Design, simulation, and 3D-printing of new auxetic metamaterials considering sensitivity analysis under impact loadings

Abstract

In this study, a new class of three-dimensional (3D) printed metamaterials with auxetic behavior was designed, fabricated, and experimentally analyzed under impact loading conditions. The metamaterials were created by combining multiple unit cells—honeycomb, tetra-chiral, and cubic—in different configurations to enhance their energy absorption capability. All samples were fabricated using the Fused Deposition Modeling (FDM) method with polylactic acid (PLA) material. Impact tests were conducted at three drop heights of 1, 3, and 5 cm to evaluate parameters such as maximum acceleration, absorbed energy, and specific energy absorption. The honeycomb–tetra-chiral–cubic configuration showed the best performance, achieving a specific energy absorption of 5.13 J/g, which was approximately 18% higher than the other designs. The maximum acceleration increased nonlinearly with drop height, indicating a strong dependence on impact energy. A sensitivity analysis performed using Design Expert software demonstrated that impact height was the most influential parameter, followed by Poisson’s ratio and unit-cell type. These findings confirm that combining auxetic geometries in hybrid lattice structures significantly improves energy absorption efficiency, making them promising candidates for use in protective equipment and lightweight structural applications.

Introduction

Materials with properties beyond those found in nature due to their unique geometry are referred to as metamaterials. Metamaterials are composed of repetitive arrangements of unit cells and can be classified into four main categories: electromagnetic (EM), thermal (TM), acoustic (AMM), and mechanical metamaterials (MM)¹. Mechanical metamaterials include auxetic², pentamode³, and those exhibiting negative compressibility and stiffness⁴. Various auxetic structures have been designed and fabricated, such as honeycomb, chiral, star-shaped, fractal, origami-kirigami, incomplete gear, and rotational structures, in both two- and three-dimensional forms. They are designed through mathematical methods⁵, topology optimization, or nature-inspired approaches⁶, and due to complex geometries, are often manufactured using additive manufacturing.

Mangli et al.⁷ investigated gradually stiff structures exhibiting a negative Poisson’s ratio (NPR), designed via topology optimization and fabricated by additive manufacturing. Kin et al.⁸ developed unit cells with NPR ranging from − 0.3 to −0.6, validated experimentally. Hamzei et al.⁹ studied octagonal 2D auxetic metamaterials under compression, fabricated additively, and calculated Poisson’s ratios.

Recent research on architected lattice structures has emphasized combined experimental and numerical investigations to optimize mechanical performance metrics such as specific energy absorption (SEA), plateau stress, and stiffness under quasi‑static and dynamic loading. Dara et al. conducted a systematic study of 2.5‑dimensional nature‑inspired infill structures, showing that novel snowflake‑inspired configurations fabricated via additive manufacturing significantly improve the mean crush force and SEA under out-of-plane quasi‑static compression compared to six conventional infill geometries, as validated through ANSYS simulations and experimental tests¹⁰. In a complementary effort, open lattice cellular designs with tailored crystalline arrangements demonstrated enhanced stiffness and greater energy absorption than traditional honeycomb structures, underscoring the influence of topology and relative density on mechanical response¹¹. Furthermore, the Different Conformal Lattice Structure (DCLS)—a tessellated combination of SC, BCC, and FCC lattices—was experimentally and numerically shown to achieve exceptionally high SEA under both quasi‑static and dynamic conditions, with simulation results closely matching Split Hopkinson Pressure Bar tests¹². Studies on open-flower-inspired lattices revealed that optimized topologies reduce stress localization, distribute load more uniformly, and elevate plateau stresses, thereby enhancing both static and dynamic compression behavior¹³. Additionally, investigations into interpenetrated tessellated cellular structures have highlighted their controlled energy transfer and improved load sharing, contributing to more efficient energy absorption during compression¹⁴. Collectively, these works demonstrate the critical role of geometric design, topology tessellation, and additive manufacturing techniques in advancing the mechanical efficiency of lattice metamaterials.

