Axial compression characteristic of hollow auxetic cylinder lattice structures

Abstract A hollow auxetic cylinder lattice structure (HCLS) is proposed with tapered three-dimensional reentrant cells (TRCs) and connecting plates. The compressive performance of HCLSs is studied by finite element analysis (FEA) and compression test. HCLSs show excellent energy absorption and negative Poisson’s ratio properties as well as unique rotational deformation mode during compression. The influences of structural parameters on the compression performance of HCLS are further studied. The values of the maximum rotation angle of H-hs2.0 and minimum negative Poisson’s ratio (MPR) of H-la.3 are 37.32 degree and −0.653, respectively. HCLSs exhibit various functional properties and have great potential for application.


Nomenclature
the maximum width of the large plane hexagonal reentrant cell in TRC w s the median width of the large plane hexagonal reentrant cell in TRC a the angle between the lower surface of the fourth layer of cells and the lower platen h the angle between two adjacent cells in one layer

Introduction
Currently, auxetic lattice structure is a research hotspot in the field of new materials/structures.Auxetic lattice structures [1][2][3][4][5][6][7] usually have excellent toughness [8], stable compression resistance [9][10][11][12], and ultra-light structural mass (SM) [13][14][15] characteristics.They have great application potential in spacecraft, trains, and automobiles [16][17][18][19][20][21][22].The microstructure design is the key to realize the negative Poisson's ratio effect [23][24][25][26][27] and novel mechanical properties of auxetic lattice structure, and the common auxetic microstructures include double-arrowed [28,29], reentrant [30,31], star-shaped [32], and chiral-shaped [33] structures.Scholars have studied different configurations of auxetic lattice structures to improve their mechanical properties.Gibson et al. [34] studied the reentrant mechanism in 1982.Wang et al. [35] proposed a new cylindrical mechanical metamaterial consisting of an array of reentrant cells that showed excellent compression-torsion property.Guo et al. [29] proposed a three-dimensional double-arrowed lattice structure.Compression experiments and numerical analysis showed that the three-dimensional lattice structure had good impact resistance.Tan et al. [36] proposed two reentrant hierarchical honeycombs, and numerical simulations showed that the specific energy absorption (SEA) of the two hierarchical honeycombs was higher than that of classical reentrant honeycomb.Wang et al. [37] proposed a new reentrant honeycomb model with octagon microstructures and reentrant angles.The experimental results were in good agreement with FEA.The effects of various parameters on its negative Poisson's ratio and energy absorption capacity were investigated.Osman et al. [38] presented a new type of stretched cell truss lattice.The quasi-static compression experiments showed that its yield strength was higher than that of conventional truss lattice.A three-dimensional hybrid double-arrowed structure with convex and concave quadrilateral components was proposed by Guo et al. [39].Its crushing behavior and energy absorption performance were studied by experiments and numerical analysis, and the effects of geometrical parameters under different crushing conditions were revealed.Wang et al. [40] proposed a star-shaped arrow honeycomb structure.The simulation results showed that this structure could absorb more energy than the star-shaped honeycomb structure at low velocity impact.Zhang et al. [41] studied the effect of temperature on the out-of-plane compression properties of a new hollow pyramidal lattice sandwich structure with truncated square honeycomb reinforcement.The results showed that its initial breaking strength and peak strength at different temperatures are higher than those of the conventional hollow pyramidal lattice structure of the same mass.Wang et al. [42] presented a sandwich panel with three-dimensional double-v structural cores.The parameter analysis results demonstrated that the structure outperformed the solid panel in terms of both lightweight and protective properties.
Besides the design of microstructures, auxetic structures have also been shown to contribute to the energy absorption properties [14,[43][44][45][46][47][48][49][50][51][52][53][54] against compressive loads [55][56][57].Mohsenizadeh et al. [58] studied the effect of toughened foam on the crushing force and energy absorption characteristics of square section tubes under uniaxial quasi-static loading.The preparation of polymeric plastic foams by quasi-triaxial compression was presented by Mohsenizadeh et al. [59], and the plastic foam materials with negative Poisson's ratio were found to have great potential as complementary materials for energy absorption and structural applications.Lee et al. [60] placed reentrant units on a tube wall to alleviate impact load, and the results showed that the auxiliary tube exhibited higher specific energy absorption compared with conventional structure.The effect of cell configurations on the energy absorption of toughened structures under dynamic loading was investigated by Shokri Rad et al. [61].It was found that the stiffened structures outperformed the non-stiffened structures in terms of energy absorption.Ji et al. [62] designed and prepared a microporous lattice structure with a periodic inner core.FEA and quasi-static compression tests showed that the structure could be used as a buffer material for energy absorption applications.Yao et al. [63] designed a square conical frame structure, and its energy absorption properties were analyzed by quasi-static compression experiments and simulation analyses.Ma et al. [64] designed and fabricated new chiral cylindrical shells.The results showed that both the antichiral and chiral axial shells could produce additional behavior that was beneficial to the energy absorption and   vibration isolation.The samples of the sandwich panels with BCC lattice, octet-truss, and reentrant auxetic cores were prepared by Beharic et al. [65].It was found that the geometric design of the honeycomb cores had a significant effect on the impact energy absorption.Yazdani Sarvestani et al. [66] performed low velocity impact tests to investigate the mechanical properties of sandwich panels.The experimental and numerical results showed that the auxetic sandwich panels have high energy absorption capacity.
Although much work has been done on auxetic structures, the search for innovative auxetic structures with better mechanical properties, more novel deformation modes and better energy absorption remains a long-term pursuit.In this work, two four-layer HCLSs consisting of tapered threedimensional reentrant microstructures and connecting plates are proposed, and their geometry is carefully presented.The 3D printing technology is applied to manufacture the test specimen of HCLS.The deformation patterns and compression forces of HCLS are studied by quasi-static compression test and FEA, and the novel mechanical properties of HCLS are presented.In addition, the effects of several structural parameters on the compression characteristics of HCLS are systematically studied.
The contents of the following sections are summarized as follows.In Section 2, the model and specimen of HCLS are presented.Section 3 shows the FEA and compression test of HCLS.Section 4 depicts the influences of several structural parameters on the compression properties of HCLS.In Section 5, the conclusions of this study are shown.

