1. Introduction
In the past few centuries, with the amazing advances in science and technology, the requirements for engineered materials have become more and more specific and novel. Fortunately, the required mechanical properties can be achieved through certain processing programs and structural designs. The idea has effectively implemented negative Poisson’s ratio (NPR) lattice structures in many fields. Mechanical metamaterials are a class of architected materials with superior physical properties beyond nature. Through the rational design of their microstructures and material, they have counterintuitive properties that can achieve unique and uncommon mechanical properties such as negative Poisson’s ratio, graded stiffness, nonreciprocal response, negative coefficient of thermal expansion, etc. Mechanical metamaterials are an innovative field focusing on the interplay between material and structure. The development has shifted towards 2D and 3D metamaterials, allowing programmability and unique properties. These materials can be adjusted through deformation, controlled by internal or external stimuli, enhancing flexibility and adaptability. Using modular structures is highly effective for creating metamaterials with features like adjustable negative Poisson’s ratio, demonstrating the vast potential of this emerging field. 
As a special type of mechanical metamaterials, auxetic metamaterials with negative Poisson ratio can expand their volume when stretched, and vice versa. The first NPR materials were reported in the 1870s. Voigt’s experimental calculation determined that the Poisson’s ratio of pyrite is υ ≈ - 1/7. （标注引用） Lake et al. designed artificial NPR structures in foam materials and computed the Poisson’s ratio as υ ≈ - 0.7 for the first time in 1987. Evans et al. then began using the proprietary term “auxetic”, which means that the material expands when stretched but does not signify “compression” in the literal sense. Since this term is universally accepted, it can be also referred to NPR structural materials as “auxetic metamaterials”. The mechanical properties of the material are superior to conventional materials, such as buffering properties, fracture resistance and energy absorption. Auxetic materials show potential in the biomedical field, aerospace, protective devices, strain sensors and fasteners. According to the geometrical relations of the auxetic unit cell, the preponderant topological NPR lattice structures are re-entrant and chiral structures. Although the rotating rigid square elements cannot be referred to as lattices, the structure can also be considered to originate from a unit cell array and exhibit NPR behavior.
Auxetic materials, known for their negative Poisson's ratio, have significant advantages in impact resistance, energy absorption, and vibration isolation, playing important roles in various fields. For example, in medical and biomedical applications, auxetic metamaterials hold great promise, particularly as materials for vascular stents. Traditional vascular stent materials have certain limitations and deficiencies, whereas auxetic metamaterials, with their excellent mechanical properties and biocompatibility, can offer better elasticity and adaptability. This can reduce complications and restenosis after stent implantation, thereby improving patients' quality of life. In the aerospace sector, auxetic metamaterials can be used as sandwich materials to enhance the stability and structural rigidity of aircraft wings. Their unique properties allow them to provide additional support and stability when wings are subjected to wind loads, helping to reduce vibrations and instabilities during high-speed flights and thus improving flight safety and efficiency. In the automotive industry, auxetic metamaterials can be employed to construct sandwich beams as protective devices, safeguarding drivers and passengers. The design and manufacture of these beams can leverage the energy absorption and distribution characteristics of auxetic metamaterials, effectively cushioning and reducing the impact forces in vehicle collisions. This would significantly enhance vehicle safety and reduce injury in accidents. Auxetic metamaterials also have immense potential in mitigating explosive impacts. Sandwich panel structures can effectively absorb and disperse energy, thus reducing damage to the surrounding environment and structures from blasts. The unique properties of auxetic metamaterials make them ideal for designing and manufacturing explosion-proof equipment and structures, enhancing safety and minimizing the damage of blast incidents.
The unit cells of metamaterials can be similar while the assembled structures in the overall phase are relatively different. In the unit phase, some existing unit cell structures can be chosen, such as re-entrant hexagon structure, double arrow structure, star-shaped structure and missing rib structure, etc. For design optimization, or to create a new type of unit cell structure to achieve specific goals, the microstructures determine local performance. In the overall phase, the performance resembles homogenous materials. Specific mechanical behaviors can be achieved through different combinations of the same unit cell or particular combinations between different units. And metamaterials’ mechanical performance is between the natural materials dominated by their intrinsic material properties and the artificial structures influenced by their structural characteristics. Considering that the performance of mechanical metamaterials is mainly determined by unit cells, their tunability is typically achieved by rational design and optimization of the unit cells. This makes it possible to obtain structures with desirable mechanical properties through designing models. Therefore, the mechanical properties of mechanical metamaterials are usually satisfied through the design and optimization of their microstructure units, as well as different combinations. 
While beam-based lattice structures have been the mainstay in the field of mechanical metamaterials over the past two decades, their structural efficiency has been less than optimal, preventing them from reaching the theoretical stiffness and strength limits, known as the Hashin-Shtrikman and Suquet upper bounds. These bounds represent the maximum potential for any isotropic cellular topology. Plate-based designs, on the other hand, are theoretically capable of reaching these upper bounds, but practical implementation has been hindered by substantial manufacturing obstacles. Furthermore, its most notable characteristic is the authenticity exhibited in the wall thickness. Upon vertical compression of the tubular structure, a contraction of the tube wall occurs, leading to a reduction in the outer diameter and an increase in the inner diameter. This novel characteristic of the proposed auxetic tubular structures significantly broadens the potential scope of applications for such structures.
In this research, a plate-based auxetic cylindrical metamaterial was proposed based on the star-shaped structure. The auxetic behavior, effective stiffness, and effective yield strength of the cylinder were validated through in situ compression tests and Finite Element Method (FEM) simulations. In comparison to beam-based lattice structures, the plate-based auxetic cylindrical metamaterial was demonstrated to have considerable enhancement in both Young's modulus and strength. 

