The novel tubular metamaterials were fabricated using selective laser sintering (SLS) with an S-TPU 500 3D printer (Suzhou, China). Prior to fabrication, a computer-aided design (CAD) model was created using SolidWorks software (v2023, SolidWorks Inc., Massachusetts, USA). The resulting geometry was exported in STEP format and subsequently imported into the printer’s control system for processing. The specimens were printed with a nozzle tip diameter of 0.2 mm, a print speed of 30 mm/s and a layer height of 0.15 mm. Thermoplastic polyurethane (TPU) with a Shore hardness of 95 A, supplied by SUNLU (Zhuhai, Guangdong Province, China), was used as the base material. The geometric parameters of the fabricated 3D-printed structures are summarized in Table 1.

The primary material properties were determined through uniaxial tensile tests. A standard dumbbell-shaped TPU specimen was modeled in accordance with the ASTM D638 international standard and fabricates as described in the previous section. The design and dimensions of the dumbbell specimen are shown in Figure 2A. As illustrated in Figure 2B, uniaxial tensile tests were performed on three specimens using an MTS universal testing machine (Model: MTS Criterion 43.104) equipped with a 10 kN load cell, which had a maximum measurement deviation of less than 1% of the indicated force. The tests were conducted at a constant crosshead speed of 10.0 mm/min, while load and displacement data were continuously recorded. The samples were held using self-locking wedge clamps, which ensured reliable fixation through the wedging action induced by the applied tensile load. The nominal stress-strain curves were derived from the measured load-displacement data and the initial specimen geometry. The nominal stress-strain curves were derived from the measured load-displacement data and the initial specimen geometry. A hyperelastic material constitutive model was then fitted to the experimental results. Several hyperelastic models–namely, Mooney-Rivlin, Polynomial, Neo-Hookean, Arruda-Boyce, and Yeoh–were compared with the experimental data. Among these, the Polynomial model (N = 2) provided the best fit to the experimental stress-strain curve, as shown in Figure 2C. Its strain energy density function is presented in Equation 1: W = C 10 I ¯ 1 − 3 + C 01 I ¯ 2 − 3 + C 20 I ¯ 1 − 3 2 + C 11 I ¯ 1 − 3 I ¯ 2 − 3 + C 02 I ¯ 2 − 3 2 + 1 D 1 J − 1 2 + 1 D 2 J − 1 4 (1) where I ¯ 1 and I ¯ 2 are the strain invariants, J determinant of the deformation gradient, and C 10 , C 01 , C 11 , C 20 , C 02 , D 1 , and D 2 are the hyperelastic constants. The identified material parameters from the Polynomial (N = 2) model were subsequently implemented in Abaqus (v2021, Dassault Systèmes SIMULIA, USA) to define the TPU material behavior. The resulting parameters are summarized in Table 2.

The quasi-static mechanical behavior of the proposed structure was evaluated under compressive loading. During compression, it was observed that the exposed structural walls could come into direct contact with the pressure plates, and their bending deformation might induce premature structural failure. To mitigate this effect, 1.0 mm thick plates made of the same material were attached to the top and bottom surfaces of the specimen to ensure uniform load distribution and reduce stress concentrations during testing, as illustrated in Figure 3A. The printed specimens had an out-of-plane thickness of 10.0 mm, which was sufficient to prevent out-of-plane instability under the applied compressive loads.

Compressive loads were applied to the specimens at a constant rate of 2.0 mm/min, and the corresponding load-displacement data were recorded. In the test setup, two flat platens were positioned above and below the specimens, with the specimens centered on the platform. The force and displacement sensors were connected to the upper and lower platens to collect the reaction forces and displacement data from the experiment during the compression process. The tests were conducted using an MTS universal testing machine equipped with a 10.0 kN load cell and a maximum force measurement error of less than ±1%. Each specimen was compressed to a displacement of 52.0 mm, corresponding to an engineering strain of 0.5. A high-resolution camera was positioned in front of the experimental setup to capture the deformation of the structures throughout the testing process. The recorded images were subsequently analyzed to derive the compressive stress-strain curves.

Details regarding the calculation of stress-strain response, energy absorption, and specific energy absorption are provided in the following section. The quasi-static uniaxial compression test setup is shown in Figure 3B.