The models of HCLSs
In this section, the models of HCLS and HCLS-R as well as the test specimen of HCLS are presented.

The design and fabrication of HCLS
Figure 1 shows that the details of the HCLSs.Figure 1(a) depicts a plane hexagonal reentrant cell.Figure 1(b) is two plane hexagonal reentrant cells, and the left cell is smaller than the right one.As shown in Figure 1(c), a tapered TRC is generated by connecting the two plane hexagonal reentrant cells.The TRC is symmetrical up and down along its mid-plane.Figure 1(d) shows the HCLS, which is composed of horizontal connecting plates (HCPs), vertical connecting plates (VCPs) and TRCs.Thus, TRCs are connected by VCPs and HCPs.The shapes of VCP and HCP are isosceles trapezoids.In HCLS, there are four layers of cells, each layer has 10 TRCs and ten HCPs, and there are 10 VCPs between two adjacent layers of cells.In addition, with the material nylon 11, the 3D printing technology is also applied to fabricate the specimen of HCLS. Figure 1(e) shows the sample of HCLS, and its SM is 36.55 g. Figure 1(f) shows the second HCLS model, and it is named the hollow auxetic cylinder lattice structure with reverse lattice (HCLS-R).HCLS-R also contains four layers of TRCs, four layers of HCPs and three layers VCPs.The adjacent TRCs in each layer are oriented in opposite directions.

The structural parameters of HCLSs
Figure 2 shows the detailed geometric parameters of HCLS.In Figure 2, the outer and inner diameters of HCLS are defined as D l and D s , and its wall thickness and height are defined as D o and H.In TRC, the median width, maximum width and height of the large plane hexagonal reentrant cell are defined as w s , w l , and h l , respectively, and the median width, maximum width and height of the small plane hexagonal reentrant cell are defined as d s , d l , and h s .In VCP, the short and long side lengths are defined as f s and f l , respectively.In HCP, its short and long side lengths are defined as S s and S l , respectively.The angle between two adjacent cells in one layer is defined as h.The thicknesses of TRC, VCP, and HCP are defined as t.Table 1 depicts the detailed dimensions of HCLS and HCLS-R.

Compression experiment and FEA of HCLSs
The FEA and axial compression test of the HCLSs are performed in this section.slightly to eliminate the initial gap.The testing machine compresses the specimen, and the speed and compression distance are set to 2 mm/min and 40 mm, respectively.

The preparation of FEA and test
The test machine is CMT-5105GL, and this machine will automatic record the test results.The deformation of the sample is recorded by high-speed camera, and the measurement method of Poisson's ratio is presented in the Supporting Information.
Figure 3(b) shows the simulation model, and the HCLS is placed between the lower and upper platens.The lower platen is completely fixed, and the five degrees of freedom of the upper platen are constrained so that it can only move downward to compress the sample at a constant speed of 0.5 mm/ms.The duration of the FEA is 80 ms.The FEA is performed with the commercial software LS-DYNA, and the material model of the HCLS is MAT-24 from the material library of LSDYNA.The contact algorithms between the lower and upper platens and the HCLS are surface-to-surface contact, and the HCLS itself applies self-contact.The coefficients of kinetic friction and static friction are 0.2 and 0.3, respectively.Four-node fully integrated elements are applied to build the finite element model of HCLS, and there are five integration points in the element thickness direction.The mesh size is 2 mm, and the effect of grid size on the compression performance of the HCLS is presented in the Supporting Information.In addition, the influence of loading speed on the FEA is determined by calculating the ratio of kinetic energy to total energy, and the ratio is far less than 10% [67], indicating that the loading speed is appropriate.
The compression performance of HCLS is evaluated by the indicators of energy absorption (EA), mean compression force (MCF), peak compression force (PCF), specific energy absorption (SEA), MPR, and maximum rotation angle (MRA).The mathematical models of EA, SEA, and MCF are shown as follows: where g is the maximum compression distance and f(x) is the compression force.
To obtain the stress-strain curves of nylon 11, the tensile tests are performed with three tensile specimens manufactured by 3D printing.Figure 4 shows the test specimens and the preparation of the tensile test.The machine CMT-5105GL is also used to perform the tests with a speed of 2 mm/min at room temperature.The strains of specimens are recorded by an extensometer.Figure 4(a) depicts the test specimens before and after experiments.Figure 4(b,c) shows the preparation of the tensile test and the stress-strain curves of nylon 11, respectively.The density, Young's modulus, and Poisson's ratio of nylon 11 are obtained as 1.03 Â 10 3 kg/m 3 , 1550 Mpa, and 0.03, respectively.The engineering stress-strain curves should be converted into effective stress-strain curves, and the mathematical models of the conversion process are shown as follows: where e e , e t , and e eff are the engineering, true, and effective strains, and r e , r t , and r eff are the engineering, true, and effective stresses.

