following equations:(2)Fy1=sbN+l2Sinφ,(3)Fy2=sbN+l1Sinθ,where S is the stress caused by the external forces. Deflection d1 on the long oblique bar and deflection d2 on the short oblique bar can be expressed as follows:(4)d1=sN+l2sinϕl13sinθET3,(5)d2=sN+l2sinθl23sinφET3.Because the structure is compressed in the vertical and horizontal directions, the strain of the unit cell along both directions can be expressed as follows:(6)εy=d1sinθ+d2sinϕl1cosθ−l2cosϕ=sl12sin2θN+l1sinθsinϕ+sinθET3sinϕcosθ−sinθcosϕ,(7)εx=−d2cosϕ+d1cosθN+l1sinθ=−sl13sinθsin2θcosϕ−sin2ϕcosθET3sin2ϕ.According to the definition of the equivalent performance of a negative Poisson ratio structure, the equivalent Poisson ratio and equivalent elastic modulus of the unit cell structure can be obtained as follows:(8)v¯=εxεy=−l1sin2θcosϕ−sin2ϕcosθsinϕcosθ−sinθcosϕsinθsin2ϕN+l1sinθsinϕ+sinθ,(9)E¯=sεy=ET3sinϕcosθ−sinθcosϕl12sin2θN+l1sinθsinϕ+sinθ.A negative Poisson ratio structure has variable stiffnesses. The stiffness of the structure will become progressively larger as external forces are applied, and the resistance to deformation will become progressively stronger. The mechanical properties of structures with different parameters can be predicted using an equivalent mechanical model, which provides a theoretical basis for subsequent tests and analyses.The relative density of a structure refers to the ratio of the density of the negative Poisson ratio structure to that of the matrix material. The relative density is an important index for describing the performance of a negative Poisson ratio structure. Because a negative Poisson ratio structure has a honeycomb structure, the relative density is between 0 and 1, according to the definition of the relative density. When the clearance between the unit cells of the structure is reduced and the cell walls are filled with matrix materials, the density of a negative Poisson ratio structure will approach the density of the matrix materials; that is, the value is close to 1. When the gaps between the unit cells of a negative Poisson ratio structure are large and there is no filling material between the structures, the relative density of a negative Poisson ratio structure will approach 0. There is no direct relationship between the size of the unit cell and its relative density. Instead, it mainly depends on the shape of the unit cell.The relationship between the relative density of the structure and the parameters of the unit cell structure is defined as follows:(10)ρR=Tbl1+2l2+2/sinθ3l2cosϕN+l1sinθ2.The relative density values of different structures can be obtained from a mechanical model describing the relative density and parameters of cellular structures, which provides a basis for subsequent simulations and analyses.LetH be the monocyte height in Figure 2(b), let the rod width be T, and let the combined angle be θ-φ. Refer to previous literature [41]; the selected parameters are presented in Table 1.Table 1
Structural parameters.
Unit cell heightBar widthAngleH (mm)T (mm)θ-φ (°)16140–55181.545–6020250–65The structural parameters of nine samples were determined from orthogonal test results and are listed in Table2.Table 2
Test scheme.
Sample numberUnit cell heightBar widthAngleH (mm)T (mm)θ-φ (°)116140–55218145–60320150–654161.545–605181.550–656201.540–55716250–65818240–55920245–60


## 2.2. Flat Compression Test

### 2.2.1. Fabrication of Experimental Samples

A basic single-cell diagram for each group of samples was based on the single-cell structure shown in Figure2(b) and Table 2, as shown in Figure 3. Aluminum alloy 6061 was used as the structural core material, and its basic mechanical properties are listed in Table 3. If a combined row of four unit cells is regarded as a complete piece of material to be processed, it can be seen from Figure 2(a) that 24 pieces were required to construct a sample model for assembly. The assembled samples are shown in Figure 4.Figure 3
Unit cell plots of different samples.Table 3
Basic properties of aluminum alloy 6061.
Aluminum alloy 6061DensityPoisson’s ratioElasticity modulusYield strengthValue2.75 g/cm30.33068.9 GPa110 MPaFigure 4
Samples of core material after assembly: samples (a) 1–3, (b) 4–6, and (c) 7–9.
(a)(b)(c)