Zhao et al.¹⁵ studied 4D printed shape-memory polymer metamaterials, evaluating energy absorption under compression at different temperatures. Zheng et al.¹⁶ proposed a 3D auxetic metamaterial allowing tuning of Poisson’s ratio and heat transfer. Montazeri et al.¹⁷ examined asymmetric re-entrant auxetic cells versus other types, showing higher energy absorption and deformation stability. Nina et al.¹⁸ designed nature-inspired graded metamaterials (FGLBs) with high stiffness and energy absorption.

Recent research in hybrid lattice metamaterials has focused on combining different basic lattice types to achieve enhanced mechanical performance and energy absorption. For example, hybrid lattice designs with combined reentrant-surface lattice structures have demonstrated increased stiffness and specific energy absorption compared to single lattices, highlighting the potential of hybridization for tailored mechanical properties¹⁹. Similarly, voxel alternation and layered transition hybrid forms have shown improved compressive resistance and deformation behavior²⁰. Other recent studies explore combining auxetic and non-auxetic elements to tune Poisson’s ratio and dynamic response²¹. Moreover, multidisciplinary reviews on hybrid lattice strategies provide insights into design guidelines and multifunctionality enabled by additive manufacturing²². Zhou et al.²³ investigate the mechanical performance of functionally graded and auxetic lattice structures fabricated via additive manufacturing. Functionally graded rotating lattices demonstrate that spatial variation in geometry and relative density enables effective tuning of stiffness, strength, and energy absorption. In parallel, 3D re-entrant honeycomb auxetic structures exhibit negative Poisson’s ratio behavior, leading to unconventional deformation mechanisms and enhanced energy absorption. Both works highlight the strong influence of lattice topology and geometric parameters on mechanical response. Together, they establish graded and auxetic architectures as key design elements for hybrid lattice metamaterials with tailored mechanical properties²⁴. Masoumi et al.²⁵ investigated auxetic bones with different porosities via FDM, showing lighter structures with significant mechanical improvements.

Therefore, the novelty of this work lies in extending the previously proposed auxetic metamaterial structures from quasi-static compression to dynamic impact loading, where their mechanical response, deformation mechanisms, and energy absorption behavior are fundamentally different. For the first time, the impact performance of these structures is systematically investigated using a DOE-based approach, enabling quantitative assessment of the individual and interactive effects of key geometric and loading parameters. This study demonstrates that the optimal structure under static compression is not necessarily optimal under impact conditions and provides practical design guidelines for impact energy absorption applications.

Materials and methods

Mechanical properties of Poly lactic acid

PLA polymer was selected as the material for fabricating the metamaterial specimens. According to previous studies, the density and Poisson’s ratio of PLA were considered to be 1240 kg/m³ and 0.36, respectively²⁶.

Compression tests were conducted by the ASTM D695 standard at three different strain rates. In addition, although the standard ASTM D695 is primarily developed for rigid plastic materials, it has also been widely adopted by many researchers for evaluating the compressive properties of 3D-printed polymers, including PLA. The use of this standard in this study allows direct comparison of results with previous works employing the same methodology and provides a consistent baseline for mechanical evaluation²⁷.

The test specimens were fabricated using FDM with 100% infill, a layer height of 0.2 mm, and a nozzle temperature of 180 °C. Figure 1 shows the 3D printed specimens before compression testing.

Fig. 1.

The compression test samples²⁷.

Design of metamaterial

This section aims to design a metamaterial structure considering its energy absorption properties under both quasi-static and impact loading conditions. In this study, using previously developed unit cells optimized for energy absorption and mechanical behavior under compression and impact, the design process of a new structure has been carried out.

According to the literature, honeycomb²⁸, tetra chiral, and cubic cells^29,30 were selected as unit cells due to their superior mechanical performance in compression or impact tests. These cells and their geometrical dimensions are illustrated in Fig. 2.

Fig. 2.

Selected unit cells and their dimensions (mm): (a) cubic cell, (b) tetra chiral cell, and (c) honeycomb cell.