The results of FEA and experiment
Figure 5 and Table 2 show the results of the experiment and FEA.contact with the edges of cells in the first and fourth layers, respectively, in the initial stage.At 0 mm, the angle of a is 17.75 degree, and a is the angle between the lower surface of the fourth layer of cells and the lower platen.As the compression distance increases, the cells in the first and fourth layers rotate inward along the contact edges, and the value of a decreases.In addition, since the value of h l is significantly larger than that of h s , the strength on the outside of the cells is weaker than that on their inside.Thus, at 5 mm, the outer parts of cells are concavely deformed due to the extrusion of the VCPs.At 7.5 mm, the rotation ends, the value of a is reduced to 0 degree, and the lowermost and uppermost cells are in surface contact with the platens.Therefore, the inward rotation of the cells in the first and fourth layers and the deformation of the outer parts of cells cause the cells in the middle to contract inward.
In Figure 5(a), the cells are subjected to the compression forces of VCPs and HCPs at 20 mm, and they undergo torsional deformation.The cells start to squeeze each other, and the cells in the middle two layers have significant rotational deformation along the structural center of HCLS.The main reason of this rotation is that the values of f l of these VCPs are large, and they approximate slender rods.After 10 mm, the compression forces of VCPs increase significantly and they begin to buckle.The buckling of VCPs drives the cells to rotate a small angle along the center of HCLS.As compression distance increases, the bending of VCPs increases, the rotation angle of cells increases, and the longitudinal and lateral compression forces on the cells become uneven.As a result, the cells begin to have local torsional deformations, and these torsional deformations further promote the rotation of these cells.At 30 mm, the cells in the first and fourth layers also have significantly torsional deformation, and the rotation angle of the cells in the middle two layers is further increased.At 40 mm, the structure tends to be compacted and the angle of rotation reaches its maximum.
Figure 5(c,d) are the compression force and EA curves of FEA and test.In Figure 5(c), the compression force curves of the test and HCLS are very similar in waveform and  amplitude.As compression distance increases, the compression forces of the test and HCLS increase first, tend to stabilize after 13 mm, and continue to increase after 30 mm.This behavior is consistent with the deformation mode of HCLS.In the initial state, the HCLS is in line contact with the upper and lower platens.The initial compressive resistance of HCLS is low.Therefore, the initial slope of the compression force curve is small.After 7.5 mm, the contacts between HCLS and the upper and lower platens become surface-to-surface.After 30 mm, the deformation of cells is already very significant, the interaction between cells increases significantly, and the compression force starts to increase rapidly.From Figure 5(d), the EA curves of the FEA and test are also very similar.The slope of their EA-deformation curves is small before 7.5 mm, after which they increase and level off.After 30 mm, their slope increases further.In Table 2, the numerical results of the test and HCLS are also very close.These indicate that the compression performance of HCLS and the test is very consistent, and the simulation model has sufficient accuracy.HCLS shows excellent energy absorption, significant negative Poisson's ratio, and novel compression-rotation properties.Figure 6 and Table 2 show the results of HCLS-R.The trend of the compression force curve of HCLS-R is similar to  that of HCLS.Its compression force is greater than that of HCLS before 10 mm and after 20 mm, and less than that of HCLS between 10 and 20 mm.The energy absorption efficiency of HCLS-R is better than that of HCLS, while its absolute value of MPR and its value of MRA are both smaller than those of HCLS. Figure 6(a,b) depict the strain nephograms and rotational properties of HCLS and HCLS-R.From Figure 6(a), HCLS-R is also in contact with the lower and upper platens through the long sides of the uppermost and lowermost cells at the beginning.At 10 mm, the upper surfaces of the first layer of cells and the lower surfaces of the fourth layer of cells are in surface contact with the lower and upper platens, respectively.The cells between the layers begin to touch each other.The inward contraction of cells in the middle region of HCLS-R is not obvious.Because the rotation directions of adjacent cells within the first and fourth layers along their contact edges are different.The inward rotation causes the cells in the middle to contract, and the The reason for the rotation of the cells of HCLS-R is the same as that of HCLS, which is the buckling of VCPs and the local torsion of cells.HCLS-R and HCLS differ in the direction of cell rotation in compression, and they rotate to the left and right, respectively.The torsional deformation of the lower three-layer cells of HCLS is larger than that of its first layer, and the torsional deformation of the upper three-layer cells of HCLS-R is larger than that of its fourth layer.At 30 and 40 mm, the cells of HCLS and HCLS-R continue to be squeezed and continue to rotate along their centers.The values of MRA of HCLS and HCLS-R are 34.53 and 29.23 degrees, respectively.The cells in the third layer of HCLS have the largest rotation angle, and the second layer cells of HCLS-R have the largest rotation angle.Furthermore, after 20 mm, the cells and connection plates of HCLS-R are compressed more tightly compared with HCLS during the rotation of cells.Therefore, the compression force of HCLS-R is greater than HCLS after 20 mm.
Figure 6(c,d) depict the Poisson's ratio and rotation angle curves of HCLS and HCLS-R.From Figure 6(c), with the increase of deformation, HCLS's Poisson's ratio value first decreases and then increases, and the Poisson's ratio value of HCLS-R tends to decrease continuously.The negative Poisson's ratio effect of HCLS is significantly stronger than that of HCLS-R.From Figure 6(d), the rotation angle of cells is not evident in the early stage of compression.The rotation angle of cells starts to increase significantly after 10 mm.The rotation angle of the cells of HCLS is smaller than that of HCLS-R before 30 mm.After 30 mm, the growth rate of a rotation angle of HCLS and HCLS-R gradually slow down and stabilize.Because the cells and connecting plates of HCLS-R are compressed more tightly in the later stage compared with HCLS, the rotational resistance of its cells is greater, and its axial compressive strength is also greater.In addition, it is found that the plateau period of the compression force of HCLS and HCLS-R overlaps with the high-speed growth area of the rotation angle, which indicates that the rotation deformation of lattice structures is conducive to obtaining a stable compression force.