### 2.2.2. Test Process

To obtain the strength and stiffness of this structure, we performed a flat compression test on each sample. Each sample was analyzed three times to ensure the repeatability of the measurements. A Tianchen WAW-10 microcomputer-controlled hydraulic universal test machine was used for this test. The flat compression test rate was 0.2 mm/min, which was continued after the test specimen reached a limit load (maximum load), and the test ended when the load dropped to 60% of the limit load. A comparison of each structure before and after the experiment is shown in Figure5.Figure 5
Comparison of samples before and after tests. (a) Sample 1 before and after test. (b) Sample 2 before and after test. (c) Sample 3 before and after test. (d) Sample 4 before and after test. (e) Sample 5 before and after test. (f) Sample 6 before and after test. (g) Sample 7 before and after test. (h) Sample 8 before and after test. (i) Sample 9 before and after test.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 2.2.1. Fabrication of Experimental Samples

A basic single-cell diagram for each group of samples was based on the single-cell structure shown in Figure2(b) and Table 2, as shown in Figure 3. Aluminum alloy 6061 was used as the structural core material, and its basic mechanical properties are listed in Table 3. If a combined row of four unit cells is regarded as a complete piece of material to be processed, it can be seen from Figure 2(a) that 24 pieces were required to construct a sample model for assembly. The assembled samples are shown in Figure 4.Figure 3
Unit cell plots of different samples.Table 3
Basic properties of aluminum alloy 6061.
Aluminum alloy 6061DensityPoisson’s ratioElasticity modulusYield strengthValue2.75 g/cm30.33068.9 GPa110 MPaFigure 4
Samples of core material after assembly: samples (a) 1–3, (b) 4–6, and (c) 7–9.
(a)(b)(c)


## 2.2.2. Test Process

To obtain the strength and stiffness of this structure, we performed a flat compression test on each sample. Each sample was analyzed three times to ensure the repeatability of the measurements. A Tianchen WAW-10 microcomputer-controlled hydraulic universal test machine was used for this test. The flat compression test rate was 0.2 mm/min, which was continued after the test specimen reached a limit load (maximum load), and the test ended when the load dropped to 60% of the limit load. A comparison of each structure before and after the experiment is shown in Figure5.Figure 5
Comparison of samples before and after tests. (a) Sample 1 before and after test. (b) Sample 2 before and after test. (c) Sample 3 before and after test. (d) Sample 4 before and after test. (e) Sample 5 before and after test. (f) Sample 6 before and after test. (g) Sample 7 before and after test. (h) Sample 8 before and after test. (i) Sample 9 before and after test.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 2.3. Three-Point Bending Test

### 2.3.1. Fabrication of Experimental Samples

When fabricating the samples, different core materials were required to fabricate different panels. Carbon fiber prepreg cloth was cut to match the sizes of the different core structures. Four layers of carbon cloth were placed so that the fibers had the same orientation. A small piece of felt was dipped in an appropriate amount of demolding agent, the surface of the steel plate was wiped clean, and the cut carbon cloth was spread on the steel plate. After its placement, the cloth was covered with a vacuum bag, which was glued to the steel plate with sealant, and the plate was placed in a curing furnace. After curing, the carbon fiber panel was bonded to the core material of the corresponding size. The complete structure is illustrated in Figure6.Figure 6
Completed samples. (a) Samples 1–3. (b) Samples 4–6. (c) Samples 7–9.
(a)(b)(c)


### 2.3.2. Test Process

A Tianchen WAW-300 microcomputer-controlled hydraulic universal testing machine was used to test the samples at a rate of 5 mm/min. Each sample was analyzed three times to ensure the repeatability of the measurements. Clamps were placed on the lower surface of the test bed and leveled, and a carbon fiber-laminated plate sample was placed on the clamps. To ensure the consistency of the test and comparability of the results, the supporting points of the fixture were aligned with two outer pieces of the corresponding plate core material when determining the span because the samples had different sizes, as shown in Figure7. After determining the span for each sample, the upper pressure head was placed on the laminated plate. Figure 8 shows each structure before and after the experiment.Figure 7
Three-point bending diagram.Figure 8
Comparison of samples before and after experiments. (a) Sample 1 before and after experiment. (b) Sample 2 before and after experiment. (c) Sample 3 before and after experiment. (d) Sample 4 before and after experiment. (e) Sample 5 before and after experiment. (f) Sample 6 before and after experiment. (g) Sample 7 before and after experiment. (h) Sample 8 before and after experiment. (i) Sample 9 before and after experiment.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 2.3.1. Fabrication of Experimental Samples