The wall thicknesses and unit cell dimensions were determined based on previously reported geometries in the literature and practical manufacturing constraints^28–30. Previously designed unit cells (honeycomb, tetrachiral, and cubic) were used as the foundation for developing the new structures.

In most previous works, additive manufacturing has been employed to fabricate these metamaterials, typically using FDM, Selective Laser Sintering (SLS), or Selective Laser Melting (SLM).

Considering the available equipment and the objective of this study, the FDM method was selected due to its accessibility and reliability for polymer-based structures. The nozzle diameter of the employed 3D printer is 0.2 mm, which restricts the minimum printable feature size and prevents the fabrication of excessively thin walls. Therefore, the selected wall thicknesses (0.49, 0.9, and 1.4 mm) were chosen within the printable range reported in the literature and consistent with the minimum feature sizes (> 0.2 mm) achievable by FDM.

After fabrication, the dimensions of all printed specimens were measured, showing a high level of accuracy (within ± 0.05 mm).

These considerations collectively justify the selected wall thicknesses and geometric differences between unit cells.

The standard specimen for the compression test (ASTM D695) can be a flat plate. Based on the selected unit cells, four different metamaterial structures were designed. These four structures include combinations of two or three of the selected unit cells. It is worth noting that the cells are arranged in a regular pattern within a rectangular cuboid volume. The occupied volume in all four structures is identical. The designs of these four structures are shown in Fig. 3, and their corresponding names are presented in Table 1. All unit cells were arranged within the same overall dimensions (constant length, width, and thickness), ensuring that the relative density remained identical among the four designs.

Fig. 3.

Standard compression test specimens consisting of combined unit cells (mm): (a) honeycomb–cubic, (b) honeycomb–tetra chiral, (c) cubic–tetra chiral, and (d) honeycomb–tetra chiral–cubic.

Table 1.

Names and descriptions of the designed metamaterial structures.

Quantity	Code	Metamaterial structure
3	HT	Honeycomb – Tetra chiral
3	HC	Honeycomb – Cubic
3	TC	Tetra chiral – Cubic
3	HTC	Honeycomb – Tetra chiral – Cubic

Although the cubic and tetra-chiral unit cells are conventionally classified as non-auxetic, several studies have shown that honeycomb structures can exhibit auxetic behavior. Moreover, according to other research³¹, combining non-auxetic and auxetic lattices can lead to an overall auxetic response due to the interaction between adjacent unit cells. Therefore, the selected combination of cubic, honeycomb, and tetra-chiral patterns was intended to investigate such potential synergistic effects on the mechanical performance.

Fabrication of designed metamaterials via 3D printing

As previously mentioned, the FDM process was employed to fabricate the designed metamaterial structures. The printing device used in this study was the Creality K1, manufactured by Sizan Company. This printer is capable of producing samples with maximum dimensions of 220 × 220 × 250 mm. The maximum print speed is 250 mm/s, and it offers a dimensional accuracy of 0.1 mm in both the longitudinal and transverse directions. The printer supports layer heights ranging from 0.1 mm to 0.35 mm, with a maximum nozzle temperature of 300 °C and a build plate temperature adjustable up to 100 °C.

Considering the capabilities of the selected printer, 3D printing parameters were chosen based on their influence on mechanical properties, as reported in the literature^26,27. These parameters are summarized in Table 2. Identical fabrication parameters were used for the impact test specimens as well. All 3D printing settings, including print pattern, temperature, speed, nozzle configurations, and others, were kept consistent across all specimens fabricated and tested in this study. Figure 4 shows the 3D printed specimens before experimental testing.

Table 2.

3D printing Parameters^26,27.

Material	Nozzle diameter [mm]	Layer height [mm]	Infill [%]	Nozzle Temp [°C]	Retraction	Print Speed [mm/s]
PLA	0.2	0.2	100	180	1.2	20

Fig. 4.

The 3D printed specimen before experimental testing.

Impact testing

To evaluate the energy absorption characteristics of materials, various loading conditions have been employed in the literature, such as impact, compression, tension, three-point bending, localized compression, torsion, shear, localized torsion, combined compression-shear, and combined compression-torsion. Accordingly, in the present study, impact loading was selected to investigate the energy absorption behavior.