The compression performance of HCLS with various geometric parameters
The effects of parameters D s , h s , h l , w s , and layer number on the axial compression properties of HCLS are investigated in this section.The compression loads are the same as those in Section 3.

The effect of parameter h l
In this section, the values of h l are set as 4.8, 6.4, 8, 9.6, and 11.2 mm, respectively, and these values are 0.375, 0.5, 0.625, 0.75, and 0.875 times the original h l value (12.8 mm).These models are named H-h l 0.375, H-h l 0.5, H-h l 0.625, H-h l 0.75, and H-h l 0.875.Figure 7(a-f) show the geometric models of H-h l 0.375, H-h l 0.5, H-h l 0.625, H-h l 0.75, H-h l 0.875, and HCLS.The detailed geometric parameter values of these models are shown in Table 3. Table 4 and Figure 8 show the results of these HCLSs.With the increases of h l , the value SEA first increases and then decreases.The values of EA, SM, MCF, and q e increase with h l .H-h l 0.75 and HCLS have the best energy absorption and compression resistance characteristics, respectively.In  addition, the values of the MPR of these models decrease as h l increases.The values of MRA of these HCLSs decrease first and then increase as h l increases.The range of q e of these models is 266.61 kg/m 3 to 297.23 kg/m 3 , and this shows that these HCLSs are lightweight.Figure 8(c,d) depict the compressive force and EA curves.From Figure 8(c), the compression force curves of H-h l 0.375, H-h l 0.5, and H-h l 0.625 have two obvious peaks.This is because of their small cell size and high stiffness, and the deformation of their cells in compression is small.Thus, their VCPs directly bear large compression forces and buckle around 2-5 mm and 20-25 mm. Figure 9(a) shows the buckling behavior of H-h l 0.375, H-h l 0.5, and H-h l 0.625.For H-h l 0.375 and H-h l 0.625, the first time is the simultaneous buckling of the three-layer VCPs, and the second is the buckling of the second-layer VCPs.For H-h l 0.5, the first time is also the buckling of the first and second layers VCPs, and the second is the buckling of the third-layer VCPs.With the increases of deformation, the compression force of H-h l 0.75 first increase, then, decreases slightly, then increases slowly, and increases rapidly after 30 mm.The compression forces of H-h l 0.875 and HCLS first increase, then basically stabilize, and then, increase rapidly after 30 mm.From Figure 8(d), the energy absorption of HCLS increases as h l increases.Figure 8(e,f) are the Poisson's ratio and rotation angle curves of these models.It is found that as h l increases, the negative Poisson's ratio effect increases and the MRA first decreases and then increases.Furthermore, as h l decreases, the onset moment of cell rotational movement is advanced.
The deformation modes of these HCLSs are illustrated in Figure 9, and Figure 9(b) is the schematic of the deformation modes of these models at 10 mm.The deformation modes of these five models are significantly different.When the distance is 10 mm, the three-layer VCPs of H-h l 0.375 bend, and its first layer is fully bent.The cells in the second and third layers rotate to the right.For H-h l 0.5, its first and second layers of VCPs bend, and the second layer of cells rotate to the right.Regarding H-h l 0.625, its first and third layers of VCPs bend, and the second and third layers of cells rotate to the right.For H-h l 0.75, its three-layer VCPs bend, and its cells in the second and third layers rotate to the right during compression.Obviously, the VCPs of H-h l 0.375, H-h l 0.5, H-h l 0.625, and H-h l 0.75 have already undergone  buckling before 10 mm.The contraction of cells in the middle region of these models is not Regarding H-h l 0.875, its VCPs are undergoing buckling, and its second and third layers of cells rotate to the left.The direction of rotation of its cells is opposite to that of H-h l 0.75.Regarding HCLS, its VCPs have not yet buckled, and its cells in the middle region contract inward significantly during compression.It is found that in the initial state of compression, the decreases of h l enhance the bending of VCPs as well as the rotation of cells, and reduces the negative Poisson's ratio effect of HCLS.The main reasons for this phenomenon are summarized as follows.When the value of h l is small, the size of the cells is small, and the length of VCP is large.The strength of the cells is much greater than that of the VCPs.Therefore, the VCPs are approximately slender rods.When subjected to compression loads, the VCPs first buck and then bend.These VCPs drive the cells to  rotate along the center of these HCLSs.As h l increases, the size of cells increases and their strength and the size of VCPs decreases and their strength increases.The moment of buckling of these VCPs will be delayed.Furthermore, with a small h l , both the inward rotational angle of cells in the first and fourth layers and the compressive deformation of the outer parts of the cells are small in the initial state.As a result, the negative Poisson's ratio effect of HCLS with small h l is not obvious.At 20 mm, the deformation and rotation of VCPs and cells are more pronounced than at 10 mm.The VCPs of the first and third layers of H-h l 0.375 and H-h l 0.625 are completely bent, and the cells of the second and third layers continue to rotate.The VCPs of the first and second layers of the H-h l 0.5 are fully bent, and the second layer of cells continues to rotate to the right.Furthermore, the second layer of VCPs of H-h l 0.375 and H-h l 0.625 as well as the third-layer of VCPs of H-h l 0.5 do not bend at 20 mm.This predicts that as the deformation increases, their VCPs will suffer from compression instability.That is, they will produce sharp peak compression forces.The bending and rotation of the VCPs and cells of H-h l 0.75, H-h l 0.875, and HCLS are more complex, and their cells also begin to undergo rotational deformation.At 30 and 40 mm, the rotation and deformation of cells and the bending of VCPs are further strengthened, and the compression forces of these models begin to rise quickly.

