Developing chiral cuboids with a wholly auxetic response is essential for several reasons. These 3D auxetic structures can achieve an NPR under both tension and compression. Thus, they are capable of responding effectively to both tensile and compressive loading conditions. They expand in both directions when stretched and uniformly densify under compression. This dual-response capability holds the potential for designing structures that can withstand a broader range of mechanical stresses. Moreover, such coupling chiral cuboids could theoretically demonstrate improved energy absorption and dissipation properties compared to materials exhibiting only one-sided auxetic behavior. These characteristics can pave the way for innovative solutions in applications where impact resistance and shock absorption are crucial, such as protective gear and structural reinforcements, or in scenarios where a more natural stress response is desired, like in biomedical implants. Figure 1 illustrates the structural design of the proposed coupling chiral cuboids from units to mechanical metamaterials. Figure 1A presents a coupling chiral unit composed of a hexagonal cell and 4 Z-shaped ligaments. The hexagonal cell is obtained by cutting the square with the diagonal length of L₁, and the length of cut portions is equal to the thickness of chiral unit γ. The effective length and thickness of Z-shaped ligaments are g and t. The chiral plates are formed by periodically expanding the chiral units that are connected by Z-shaped ligaments. Figure 1A demonstrates a 5 × 5 chiral plate. The chiral plates are verified with the wholly auxetic response by numerical simulations (Note S1). In order to constitute the chiral cuboids, the new chiral unit is designed by assembling 2 orthogonal units of chiral plates (Fig. 1B). The number of units of chiral cuboids in the x, y, and z directions are N_x, N_y, and N_z, and the length (x direction), width (y direction), and height (z direction) are L, W, and H, respectively. A 5 × 5 × 5 coupling chiral cuboid is displayed in Fig. 1B.

Figure 2 presents the structural deformations of Z-shaped ligament, coupling chiral unit, and assembled metamaterial structure under axial tension and compression. According to the analysis, the wholly auxetic response of chiral cuboids mainly stems from the bending of area 2 of Z-shaped ligaments. In particular, the bending will result in the elongation (shortening) of Z-shaped ligaments under axial tension (compression) (Fig. 2A), which leads to the lateral expansion (contraction) of chiral units and overall structure (Fig. 1B and C). Theoretical modeling of elastic modulus and Poisson's ratio of chiral cuboids are conducted to quantitatively characterize the mechanical response in Methods. In summary, to address the challenges of auxetic materials and structures in achieving 3D systems and a wholly auxetic response, we propose 2 innovative structural design strategies. First, the ligaments are designed in a Z shape, enabling the assembled metamaterial structures to achieve a wholly auxetic response through their properties of elongation and shortening under tension and compression, respectively. Second, the 3D units of coupling chiral cuboids are constructed by assembling 2 orthogonal 2D units of chiral plates, thereby achieving a breakthrough from 2D systems (chiral plates) to 3D systems (coupling chiral cuboids).

Experiments and numerical simulations are conducted to validate the wholly auxetic response of coupling chiral cuboids. The results are compared with the theoretical models. A parametric analysis is subsequently performed. Figure 3A presents the fabrication principle of the coupling chiral cuboid samples using selective laser sintering (SLS) technology. Soft nylon is used for printing the samples. The Young's modulus (E) of soft nylon material is determined as 33.46 MPa by the tensile tests of soft nylon dumbbell samples (Fig. 3B). The structural parameters of soft nylon dumbbell samples and experimental setup of tensile tests are summarized in Note S2. The material Poisson's ratio is taken as 0.4 (i.e., the Poisson's ratio of nylon), and the material density is measured as 1.11 g/cm³. Figure 3C demonstrates the fabricated chiral cuboid samples with the structural parameters of γ = 2mm, L₁ = 10mm, g = 4mm, t = 1mm, N_x = 5, N_y = 5, N_z = 5, L = 45.3mm, W = 45.3mm, and H = 62.6mm. Figure 3D displays the experimental setup of axially tensile and compressive tests of chiral cuboid samples. The square ends are bonded to the 3D-printed polylactic acid (PLA) rigid fixtures using the instant-drying adhesive to ensure the uniform distribution and stability of applied axial displacement. The instruments and operating procedures for preparation and mechanical tests are detailed in Methods.

