In-plane compression behavior of meta-tetrachiral and common auxetic structures

Meta-tetrachiral structure, as a novel negative Poisson’s ratio auxetic materials, combines the topological features of tetrachiral and anti-tetrachiral structures. Under the in-plane compression, a comparative study on the mechanical properties of the meta-tetrachiral and the common tetrachiral and anti-tetrachiral structures is investigated experimentally and numerically. Three types of specimens are fabricated by using Stereo Lithography Apparatus (SLA) technology; numerical models are established by using FEA software and verified by quasi-static compression tests. The simulated and experimental results together indicate that the in-plane deformation patterns of the meta-tetrachiral are limited by the different loading directions. The deformation mechanism of the meta-tetrachiral loaded along different directions is summarized by the deformation behavior of the specimens. The Young’s modulus of the meta-tetrachiral loaded along different directions is larger than that of the anti-tetrachiral structure. The auxetic properties (NPR) of the meta-tetrachiral structures loaded along different directions are superior to that of the tetrachiral structure. In addition, it is found that the meta-tetrachiral loaded along the Y-direction shows a better performance in terms of energy absorption compared to the other structures. The effect of the meta-tetrachiral structures with different wall thicknesses on energy absorption was investigated.


Introduction
Auxetic material is an artificial structure functional material with a negative Poisson's ratio effect.The concept of 'auxetic' was first introduced by Evans [1] in 1991, which specifically refers to the phenomenon of expansion or contraction when the material is stretched or compressed axially.In a mechanical sense, this peculiar negative Poisson's ratio effect not only gives the material a greater shear modulus, but also allows it to achieve stronger fracture toughness and energy absorption capacity.In recent years, the unique mechanical properties of auxetic materials have received a lot of attention in various fields, and applications related to them are emerging.In the macroscopic field, it can be used as sandwich panel cores for aerospace components [2,3] and energy-absorbing materials for vehicle collisions [4].In the microscopic field, it can also be used as scaffolds [5,6], micro smart sensors [7], molecular sieves [8,9] and filters [10].In the textile industry, it can be used as a special fabric for new types of clothing [11][12][13].
The design of the unit structure of auxetic materials is related to the overall mechanical properties of the material, which has been the focus of research.Up to now, a variety of material structures have been derived [14][15][16][17][18].The more intensively studied structures with relatively mature technology include bucklicrystals structures [19][20][21][22], auxetic foam structures [23], re-entrant honeycombs structures [24][25][26][27], 3-4-6 star array structures [28], chiral [29][30][31][32] and anti-chiral [33][34][35][36][37][38] structures and novel hybrid re-entrant and chiral structures [39][40][41][42][43]. Zhang et al [44] proposed an in-plane impact resistance and energy absorption properties of the material at constant velocities using numerical simulations based on a bionic re-entrant honeycomb model.Kadir et al [45] compared the mechanical characteristics of the anti-tetrachiral structure with the re-entrant honeycomb structure by numerical analysis and in-plane quasi-static compression experiments, and concluded that the anti-tetrachiral structure has a stronger energy absorption capacity than the re-entrant honeycomb structure.Based on the excellent mechanical properties of the anti-tetrachiral structure, Wu et al [46] designed a new anti-chiral non-pneumatic tire and studied its load carrying capacity under compression using the finite element method.Chen et al [47] focused on anisotropic anti-tetrachiral lattice materials and obtained the effect of ligament transverse to longitudinal ratio on Young's modulus and Poisson's ratio by theoretical and numerical analysis.In addition, research on chiral structures is ongoing.Lorato et al [48] studied the elastic properties of hexachiral, tetrachiral and trichiral structures using finite element and experimental methods.Alderson et al [49] measured Young's modulus and Poisson's ratio of hexachiral, tetrachiral and trichiral structures in-plane, where Young's modulus increases with the number of ligamentous allotments and yielded a Poisson's ratio close to −1 for tetrachiral structure.Zhang et al [50] proposed a novel composite tetrachiral material and investigated the deformation characteristics and energy absorption capacity of the specimens by quasi-static compression tests and numerical simulations.
Based on the excellent mechanical properties of tetrachiral and anti-tetrachiral structures, a new type of chiral structure has recently emerged, which combines the connection of chiral and anti-chiral ligaments, called meta-tetrachiral structure [51].Up to now, a large number of studies have been reported.Li et al [52] established an analytical expression for the in-plane modulus of meta-tetrachiral structure and tested its in-plane tensile properties through experiments and finite element simulations.Niu et al [53] investigated the effect of different missing ligament ratios on the elastic modulus and Poisson's ratio of the meta-tetrachiral material system.Qi et al [54] designed a three-dimensional re-entrant structure based on the meta-tetrachiral, and investigated the coupled tension and rotation deformation mechanisms.However, all of them lack research on energy absorption of materials.Qi et al [55] proposed a meta-tetrachiral structure of geometric gradient type, studied the in-plane dynamic deformation mechanism and energy absorption capacity through finite element analysis.Liu et al [56] designed a flexible porous chiral tracheal stent based on the meta-tetrachiral, experiments and simulations indicated that it has desirable auxetic performance.However, none of them have investigated the quasi-static mechanical response and energy absorption capacity in the face of meta-tetrachiral materials.
In this paper, the in-plane mechanical behavior of meta-tetrachiral structure is investigated and common tetrachiral and anti-tetrachiral structures are introduced for comparison.For asymmetric meta-tetrachiral specimens, quasi-static compression tests were performed in both in-plane X and Y directions, respectively, while for symmetric tetrachiral and anti-tetrachiral specimens, the tests were performed in one direction only.The deformation modes and deformation mechanisms and Poisson's ratio of each structure during compression are discussed using a combination of numerical models and experimental tests.In addition, the energy absorption capacity of the material is one of our concerns.This paper aims to investigate the in-plane compression properties of meta-tetrachiral structure, and the experimental conclusions and research results obtained in the article can provide scientific references for the application of meta-tetrachiral structure in related fields in the future.