The influences of parameter h s
The values of h s are taken as 5.60, 6.72, 7.84, and 8.96 mm in this section, and these values are 1.25, 1.5, 1.75, and 2.0 times the original h s value (4.48 mm), respectively.Thus, these models are named H-h s 1.25, H-h s 1.5, H-h s 1.75, and H-h s 2.0. Figure 10(a-e) depict the geometric models of HCLS, H-h s 1.25, H-h s 1.5, H-h 1.75, and H-h s 2.0.Table 5 depicts the geometric parameter values of these HCLSs.
Figure 11 and Table 6 depict the results of these HCLSs.In Table 6 and Figure 11(a,b), as h s increases, the values of q e and SM increase, and the values of MCF, PCF, and EA increase first, then, decrease and finally increase.The value of SEA first increases, then decreases, then increases and then decreases.Based on the results of SEA and MCF, it is found that the energy absorption characteristic of H-h s 1.25 is the best, and H-h s 2.0 has the best compression resistance.As h s increases, the absolute value of MPR increases first and then decreases, and the value of MRA decreases first and then increases.The results for MPR and MRA tend to Figure 11(c,d) illustrate the compression force and EA curves of these HCLSs.During the compression process, their compression forces first increase, then remain roughly stable and finally increase.The EA curves of these models have a low slope for the first 10 mm and then increase nearly linearly, and increase exponentially after 30 mm. Figure 11(e,f) are the Poisson's ratio and rotation angle curves.It is found that the MPR values of these models are obtained in the interval 10 À 15 mm, and H-h s 1.25 has the most obvious negative Poisson's ratio effect.From Figure 11(f), the cells of these models show little rotation in the first 10 mm.After that, their rotation angles increase rapidly and reach the maximum values at 40 mm.
Figure 12(a,b) depict the deformation modes and rotation angles of these HCLSs.At 10 mm, the cells between adjacent layers deform and begin to touch each other, and the middle two layers of cells and connecting plates twist slightly.At 20 mm, the deformation of the cells of these models is uniform and symmetrical, and their middle two layers of cells have angular torsional deformation.The torsional deformation of H-h s 2.0 and HCLS is the most obvious.At 30 mm, the lower and upper cells are twisted, and the deformation in the middle layers becomes more severe.The rotation of the cells of H-h s 1.25 is clockwise, and the other models rotate counterclockwise.At 40 mm, the torsional deformation of these models reach the maximum, and the connecting plates and cells of each layer collapses.13 and Table 7 show the top plane views and the geometric parameter values of these HCLS models, respectively.q e (kg/m 3 ) MPR MRA (degree) Figure 14 and Table 8 depict the results of HCLS, H-D s 0.91, H-D s 0.83, H-D s 0.77, H-D s 0.71, and H-D s From Table 8 and Figure 14(a,b), as the value of D s decreases, the values of PCF, SM, EA, MCF and q e increase, and the value of SEA trends to increase.This means that the compression performance of HCLS is improved with the decreases of D s .From the results of MPR and MRA, as D s decreases, the value of MPR tends to increase and the value of MRA tends to decrease first, and then, keep stable.
Figure 14(c,d) illustrate the compression force and EA curves.From Figure 14(c    inward.After 10 mm, the cells begin to produce obvious rotational deformation, and the rotational deformation ensures the stability of compression force of HCLS.Therefore, the compression of these models remains stable or decreases slightly.At 20 mm, the inward contraction magnitude of the cells in the middle region further increase.The torsional deformation of these models' cells in the middle two layers occurs.At 30 mm, the torsional deformations of these models are more severe.The rotational speed of these HCLSs is gradually reduced, and their compressive resistance strength increases.At 40 mm, the torsional deformation amplitude of these models reaches the

