A novel tubular structure with negative Poisson's
ratio based on
Plate-based cylinder metamaterial with negative Poisson’s ratio
and outstanding mechanical performance
A novel type of tubular structure with auxeticity 
摘要
1. Introduction
As science and modern technology improves by leaps and bounds over the past centuries, an increasing number of requirements for engineering materials are becoming ever more specific and more 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 various fields. 
Mechanical metamaterials are a type of architected materials with superior physical properties beyond nature. They have counterintuitive properties that can achieve unique and uncommon mechanical properties through the rational design of their microstructures and material. e.g., negative Poisson’s ratio, graded stiffness, nonreciprocal response, negative coefficient of thermal expansion, etc. Mechanical metamaterials represent an innovative field, leveraging the unique relationship between material and structure to introduce functionalities beyond traditional mechanical properties. The development has recently pivoted to two-dimensional (2D) and three-dimensional (3D) metamaterials, which offer programmability and unconventional physical properties. These materials can be tuned via deformation, managed by internal or external stimuli, enhancing their flexibility and adaptability. Modular and traditional origami structures are particularly effective in creating such metamaterials, allowing for various deformation pathways and properties, such as adjustable negative Poisson’s ratio, highlighting the expansive potential of this emerging scientific area.
As a special type of mechanical metamaterials, auxetic metamaterials with negative Poisson ratio can expand its volume when stretched, and vice versa. The first NPR materials were reported in the 1870 s. Voigt’s experimental calculation determined that the Poisson’s ratio of pyrite is υ ≈ - 1/7. Lake et al. designed artificially NPR structures in foam materials and computed the Poisson’s ratio as υ ≈ - 0.7 for the first time in 1987. Then Evans et al. begin to use 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 commonly recognized, it can be also referred to NPR structural materials as “auxetic metamaterials”. The mechanical properties possessed by auxetic metamaterials is superior to conventional materials, such as buffering , fracture resistance and energy absorption. Auxetic materials shows potential in the biomedical field, aerospace, protective device, strain sensor and fastener.  According to the geometrical relations of 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.
The unit cells of metamaterials can be similar while the assembled structures in the overall phase are relatively different. In the unit phase, there are 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 create a new type of single 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 auxeticity 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, we proposed a plate-based auxetic cylindrical metamaterial, based on the star-shaped structure. The auxetic behavior, effective stiffness, and effective yield strength of this proposed plate cylinder were validated through in situ compression tests and Finite Element Method (FEM) simulations. In comparison to beam-based lattice structures, our proposed auxetic cylinder demonstrated a considerable enhancement in both Young's modulus and strength. The succeeding sections will elucidate the underlying mechanism contributing to the auxetic behavior as well as the augmentation in Young's modulus and strength.
.
2. Design of auxetic cylinder structures
2.1 Generation approach for the proposed auxetic cylinder
The auxetic plate cylinder was 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. And θ represents the angle between the adjacent star’s cell walls (see Figure 1 (a)). The yellow square 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 (Color online) (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 Manufacturing of plate-based auxetic cylinder
The 3D solid model of the auxetic plate cylinder is modeled using SOLIDWORKS (2022) and then photopolymerization printing is performed using Form 2. Form 2’s printing resolution 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 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 as shown 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 really small. It has been studied in 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 paper. From Figures 2 (a) and (b), it can be seen that the model has been printed completely and is suitable for conducting mechanical experiments. 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 9mm/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. 
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 box to select the boundary of the model, 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. Afterwards, every 100 frames, record the coordinate values of each vertex and use them to calculate the displacement. Calculate the Poisson's ratio of the sample based on the recorded video screenshot, the Poisson's ratio of the cylinder can be calculated by the radial displacement of twelve points A, B, C, D, E, F and their corresponding points on the other side of them (see Figure 3 (b)). If uniform compression is applied to the top of the cylinder, the strain in the y-direction of the cylinder y can be expressed as
                                                                   (1)
 is the vertical 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)
The corresponding displacements of the six points A, B, C, D, E and F were UA, UB, UC, UD, UE, and UF respectively.
Hence, the effective Poisson's ratio of the cylinder can be formulated as
                                                                   (3)
  
Figure 2 (Color online) (a) 3D printed star-shaped honeycomb (SSH); (b) and its vertical view;(c) Layout of in situ compression experiment site.
 
Figure 3 (Color online) (a) The deformation process of SSH; (b) vertex used to record the displacement.

2.4 Comparison between experiments and simulations
During the experiment, in situ compression experiments were conducted on five different angle models by changing the angle  of the star shaped structure in SSH, and then the experiments were compared with simulations. In this experimental stage, SSH has 5 axial layers and 2 radial layers with a wall thickness of 0.9mm. This is because the excessively thin wall thickness is difficult to print, and printing defects can have a significant impact. Further more, 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 3, where only the agreement between experiments and simulations will be described. The star-shaped points represent experimental data, while the other points represent simulation results. Define the normalized Young’s modulus and yield strength as E* and *. And υ is the Poisson's ratio of SSH. We find that the FEM results agree well with the experiment results (see Figure 4). The reliability of the simulation was verified, and the same parameters and boundary conditions were used in subsequent simulations.
 