2. Design of auxetic cylinder structures
2.1 Generation approach for the proposed auxetic cylinder
The plate-based auxetic cylinder is generated based on a suitable primitive unit star-shaped cellular structure. The star-shaped honeycombs are composed of square re-entrant corners of length a and thickness ts, joined by straight ribs or ligaments with length l and thickness tp. The θ represents the angle between the adjacent star’s cell walls (see Figure 1 (a)). The yellow square in Figure 1(b) represents the original unit cell. 2D star-shaped honeycombs (SSH) were obtained by the periodic arrangement of the star-shaped cellular structure (see Figure 1(b)). By rotating the 2D SSH around the symmetry axial along the y direction, the plate-based cylinder was obtained. After rotating, the inclined ligaments became plates (see Figure 1(c)). The 1/4 model formed after rotation is shown in Figure 1 (d), where A, B, C, and D are the outer corner points of the SSH star-shaped part, used for calculating Poisson's ratio later.
 
Figure 1 (a) The selected primitive unit cell; (b) 2D star-shaped honeycomb (SSH); (c) 3D star-shaped honeycomb structure obtained by rotating 2D honeycomb structure along the central axis; (d) SSH model and selection of monitoring points.
2.2 3D printing of plate-based auxetic cylinder
The 3D solid model of the auxetic cylinder is modeled using SOLIDWORKS (2022) and then photopolymerization printing is performed using Form 2. The printing resolution of Form 2 is ∼50 μm. Photopolymerization stereolithography was used to print the samples. During the printing, the layer height, energy intensity of the laser, and the wavelength of the diode violet laser were fixed as 50 μm, 250 mW/cm2, and 405 nm, respectively.
The photosensitive resin Tough1500 is produced by Formlab and used for this printing process. Due to the enclosed space inside the model, the photosensitive resin will be filled inside the model after printing. Therefore, some small holes were opened on the side walls of the model to clean the unpolymerized resin inside. 
2.3 Mechanical testing of auxetic cylinder
The printed model can be seen in Figure 2 (a), with small holes arranged in a regular pattern and some printing defects visible on the outer wall of the model which are small. It has been studied in the literature that punching small holes in thin-walled structures has little effect on the overall mechanical properties, so the influence of small holes and defects will be ignored in this research. From Figures 2 (a) and (b), it can be seen that the model has been printed completely and is suitable for conducting mechanical experiments. We also printed 3/4 scale models to verify the accuracy of the model printing. As illustrated in the figure 2 (d), the SSH has been printed correctly. The experimental setup is shown in Figure 2 (c), all mechanical tests were conducted using the DDL100 electronic universal testing machine, with a force measurement accuracy of ± 0.5% of the indicated value and a deformation measurement accuracy of ± 0.5% of the indicated value. Use polished steel plates to compress metamaterials at a strain rate of approximately 9 mm/min. At least three samples were measured for the mechanical performance of each report. The deformation process of SSH during in situ compression testing was recorded using a high-speed camera. The stress-strain curve was obtained through the load-displacement curve of the in situ compression test, as shown in Figure 3. （图4怎么到图3前面了？建议两幅图合并）
Take an image every 100 frames from the experimental video to calculate Poisson's ratio. To better illustrate the deformation of SSH, take an image every 300 frames for demonstration purposes only, as shown in Figure 3 (a). First, use a green rectangular box to select the model's boundary, then fix the boundary, and use it as the initial calculation data with the deformation of the model. Record the pixel coordinate values of each vertex. Afterward, for every 100 frames, record the coordinate values of each vertex and use them to calculate the displacement (as shown in Figure 3 (b)). The radial displacement of the sample be calculated by the six points A, B, C, D, E, F and their corresponding points on the other side of them. If uniform compression load is applied to the top of the cylinder, the strain y in the y-direction of the cylinder can be expressed as
                                                                   (1)
where  is the axial displacement (y direction) of the structure, and y is the initial height of the model.
The strain in the radial direction (x direction) of the cylinder x can be expressed as
         (2)
where UA~UF and UA'~UF' are the radial displacements of the six points A, B, C, D, E, F, and their corresponding points on the other side of them, respectively. D is the outside diameter of the cylinder.
Hence, the effective Poisson's ratio of the cylinder can be obtained by
                                                                    (3)
Three in situ compression tests were conducted on the SSH at 150°. These repeated experiments were performed to minimize errors arising from printing and the influence of the experimental environment on the results. As shown in Figure 4, the stress-strain curves from the three experiments display minimal differences, with an error margin of 5%, which is considered acceptable. The overall progression aligns with expectations and can be broadly categorized into three stages: the elastic deformation stage, the plateau stage, and the densification stage.
   
Figure 2 (a) 3D printed star-shaped honeycomb (SSH); (b) and its vertical view;(c) Layout of in situ compression experiment site; (d) Section view of SSH.
 
Figure 3 (a) The deformation process of SSH; (b) vertex used to record the displacement.
 
Figure 4 Stress-strain curves of three in situ compression experiments of SSH at 150°
 