The influence of parameter w s
In this section, the values of w s are set to 5.25, 6, 6.75, 8.25, and 9 mm, respectively, and these values are 0.7, 0.8, 0.9, 1.1, and 1.2 times of the original w s value (7.5 mm).These models are named H-w s 0.7, H-w s 0.8, H-w s 0.9, H-w s 1.1, and H-w s 1.2. Figure 16(a-f) depict the geometric models and parameter values of H-w s 0.7, H-w s 0.8, H-w s 0.9, HCLS, H-w s 1.1, and H-w s 1.2, and Table 9 shows the detailed geometric parameter values of these HCLSs.
Figure 17 and Table 10 depict the FEA results of these HCLSs.As w s increases, the values of q e and SM increase, the values of EA, MCF, PCF, and SEA first decrease, and then, increase.H-w s 1.2 has the best energy absorption characteristic.As w s increases, the negative Poisson's ratio effect of HCLS decreases, and the value of MRA tends to increase first and then decrease.The main reason is that as w s decreases, the width of cells decreases and the interaction between cells weakens, which allows for greater contraction of cells in the middle region.Figure 17(c,d) depict the compression force and EA curves.With the increase of compression distance, their compression forces first increase, then remain roughly stable, and then continue to increase.An interesting phenomenon is discovered that the models with smaller w s values have greater compression forces before 10 mm, and the HCLSs with larger w s values tend to have greater compression forces after 10 mm.From Figure 17(d), the slope of the EA curves of these models with smaller w s values is larger in the initial state and tends to be smaller than other models in the later stage.This means that the EA values of these HCLSs are consistent with their compression  force performance.The main reason is that the cells with smaller values of w s have larger vertical stiffness.When subjected to compression loads, these cells have strong compression resistance, so the initial compression force of the HCLS with small w s is large.As the compression progresses, the VCPs gradually become unstable and the cells rotate.The lateral widths of cells with smaller w s are smaller, and the interaction between cells is weak, so the later compression force increase of the model with smaller w s is worse than that of other models.17(e), it is found that the negative Poisson's ratio effect decreases as the value of w s increases.This is because the interaction strength between cells decreases as the value of w s decreases, and the resistance of cells to contract inward is small.This also means that the compressive resistance of the structure is reduced.Therefore, compressive strength and negative Poisson's ratio effects tend to contradict each other.From Figure 17(f), the MRAs of HCLSs first increase and then decrease as w s increases.The main reason is that the torsional stiffness of cells with small w s is large, and it is not easy to produce rotational motion.Conversely, the cells with larger w s value are wider, the interaction between cells is stronger, and the resistance of the rotational movement of cells is also greater.
Figure 18 shows the deformation modes and rotation angle of these HCLSs.At 10 mm, their cells are compressed and deformed axially.The cells in the middle region contract inward.The rotation angles of these models are very small.In addition, the VCPs of H-w s 0.7 produce buckling behavior, and its second and third layers of cells begin to twist.Other models will also have buckling behavior when the compression distance exceeds 10 mm.Before buckling, the compression resistance of the structure increases with the compression distance.Once a structure is buckled, its compression strength will remain stable or decrease slightly.This is consistent with their compression force performance.At 20 mm, the contraction effect of cells is further enhanced.The VCPs of these models bend and the cells of these models undergo torsional deformation.At 30 and 40 mm, the torsional deformation of connecting plates and cells is further increased.The cells of H-w s 1.2 rotate clockwise, and the other models rotate counterclockwise.The rotation angle of H-w s 0.7 is significantly smaller than other models, and the torsional deformation directions of adjacent cells within a same layer are opposite.