The mechanical performance of the tubular metamaterials was characterized using several key parameters, among which the effective Poisson’s ratio is particularly important, as it quantifies the material’s transverse strain response under axial loading. To evaluate variations in the global Poisson’s ratio of the structures, eleven markers were strategically placed on each experimental specimen and its corresponding FE model, as illustrated in Figure 4. Given the non-uniform distribution of orthogonal deformation across the specimen surface, the local Poisson’s ratio at each point was calculated, as shown in Equation 2: ν i = − Δ x i R Δ y 0 H   1 ≤ i ≤ 10 (2)

The average Poisson’s ratio was determined by calculating the mean value of the effective Poisson’s ratio obtained from measurements at these characteristic points is expressed in Equation 3: ν ¯ = 1 10 ∑ 1 10 ν i 1 ≤ i ≤ 10 (3) where Δ x i denotes the deformation of point N i in the diametral direction, and Δ y 0 represents the displacement of point N 0 along the height direction.

The tunable stiffness introduced in this study was characterized by its dependence on deformation and was quantitatively evaluated using the effective Young’s modulus. The effective Young’s modulus ( E ) was defined as the ratio of the nominal stress applied to the structure to the corresponding nominal axial strain of the tube, measured within the initial linear portion of the stress-strain curve. This parameter was calculated using the following Equation 4: E = F A Δ h H (4) where F denotes the reaction force exerted by the structure under compressive displacement along the y-axis, A represents the cross-sectional area of the tube, ​ Δ h refers to the displacement of the upper boundary of the structure in the y-direction, and H is the initial height of the tube.

During the initial loading phase, the tubular metamaterial exhibited elastic behavior characterized by a linear relationship between the applied stress and strain. The slope of the stress-strain curve in this region corresponded to the elastic modulus, a fundamental parameter that reflects the stiffness of the tubular metamaterial.

Once the yield limit was reached, localized plastic deformation initiated within the metamaterial structure. This marked the onset of the stress plateau region, characterized by a stress level that remains approximately constant or increases gradually with further strain. The average stress within the plateau region, denoted as σ p l , was calculated as shown in Equation 5: σ p l = ∫ ε y ε s d σ ε d ε ε s d − ε y (5) where ε y and ε s d represent the yield strain and strain at densification, respectively.

The plateau region plays a critical role in energy absorption, as it allows substantial deformation to occur at an approximately constant stress level. This stage represents the primary phase of energy dissipation within the metamaterial. The effective compression stroke for energy absorption extends from the onset of external loading to the point of full densification. Beyond this densification threshold, additional strain leads to intense compaction of the structural elements, resulting in a sharp rise in stress. In this densification regime, the stress-strain curve exhibits a pronounced steepening, reflecting the transition of the metamaterial toward behavior characteristic of a solid body. This stage typically corresponds to the maximum energy absorption capacity of the structure.

The specific energy absorption (SEA) capacity is one of the most important criteria for evaluating the performance of metamaterials. The energy absorption (EA) was calculated as the area under the stress-strain curve up to a specified strain, as expressed by the following Equation 6: E A = ∫ 0 ε s d σ ε d ε (6) where ε s d is the strain at densification, and σ ε corresponds to the stress associated with a given strain.

The specific energy absorption (SEA) was defined as the absorbed energy per unit mass and is given by Equation 7: S E A = E A ρ m ρ v (7) where ρ m denotes the density of the base material, and ρ v represents the relative density of the undeformed structure.

The energy absorption efficiency (EAE) was evaluated based on the extent to which the energy absorption potential is utilized at different stages of deformation, and is defined as shown in Equation 8: η ε = ∫ 0 ε σ ε d ε σ ε (8)

The maximum efficiency value was achieved under the condition from Equation 9: d η ε d ε ε = ε s d = 0 (9)

Accordingly, this analysis enabled a comprehensive characterization of the mechanical behavior of the tubular metamaterials under loading, encompassing both elastic and plastic deformation stages as well as the overall energy absorption processes.

One of the key geometric parameters influencing the performance of the tubular metamaterial under compressive and tensile loading is the gap between the horizontal struts of the concave unit cells, denoted as g . To generalize the parametric analysis and facilitate comparison across different scales, a dimensionless parameter g ¯ ​is employed. To investigate the effect of this parameter under in-plane compression and tension, six models with different gap values ( g = 0.05 mm, 0.1 mm, 0.2 mm, 0.3 mm, 0.4 mm and 0.5 mm) were designed and analyzed using the validated FE model, as shown in Figure 9A.