Figure 4 (Color online) Normalized Young’s modulus, yield strengths and Poisson’s ratio of simulated and experimental cylindrical results with angle  variation. 
3 Simulations
For finite element simulation, ABAQUS/ Explicit commercial package was used to perform FEM on models of different structural sizes. In the compression experiment, the 8-node quadratic tetrahedron element C3D8 was employed in the FEM. A quarter of a complete cylinder was used in the simulation, shown in Figure 5(a). This quarter model is equivalent to the whole cylinder by setting certain 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. The upper and lower plates in the simulation are set as rigid bodies to simulate the compression testing machine in the experiment. Apply fixed constraints to the lower reference point so that it remains fixed and does not move. By providing smooth displacement to the upper reference point, qualitative and quantitative analysis can be conducted by observing the deformation mode of the structure and calculating the displacement of the nodes. In our simulations, elastic-perfectly plastic behaviors were used for the basic materials.
According to the deformation processes, the stress is proportional to the strain at first, that is, it obeys the general Hook’s law, and the slope is Young's modulus. In this region, 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 each adjacent row of cells contact, and compressive densification occurs eventually. And SSH has been completely compacted, as shown in Figure 5(b). 
In our study, we dived into the influence of mesh size on the results obtained through FEM simulations. 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 respective element counts for these models were 14364, 27264, 68160, and 368448. Regarding material properties, the Young’s modulus were 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, leading to the selection of the d/ts = 2 model for further analysis. To compute the effective yield strength of the plate-based auxetic cylinders, a 1% offset was applied in the FEM, we can see from Figure 5(c).

 

Figure 5 (Color online) (a) Setting of boundary conditions; (b) typical stress-strain curve of model SSH; (c) calculation of the effective Young’s modulus E and effective yield strength Y
4 Results and discussion
4.1 Deformation mechanism of auxetic plate cylinder
As shown in Fig. 6(a), the tubular structure was formed by rotating the sections with 5 cell layers. To clearly illustrate the proposed negative Poisson's ratio mechanism of circular tubes, and we take a five layers SSH structure as an example. 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 displacement changes between different layers are comparable, 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 6(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 6(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 6(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 6 (Color online) (a) Definition of layers and points; (b) and the displacement of node i and i ' ; (c) and the displacement of 5 layers SSH’s 5th node; (d) and 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 7. In the simulation, the angle   varies within the range of 100 °to 150 °.
The SSH’s normalized Young’s modulus, yield strength and Poisson’s ratio shows a trend of first decreasing and then increasing with the increase of angle  (see Figure 7). It is worth noting that compared to Poisson's ratio, the normalized Young's modulus and yield strength do not change significantly under the influence of angle . This means that in the mechanical properties of a cylinder, the influence of angle  on the Young's modulus and yield strength is relatively small, while the influence on Poisson's ratio is more significant. 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. And it can be seen from the graph that they basically reach their minimum values between 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 presence of a certain wall thickness.（加个孔隙率公式？星形实际占的面积，乘壁厚（体积）/矩形（管））（2022-10/反蜂窝1）During the deformation process, internal compression and friction 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 7 (Color online) Normalized Young’s modulus, yield strengths and Poisson’s ratio of SSH cylinders with angle  variation. 

4.2 The value of l
By changing the ligament length l, as shown in Figure 8. Normalized Young’s modulus curve with little fluctuation indicating that the deformation behavior of each structural part is relatively consistent at this stage, resulting in no significant difference in Young's modules. And as l increases, Poisson's ratio will decrease 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.
At first, 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. And 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 8 (Color online) Normalized Young’s modulus, yield strengths and Poisson’s ratio of SSH cylinders with ligament length l variation.
4.3 The value of ts and tp
By changing the value of the star-shaped wall thickness ts, as shown in Figure 9, studied the influence of different values of ts on the performance of tubular structures. The changes in Normalized Young’s modulus and yield strength are basically proportional to ts, This means that as the ts value increases, the normalized Young’s modulus and yield strength will also increase accordingly. This proportional relationship indicates that within a certain range, as the ts value increases, the strength and stiffness of the SSH will also increase accordingly, thereby improving the compressive performance and bearing capacity of the SSH. Figure 9 (b) shows the variation of Poisson's ratio of tubular 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 9 (b) yellow star-shaped dots). Introducing transition points as reference points can provide an additional reference standard, helping us better understand the performance changes of structures at different stages. Because sometimes the Poisson's ratio may not differ significantly, but the ability to absorb energy can vary greatly. As shown in Figure 9 (b), with the increase of ts value, the strain range that can maintain negative Poisson's ratio gradually decreases, which means that the negative Poisson's ratio characteristics of SSH under these conditions will be limited, and its ability to absorb energy will gradually weaken. At the same time, as the ts value increases, the stiffness and strength of the structure will gradually increase (see Figure 9 (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 value 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 properties can be selected according to specific needs to meet design requirements.
 