The influence of layer number
The layer number are set to 3, 5, 6, 7, 8, 9, and 10.These models are named H-la.3,H-la.5, H-la.6, H-la.7,H-la.8, Hla.9, and H-la.10, and Figure 19 shows these geometric models.The detailed sizes of the geometric parameters of these models are illustrated in Table 11.
Figure 20 and Table 12 depict the FEA results of these HCLSs.As the number of layers increases, the following trends can also be obtained.First, the values of EA, MCF, SM, and q e increase, and the value of SEA tends to decrease first and then increase.Second, the value of MPR increases, and the value of MRA increases first, then decreases, and then, increases.The reason for the increase of MPR value is shown as follows.As the number of layers increases, the value of h l reduces and the strength of cells increases.The angle of inward rotation of the lowermost and uppermost cells and the deformation of the outer parts of cells is reduced, so that the contraction of the cells in the middle is reduced.Figure 20(c,d) depict the compression force and EA curves.In Figure 20(c), the compression forces of H-la.4 -H-la.10first increase, then stabilize and then continue to increase.The compression force of H-la.3 increases first, and then, has a sharp drop around 13 mm.This is because its VCPs is buckling and its cells undergo torsional deformation.In Figure 20(d), with the increases of compression distance, the EA values of these models consistently increase.As the layer number increases, the amplitude of compression force curves and the slope of EA curves tend to increase.This is because that the increases of layer number enhances the interaction between cells.Figure 20(e,f) shows the graphs of Poisson's ratio and rotation angle of these lattice structures.H-la.3's negative Poisson's ratio effect is the most obvious, and H-la.5 obtain the MRA among these models.
Figure 21 depicts the deformation modes and rotation angle of these HCLSs with different layer number.At 10 mm, the lowermost and uppermost cells are in contact with the platens through their upper and lower surfaces.Only H-la.3 and HCLS contract significantly inward.At 20 mm, the middle two layers of cells of these HCLSs undergo torsional deformation, and their VCPs bend.As layer number increases, the MRA of HCLS increases first, then decreases and then increases.The deformation directions of the upper two layers and the lower two layers of H-la.6 and H-la.7 are opposite.Therefore, the MRAs of the two models are smaller than other models.At 30 mm, the torsional deformation of cells and the bending of connecting plates continue to increase.At 40 mm, almost every layer of cells and connecting plates collapse.The cells in HCLS, Hla.5, and H-la.7 that obtain the MRAs rotate counterclockwise, and the rest models' rotation is clockwise.

Conclusion
In this study, a hollow auxetic cylinder lattice structure (HCLS) and a HCLS with reverse lattices (HCLS-R) are proposed based on tapered three-dimensional reentrant cells (TRCs).With the material nylon 11, the 3D printing technology is applied to manufacture the test sample of HCLS.Their structural deformation patterns and energy absorption properties are investigated through finite element analysis (FEA) and quasi-static compression test.The results of FEA are very similar to those of experiment, and HCLSs exhibit excellent energy absorption, unique compression rotation and significantly negative Poisson's ratio properties.The rotational deformation of cells ensures the stability of compression force of HCLSs.
The influences of five structural parameters on the compression characteristics of HCLS are further investigated.As h l increases, the negative Poisson's ratio effect increases and the MRA first decreases and then increases.With the value of h s increases, the absolute value of minimum negative Poisson's ratio (MPR) increases first and then decreases, and the value of MRA decreases first and then increases.As D s decreases, the value of MPR tends to increase and the value of MRA tends to decrease first and then keep stable.As w s increases, the negative Poisson's ratio effect decreases, and the value of MRA tends to increase first and then decrease.With the increases of layer number, the value of MPR increases, and the value of MRA increases first, then, decreases and then increases.In addition, H-h s 2.0 and H-la.3 obtain the most significant rotational and negative Poisson's ratio properties separately, and the values of the MRA of H-h s 2.0 and the MPR of H-la.3 are 37.32 degree and À0.653, respectively.

Disclosure statement
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Figure. 1 .
Figure. 1.The geometric structures of HCLSs.(a) A plane hexagonal reentrant cell.(b) Two plane hexagonal reentrant cells, and the right cell is a larger one.(c) A tapered TRC.(d) The details of HCLS.There are four layers of cells in HCLS, and they are layer 1, layer 2, layer 3, and layer 4 from top to bottom.(e) The sample of HCLS, and (f) The geometric model of HCLS-R.

Figure. 2 .
Figure. 2. The details of HCLS.(a) The geometric parameters of TRCs with VCP and HCP.The geometric parameters of HCLS in (b) top plane and (c) isometric views.

Figure. 3 .
Figure. 3. The preparation of FEA and compression test.(a) The quasi-static compression test of HCLS and (b) The simulation model of HCLS.

Figure 3 (
Figure 3(a) illustrates the preparation of compression test at room temperature.The specimen of HCLS is placed on the lower platen.The upper platen is slowly approached to the sample until their surfaces touch

Figure. 4 .
Figure. 4. The tensile testing of the samples of nylon 11.(a) The test specimens before and after experiments.(b) The preparation of tensile test.(d) The engineering stress-strain characteristics obtained from the experiments.

Figure. 5 .
Figure. 5.The results of test and FEA.(a) The deformation modes of test and FEA.(b) The compression deformation of cells in the initial stage.Plots (c) compression force and (d) EA of FEA and test.

Figure 5 (
Figure 5 and Table2show the results of the experiment and FEA.Figure5(a) is the deformation modes of the test and FEA.From Figure 5(a), the deformation modes of the FEA and test are very similar.At 10 mm, the cells are squeezed by the VCPs, and the areas connected to the VCPs on the upper and lower surfaces of the cells are concave inward.In FEA and test, the cells in the adjacent layers start to touch each other, and the cells in the middle region contract inward.The reason for the contraction is shown as Figure 5(b).From Figure 5(b), the upper and lower platens are in

Figure. 6 .
Figure. 6.The deformation modes of HCLS and HCLS-R at different compression distances.(a) The strain nephograms in left plane view and (b) the rotation deformation modes in isometric view.The plots of (c) Poisson's ratio-deformation and (d) rotation angle-deformation.

Figure. 7 .
Figure. 7. The geometric structure of HCLS with different h l .(a) The left plane and single-cell views of H-h l 0.375.The left plane and single-cell views of (b) H-h l 0.5, (c) H-h l 0.625, (d) H-h l 0.75, (e) H-h l 0.875, and (f) HCLS.