When fabricating the samples, different core materials were required to fabricate different panels. Carbon fiber prepreg cloth was cut to match the sizes of the different core structures. Four layers of carbon cloth were placed so that the fibers had the same orientation. A small piece of felt was dipped in an appropriate amount of demolding agent, the surface of the steel plate was wiped clean, and the cut carbon cloth was spread on the steel plate. After its placement, the cloth was covered with a vacuum bag, which was glued to the steel plate with sealant, and the plate was placed in a curing furnace. After curing, the carbon fiber panel was bonded to the core material of the corresponding size. The complete structure is illustrated in Figure6.Figure 6
Completed samples. (a) Samples 1–3. (b) Samples 4–6. (c) Samples 7–9.
(a)(b)(c)


## 2.3.2. Test Process

A Tianchen WAW-300 microcomputer-controlled hydraulic universal testing machine was used to test the samples at a rate of 5 mm/min. Each sample was analyzed three times to ensure the repeatability of the measurements. Clamps were placed on the lower surface of the test bed and leveled, and a carbon fiber-laminated plate sample was placed on the clamps. To ensure the consistency of the test and comparability of the results, the supporting points of the fixture were aligned with two outer pieces of the corresponding plate core material when determining the span because the samples had different sizes, as shown in Figure7. After determining the span for each sample, the upper pressure head was placed on the laminated plate. Figure 8 shows each structure before and after the experiment.Figure 7
Three-point bending diagram.Figure 8
Comparison of samples before and after experiments. (a) Sample 1 before and after experiment. (b) Sample 2 before and after experiment. (c) Sample 3 before and after experiment. (d) Sample 4 before and after experiment. (e) Sample 5 before and after experiment. (f) Sample 6 before and after experiment. (g) Sample 7 before and after experiment. (h) Sample 8 before and after experiment. (i) Sample 9 before and after experiment.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 2.4. Low-Velocity Impact Test

To test the impact resistance of the designed lattice structure with a negative Poisson ratio, an impact testbed was built, as shown in Figure9. The dimensions of the steel plate used in the impact test bed were 230 mm × 230 mm × 10 mm, and the falling height was 50 cm. After electromagnet adsorption, the steel plate moved freely after the power was cut off, and the lattice structure was impacted at a low speed. The impact resistance was demonstrated by measuring the compression deformation before and after impact. To facilitate the measurement of the compression deformation of the specimen before and after the impact, an aluminum plate was placed on the specimen during the measurement. The structure before and after the low-speed impact experiment is shown in Figure 10 (sample 8). The test was repeated three times.Figure 9
Impact testbed.Figure 10
Sample 8 (a) before and (b) after low-speed impact experiment.
(a)(b)