Impact testing is a widely used method to assess the resistance of materials against sudden and transient loads. This test typically involves dropping a projectile from a specific height, where the kinetic energy of the projectile upon impact causes deformation, fracture, or damage in the test specimen. The primary objective of impact testing is to analyze the mechanical response of the material under dynamic loading conditions and determine key indicators such as absorbed energy, extent of deformation, and fracture mode (brittle or ductile).

The selection of the drop heights was based on the results of a previous quasi-static compression study²⁷. In that study, the absorbed energy (E) of each specimen was determined experimentally. Using the energy equivalence based on Eq. 1:

1

where m =7.5 kg is the mass of the impactor and g is the gravitational acceleration, the equivalent drop height was calculated for each specimen.

Since four different specimens were investigated, four corresponding drop heights were obtained. To ensure that all specimens were tested under identical impact conditions, the minimum calculated height among the specimens was selected as the reference drop height. Based on this value, drop heights of 1, 3, and 5 cm were chosen to represent increasing impact energy levels while remaining within the energy range determined from the quasi-static tests.

The standard employed for impact and energy absorption testing of polymers was ASTM D5420³². The acceleration measurements were obtained using a calibrated commercial accelerometer. Sensor calibration was performed according to the manufacturer’s specifications prior to testing. Moreover, the accelerometer has a stated accuracy of ± 2%, which defines the uncertainty range of the reported acceleration values.

Simulation

The impact simulations were conducted using the Dynamic/Explicit module of ABAQUS. The projectile was modeled as a rigid plate positioned above the specimen. A mass of 7.5 kg was applied at the center of the projectile, and surface contact between the projectile and the specimen was defined with a friction coefficient of 0.3. Its initial velocity was calculated based on a drop height of 1 cm using Eq. 2.

2

In this relation, v is the velocity in (m/s), g is the gravitational acceleration in (m/s²), and h is the drop height in (m). Based on Fig. 5, which illustrates the mesh convergence with respect to von Mises stress, a mesh size of 1 mm was selected.

Fig. 5.

Mesh convergence diagram for the HTC structure.

Poisson’s ratio was modeled as an independent parameter because it provided a direct measure of the lattice’s transverse contraction/expansion behavior, which could be compared across different hybrid geometries. While geometric descriptors can predict Poisson’s ratio for idealized unit cells, in practice, imperfections, layer stacking, and hybridization effects lead to deviations from ideal behavior. Modeling Poisson’s ratio as an independent parameter ensures a more accurate representation of the mechanical response in the simulations.

Sensitivity analysis

Following the design of four structures with different Poisson’s ratios and the execution of impact tests, it is necessary to examine the influence of variable parameters on the output results. Therefore, a sensitivity analysis was performed using Design Expert software. The effects of input factors on the output—specifically, the maximum acceleration recorded during impact testing—are presented in Tables 3 and 4. The primary objective of this analysis was to determine whether the cell type or the Poisson’s ratio of the structure had a significant effect on energy absorption, and if so, which factor had a greater impact.

Table 3.

Input Parameters.

Factor	Name	Units	Type	SubType	Minimum	Maximum	Mean	Std. Dev.
A	Poisson’s ratio	—	Numeric	Continuous	1.0000	1.70	1.36	0.2791
B	h	cm	Numeric	Continuous	1.0000	5.00	2.33	1.37
C	Cell	—	Categorical	Nominal	HT, HTC	—	—	—

Table 4.

Output Response.

Response	Name	Units	Observations	Minimum	Maximum	Mean	Std. Dev.	Ratio
R1	a	m/s²	18	128	285	173.39	45.54	2.23

Among the available models, the Cubic model was selected for the sensitivity analysis. The related data are presented in Table 5. According to the table, the standard error values for all input factors are relatively low, indicating the accuracy of the model.

Table 5.

Cubic model details for the second case Study.