Design and fabrication of lattices
2.1.Lattices geometry Figure 1 illustrates the unit cell geometric models of the meta-tetrachiral and tetrachiral and anti-tetrachiral structures.As can be seen from the figure, the axial ligaments (purple) and transverse ligaments (yellow) of the meta-tetrachiral structure are connected in the same way as those of the tetrachiral and anti-tetrachiral structures, respectively.In the same single cell, some of the ligaments choose to connect to the heterolateral side of the cylinder, while the rest of the ligaments choose to connect to the ipsilateral side of the cylinder.Four independent parameters, L x , L y , r and t, are used to describe the geometry of the unit structure, which represent the transverse axis distance of adjacent cylinders, axial axis distance, outer radius of the cylinder and material wall thickness, respectively.For convenience, the above practical geometric parameters can be further simplified to dimensionless parameters: α x = L x /r, α y = L y /r, β = t/r, γ = L y /L x .To analyze the in-plane compression properties of the meta-tetrachiral structure, the cylinder is assumed to be a rigid rotating node with a linear elastic bending deformation inside the tangential ligament.Based on the rigid cylinder assumption, strain energy analysis and linear elastic Bernoulli-Euler beam theory, the in-plane elastic modulus of the heterogeneous metatetrachiral unit structure was derived [52] as follows: where E c indicates the elastic modulus of the sample's base material.Considering that the experiments need to focus on the connection of the meta-tetrachiral with the tetrachiral and anti-tetrachiral structures in the deformation mode, the same values of E x and E y are specified here for convenience, i.e., let L x = L y .In addition, due to the limitation of 3D printer manufacturing accuracy, we chose a more precise unit size as possible while ensuring the structure has the right modulus.As a comparison, the same values are used to specify the unit and overall dimensions of the three kinds of structures.The overall length, height, and thickness of the material are indicated by L, H, and T (not labeled in the figure), respectively, and the specific dimensional values of the unit and the whole are covered in table 1.