## 3. Results and Discussion

### 3.1. Flat Crush Test

The force–displacement curves for the nine groups of experimental results were recorded after completing the tests on the test machine; the results are shown in Figure11. The Poisson ratio, strength, and stiffness of the pattern could be calculated based on the mechanics of the materials. The results for all the samples are summarized in Table 4.Figure 11
Flat crush force–displacement curves for samples.Table 4
Flat crush results for samples.
Sample numberUltimate load (N)Ultimate displacement (mm)Sectional area (mm2)Strength (MPa)Stiffness (MPa)Poisson’s ratio14471.30275.131.6210.4−0.4724591.78330.511.393.1−0.4632782.36363.000.771.4−0.35411362.62298.513.8125.0−0.2856682.58347.001.937.1−0.3868311.78371.132.2417.5−0.22714451.90459.003.1534.9−0.24811782.88307.133.8476.9−0.21910952.17426.512.5738.7−0.45The results from the flat compression experiment showed that the specimens deformed under pressure. As the load increased, the degree of bending of the specimen gradually increased, and the top and bottom layers of the aluminum sheet were considerably deformed. When the ultimate load was reached, a degumming failure occurred between the aluminum sheet at the top or bottom and the aluminum sheet in the other layer, but the overall structure did not experience large deformations.To further study the influences of different structural parameters on the performance of the laminated plates, a flat compression simulation study was conducted on these samples, as shown in Figure12. In this flat compression simulation study, the upper and lower plates were 5 mm-thick steel plates, and the material properties are listed in Table 5. The core material was aluminum alloy 6061, and its properties are listed in Table 3. Binding constraints were applied to the core and the two plates. The surface of the lower plate was fixed, and a load of 500 N was applied to the surface of the upper plate. The effect of the adhesive layer was not considered in the numerical simulations. Therefore, a surface-based tie constraint was applied to the interfaces between the core and panel. The panel mesh size was 3 mm, the core mesh size was 0.5 mm, the mesh type of the panel and core was hexahedral mesh, and the maximum number of grids for the nine models was 252,772.Figure 12
Flat pressure simulation.Table 5
Basic properties of steel.
SteelDensityPoisson’s ratioElasticity modulusYield strengthValue7.75 g/cm30.310193 GPa207 MPaThe results are shown in Figure13. The SEA (Specific Energy Absorption) values for the real test and simulation are listed in Table 6. From Table 6, it can be observed that the SEA values were relatively close.Figure 13
Simulated flat crushing results. (a) Comparison of total deformations of samples, (b) stress and (c) strain of each sample, and (d) Poisson’s ratio for each sample.
(a)(b)(c)(d)Table 6
SEA values in real tests and simulations.
Real test (MJ)Simulation (MJ)Sample 15.365.52Sample 24.296.98Sample 36.558.03Sample 418.2514.45Sample 513.1414.87Sample 611.1311.80Sample 717.7420.85Sample 822.8628.67Sample 912.2212.65A comparative analysis of the above results showed that the total deformation, stress, and strain of each sample exhibited the same trends. The highest point was seen for sample 3, the lowest point was seen for sample 8, and the highest and lowest points differed by a factor of 4–5. Sample 3 was the weakest, which was consistent with the original prediction. Because sample 3 had the largest cell size and smallest bar width, its performance should have been weaker than those of the other structures. A comparison of samples 1–3, 4–6, and 7–9 showed that the low-angle samples in the three groups performed better than the high-angle samples, and the effect of the unit cell height was nearly insignificant. The values for sample 8 were the smallest, indicating that sample 8 had the highest strength and stiffness under the same load, which was consistent with the test results.


### 3.2. Three-Point Bending Test

Nine groups of force–displacement curves were recorded after conducting the tests with the test machine, and the fitting results are shown in Figure14.Figure 14
Three-point bending force–displacement curve for each sample.The ultimate loads for all the samples are summarized in Table6.Because of problems with the testing machine and the structure of the samples, the bending test results were not ideal. The panel began to bend and deform as the load increased, and the core material inclined to the left or right. When the load increased to the limit load, most of the samples were destroyed as a result of a degumming failure between the core material and panel. The force–displacement curves showed that the load dropped sharply at the moment of sample failure, and the curves rose again because the panel was not completely detached after failure in some samples.Similarly, simulated three-point bending tests were conducted as shown in Figure15. In the three-point bending simulation, the simulation model shown in Figure 7. To simulate the contact area between the cylinder and the plate, the surface was divided, assuming that the width of the contact surface was 1 mm. A load of 500 N was applied to the contact area between the upper plate and cylinder, and a fixed constraint was applied to the contact area between the lower plate and cylinder. The upper and lower plates were 0.8 mm thick carbon fiber, with a total n-layer, and the paving angle was 0/90/0/90. The ACP module was used to complete the carbon fiber modeling, and the material properties of each layer are listed in Table 7. The mesh size used for all the parts was 0.5 mm, the mesh type was hexahedral, and the maximum number of meshes was 246,248.Figure 15
Simulated three-point bending test results. (a) Deformation in each sample and maximum (b) stress and (c) strain in each specimen.
(a)(b)(c)Table 7
Three-point bending ultimate loads and displacements of samples.
Sample numberUltimate load (KN)Ultimate displacement (mm)Bending strength (N/mm2)10.51.300.26120.451.780.16030.32.360.09440.62.620.28750.552.580.17260.51.780.15670.71.900.29980.452.880.17790.252.170.093The results are shown in Figure15.A comparison of the broken line graphs for the samples shows that sample 3 exhibited the maximum deformation, stress, and strain, indicating that it had low structural stability and was more prone to deformation and stress concentration under the action of a bending moment. The angle between the inclined bar and monocyte axis of symmetry gradually increased for a given bar width, the deformation and stress of the sample increased, and the flexural strength of the sample decreased, which was consistent with the test results. Samples 4 and 8 exhibited similar stress and strain values, but the deformation of sample 8 was smaller. Thus, sample 8 contained the structure with the optimal performance in the three-point bending test.