Term	Standard error	VIF	Ri²	Power
A	0.7144	5.51852	0.8188	42.4%
B	0.7638	4.66667	0.7857	65.5%
C1	0.7071	—	—	7.6%
C2	ALIASED	—	—	—
C3	0.6124	—	—	—
AB	1.0100	5.98148	0.8328	14.4%
B²	0.8660	3.33333	0.7000	54.0%

The Variance Inflation Factor (VIF) indicates the degree of multicollinearity among independent variables. A VIF of 1 means no multicollinearity exists (ideal condition). If VIF is between 1 and 5, it indicates low to moderate multicollinearity, which is acceptable. A VIF between 5 and 10 suggests considerable multicollinearity, and values above 10 indicate severe multicollinearity, which could make the model’s results unreliable. According to the results, the VIF in the second case is around 4, indicating that the parameters do not have a significant mutual influence.

The R_i² value is above 0.8, which confirms the effectiveness of the input factors in the model. Additionally, the Power values show that the model successfully identifies the effect of each factor. Thus, the cubic model is deemed suitable for sensitivity analysis.

In this analysis, certain factors were aliased, meaning their individual effects could not be determined and were therefore disregarded.

Results and interpretation

Results of compression testing

The results of the compression test to determine the mechanical properties of PLA are illustrated in Fig. 6.

Fig. 6.

Stress–strain curve for standard specimens from compression testing.

Compression tests were carried out at strain rates of 1, 1.3, and 1.6 mm/min to evaluate the material behavior at closely spaced rates, similar to previous studies on 3D-printed PLA³³. The stress–strain curves showed that yield stress remained nearly constant across lower strain rates, with minimal differences in the initial peak.

Although both loading rates (1 and 1.6 mm/min) are relatively low, the observed trend was consistent across repeated tests, confirming that even small variations in strain rate can slightly influence the strain-hardening behavior of PLA, as reported in previous studies³³. Higher strain rates increased compressive strength at larger strains³⁴, consistent with other studies covering a wide range of rates^35–38.

PLA, a semi-crystalline, amorphous polymer, exhibits viscoelastic behavior, making its mechanical response rate-dependent. At higher strain rates, limited molecular rearrangement increases stiffness and resistance to deformation³⁹. For the standard rate of 1.3 mm/min, Young’s modulus was 3089 MPa, and the yield stress was 67.53 MPa, comparable with values reported in previous studies^27,40.

Results of impact testing

As previously described, the impact test was conducted on the fabricated samples at three different drop heights: 1 cm, 3 cm, and 5 cm. Each test was repeated twice per sample. At the 1 cm and 3 cm heights, all specimens remained intact. In a few cases, the outer frame of the samples experienced minor damage; however, the cellular structures remained structurally sound. In contrast, at the 5 cm drop height, only the HTC samples remained undamaged. All other samples experienced complete failure. As anticipated, the HTC structure exhibited superior energy absorption performance. No data are shown for HT, HC, and TC at 5 cm drop height due to premature failure of the samples. The acceleration-time plots obtained from these tests are shown in Fig. 7.

Fig. 7.

Acceleration-time plots at drop heights of (a) 1 cm, (b) 3 cm, and (c) 5 cm.

One of the key pieces of information extracted from the Fig. 7 is the peak acceleration value. According to Newton’s second law, F = ma, this corresponds to the maximum instantaneous force applied to the sample, where m is the mass of the pendulum and a is the peak acceleration measured. Table 6 presents the maximum acceleration values for each sample. As shown, the HC sample exhibited the highest peak acceleration.

Table 6.

Maximum acceleration values recorded during impact tests.

Sample	HT	HC	TC	HTC
Acceleration (m/s²) at 1 cm	128	159	139	133
Acceleration (m/s²) at 3 cm	199	280	171	161
Acceleration (m/s²) at 5 cm	–	–	–	181.65

To compute the absorbed energy, velocity–time curves were first obtained by numerically integrating the acceleration-time data. Displacement was then calculated by integrating the velocity–time curves. Force at any moment was derived using F = ma, as previously mentioned. Finally, the energy absorbed was estimated from the area under the force–displacement curve. These force–displacement graphs are shown in Fig. 8, and Fig. 9 illustrates the energy for different structures (HT, HC, TC, and HTC) at drop heights of 1 cm, 3 cm, and 5 cm, while Fig. 10 presents the variation of SEA with drop height. These plots clearly highlight the superior energy absorption performance of the HTC structure.