3 FEM Simulation and Experimental Validation
3.1 FEM Simulation and Model Analysis
The ABAQUS/ Explicit commercial package was used to perform FEM simulation on the models of different structural sizes. In the FEM model, the 8-node quadratic tetrahedron element C3D8 is employed. To improve computational efficiency, a quarter of a complete cylinder is used in the simulation, as shown in Figure 5(a), which is equivalent to the whole cylinder by setting symmetric boundary conditions. The plane perpendicular to the x and z axes are defined as the x and z plane, and symmetric displacement constraints are applied to the x and z planes to prevent them from moving along the normal direction. In the simulation, the flat plate located at the upper and lower ends of the cylinder is set as a rigid body to simulate the compression testing machine in the experiment. Apply fixed constraints to the lower flat plate so that it remains fixed and does not move. By providing smooth displacement to the upper flat plate, qualitative and quantitative analysis can be conducted by observing the deformation mode of the structure and calculating the displacement of the nodes. In the simulations, the elastic-perfectly plastic constitutive relation is adopted for the basic materials.
The compressive stress-strain curve of the SSH model is shown in Figure 5(b). Similar to the experimental results, it can be divided into three stages according to the deformation processes: elastic stage, platform stage and densification stage. In the elastic stage, the stress is proportional to the strain, i.e., it obeys the Hook’s law, and the slope is Young's modulus. In this stage, the deformation of the SSH is elastic and uniform. The corner points of the SSH begin to rotate as the main load-bearing part, while the connecting plate part mainly undergoes linear compression deformation, absorbing a small amount of energy. In the platform stage, as the main stage of energy absorption, as the structure deforms, stress increases with strain at a slope smaller than the Young's modulus. With the rotation and displacement of the SSH walls, contact begins to occur between the walls, resulting in compression and friction to prevent further deformation. When the bending moments of its cell walls exceed the plastic limit, the plastic collapse will take place in the honeycomb, the compressive stress increases sharply with the increase of strain. And the slope of stress-strain curve is almost a constant. Because the adjacent rows of cells contact, and compressive densification occurs eventually. Then the SSH has been completely compacted. 
In the FEM simulation, the influence of mesh size on the results was taken into account. We specifically examined FEM models featuring mesh sizes with d/ts ratios of 4, 3, 2, and 1, where d denotes the mesh size and ts the thickness of the star-shaped plates. The number of elements in these models is 14364, 27264, 68160, and 368448, respectively. Regarding material properties, the Young’s modulus is recorded as 49.2, 48.8, 48.1, and 47.4 MPa, with corresponding strengths of 4.9, 4.8, 3.8, and 3.5 MPa. Notably, the FEM models with d/ts ratios of 2 and 1 showcased very similar Young’s modulus, so the model with the d/ts = 2 was selected for the subsequent simulation. To compute the effective yield strength of the plate-based auxetic cylinders, a 1% offset is applied in the FEM, as shown in Figure 5(c).

  