### 3.3. Low-Speed Impact Test

To further study the influences of different structural parameters on the impact performance, a low-speed simulation study was conducted for these samples, as shown in Figure16.Figure 16
Low-speed impact simulation. (a) Loading and (b) boundary condition.
(a)(b)According to the formula for a free-falling body, the impact energy of a steel plate falling from 50 cm is equivalent to its impact energy when it has an initial velocity of 3.13 m/s. Therefore, the impact speed of the steel plate was set to 3130 mm/s. A total of 32 surfaces of the core in contact with the ground were fixed. The steel panel mesh size was 5 mm, and core mesh size was 0.5 mm. The mesh type of the panel and core was hexahedral mesh, and the maximum number of grids for the nine models was 252,772.The deformation compressions of the nine groups of lattice structures after impact in the finite element method (FEM) simulations and experiments are listed in Table8, and the deformation of sample 8 after impact is shown in Figure 17. In Figure 18, although the maximum displacement of the lattice structure was 19.448 mm, its steel plate displacement was only 15.65 mm. We considered the displacement of the steel plate to be the overall deformation value for the lattice structure, which prevented the simulated deformation caused by the displacement of the outer suspension core bracket from being too large.Table 8
Low-speed impact displacements of samples.
Sample numberExperiment displacement (mm)FEM displacement (mm)Error (%)123.920.85−12.76%226.124.89−4.64%327.225.51−6.21%418.316.06−12.24%525.724.34−5.29%622.319.05−14.57%717.616.96−3.64%817.015.65−7.95%918.918.64−1.38%Figure 17
Simulated low-speed impact results. (a) Maximum stress and (b) strain in all specimens.
(a)(b)Figure 18
Deformations of sample 8 after impact in (a) experiment and (b) simulation.
(a)(b)The maximum stress and elastic strain of the low-speed impact obtained from the simulation are shown in Figure17.According to Table9, the absolute value of the error between the test and simulation deformation results was 14.57% at the maximum and 3.64% at the minimum, indicating the reliability of the simulation results. A comparison of the broken line graphs for the samples shows that the bar width gradually increases, and the total deformation of the structure becomes progressively smaller, indicating that its resistance to impact gradually increases. Sample 3 exhibited the maximum stress, deformation, and strain, which could have been related to the fact that it had the minimum relative density. Thus, it had the weakest resistance to impact and the most obvious damage for a given impact energy. Sample 8 exhibited the minimum deformation and strain, and it had the maximum relative density. Thus, sample 8 was found to be the optimal structure for resisting damage due to impact.Table 9
Basic properties of carbon/epoxy laminate.
DensityExEyEzvxyvxzvyzGxyGxzGyz1.6 g/cm3131 GPa8 GPa8 GPa0.310.310.35400 MPa5400 MPa4500 MPa