The tensile and compressive response of coupling chiral cuboids with different t are numerically simulated in Abaqus/CAE using the static/general solving algorithm. The structural and material parameters used in numerical simulations are summarized in Note S6. It is worthwhile pointing out that t is the uniquely tunable structural parameter when the overall dimensions of chiral cuboids (i.e., L, W, and H) are fixed. Figure 3E presents the meshing, boundary, and loading conditions. In numerical simulations, the bottom parts are clamped, and the upper parts are subjected to an axial displacement. In particular, the displacement is set as 9.39 mm (i.e., axial strain is 15%) under axial tension and 6.26 mm (i.e., axial strain is 10%) under axial compression. For the chiral cuboids with different t, the models are meshed into 193,437 (t = 0.5mm), 226,225 (t = 1mm), 214,198 (t = 1.5mm), 247,896 (t = 2mm), and 265,492 elements (t = 2.5mm) with Tet shape and Free technique, respectively.

Figure 4 compares the results of experiments, theoretical analysis, and numerical simulations. Figure 4A presents the deformed configurations of the fabricated coupling chiral cuboid with the structural parameters of γ = 2mm, L₁ =10mm, g = 4mm, t = 1mm, N_x = 5, N_y = 5, and N_z = 5 under axial compression and tension in experiments and numerical simulations. The chiral cuboid exhibits a fully auxetic response, meaning it expands laterally under axial tension and contracts laterally under axial compression, as observed in both experiments and numerical simulations. In order to more prominently observe the wholly auxetic response of chiral cuboid, the deformation process of chiral cuboid under axial tension and compression in numerical simulations is displayed, and the deformation scale factor is set as 2 in the lateral deformation directions (Note S3).

Figure 4B and C compares the axial force–axial displacement relationship between experiments, theoretical analysis, and numerical simulations under axial tension and compression. Figure 4D and F uses radar charts to compare the relationships between lateral expansion/axial displacement and lateral contraction/axial displacement from theoretical analysis and numerical simulations under axial tension and compression. Based on these results (Fig. 4D and F), comparisons of Poisson's ratio as it varies with axial displacement are presented for both theoretical analysis and numerical simulations under axial tension and compression (Fig. 4E and G). It is important to note that the lateral deformation of chiral cuboids under axial loads is not uniform along the axial direction, leading to different Poisson's ratios at different axial positions. Therefore, the Poisson's ratio reported in this study is an equivalent Poisson's ratio determined from the maximum lateral deformation. Detailed explanations of the calculation and measurement methods for this equivalent Poisson's ratio can be found in Note S4. Overall, the theoretical analysis, experiments, and numerical simulations demonstrate acceptable agreement in their results. The probable sources of error include the following: In experiments, the material (soft nylon) shows nonlinearity with increasing loading displacement. However, both theoretical analysis and numerical simulations assume linear elasticity for the material (Fig. 4B). In numerical simulations, the clamped ends restrict lateral deformation as loading displacement increases, a factor not accounted for in the theoretical analysis (Fig. 4D).

The coupling chiral cuboids are validated with the structure-induced Poisson's ratio sign-switching via numerical simulations. The chiral units and overall structures of chiral cuboids with different Z-shaped ligaments t are displayed in Note S6. Figure 5A and B compares the deformed configurations of chiral cuboids with different t under axial tension and compression, respectively. In order to more prominently demonstrate the Poisson's ratio characteristics of chiral cuboids, the deformation scale factor is set as 2 in the lateral deformation directions. The Poisson's ratio of the chiral cuboids exhibits a sign-switch from negative to positive as t increases. Figure 5C to H quantitatively investigates the influences of t on the elastic modulus and Poisson's ratio of chiral cuboids. Figure 5C and F presents the axial force–axial displacement and lateral expansion–axial displacement relationships of chiral cuboids with different t under axial tension, while Fig. 5D and G demonstrates the axial force–axial displacement and lateral contraction–axial displacement relationships of chiral cuboids with different t under axial compression. The elastic modulus and Poisson's ratio of chiral cuboids with different t are calculated by the results of time step 2 in Fig. 5C, D, F, and G, as shown in Fig. 5E and H. The results indicate that the elastic modulus and Poisson's ratio of chiral cuboids under axial tension and compression are positively correlated with t. In addition, the tensile and compressive elastic modulus of chiral cuboids are practically equal, while the tensile Poisson's ratio of chiral cuboids is larger than the compressive Poisson's ratio. The sign-switching of Poisson's ratio in chiral cuboids is attributed to the varying ratio of thickness to width (t/w) in Z-shaped ligaments. Specifically, as t and t/w increase, the bending deformation within area 2 of the Z-shaped ligaments diminishes, while shear deformation becomes increasingly prevalent. Consequently, the chiral units lose their characteristic expansion under axial tension and contraction under axial compression, causing the Poisson's ratio of the chiral cuboids to transition from negative to positive.