Fabrication of auxetic lattices
3D printing additive manufacturing is flexible and diverse, this paper uses Stereo Lithography Apparatus (SLA) technology to print the three different structural materials mentioned above.SLA is a technology that uses the principle of single photon polymerization to rapidly cure and mold liquid UV-photosensitive resin materials.Under computer control, the laser beam is scanned on the resin surface on a predetermined trajectory, causing the resin surface to cure to form a scanned cross-sectional layer film of the part.Once a layer is formed by curing in the resin tank, another layer of liquid resin is covered over the freshly formed layer film and continues to be cured and bonded to the surface of the previously formed part by scanning with a microscope.The iSLA900D light-curing 3D printer was used, with a UV light source wavelength of 355 nm and an output power of 330 mW.Using a high-speed scanning dual-oscillator system and layer-by-layer printing by vacuum adsorption coating technology, the slice thickness is 0.1 mm.The laser focus fills the resin surface layer by layer from point to line and from line to surface.Due to the Gaussian distribution of the laser light intensity, this may lead to non-uniform curing inside and at the edges of the scanned point, and subsequent processing to improve the curing of the samples will increase the strength of the structure while being able to ensure that the interior of the structure is in an isotropic state.Figure 2 shows the main view of the three kinds of specimens obtained by printing.
After printing, the actual sizes of the samples were randomly measured using electronic vernier calipers.the average values obtained from the measurement are the same as the design sizes.Compared with the traditional 3D printing method of fused deposition modeling, the advantages of SLA technology in printing accuracy can be more favored.Subsequently, the three structures were placed on an electronic balance and weighed, the ratio of the measured mass to the volume of the sample (100 mm × 100 mm × 50 mm) was taken as the density of the material, and the densities of the meta-tetrachiral, tetrachiral and anti-tetrachiral specimens were ultimately calculated to be 223.13kg m −3 , 220.47 kg m −3 and 225.79 kg m −3 , respectively, these values will prepare the way for exploring energy absorption capacity. 3. Experiments and finite element simulations

Tensile test of parent material specimens
To obtain mechanical parameters of the parent material (photosensitive resin), four sets of tensile specimen models (type IV) were printed according to ASTM D638 standard [57], with the specimen dimensions shown in figure 3(a).Quasi-static tensile testing of the material in the axial direction at a constant speed of 0.1 mm s −1 using an INSTRON 5967 universal testing machine.Figure 3(b) describes the test results of the four groups of experiments, and it can be seen that the trend of the four curves is the same.The specimen starts to enter the irrecoverable plastic state after the elastic stage, at which the stress is about 20 MPa and the corresponding strain is located at 0.012, and then the stress continues to increase until it reaches the maximum value of 41.73 MPa when the material fractures.After processing, the average values of the elastic modulus, plastic stress and strain of the samples themselves are listed in table 2. In addition, the material parameters such as density and Poisson's ratio of the resin material are given by the manufacturer, and their values are 1.10g cm −3 and 0.38, respectively.

Quasi-static compression experiments
The mechanical properties of the three structures were tested using an INSTRON 5967 universal test system with a load range of 30 kN.For asymmetric meta-tetrachiral structure, quasi-static compression tests are chosen to be performed in both X and Y directions separately, while for isotropic tetrachiral and anti-tetrachiral structures, tests need only be performed in the Y direction.Each structure was subjected to two replicate experiments along the desired compression direction.The indenter was set to compress along the axial direction at a constant rate of 0.1 mm s −1 before the experiment, corresponding to a strain rate of 0.001 s −1 .Due to the special auxetic properties of the structure making Poisson's ratio measurements critical, eight fixed markers were set on each sample and the scale was utilized to track the change in displacement of the marker points in both the X and Y directions after the test was initiated.A computer connected to the equipment automatically records parameters such as compression time, displacement, load, and engineering stress and strain of the material, where the quasi-static engineering stress (σ) is defined as the ratio of the instantaneous force to the area (L * T) of the specimen under stress in the compression direction, and the quasi-static engineering strain (ε) is defined as the ratio of the instantaneous displacement in compression to the original height (H) of the specimen in that compression direction, where L, T and H are all mentioned in section 2.1.The entire compression process is recorded by a camera.Once the material is densified, the test is over.