## 3.1. Flat Crush Test

The force–displacement curves for the nine groups of experimental results were recorded after completing the tests on the test machine; the results are shown in Figure11. The Poisson ratio, strength, and stiffness of the pattern could be calculated based on the mechanics of the materials. The results for all the samples are summarized in Table 4.Figure 11
Flat crush force–displacement curves for samples.Table 4
Flat crush results for samples.
Sample numberUltimate load (N)Ultimate displacement (mm)Sectional area (mm2)Strength (MPa)Stiffness (MPa)Poisson’s ratio14471.30275.131.6210.4−0.4724591.78330.511.393.1−0.4632782.36363.000.771.4−0.35411362.62298.513.8125.0−0.2856682.58347.001.937.1−0.3868311.78371.132.2417.5−0.22714451.90459.003.1534.9−0.24811782.88307.133.8476.9−0.21910952.17426.512.5738.7−0.45The results from the flat compression experiment showed that the specimens deformed under pressure. As the load increased, the degree of bending of the specimen gradually increased, and the top and bottom layers of the aluminum sheet were considerably deformed. When the ultimate load was reached, a degumming failure occurred between the aluminum sheet at the top or bottom and the aluminum sheet in the other layer, but the overall structure did not experience large deformations.To further study the influences of different structural parameters on the performance of the laminated plates, a flat compression simulation study was conducted on these samples, as shown in Figure12. In this flat compression simulation study, the upper and lower plates were 5 mm-thick steel plates, and the material properties are listed in Table 5. The core material was aluminum alloy 6061, and its properties are listed in Table 3. Binding constraints were applied to the core and the two plates. The surface of the lower plate was fixed, and a load of 500 N was applied to the surface of the upper plate. The effect of the adhesive layer was not considered in the numerical simulations. Therefore, a surface-based tie constraint was applied to the interfaces between the core and panel. The panel mesh size was 3 mm, the core mesh size was 0.5 mm, the mesh type of the panel and core was hexahedral mesh, and the maximum number of grids for the nine models was 252,772.Figure 12
Flat pressure simulation.Table 5
Basic properties of steel.
SteelDensityPoisson’s ratioElasticity modulusYield strengthValue7.75 g/cm30.310193 GPa207 MPaThe results are shown in Figure13. The SEA (Specific Energy Absorption) values for the real test and simulation are listed in Table 6. From Table 6, it can be observed that the SEA values were relatively close.Figure 13
Simulated flat crushing results. (a) Comparison of total deformations of samples, (b) stress and (c) strain of each sample, and (d) Poisson’s ratio for each sample.
(a)(b)(c)(d)Table 6
SEA values in real tests and simulations.
Real test (MJ)Simulation (MJ)Sample 15.365.52Sample 24.296.98Sample 36.558.03Sample 418.2514.45Sample 513.1414.87Sample 611.1311.80Sample 717.7420.85Sample 822.8628.67Sample 912.2212.65A comparative analysis of the above results showed that the total deformation, stress, and strain of each sample exhibited the same trends. The highest point was seen for sample 3, the lowest point was seen for sample 8, and the highest and lowest points differed by a factor of 4–5. Sample 3 was the weakest, which was consistent with the original prediction. Because sample 3 had the largest cell size and smallest bar width, its performance should have been weaker than those of the other structures. A comparison of samples 1–3, 4–6, and 7–9 showed that the low-angle samples in the three groups performed better than the high-angle samples, and the effect of the unit cell height was nearly insignificant. The values for sample 8 were the smallest, indicating that sample 8 had the highest strength and stiffness under the same load, which was consistent with the test results.


## 3.2. Three-Point Bending Test

Nine groups of force–displacement curves were recorded after conducting the tests with the test machine, and the fitting results are shown in Figure14.Figure 14
Three-point bending force–displacement curve for each sample.The ultimate loads for all the samples are summarized in Table6.Because of problems with the testing machine and the structure of the samples, the bending test results were not ideal. The panel began to bend and deform as the load increased, and the core material inclined to the left or right. When the load increased to the limit load, most of the samples were destroyed as a result of a degumming failure between the core material and panel. The force–displacement curves showed that the load dropped sharply at the moment of sample failure, and the curves rose again because the panel was not completely detached after failure in some samples.Similarly, simulated three-point bending tests were conducted as shown in Figure15. In the three-point bending simulation, the simulation model shown in Figure 7. To simulate the contact area between the cylinder and the plate, the surface was divided, assuming that the width of the contact surface was 1 mm. A load of 500 N was applied to the contact area between the upper plate and cylinder, and a fixed constraint was applied to the contact area between the lower plate and cylinder. The upper and lower plates were 0.8 mm thick carbon fiber, with a total n-layer, and the paving angle was 0/90/0/90. The ACP module was used to complete the carbon fiber modeling, and the material properties of each layer are listed in Table 7. The mesh size used for all the parts was 0.5 mm, the mesh type was hexahedral, and the maximum number of meshes was 246,248.Figure 15
Simulated three-point bending test results. (a) Deformation in each sample and maximum (b) stress and (c) strain in each specimen.
(a)(b)(c)Table 7
Three-point bending ultimate loads and displacements of samples.
Sample numberUltimate load (KN)Ultimate displacement (mm)Bending strength (N/mm2)10.51.300.26120.451.780.16030.32.360.09440.62.620.28750.552.580.17260.51.780.15670.71.900.29980.452.880.17790.252.170.093The results are shown in Figure15.A comparison of the broken line graphs for the samples shows that sample 3 exhibited the maximum deformation, stress, and strain, indicating that it had low structural stability and was more prone to deformation and stress concentration under the action of a bending moment. The angle between the inclined bar and monocyte axis of symmetry gradually increased for a given bar width, the deformation and stress of the sample increased, and the flexural strength of the sample decreased, which was consistent with the test results. Samples 4 and 8 exhibited similar stress and strain values, but the deformation of sample 8 was smaller. Thus, sample 8 contained the structure with the optimal performance in the three-point bending test.