Figure 5 (a) Setting of boundary conditions; (b) typical stress-strain curve of the SSH model; (c) the calculations of the effective Young’s modulus E and effective yield strength Y

3.2 Experimental Verification of Simulation Results
During the experiment, in situ compression measurements were conducted on five different models by changing the angle  of the star shaped structure in SSH, and then the experiments were compared with simulations. In the experiment, the SSH has 5 layers in the axial direction and 2 layers in the radial direction, and the wall thickness is 0.9 mm. This is because the excessively thin wall thickness is difficult to print, and printing defects will not be ignored. Furthermore, if the model is too high, it will be difficult to be supported during printing, so only 5 axial layers were printed. In addition, due to the fact that SSH becomes solid when the star angle  is less than 125 ° for this type of model, the angle was not further reduced for simulation and experimentation. The impact of specific parameters and trends on the model will be discussed in section 4. Figure 6 illustrates the comparison between the experimental results and the simulation results, where E*, * and υ are normalized Young’s modulus, normalized yield strength and Poisson's ratio of SSH, respectively. It can be found from the figure that the FEM results agree well with the experiment results. The reliability of the simulation was verified, and the same parameters and boundary conditions were used in subsequent simulations.
 
Figure 6 The variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of the cylinder with angle 

4 Results and discussion
4.1 Deformation mechanism of auxetic plate cylinder
As shown in Fig. 7 (a), a cylindrical structure is formed by rotating a cross section with 1~5 cell layers so that the mechanism of negative Poisson's ratio of the cylindrical structure can be analyzed. The definitions of the nth layer and the i(i)th node are shown in Figure 6(a). Here we will discuss the radial displacement (U3) of different layers and their nodes. During the compression of the plate-based SSH, the overall contraction is observed, and the radial displacement changes of different layers are different, resulting in uneven deformation of the multi-layer auxetic cylinder. The displacement distribution of the i(i)th node in the radial direction of different layers of SSH is shown in Figure 7(b). For multi-layer SSH, the displacement of the ith node is positive, while the displacement of the i th node is negative. However, for both the ith node and the i th node, the absolute displacement of nodes in the inner and outer layers of the tube is always less than that of nodes in the middle layer. This rule also applies to four layers SSH. For SSH with 5 layers, the outermost displacement (i.e. 5Layers 5node) is shown in Figure 7(c)，and only focusing on the stage before the failure of the structure, it can be seen that the displacement shows a decreasing trend with the overall deformation. This is because under the action of axial load, the star shaped structure rotates radially inward to drive the nodes to move inward. Meanwhile, the displacement of the innermost node, as shown in Figure 7(d), will exhibit a phase of first increasing and then decreasing with strain. The first upward trend is due to the node start to rotate normally, the node moves radially outward, and when adjacent plates come into contact, it prevents further rotation and displacement, resulting in a downward trend.
 