Figure 9 (Color online) (a) Normalized Young’s modulus and yield strengths of SSH cylinders with star-shaped wall thickness ts variation; (b) Poisson’s ratio and the transition point changes with ts variation.
As shown in Figure 10 (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. Indicating that in the initial compression stage, the normalized Young's module of SSH is not sensitive to changes in plate thickness and has not begun to play a major role. Therefore, when considering the elastic stage of the main application model, there is no need to pay too much attention to the changes in tp. As tp increases, normalized yield strength strengthens, indicating 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. As the tp value increases, as shown in Figure 10 (b), the strain range within which the material can maintain a negative Poisson's ratio decreases, which is not conducive to maintaining the energy absorbing state. However, SSH shows negative Poisson’s ratios even when the compression strain is 60%. This shows the high energy absorption and high toughness of the proposed cylindrical metamaterial. Metamaterials with negative Poisson's ratio exhibit special volume expansion characteristics under stress, and an increase in tp value limits the range of this characteristic, resulting in certain limitations on the material's negative Poisson's ratio strain range. Therefore, in designing models, if one wants to maintain a large strain range of negative Poisson's ratio, tp should be limited within a reasonable range.

 
Figure 10 (Color online) (a) Normalized Young’s modulus and yield strengths of SSH cylinders with  the connecting plate thickness tp variation; (b) Poisson’s ratio and the transition point changes with tp variation.

4.4 The value of r
Figure 11 (a) shows the model of SHH when the value of r is 0. There is no gap at the center of the tube, and the structure is exactly cylindrical. And as shown in figure 11 (b), d is the outer diameter of the entire model, and d is the inner diameter of the model.
As the radius value increases, the Poisson ratios of the tubular structurer became unstable early. This is related to the uneven radial deformation of the cross-section caused by a large radius, where there is no space for deformation of the internal material. This uneven internal compression is the reason of instability. Therefore, when customizing the model, the value of r should not be too large. This can ensure the stability of the tubular structure to a certain extent, and also maintain its negative Poisson's ratio performance within the widest possible range. From Figure 11 (c), it can be seen that the Poisson's ratio 𝜐 basically increases with the increase of r value, 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 stabilize. Similar to other plate-based lattices, the high stiffness of negative Poisson's ratio cylinders with smaller radius may contribute to the plate design of cylinders and maximize their energy absorption properties.
 
Figure 11 (Color online) (a) Top view of SSH with inner diameter r=0; (b) and definition of inner diameter d and outer diameter d; (c) normalized Young’s modulus, yield strengths and Poisson’s ratio of SSH cylinders with inner diameter r change.

4.5 Number of layers N
As shown in Figure 12, N is used here to represent the number of radial layers. Due to the different deformation mechanisms of single-layer and multi-layer cylinders, different N values are used for comparison. When N equals to 1，shrinkage can be observed during the compression process. However, during the compression process of the multi-layer negative Poisson's ratio cylinder, the expansion of the central layer and the contraction of the outer layer can be observed (see Figure 12).
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. The results indicate 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.
  
Figure 12 (Color online) Normalized Young’s modulus, yield strengths and Poisson’s ratio of SSH cylinders with number of layers N change.

4.7 Material integration in heterogeneous component design 
By changing the material properties of different parts of the structure, the overall strength and stiffness of the structure can be adjusted. Achieving controllable lightweight design, improving its performance and efficiency, and adapting it to different stress and environmental requirements. As shown in Figure 13 (a), the model is divided into star-shaped parts and plate marked in the picture, Es is the Young's modulus of the star-shaped parts material, and Ep is the Young's modulus of the plate material. The influence of material properties of different structural parts on the overall performance of tubular structures was studied. It shows a comparison of the stiffness and strength of plate parts under enhanced or weakened conditions, respectively. And in figure 13(b), we can see how normalized Young’s modulus, yield strengths and Poisson’s ratio of SSH cylinders changes. 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 13 (b), it can be seen that the normalized yield strength of the strengthened and general structures is approximately equal.
 
Figure 13 (Color online) (a) How to separate the star-shaped parts and plate; (b) normalized Young’s modulus, yield strengths and Poisson’s ratio of SSH cylinders with relative Young's modulus Es/Ep change.
Figure 14(a) shows the stress-strain curve of SSH. 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. 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 weakened plate structure, the model 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 14 (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 strengthened SSH 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 weakened conditions (see Figure 14 (c)). This indirectly explains why the stress-strain curves of the 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 the strengthened and weakened structures 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 conventional structure has a relatively uniform transition and does not have particularly rapid transforming points. Subsequently, when the strain of the strengthened SSH reaches around 0.3, it reaches plateau region Ⅱ，and resumed its energy absorbing function again. The weakened board model will also reach the plateau region Ⅱ later. At this point, SSH has entered the densification stage and has begun to become unstable. The weakened structure shows a relatively flattened deformation, while the strengthened plate is less prone to deformation compared to the star-shaped part, resulting in rotational deformation of the connecting plate. Since this stage is not the focus of this article, we didn’t dive deeply into the research. 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.
The reinforced 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 period of time because star-shaped parts have better support and connection methods, 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 part that weakens the material properties is 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 state during the stress process, delay the deformation and fatigue 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 (Color online) (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

Help me to get the Conclusions.