Finite element analysis
The quasi-static compression process was simulated using ABAQUS finite element analysis software.Figures 4(a) and (b) show a schematic diagram of the compression simulation of the meta-tetrachiral structure based on the Y-direction.To match the experimental tests, eight markers of the same position were created on the simulated specimen and their displacement changes were output.The upper and lower plates are defined as rigid thin plates, and the degrees of freedom of the constrained lower plate are completely fixed, since the simulation using a loading speed of 0.1 mm s −1 is very time-consuming, A constant speed of 100 mm s −1 is given to the upper plate to improve computational efficiency.Preliminary simulations after comparison indicate that the stress-strain curves are not yet sensitive to a loading velocity of 100 mm s −1 , as shown in figure 4(c).To further save computational costs, the analysis step mass scaling factor was set to 100.In general, the reasonableness of mass scaling is judged by whether the ratio of ALLKE to ALLIE is less than 5% when processing the results.Taking the meta-tetrachiral structure loading along the Y direction as an example, the ratio of ALLKE to ALLIE is much less than 5%, i.e., the chosen mass scaling factor satisfies the requirement.The plate is defined as an automatic general contact with the material to maximize the ease of solution, a coefficient of friction value of 0.25 for tangential behavior as measured by using the inclined plane method [58], and the hard contact is defined in the normal direction to avoid unnecessary penetration.Edit each mechanical parameter in table 2 as a material property.In addition, the mesh division plays an important role in the convergence performance.To improve the mesh quality and to determine the appropriate cell size, four values of 0.5 mm, 0.63 mm, 1 mm and 2 mm are selected as cell sizes for mesh delineation, and it can be seen from the stress-strain curves in figure 4(d) that the difference between cell sizes of 0.5 mm and 0.63 mm before 40 percent strain is small, showing convergent solutions.Therefore, the cell size of 0.63 mm was used for the creation of the finite element model.

Results and discussion
4.1.Deformation modes 4.1.1.Meta-tetrachiral in the X-direction and tetrachiral structures Figure 5 shows the compressive deformation process of the meta-tetrachiral structure compressed along the X-direction and the tetrachiral structures, and the comparison shows that the experimental results agree well with the simulation results in the same state.By comparing the deformation processes of the two structures, it can be seen that the two materials exhibit nearly identical deformation patterns with continuous downward pressure of the indenter.Figure 6 depicts the stress-strain curves obtained from experiments and simulations, which combined with the curve trend can reflect the deformation pattern of the materials in more depth.Taking the experimental curve in figure 6(a) as an example, initially, the stress in the meta-tetrachiral structure compressed along the X-direction increases violently in a linear trend, which corresponds to the material being in an elastic phase that has not yet yielded.As the force continues to load, when the strain is around 0.025, the stress rises relatively sluggishly and the elastic state thus ends, at which time the distortion of the cylinder becomes more and more obvious, resulting in the material showing a tendency to shrink in the transverse direction, and then a layer of the material starts to extrude and deform to cause the layer to densify, and the stress also drops rapidly from the maximum value of 0.76 MPa to 0.12 MPa.After experiencing the collapse of one layer, the stress continues to increase and then decreases, and the layer adjacent to the material continues to deform until it collapses, while the other layers are almost unaffected by the deformation.The next compression scenario is repeated with continuous oscillation of the curve, causing the material to collapse layer by layer until the overall densification.In figure 6(b), the stress curve of the tetrachiral structure shows a similar trend as the former.However, it undergoes a shorter elastic phase, and the slope can determine that the tetrachiral structure has a greater Young's modulus.When the stress reaches 0.82 MPa, a layer of the material flexes, followed immediately by a cliff-type reduction in stress, which is due to the poor toughness of the material itself in practice, resulting in the fracture of the specimen after it has withstood the stress limit.Compared to the experimental results, the simulated curves showed less pronounced periodic variations but were numerically approximately the same as the experimental curves in terms of stress averages.
It is noteworthy that the meta-tetrachiral compressed along the X-direction and the tetrachiral structure in the experiment have their axial eight-layer structures densified sequentially, which just corresponds to the eight periodic oscillations occurring one after another in the two curves in figure 6, which means that the oscillation of one cycle from peak to trough in the image just corresponds to the collapse of one layer of the material.Moreover, the slope of the stress corresponding to each collapse during compression is approximately the same, which better explains the fact that the deformation of the material under stress is limited to that layer, while the other layers are almost unaffected.Compared with the simulated results, although the experimentally obtained stress curves have larger oscillations due to material brittleness, the exaggerated curve trends of the two structures can better illustrate the deformation pattern of layer-by-layer collapse of both.