The proposed coupling chiral cuboids can serve as the inner cores of impact-resistant sandwich panels to develop the overlays for the ocean engineering equipment (Fig. 6A). The ball rebounding tests are conducted to verify the impact-resistant performance of sandwich panels based on the chiral cuboids and deeply study the effect of Poisson's ratio on the impact energy absorption. Figure 6B presents the experimental setup of ball rebounding tests; in particular, the sandwich panel samples with different thickness of Z-shaped ligaments t (1, 2, and 2.5 mm) and solid sample with the identical dimensions are respectively placed at the bottom of 3D-printed PLA fixture, and a silicone ball with a diameter of 20 mm is dropped from the round hole at the top of fixture (the drop height is 150 mm). The sandwich panel and solid samples are fabricated by the SLS technology using soft nylon. Figure 6C and D compares the rebound height and energy absorption rate between sandwich panel and solid samples. It can be seen that the sandwich panel samples demonstrate a smaller rebound height and a larger energy absorption rate than the solid sample, which certifies that the sandwich panels with the inner cores of chiral cuboids possess superior impact-resistant performance than the solid materials. Additionally, the rebound height and energy absorption rate are positively and negatively correlated with t, respectively. The influences of effective Poisson's ratio on the average rebound height and energy absorption rate are obtained from the results of Fig. 6C and D (Fig. 6E). The effective Poisson's ratios of sandwich panel and solid samples used in the ball rebounding tests are summarized in Note S7. The results reveal that the effective Poisson's ratio proportionally affects the rebound height, while anti-proportionally influences the energy absorption. Moreover, compared with the solid sample, the sandwich panel sample can achieve a 49.3% growth in the energy absorption rate. It is worth pointing out that more durable and corrosion-resistant materials, such as aluminum alloys, titanium alloys, and fiber-reinforced composites, are necessary to fabricate sandwich panels based on chiral cuboids for engineering applications. The challenges posed by these materials may include (a) a lack of large-scale preparation technologies to manufacture the overlays as a single unit for the proposed sandwich panels and (b) preparing the local structures of sandwich panels separately and assembling them to form the overlays, which would significantly increase manufacturing time and cost.

This section theoretically characterizes the elastic modulus (E_z) and Poisson's ratio (_v) of coupling chiral cuboids. Assuming the chiral cuboids are subjected to an axial force and reach the equilibrium, the responsive force in area 1 of Z-shaped ligaments F₁ is:where N_x and N_y present the number of chiral units in the x and y directions, respectively. The free body diagram (FBD) that explains the relationship between F and F₁ given in Eq. 1 is drawn in Note S8. Note that F takes a positive number under axial tension, while F is negative under axial compression. Area 2 of Z-shaped ligaments can be considered as the beams clamped at both ends so that the bending-induced deflection of Z-shaped ligaments can be expressed as:where I=\frac{\gamma t^3}{12} denotes the cross-section moment of inertia of area 2 of Z-shaped ligaments. Integrating the deformation of Z-shaped ligaments, the lateral deformation in the x and y directions (D_x and D_y), and axial deformation (D_z) of chiral cuboids can be written as:

Considering that the displacement control method is chosen in experiments and numerical simulations, we assume the axial displacement D_z as the independent variable. Substituting Eq. 2 into Eq. 3, F can be obtained as:

Applying D_z to express D_x and D_y, we have

Consequently, E_z and ν in the x and y directions (ν_x and ν_y) can be calculated as:

Equation 6 reveals that the E_z and ν of chiral cuboids can be uniquely determined by the material and structural parameters. The developed theoretical models are solely applicable to the small-deformation conditions. In addition, the accuracy of theoretical models can be improved by reducing the thickness of Z-shaped ligaments t or enlarging the number of chiral units. The influence of t on the effectiveness of theoretical models is discussed in Note S9.

The coupling chiral cuboid and dumbbell samples in experiments are fabricated using soft nylon material. The samples were printed using an industrial-grade SLS printer (i.e., HT403P, FARSOON Technologies Co. Ltd., Changsha, China). The 3D printer uses a laser to fuse the soft nylon powder into the samples. In particular, the laser traces the pattern of each cross-section of sample models and replicates it onto the powder bed. Once one layer is built, the build chamber is lowered and construction of new layer begins on the top of previous layer. The PLA fixtures are prepared using fused deposition modeling (FDM) technology. A dual-extruder 3D printer (i.e., Rasie3D Pro2) is used for this purpose. The platform temperature, printing temperature, filling speed, and filling rate are separately set as 60 °C, 205 °C, 60 mm/s, and 50% during the printing. For the mechanical tests, a computerized electronic fatigue testing machine (manufactured by the Han Shen Automation Co. Ltd., Jinan, China) was used to apply the axial loads in the displacement control mode. The loading speed is set as 20 and 5 mm/min for the mechanical tests of soft nylon chiral cuboid and dumbbell samples, respectively.