## 3.3. Low-Speed Impact Test

To further study the influences of different structural parameters on the impact performance, a low-speed simulation study was conducted for these samples, as shown in Figure16.Figure 16
Low-speed impact simulation. (a) Loading and (b) boundary condition.
(a)(b)According to the formula for a free-falling body, the impact energy of a steel plate falling from 50 cm is equivalent to its impact energy when it has an initial velocity of 3.13 m/s. Therefore, the impact speed of the steel plate was set to 3130 mm/s. A total of 32 surfaces of the core in contact with the ground were fixed. The steel panel mesh size was 5 mm, and core mesh size was 0.5 mm. The mesh type of the panel and core was hexahedral mesh, and the maximum number of grids for the nine models was 252,772.The deformation compressions of the nine groups of lattice structures after impact in the finite element method (FEM) simulations and experiments are listed in Table8, and the deformation of sample 8 after impact is shown in Figure 17. In Figure 18, although the maximum displacement of the lattice structure was 19.448 mm, its steel plate displacement was only 15.65 mm. We considered the displacement of the steel plate to be the overall deformation value for the lattice structure, which prevented the simulated deformation caused by the displacement of the outer suspension core bracket from being too large.Table 8
Low-speed impact displacements of samples.
Sample numberExperiment displacement (mm)FEM displacement (mm)Error (%)123.920.85−12.76%226.124.89−4.64%327.225.51−6.21%418.316.06−12.24%525.724.34−5.29%622.319.05−14.57%717.616.96−3.64%817.015.65−7.95%918.918.64−1.38%Figure 17
Simulated low-speed impact results. (a) Maximum stress and (b) strain in all specimens.
(a)(b)Figure 18
Deformations of sample 8 after impact in (a) experiment and (b) simulation.
(a)(b)The maximum stress and elastic strain of the low-speed impact obtained from the simulation are shown in Figure17.According to Table9, the absolute value of the error between the test and simulation deformation results was 14.57% at the maximum and 3.64% at the minimum, indicating the reliability of the simulation results. A comparison of the broken line graphs for the samples shows that the bar width gradually increases, and the total deformation of the structure becomes progressively smaller, indicating that its resistance to impact gradually increases. Sample 3 exhibited the maximum stress, deformation, and strain, which could have been related to the fact that it had the minimum relative density. Thus, it had the weakest resistance to impact and the most obvious damage for a given impact energy. Sample 8 exhibited the minimum deformation and strain, and it had the maximum relative density. Thus, sample 8 was found to be the optimal structure for resisting damage due to impact.Table 9
Basic properties of carbon/epoxy laminate.
DensityExEyEzvxyvxzvyzGxyGxzGyz1.6 g/cm3131 GPa8 GPa8 GPa0.310.310.35400 MPa5400 MPa4500 MPa


## 4. Conclusion

In this study, the mechanical properties of a negative Poisson ratio lattice structure were experimentally investigated using simulations. First, a new negative Poisson ratio lattice structure was designed and manufactured based on the interlocking assembly process. Second, a carbon fiber panel was fabricated to prepare for the bending test. Finally, a flat compression test, three-point bending test, and low-speed impact simulation were conducted for this structure.Among all the structural parameters in the current test scheme, a comprehensive comparison of sample 8 shows that it had the optimal structure under various test conditions. It was found that the width and angle of the oblique bar had the greatest influence on the overall performance of the structure. According to an analysis of the established mechanical model, on the premise of ensuring that the range of structural parameters yielded a negative Poisson ratio, a larger rod width provided a structure with greater strength, stiffness, and resistance to deformation. The angle of the oblique rod affected the deformability of the structure. A smaller angle between the oblique rod and unit cell axis of symmetry produced a more obvious negative Poisson ratio. However, this affected the overall thickness of the core material. The density of the core material had a significant influence on the impact resistance, which improved when the density of the core material was higher.The results showed that the main failure mode of the core in the flat compression test was not buckling of the bar but rather bending deformation of the core material and a degumming failure between the interlocking slots. This was due to machining errors and slight bending deformation during the assembly process, which led to a pressure transfer failure along the ideal direction during the test. In the three-point bending test, large-surface core degumming occurred, in addition to the same failure condition as that occurring under flat compression. This occurred before the deformation of the core material. Thus, the test results did not reflect the true bending strength of each sample.



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*Source: 3726964-2022-07-30.xml*