Figure 7 (a) Definition of layers and points; (b) the displacement of node i and i '; (c) the displacement of 5 layers SSH’s 5th node; (d) the displacement of 5 layers SSH’s 1st node.

4.2 The value of 
The effective mechanical properties of SSH cylinders obtained through finite element analysis are shown in Figure 8. In the simulation, the angle  varies within the range of 100°to 150°.
The SSH’s normalized Young’s modulus, normalized yield strength and Poisson’s ratio show a trend of decreasing first and then increasing with the increase of angle  (see Figure 8). Poisson's ratio is more influenced by the volume deformation and deformation characteristics of the material, so it is more likely to undergo significant changes when the angle  changes. It can be seen from the figure that they basically reach their minimum values when the angle  is in the range of 120 ° -140 °. When the angle  is less than 100 ° or greater than 150 °, it can be observed that the Poisson's ratio value is close to 0. This is because when the angle  is less than 100 °, the initial porosity of the star shaped structure will be too small due to the wall thickness. Hence, during the deformation process, internal compression and friction will occur earlier and more easily. The negative Poisson's ratio effect of SSH is due to the rotation of star shaped corner points caused by the contraction of internal pores. If the porosity is too small, the structure cannot undergo significant deformation, resulting in a high Poisson's ratio. When the angle is bigger than 150°, due to the high porosity, the outermost interior corner of the SSH will rotate outward while the outer corner will rotate inward, offsetting the negative Poisson's ratio phenomenon caused by the rotation of the outer corner point. Therefore, only the mechanical properties of angle  within the range of 100°to 150°were studied here.
 
Figure 8 The variations of normalized Young’s modulus, normalized yield strength and Poisson’s ratio of SSH cylinders with angle . 

4.2 The value of l
By changing the ligament length l, the variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of SSH cylinders are illustrated in Figure 9. Normalized Young’s modulus curve with little fluctuation indicates that SSH with different ligament lengths l deform uniformly and consistently during the elastic stage, resulting in no significant difference in Young's modules. As l increases, Poisson's ratio decreases accordingly. This is because Poisson's ratio reflects the volume change of materials under stress. When l increases, the deformation and stress distribution of the structure will change, leading to a decrease in the Poisson's ratio value. It is precisely due to the increase in the length of the connecting plate that the star shape can rotate sufficiently, more effectively maintaining negative Poisson's ratio, while delaying the yield and failure of the model, resulting in an increase in normalized yield strength.
When the star angles do not collide, the normalized yield strength of SSH will increase with the increase of l. This is because as l increases, the load-bearing area of the structure increases, thereby sharing more load and leading to an overall increase in stress. The exception is when star angles collide with each other (l<7mm)，this may lead to local stress concentration and uneven structural deformation, thereby affecting the overall yield strength. This is because the negative Poisson's ratio property of SSH is mainly generated by the rotation of star-shaped corners. When l is too small, under small strain conditions, the corner points will collide, preventing further rotational deformation and causing instability. As l increases, the Poisson's ratio gradually decreases, the energy absorption capacity gradually weakens. Therefore, in typical design and application, l should not be designed too small. This phenomenon indicates that the design and size of the structure have a significant impact on its stress performance, and reasonable size design can help optimize the stress performance and stability of the structure.
  