Meta-tetrachiral in the Y-direction and anti-tetrachiral structures
The experimental and simulated deformation processes of the meta-tetrachiral compressed along the Y-direction and anti-tetrachiral structures are shown in figure 7. A comparison of the compression behavior shows that there are similarities and differences in the deformation patterns of the latter two.Compared to the first two, both of the latter are no longer confined to a single layer during compression, but instead exhibit a more holistic approach to deformation.However, the meta-tetrachiral loaded along the Y-direction exhibit inferior stability to the anti-tetrachiral structure, which is caused by the fact that the ligaments of the metatetrachiral structure are tilted along the loading direction.As the compression continues, the load will first be applied to the upper left and lower right cylinders as the stress points (marked in red in figure 7), causing the ligaments connected to that cylinder to yield, followed by the surrounding ligaments to bend and deform due to inertial instability leading to a partially localized deformation of the material.Whereas for the anti-tetrachiral structure, the ligaments in the loading direction will be more stable to deformation due to their vertical distribution.During compression, the friction between the material and the contact surface restricts the deformation of most of the distal units, and the material deformation is concentrated towards the interior.Figure 8 depicts the stress-strain curves of the two structures under experimental and simulated conditions, and the accuracy of the simulation results can be well verified by the curve trend.Combined with the experimental curves in figure 8(a), it can be found that the elastic state of the meta-tetrachiral structure compressed along the Y direction ends at a strain of about 0.032, and as the stress continues to increase, the specimen begins to enter into an irrecoverable plastic state and some of the units begin to twist and deform.When the curve reaches the maximum value of 0.67 MPa, it decreases slightly and then stabilizes for a long time.In figure 8(b), the antitetrachiral structure has a much flatter linear growth in the elastic range, and the overall stiffness of the material is less.The stress reaches a maximum value of 0.44 MPa when the strain is 0.076.Because some internal units of the material begin to yield, resulting in a gradual decrease in the stress, the plateau region of the antitetrachiral appears a little later compared to the meta-tetrachiral structure that compressed along the Y-direction.As the compression continued, the curve trended more gently and the deformation pattern of the  material was more stable.Compared to the simulated curves, all experimental curves have a later onset of densification strain, which is due to some unavoidable fracture phenomena in the real situation.
Taken together, the meta-tetrachiral structure compressed along the Y-direction show less stability in deformation than the anti-tetrachiral structure, but the similar curve trends in the stress-strain images of the two provide strong evidence to indicate the similarity in their deformation modes, which will be further explained by analyzing the deformation mechanism in the next section.In contrast to the ups and downs of the curves in the layer-by-layer compression mode in the plateau region, the latter two tend to have a smoother trend in the plateau phase, which further indicates a more holistic deformation pattern in the compression process.In addition, by comparing the elastic regions, it can be seen that the meta-tetrachiral structure loaded along the Y-direction have a larger elastic modulus, which would create conditions for more energy absorption.

Deformation mechanism
Tracing back to the source, different topological structures are the main reason for the difference of structural deformation modes.Figure 9 shows the deformation mechanism of each structural unit.In figure 9(a), the metatetrachiral loaded along the X-direction and tetrachiral structures are taken as the research objects.Since the ends of its transverse ligaments are connected to the opposite sides of the cylinders, the deformation mode is full-wave, and under the axial loads F 1 and F 2 , the ligaments of each unit are subjected to torsional forces in the same direction (clockwise), resulting in the pulling of the units of the same layer in the same direction, which causes the collapse of the layer; In contrast to figure 9(a), the ends of the transverse ligaments of the metatetrachiral loaded in the Y-direction and anti-tetrachiral are connected to the same side of the cylinder, and the ligaments are subjected to torsional forces in the opposite direction to produce half-wave deformation in figure 9(b).Under load, the forces in neighboring units in the same layer constrain each other, so the material will exhibit more uniform forces in the transverse direction.However, since the ligaments of the anti-tetrachiral structure along the loading direction undergo half-wave deformation, while the axial ligaments of the metatetrachiral structure loaded along the Y-direction undergo a full-wave deformation, this results in a more balanced force on the units of the anti-tetrachiral structure along the loading direction, which leads to stronger overall stability than that of the meta-tetrachiral structure along the Y-direction.