Fig. 8.

Force–displacement graphs for drop heights of (a) 1 cm, (b) 3 cm, and (c) 5 cm.

Fig. 9.

Comparison of absorbed energy among the four metamaterial designs (HT, HC, TC, and HTC) at drop heights of 1 cm, 3 cm, and 5 cm.

Fig. 10.

Comparison of the specific energy absorption (SEA) for HT, HC, TC, and HTC structures as a function of drop height.

As reported, the stiffness evolution of the auxetic structure under compression exhibits a nonlinear trend, attributed to progressive cell densification and geometric rearrangement during loading²⁷. The relative density of each lattice structure was calculated as the ratio of the lattice density to the density of the base material (PLA), based on the measured mass and the external volume of the specimens. The relative densities of the HT, HC, TC, and HTC structures are now explicitly reported in Table 7. As previously mentioned, Table 7 shows that the relative densities of all structures are approximately identical. To address this issue and ensure a fair comparison, the impact performance was primarily evaluated using the specific energy absorption (SEA), in which the absorbed energy was normalized by the specimen mass.

Table 7.

Relative densities of the HT, HC, TC, and HTC structures.

Structure	Mass (g)	Volume (mm³)	Relative density
HT	10	8495	0.94
HC	13	10,576	0.99
TC	13	10,523	0.99
HTC	12	10,456	0.93

To ensure a fair comparison, all structures were tested under identical impact conditions, including drop height, impact energy, and boundary constraints. Energy absorption (EA) was evaluated from the force–displacement response, while the specific energy absorption (SEA) was normalized by the specimen mass to eliminate the influence of density variations. This normalization enables a consistent benchmarking of impact performance across different lattice designs. The manuscript has been revised to clarify the normalization criteria and comparison methodology.

The nonlinear increase in absorbed energy with drop height reflects a transition in the dominant deformation mechanisms.

If the response were governed primarily by elastic deformation, a nearly linear relationship between absorbed energy and drop height would be expected, as the impact energy scales linearly with height. However, the observed nonlinear trend indicates the activation of additional energy dissipation mechanisms at higher drop heights.

Specifically, as the drop height increases, the response shifts from predominantly elastic behavior to plastic deformation and damage mechanisms, including material yielding, local crushing, and progressive failure. These nonlinear processes contribute to a disproportionate increase in absorbed energy, particularly at higher impact energies (e.g., HTC condition).

Therefore, the nonlinear trend observed in Fig. 10 is consistent with a damage-dominated response rather than a purely elastic one.

Based on the Fig. 10, the HTC structure demonstrated the highest specific energy absorption, confirming its superior capacity for energy absorption per unit mass. This enhancement is attributed to the increased number of cells, which improves the structure’s deformation and energy dissipation capability. Previous research⁴¹ has also emphasized that combining different geometries in a single structure can produce a synergistic effect. This phenomenon occurs when the combined performance of integrated components exceeds the sum of their individual performances.

Additionally, the Fig. 9 indicate that energy absorption increases with drop height. This can be explained by the fact that higher impact energy causes the PLA specimens to undergo more advanced failure modes, thereby absorbing more energy through increased mechanical damping mechanisms such as folding, local buckling, and irregular plastic deformation. The nonlinear increase in energy absorption with height reflects the complex mechanical behavior of the samples. A similar trend has been observed in honeycomb structures with geometrical gradients, where energy absorption increased nonlinearly with impact velocity or height, optimizing structural performance⁴².