Figure 9 The variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of SSH cylinders with ligament length l.
4.3 The values of ts and tp
When the wall thickness ts of the star-shaped cell varies, the influence of wall thickness ts on the performance of cylindrical structures is illustrated in Figure 10. It can be found that normalized Young’s modulus and normalized yield strength increase with the increase of ts. In other words, as the value of ts increases, the strength and stiffness of the SSH will also increase accordingly, thereby improving the compressive performance and bearing capacity of the SSH. Figure 10 (b) shows the variation of Poisson's ratio of cylindrical structures under different ts values, and defines the transition point as the value of strain when Poisson's ratio changes from negative to positive. This point means that the strain range within which SSH can maintain a negative Poisson's ratio state (see Figure 10 (b) yellow star-shaped dots). It can be seen from the figure that the strain range that can maintain negative Poisson's ratio gradually decreases with the increase of ts. The introduction of transition points can provide an additional reference to help us better understand the mechanical performance of structures under different conditions. For example, Poisson's ratio may not differ much, but the strain range of negative Poisson's ratio is different, and the ability to absorb energy is also different. At the same time, as the increase of ts, the stiffness and strength of the structure gradually increase (see Figure 10 (a)). This is because under this situation, the deformation behavior of the material is limited, the structure is more stable. Furthermore, the compressive performance and bearing capacity are improved. Therefore, as the ts increases, the mechanical properties of SSH will tend more towards the characteristics of traditional materials, exhibiting higher stiffness and strength, but correspondingly sacrificing a certain degree of energy absorption capacity. These changes have important guiding significance for the engineering application and design of materials, and suitable material sizes can be selected according to specific needs to meet design requirements.
 
Figure 10 (a) The variations of normalized Young’s modulus and yield strengths of SSH cylinders with the increase of ts; (b) the variations of Poisson’s ratio and the transition point with the increase of ts.
Furthermore, as shown in Figure 11 (a), with the increase of the thickness tp of the connecting plate, there is no significant change in the normalized Young's module of SSH. It indicates that in the initial compression stage, the normalized Young's module of SSH is not sensitive to the variation of plate thickness and has not begun to play an important role. Therefore, when considering the elastic stage of the cylinder, it is not necessary to pay too much attention to the effect of tp. The normalized yield strength increases with the increase of tp, which shows that during the plastic deformation stage, the plate begins to play a dominant role in bearing and absorbing energy. Therefore, the thicker the plate, the greater the normalized yield strength. It can be seen from Figure 11 (b) that the negative Poisson’s ratios of SSH is insensitive to the thickness tp of the connecting plate. However, the turning point of strain decreases with the increase of tp, which is not conducive to maintaining the energy absorbing state. Moreover, SSH keeps negative Poisson’s ratios even when the compression strain is 60%. This shows the high energy absorption of the proposed cylindrical metamaterial. Metamaterials with negative Poisson's ratio exhibit special volume expansion characteristics under compressive loading, and the increase of tp can limit the range of this characteristic, resulting in certain limitations on the material's negative Poisson's ratio strain range. Therefore, tp should be limited to a reasonable range in order to maintain a large strain range of negative Poisson's ratio.

 
Figure 11 (a) The variations of normalized Young’s modulus and yield strengths of SSH cylinders with the increase of tp; (b) the variations of Poisson’s ratio and the transition point with the increase of tp.

4.4 The value of r
The inner radius of the cylinder is defined as r. Figure 12 (a) shows the model of SHH when the value of r is equal to zero. There is no gap at the center of the tube, and the structure is exactly cylinder. If r is not zero, the structure is cylindrical shell as shown in figure 12 (b), where D and d are the outer and inner diameter of the cylindrical, respectively.
As the inner radius r increases, the cylinder became unstable early. This is related to the uneven radial deformation of SSH caused by a large radius. The overall deformation of the model is much greater than the rotation of the star shaped corner points. And the deformation of cells cannot be effective. This uneven internal extrusion is the reason of instability. Therefore, when customizing the model, r can not be too large. It can keep the stability of the cylinder to a certain extent, and also maintain its negative Poisson's ratio performance within the widest possible range. From Figure 12 (c), it can be seen that the Poisson's ratio 𝜐 basically increases with the increase of r, which means that the negative Poisson's ratio of the cylinder gradually weakens. At the same time, the normalized Young's modulus and yield strength of the cylinder will decrease accordingly. When r is greater than 10, the Young's modulus tends to be constant. 
 