Modulus of elasticity
Based on the linearity of the stress-strain curve in the elastic region, the elastic module obtained from experiments and simulations for the three types of structures is collated in table 3. The Young's modulus of the tetrachiral structure is the largest with an experimental value of 31.74MPa and a simulated value of 29.52 MPa.The elastic modulus of the meta-tetrachiral structure loaded along the Y-direction(23.13MPa) is approximately the same as that of the meta-tetrachiral structure loaded along the X-direction(25.08MPa),which follows our original design intention in section 2.1.However, the elastic modulus of the meta-tetrachiral structure loaded along the Y-direction is slightly smaller than that loaded in the X-direction, which we suspect may be due to the different deformation modes.We will continue to verify this and hope that interested readers can join us in the discussion.

Poisson's ratio
Poisson's ratio, one of the important mechanical parameters, is used to compare the elastic strain properties of materials.It is defined as the negative ratio of transverse strain to axial strain in the case of uniaxial stress.We focused on the negative Poisson's ratio properties of the meta-tetrachiral structure and calculated its Poisson's ratio loaded along different directions and compared it with two other structures.Taking the starting moment of compression as the benchmark, extracting the images in the camera every 6s, with the help of the scales on the compression platform to obtain the positional coordinates of the markers in each image, taking the average displacement of the left and right three groups of markers as the transverse displacement, and the average displacement of the upper and lower three groups of markers as the axial displacement, the displacement change of the specimen in the transverse and axial directions is obtained after calculation and organization, and then the transverse and axial strain is calculated to find out the Poisson's ratio of the specimen further according to the calculation.In actual compression, since yielding the specimen would cause difficulties in tracking the markers, in addition to the fact that obtaining and sorting out the locations of the markers would be a more complicated task, we have fixed the calculation of Poisson's ratio to be in the axial strain range of 0.15.
Figure 10 demonstrates the relationship between Poisson's ratio and axial strain for the experimental and simulated cases, the Poisson's ratios of the three structures do not differ much since their tensile expansion characteristics come from the rotation of the cylinder.The meta-tetrachiral structure loaded along the X-direction has a Poisson's ratio of −0.89 at the beginning (experimental) and −0.92 in the simulation.There is a small increase in Poisson's ratio when the strain in the experimental curve is 0.09, this situation combined with the deformation pattern in figure 5 shows that the specimen recovers a small portion of its deformation due to the buckling of the monolayer, resulting in a decrease in the amount of transverse displacement change.The Poisson's ratio remains essentially constant as the axial strain continues to increase.In figure 10(b), the Poisson's ratio for the experimental case of the meta-tetrachiral structure along the Y-direction stays around −0.92, which is slightly smaller than that of the meta-tetrachiral along the X direction.Figure 10(c) and (d) show the Poisson's ratios of the tetrachiral and anti-tetrachiral structures.The Poisson's ratios of the tetrachiral structure are 0.76 (experimental), and 0.8 (simulated).However, it was mentioned in a previous report that Poisson's ratio of tetrachiral is theoretically close to −1 [49], which may be due to its relatively small transverse strain in the present experiments.While the Poisson's ratio of the experimental curve of the anti-tetrachiral structure is close to −1 at the beginning and then gradually increases, this behavior is consistent with the deformation pattern in figure 7(b), the deformation of the cells first occurs in the interior, which makes the markers distributed in the interior of the specimen firstly deform uniformly and corresponds to a small Poisson's ratio, with the compression proceeding, the cells at the edges begin to deform, which results in a gradual decrease in the transverse strain of the internal markers, which leads to the increase of Poisson's ratio.In addition, this study ignores the shear deformation along the Z-direction during compression [59].