Analysis of the effective Poisson’s ratio-strain curves indicates that variations in the parameter g ¯ allow the Poisson’s ratio to be effectively tuned. Under uniaxial tensile loading, the Poisson’s ratio remains nearly constant across all configurations, demonstrating stable deformation behavior regardless of the gap size. Furthermore, in all examined cases, the NPR effect was consistently observed throughout the entire tensile loading process, as shown in Figure 9B.

In contrast, under compression, the effective Poisson’s ratio exhibited a pronounced dependence on the parameter g ¯ . Simulation results revealed that the sign-switchable Poisson’s ratio observed during compression arises from the onset of self-contact between the triangular struts. Increasing the gap g ¯ enlarges the initial spacing between the horizontal struts, requiring additional deformation to close the gap during the early loading phase. This delays the activation of the concave cells and, consequently, postpones the occurrence of the sign change in the Poisson’s ratio. Notably, for gap values of g ¯ = 0.05 and g ¯ = 0.1 , the Poisson’s ratio transitions almost instantaneously under both tension and compression, indicating high sensitivity and functional responsiveness of the design. The sign-switching behavior results from a two-stage deformation process. Consequently, the concave cells undergo elastic bending in the first stage, thereby preserving their auxetic response. Following the occurrence of self-contact, the structure exhibits resistance to further transverse contraction, thereby initiating a transition to conventional deformation.

Overall, it was showed that increasing the gap g ¯ enhances the NPR effect under compression, whereas smaller gap values promote an earlier onset of the sign-switchable Poisson’s ratio. These findings demonstrate that variation in g ¯ has little influence on the tensile stiffness but enables precise tuning of the compressive stiffness by adjusting this geometric parameter.

The stress-strain curves obtained from the FE simulations are presented in Figure 10. The results show that the effective elastic modulus is strongly influenced by the gap parameter g ¯ , as shown in Figure 10A. As illustrated in Figure 11, each curve displays three distinct deformation regions: (1) an initial linear elastic region, (2) a plateau region with nearly constant stress, and (3) a densification region characterized by a sharp increase in stress beyond a critical strain. The compressive behavior of the metamaterial can therefore be tailored by varying g ¯ . An increase in g ¯ results in a non-monotonic trend in stiffness, as indicated by variations in E . The maximum stiffness is observed at an intermediate gap of g ¯ = 0.2, however, further increases in g ¯ reduce the structure’s resistance to elastic deformation. A partial recovery of stiffness is observed at g ¯ = 0.4. In addition, the yield strength can be independently adjusted by varying g , allowing the onset of irreversible deformation to be delayed and the structural integrity to be maintained over a wide range of loading conditions. The analysis indicates that the yield strength increases with increasing g ¯ , reaching a maximum at g ¯ = 0.4, beyond which it gradually decreases. The variations in stiffness and yield strength for structures with different g ¯ values are summarized in Figures 12A,B.

The plateau region plays a critical role in determining the energy absorption capacity of the metamaterial. The combination of a high stress plateau and a large densification strain provide a wide range of uniform compressive response before the structure transitions into the densified state. As the densification strain increases, a corresponding rise in energy absorption is observed. The gap parameter g ¯ governs the activation sequence of the deformation mechanisms. As the gap increases, the onset of self-contact between the triangular struts of the concave cells is delayed, thereby maintaining their auxetic response up to larger strains. Once contact occurs, energy absorption is enhanced through stress concentrations at the contact zones and subsequent load transfer to the convex cells, which leads to a stable plateau stress under continued deformation. The results indicate that structures with larger gaps are capable of sustaining greater deformations before entering the densification phase. The EA and SEA—key parameters for evaluating materials used in protective and damping systems—are positively tunable with increasing gap size g ¯ , as shown in Figure 12C. The maximum EA and SEA values were obtained at g ¯ = 0.4 , corresponding to the highest densification strain, while the minimum values were recorded at g ¯ = 0.05 .

In contrast, the tensile behavior response of the structure exhibits fundamentally different characteristics. As shown in Figure 10B, all configurations demonstrate a monotonically increasing stress-strain relationship without a distinct plateau region. The curves for all gap sizes are nearly identical, indicating that the geometric gap parameter has minimal influence on the tensile mechanical response. The lack of significant variation in tensile behavior across different gap configurations is attributed to the absence of self-contact mechanisms during tensile loading, which prevents the sign-switching phenomenon observed under compression.