According to the law of physics, E = mgh, the input potential energy increases linearly with height. However, the absorbed energy by the structures typically increases nonlinearly due to shifts in mechanical behavior (from elasticity to plasticity and eventual failure) as well as localized buckling or gradual cell wall collapse. According the Fig. 8, as the applied force increases and the structure enters the yield zone, partial cell wall failure and plastic deformation cause a drop in force and an increase in displacement without a corresponding force increase. Oscillations observed in the force-displacement curves are attributed to local failures and microcracks forming in the PLA material^43–45.

As previously noted, the velocity–time graph was derived by integrating the acceleration-time data. Given that the instantaneous velocity is known and the initial sample length in the strain direction is 70 mm, the strain rate during impact testing was calculated using Eq. 3:

3

Where ε˙ is the strain rate, is the instantaneous velocity, and ​ is the initial length of the sample. The calculated strain rates at the 1 cm drop height are presented in Table 8.

Table 8.

Strain rate during 1 cm impact tests.

Sample	HT	HC	TC	HTC
Strain Rate (s⁻¹)	3571	3928	3957	3228

Results of simulation

Referring to Fig. 11, the accuracy of the performed simulations was evaluated by comparing the acceleration–time curves obtained from the simulations with the experimental results. The comparison showed that the maximum acceleration for the HT specimen differed from the experimental value by only 2.66%, while the corresponding differences for the HC, TC, and HTC specimens were 1.32%, 11.85%, and 13.07%, respectively. Furthermore, the temporal trends of acceleration were predicted with acceptable accuracy for all specimens. These results indicate that the developed simulation model is capable of reliably representing the dynamic behavior of the lattice structures.

Fig. 11.

acceleration–time curves for (a) TC, (b) HT, (c) HC, and (d) HTC.

The comparison of simulation results indicates that the energy absorption percentages of the HT, HC, TC, and HTC lattice structures are 29.0%, 16.0%, 12.0%, and 78.41%, respectively. Figure 11 shows that the HC specimen absorbed only 16.0% of the incident kinetic energy in the low-velocity impact test. This low energy absorption indicates that the HC structure primarily transmitted the impact energy elastically, exhibiting limited capacity for energy dissipation. These results demonstrate that the HC specimen performs poorly under low-velocity impacts.

In contrast, the HTC structure, which combines the honeycomb, tetrahedral, and cubic architectures, absorbed 78.41% of the impact energy, showing significantly superior performance compared to the other specimens. This remarkable improvement in energy absorption can be attributed to the more uniform stress distribution and the synergistic enhancement of mechanical properties arising from the combination of the three architectures.

These differences highlight the critical influence of geometry type and structural combination on the mechanical performance of the specimens under impact loading. Overall, the results emphasize that careful hybridization of different lattice architectures can guide the design of optimized energy-absorbing structures for industrial applications, including automotive and aerospace sectors.

The simulation results of the low-velocity impact tests indicated that the displacement distribution patterns in the 3D-printed specimens were significantly influenced by the internal geometry and layer arrangement. The displacement contours, as shown in Fig. 12, revealed that in the HC specimens, the displacement was more uniform, with fewer regions of stress concentration. This behavior is attributed to the high capability of this structure to distribute impact forces over a larger area.

Fig. 12.

displacement contours for (a) TC, (b) HT, (c) HC, and (d) HTC.

In contrast, the combination of multiple architectures in the HTC specimens, despite exhibiting more effective energy absorption, resulted in localized displacement zones due to the geometric design, indicating that the energy absorption mechanisms were primarily concentrated in specific regions. Similarly, the HT specimens exhibited more linear displacement contours, reflecting their limited ability to distribute impact energy across the specimen.

The TC specimens showed a more complex displacement distribution, indicating the simultaneous activation of multiple energy absorption mechanisms. In these structures, the displacement was distributed more effectively throughout the lattice, minimizing regions of high deformation.

These analyses demonstrate that the design of internal lattice patterns plays a critical role in improving the energy absorption performance and displacement control of 3D-printed specimens. Moreover, employing hybrid architectures, such as HTC, can enhance the mechanical and dynamic complexity of printed structures.