Energy absorption 4.5.1. Effect of structures on energy absorption
The energy absorbed per unit volume (Wv), and the energy absorbed per unit mass(SEA), are two important indicators that characterize the energy absorption capacity of a material.Among the engineering stress-strain images, the area enclosed by the curve with the X-axis is used to represent the energy absorbed per unit volume, and the energy absorbed per unit mass is calculated using equation (4) [60], whose expressions are, respectively: Where σ denotes the engineering stress, ρ denotes the material density, and ε d denotes the strain at the onset of densification of the material.In this study, the intersection of the tangents of the platform and densification states is defined as the densification strain using the method in the literature [61].The area under the experimental curve in figures 6 and 8 is used to find the value of energy absorption per unit volume, where the material absorbs a small amount of energy in the elastic phase, while the long and extensive plateau phase is the main reason for the material to absorb enough energy.Figure 11 reflects the energy absorption capacity of the three structures.For the meta-tetrachiral structure, the energy absorption capacity varies for different loading directions, which is a good demonstration of the anisotropy of the meta-tetrachiral structure.The energy absorption capacity of the meta-tetrachiral material loaded along the Y-direction is greater than that of the other three cases.Since its ligaments along the loading direction are inclined, resulting in a greater degree of deformation when the cylinder is rotated, and the energy dissipation mainly relies on the deformation of the ligaments and the cylinder, which gives it a stronger energyabsorbing capacity, with an energy absorption per unit volume of 329.15 kJ m −3 and an energy absorption per  unit mass of 1.475 kJ kg −1 .The energy absorption capacity of the meta-tetrachiral compressed in the X-direction is slightly lower than that of the anti-tetrachiral structure due to the neglect of the energy released by the fracture phenomenon during the compression process, and the energy absorption capacity of the meta-tetrachiral structure compressed in the X-direction will be even better if the material's own properties are improved.

Effect of wall thickness on energy absorption
Since the meta-tetrachiral structure has an advantage in energy absorption, striking while the iron is hot, we have investigated the energy absorption capacity of meta-tetrachiral structures with different wall thicknesses along the Y-direction.Four sets of meta-tetrachiral specimens with t = 1 mm, t = 1.1 mm, t = 1.2 mm as well as t = 1.3 mm were printed by using SLA, and the other parameters of the specimens are still referred to the dimensional values in table 1 except for the wall thickness.The densities of the four groups of specimens were 223.13 kg m −3 , 241.19 kg m −3 , 259.07 kg m −3 , and 277.61 kg m −3 .The test conditions were consistent with the previous experiments, and the stress-strain curves of the materials obtained from the tests are shown in figure 12(a).It can be seen that as the wall thickness increases, the load limit that the material can withstand in the elastic region increases significantly, in other words, the increase in thickness enhances the elastic modulus of the structure.However, the experiment also reveals a problem, under the very high load, the specimen with t = 1.3 mm has a large fracture due to its ligament reaching the load-bearing limit, resulting in the experiment can not be continued, which indicates that the enhancement of stiffness is a great test of material toughness during compression, which reminds us that we can choose a better material later to combine both the strength and toughness of the material in order to improve the energy absorption capacity.Compared to t = 1 mm, the larger range of areas under the curve for the t = 1.1 mm and t = 1.2 mm specimens creates conditions for energy absorption.Figure 12(b) demonstrates the energy absorption capacity per unit mass of the meta-tetrachiral structures in the Y-direction with different wall thicknesses, which is 1.475 kJ kg −1 when t = 1.0 mm and 1.614 kJ kg −1 when t = 1.1 mm and 1.761 kJ kg −1 when t = 1.2 mm.It can be seen that within the fracture threshold, the energy absorption capacity of the material is gradually enhanced with the increase in wall thickness, and the SEA of t = 1.2 mm specimen is 119% of t = 1 mm.Therefore, although the increase in wall thickness will lead to an increase in the mass of the structures, it also gives it a better energy absorption capacity.