Figure 12 (a) Top view of SSH with inner radius r=0; (b) and definition of inner diameter d and outer diameter d; (c) the variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of SSH with inner diameter r.

4.5 Number of layers N
As shown in Figure 13 (a), we define N as the number of radial layers of SSH. The deformation patterns of single-layer and multilayer SSH cylinders are different. For the single-layer cylinder (N= 1), shrinkage can be observed during the compression process. However, for the multilayer cylinder (N> 1), the expansion of the inner layer and the contraction of the outer layer can be observed.
Figure 13(b) shows the variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio with the number of layers N. It can be found that the Poisson's ratio of the cylinder gradually increases with the increase of the number of layers, while the normalized Young's modulus and yield strength decrease. When N=1, the auxetic behavior of SSH is most pronounced. This is because, for a cylinder, deformation is primarily determined by the deformation of the outer layer. As the number of layers increases, the radius also increases. Consequently, the relative strain decreases, and the negative Poisson's ratio effect becomes less noticeable. Moreover, when N=1, SSH will not only involve the rotation of the star-shaped points on the outer layer but also lead to significant deformation of the model itself. This allows for greater energy absorption, resulting in increased stiffness and strength. However, this relies on overall instability deformation of the model, which is not feasible for practical applications. Although it can temporarily achieve substantial energy absorption and high stiffness and strength, the model undergoes rapid deformation to the point of damage, which is detrimental to repeated use and maintaining stability. The stiffness of the multi-layer cylinder is determined by the deformation of both the central and outer layers. In the radial direction, tension counteracts the compression applied in the y-direction, resulting in high stiffness. The strength of cylinder is also a combination of two competing mechanisms: early yielding of the central layer and high elastic deformation ability of the outer layer. On the one hand, the increase of N leads to an increase in stress concentration on the central axis, resulting in yielding of the central axis at smaller strains; on the other hand, an increase in N also leads to an increase in the elastic deformation capacity of the outer layer, and an increase in degrees of freedom reduces the stiffness of the outer layer. 
  
Figure 13 (a) Definition of radial layer number N; (b)The variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of SSH with the number of layers N.

4.7 Design of material properties 
By changing the material properties of different parts of the structure, the overall strength and stiffness of the structure can be adjusted to achieve controllable lightweight design, improve its performance and adapt it to different load and environmental requirements. As shown in Figure 14 (a), the model is divided into star-shaped parts and plates marked in the figure. The star-shaped parts have different material properties from the plates. Es is the Young's modulus of the star-shaped parts, and Ep is the Young's modulus of the plates. The influence of material properties of different structural parts on the overall performance of cylinders was studied. Figure 14 (b) shows that the variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of SSH cylinders as the ratio of Young's modulus Es/Ep varies. 
As shown in Figure 14 (b), the strengthened plate structure is usually less prone to buckling because it increases the material's strength and stiffness to enhance its load-bearing capacity, thereby reducing the possibility of buckling. In addition, the negative Poisson's ratio state can be maintained for a longer time because the star-shaped parts are better supported and connected, which can effectively disperse loads and maintain structural stability, delaying the deformation process of the structure. In contrast, the weakened plate structure quickly undergoes deformation first, reaching the minimum negative Poisson's ratio value. The weakens plate is prone to deformation, while strengthened star shaped plates are less prone to deformation, resulting in the negative Poisson's ratio of the overall structure being maintained for a longer period of time. This design can help SSH maintain a relatively stable volume during the stress process, delay the deformation of the structure, and improve the durability and stability of the structure. Therefore, in structural design, the rational design and combination of strengthening and weakening parts can optimize the performance of the structure, improve its overall load-bearing capacity and stability.
 