Figure. 9 .
Figure. 9.The deformation modes of H-h l 0.375, H-h l 0.5, H-h l 0.625, H-h l 0.75, H-h l 0.875, and HCLS.(a) The left plane views of these models at different compression distances.(b) The diagrams of the deformation modes of connecting plates and cells at 10 mm for these models.(c) The rotational angles of these models at different compression distances.

Figure. 12 .
Figure.12.The deformation modes of HCLS, H-h s 1.25, H-h s 1.5, H-h s 1.75, and H-h s 2.0.(a) The left plane views of these models at different compression distances.(b) The rotational angles of these models at different compression distances.

Figure. 15 .
Figure. 15.The deformation modes of HCLS, H-D s 0.91, H-D s 0.83, H-D s 0.77, H-D s 0.71, and H-D s 0.67.(a) The left plane views of these models at different compression distances.(b) The rotational angles of the cells of these models at different compression distances.
Figure 14 and Table8depict the results of HCLS, H-D s 0.91, H-D s 0.83, H-D s 0.77, H-D s 0.71, and H-D s From Table8and Figure14(a,b), as the value of D s decreases, the values of PCF, SM, EA, MCF and q e increase, and the value of SEA trends to increase.This means that the compression performance of HCLS is improved with the decreases of D s .From the results of MPR and MRA, as D s decreases, the value of MPR tends to increase and the value of MRA tends to decrease first, and then, keep stable.Figure14(c,d) illustrate the compression force and EA curves.From Figure14(c), as compression distance increases, the compression forces of HCLS, H-D s 0.91, and H-D s 0.83 first increase, then remain basically stable and gradually increase after 30 mm.The compression forces of H-D s 0.77, H-D s 0.71, and H-D s 0.67 first increase, then decrease slightly, then increase slightly and remain stable, and finally increase gradually after 25 mm.The energy absorption performance of these HCLSs is consistent with their compression force performance.The slopes of the EA-deformation curves of H-D s 0.91, H-D s 0.83, H-D s 0.77, H-D s 0.71, and H-D s 0.67 tend to increase as the compression distance increases.Figure14(e) depicts the plots of Poisson's ratio values, and their Poisson's ratio values tend to decrease first and then increase as compression progresses.Their MPRs are obtained in the interval of 5-15 mm.In Figure14(f), the rotation angles of these models also increase significantly after 10 mm and reach their maximum values at 40 mm.Figure15depicts the deformation modes and rotational angle of these HCLSs.At 10 mm, the cells in adjacent layers touch each other, and the cells in the middle contract Figure 14(e) depicts the plots of Poisson's ratio values, and their Poisson's ratio values tend to decrease first and then increase as compression progresses.Their MPRs are obtained in the interval of 5-15 mm.In Figure 14(f), the rotation angles of these models also increase significantly after 10 mm and reach their maximum values at 40 mm.

Figure 15
depicts the deformation modes and rotational angle of these HCLSs.At 10 mm, the cells in adjacent layers touch each other, and the cells in the middle contract

Figure. 18 .
Figure.18.The deformation modes of H-w s 0.7, H-w s 0.8, H-w s 0.9, HCLS, H-w s 1.1, and H-w s 1.2.(a) The left plane views of these models at different compression distances.(b) The rotational angles of the cells of these models at different compression distances.

Figure 17 (
Figure17(e,f) are the graphs of Poisson's ratio-deformation and rotation angle-deformation of these models.From Figure17(e), it is found that the negative Poisson's ratio

Table 1 .
The values of the geometric parameters of HCLS and HCLS-R.
HCLSD s (mm)D l (mm) D o (mm) h (degree) H (mm) h s (mm) h l (mm) w s (mm) l (mm) d s (mm) d l (mm) f s (mm) f l (mm) S s (mm) S l (mm) s (mm) D l (mm) D o (mm) h (degree) H (mm) h s (mm) h l (mm) w s (mm) l (mm) d s (mm) d l (mm) f s (mm) f l (mm) S s (mm) S l (mm)

Table 2 .
The FEA results of test and FEA.

Table 3 .
The geometric parameter values of these HCLSs with different h l .

Table 4 .
The FEA results of HCLSs with different h l .

Table 5 .
The geometric parameter values of these HCLSs with different h s .

Table 6 .
The FEA results of HCLSs with different h s .

Table 7 .
The values of the geometric parameters of HCLSs with different D s .Only the values of those parameters that differ from HCLS are listed.

Table 9 .
The values of the geometric parameters of these HCLSs with different w s .
4.3.The influence of inner diameter D sIn this section, the values of D s are set to 30.909, 28.333, 26.154, 24.286, and 22.667 mm, respectively.These are 0.91, 0.83, 0.77, 0.71, and 0.67 times the original D s value (34 mm), respectively.Based on these values, these HCLS models are named H-D s 0.91, H-D s 0.83, H-D s 0.77, H-D s 0.71, and H-D s 0.67, respectively.Figure

Table 10 .
The FEA results of lattice models with different w s .

Table 11 .
The parameter values of these HCLSs with different layer number.