Conclusion
This paper focuses on a comparative study of the in-plane compression properties of meta-tetrachiral structure and common tetrachiral and anti-tetrachiral structures.Three high-precision structural specimens were successfully printed using SLA technology, effectively demonstrating the reliability of this printing method.The in-plane deformation pattern, Young's modulus, Poisson's ratio, and energy absorption capacity before densification of each specimen were investigated by a combination of numerical analysis and quasi-static compression testing.The results show that: The meta-tetrachiral structure, which combines common features of both tetrachiral and anti-tetrachiral structures, lead to anisotropy in their deformation patterns, and different loading directions can affect the deformation patterns of the materials.The transverse ligaments of the meta-tetrachiral loaded along the X direction undergo full-wave deformation, and the units in the same layer are pulled in the same direction, thus causing the structure to collapse layer by layer; the transverse ligament of the meta-tetrachiral loaded along the Y direction undergoes half-wave deformation, resulting in the mutual constraints on the stresses of the adjacent units in the same layer, the deformation of the material tends to be holistic.
The deformation pattern of the meta-tetrachiral loaded along the X-direction is similar to that of the tetrachiral structure; the deformation pattern of the meta-tetrachiral loaded along the Y-direction in the transverse direction is similar to that of the anti-tetrachiral structure, but the deformation is not as stable as the anti-tetrachiral due to the full-wave deformation of the ligaments in the crushing direction.
The Young's modulus of the meta-tetrachiral loaded along different directions is approximately the same.Although it is numerically inferior to that of the tetrachiral structure, the higher elastic stresses exhibited can be satisfactory.
The Poisson's ratio of the meta-tetrachiral loaded along the Y-direction is slightly smaller than those of loaded along the X-direction, which all exhibit stable negative Poisson's ratio behavior until an axial strain of 0.15 is applied.
Compared to the other two structures, the energy absorption ability of the meta-tetrachiral structure loaded along the Y direction is worthy of recognition.However, it should be noted that the meta-tetrachiral structure loaded in the X-direction has released some energy during compression due to the fracture phenomenon.The structure can perform better in terms of energy absorption if the material properties are improved.
As the wall thickness of the structure increases, the modulus of elasticity of the material gradually increases, and the energy absorption capacity of the material gradually increases, but at the same time, the toughness of the material will also be tested.

Figure 3 .
Figure 3. (a) CAD dimensional model of tensile specimens, (b) True stress-strain curve of parent material under quasi-static tensile test.

Figure 4 .
Figure 4. (a) Numerical simulation modeling of meta-tetrachiral structure compressed along the Y-direction, (b) Compression at a constant velocity of 100 mm s −1 , (c) Comparison of simulated and experimental stress-strain curves for different loading velocities, (d) Effect of finite element models with different cell sizes on mesh convergence.

Figure 5 .
Figure 5.Comparison of experimental and finite element simulation of deformation process in compression of 6 mm, 12 mm, 18 mm.(a) Meta-tetrachiral in the X-direction, (b) Tetrachiral.

Figure 6 .
Figure 6.Experimental and numerical analysis stress-strain curves for materials with similar deformation patterns.(a) Metatetrachiral in the X-direction, (b) Tetrachiral.

Figure 7 .
Figure 7.Comparison of experimental and finite element simulation of deformation process in compression of 6 mm, 12 mm, 18 mm.(a) Meta-tetrachiral in the Y-direction, (b) Anti-tetrachiral.

Figure 8 .
Figure 8. Experimental and numerical analysis stress-strain curves for materials with similar deformation patterns.(a) Metatetrachiral in the Y-direction, (b) Anti-tetrachiral.

Figure 9 .
Figure 9. Unit deformation mechanism of structures with similar deformation patterns.(a) Meta-tetrachiral in the X-direction and Tetrachiral, (b) Meta-tetrachiral in the Y-direction and Anti-tetrachiral.

Figure 11 .
Figure 11.Energy absorption capacity of each structure before densification.(a) energy absorption per unit volume (Wv), (b) energy absorption per unit mass (SEA).

Figure 12 .
Figure 12.(a) Experimental stress-strain curves of meta-tetrachiral structures with different wall thicknesses, (b) Energy absorption capacity per unit mass of meta-tetrachiral structures with different wall thicknesses.

Table 1 .
Three types of lattice cell size values and overall size values.

Table 2 .
Elasticity and plasticity parameters of base material under quasi-static tensile test.Young's modulus/MPa 1540

Table 3 .
Young's modulus of different specimens under simulated and experimental conditions.