Figure 14 (a) The star-shaped parts have different material properties from the plates; (b) the variations of normalized Young’s modulus, normalized yield strengths and Poisson’s ratio of SSH cylinders as the stiffness ratio of Young's modulus Es/Ep varies.
The stress-strain curves of SSH are shown Figure 15(a). It can be seen that the normalized Young's modulus E * of the overall structure is approximately equal. This is because the connecting plate has not yet played a dominant role, and the deformation is mainly generated by the star-shaped structure. The weakened part shows significantly lower and relatively uniform stress after reaching the peak stress point and decreasing, with almost no change in stress. Reflected in the Young's modulus and yield strength shown in Figure 15 (a). While the normalized yield strength of the strengthened and normal cylinders is approximately equal. At this time, the overall structure exhibits a relatively uniform elastic deformation, and the strain responses of each part are similar, resulting in the stress-strain curves approximately overlapping. In the case of plate-weakened model (Es/Ep>1), the structure first buckles and reaches the plateau region I. This is because the decrease in strength and stiffness of the structure leads to earlier concentration of local deformation, and the connecting plate first bends, causing the star-shaped parts to come into contact in pairs (see Figure 15 (b)). At the same time, a rapid increase in Poisson's ratio indicates a significant volume change in the structure under load, which may cause structural instability. Meanwhile, the stress-strain curve of the plate-strengthened model (Es/Ep<1) approximately coincides with the stress-strain curve of the normal model, indicating that their deformation modes are similar. And the deformation of each part is relatively uniform, without the occurrence of pairwise contact under plate-weakened conditions (see Figure 15 (c)). This indirectly explains why the stress-strain curves of the plate-strengthened model coincide with those of the normal model at this stage. 
In the case of weakening, the normalized yield strength shows a decreasing trend with the increase of Es/Ep. When the strain reaches around 0.25, the stress-strain curves of both plate-strengthened and plate-weakened models begin to rise with approximately equal slopes. At this point, the star plate begins to buckle, exerting an effect on the overall structural load. This shows that the star structure is beginning to bear more loads, and the overall structural stress state has changed, while the normal model has a relatively uniform transition and does not have particularly rapid transforming points. When the strain of the plate-strengthened model reaches around 0.3, it reaches plateau region Ⅱ，and still keep good energy absorbing capacity. Subsequently, the plate-weakened model will also reach the plateau region Ⅱ. This periodic change reflects the response characteristics of different parts of the structure during the loading process, and has important reference value for structural design and analysis. It can help optimize structural design, improve the load-bearing capacity and stability of the structure.
 
Figure 15 (a) Stress-strain curves under different material conditions; (b) in the case of weakened plate, the star-shaped parts in SSH are in contact with each other in pairs; (c) while in the case of strengthened plate, the overall deformation of SSH is relatively uniform.

5 Conclusions
A novel plate-based auxetic cylindrical metamaterial (SSH) was presented with an advanced star-shaped cellular structure, designed to exhibit negative Poisson's ratio (NPR) properties and enhanced mechanical performance. The study achieved significant insights into the design, manufacturing, and mechanical testing of this type of metamaterial. The notable findings from the research include:
i.	Innovative design and auxetic behavior: the developed auxetic cylinder, inspired by a star-shaped design, demonstrates significant auxetic behavior, characterized by radial contraction upon axial compression. This behavior is evident through the expansion of internal tubular diameters and contraction of external diameters. The design capitalizes on the rotational movement of star-shaped junctions, a key contributor to the NPR effect, which is efficiently maintained across different configurations.
ii.	Enhanced mechanical performance: the plate-based design shows considerable advancement over traditional beam-based lattice structures, delivering improvements in both Young's modulus and yield strength. The close alignment between FEM simulations and experimental results highlights the robustness and reliability of the proposed model.
iii.	Flexibility in design parameters: various structural parameters, such as angles, ligament lengths, wall thickness, and internal diameter, were systematically explored. The study reveals the critical role each parameter plays in influencing the mechanical properties (stiffness, strength) and auxetic properties (NPR).
In summary, this paper extensively studies promising auxetic cylindrical metamaterials, providing theoretical guidance for their application in areas demanding high energy absorption and mechanical resilience, such as aerospace, biomedical devices, and protective equipment. Advances in manufacturing, like enhanced 3D printing precision, could unlock their theoretical potential and reduce production costs, paving the way for future mass production. Further research may focus on improving material properties or exploring hybrid structures to expand these metamaterials' specialized applications.
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