Finally, the deformation mechanism observed in the proposed auxetic structure is consistent with previously reported behavior in 3D-printed auxetic metamaterials under compressive loading, where cell collapse and sliding play a dominant role in the NPR response²⁷.

A clear distinction can be observed between the deformation and failure mechanisms of the studied structures under quasi-static and impact loading conditions. Under quasi-static compression, as reported in previous study²⁷, the structural response was governed primarily by progressive unit cell collapse, cracking, and interlayer sliding, with failure mechanisms strongly influenced by test duration and boundary constraints. In particular, the HT structure exhibited extensive cell collapse and a high density of cracks, while TC and HTC samples showed partial preservation of cubic cells and localized collapse in tetrachiral and honeycomb regions.

Figure 12 illustrates the displacement contours of the TC, HT, HC, and HTC structures under vertical loading, revealing that all configurations exhibit an effective auxetic response through distinct deformation mechanisms.

As observed in Fig. 12(a), the TC structure undergoes a bending-dominated deformation, where the flexural response of the horizontal members induces rotational motion at the connecting nodes. This bending-dominated mechanism promotes a more gradual energy absorption process by distributing deformation over multiple members, thereby reducing localized stress concentrations during impact.

The HT structure shown in Fig. 12(b) exhibits a classical rotation-dominated auxetic mechanism. The inclined members experience pronounced hinge-like rotation, leading to a clear lateral expansion concurrent with axial loading. This rotation-dominated response enables rapid deformation under impact loading, influencing the rate at which impact energy is absorbed.

In the honeycomb-based HC structure (Fig. 12(c)), the collective deformation of the cell walls and the network-level rearrangement of the lattice contribute to an effective auxetic response. The combined bending and stretching of the honeycomb walls result in a relatively non-uniform energy absorption behavior, with deformation concentrated in specific regions of the lattice.

The hybrid HTC structure in Fig. 12(d) combines bending- and rotation-dominated mechanisms. The auxetic deformation of the upper re-entrant region promotes lateral expansion, which is transmitted to the lower honeycomb region, leading to a synergistic auxetic response across the entire structure. This multi-mechanism deformation pathway allows impact energy to be absorbed progressively across different structural regions, explaining the enhanced impact absorption performance of the HTC design beyond final energy values alone.

These results demonstrate that auxetic behavior in cellular structures may arise not only from idealized cell geometries but also from collective and bending-induced deformation mechanisms at the lattice level, which directly govern the impact energy absorption process.

These observations indicate that while static loading highlights ultimate collapse and damage accumulation mechanisms, impact loading emphasizes the role of dynamic deformation pathways, inertia effects, and strain-rate sensitivity in governing structural performance. Consequently, the relative effectiveness of the designs differs between static and impact conditions, demonstrating that superior quasi-static performance does not necessarily translate to enhanced impact resistance.

Poisson’s ratio is defined as the negative ratio of lateral strain to longitudinal strain. Based on this definition, the Poisson’s ratio for the designed metamaterial structures was calculated using Eq. 4:

4

In this Equation, and ​ represent the lateral and longitudinal strains, respectively. Strain is defined as the displacement divided by the initial length; therefore, displacements (δ) and initial lengths (L) were substituted into the formula. Since the structure experiences displacement on both sides in the lateral direction, the lateral displacement is multiplied by a factor of 2.

A visual representation of the displacement parameters used for the calculation is illustrated in Fig. 13, and the resulting Poisson’s ratios for the fabricated metamaterial samples are listed in Table 9.

Fig. 13.

Parameters used in Poisson’s ratio calculation.

Table 9.

Poisson’s ratio values of the fabricated metamaterial structures.

Structure	HT	HC	TC	HTC
Poisson’s Ratio	–1.5	–1.7	–1.4	–1.0

Among the structures, the HC sample exhibits the most negative Poisson’s ratio. Comparing the results with previously reported values in the literature^46–56, it is evident that the designed metamaterials in this study demonstrate more NPR than those previously introduced.

Poisson’s ratio was extracted from the finite element simulations by averaging axial and lateral strains over representative gauge lengths during the initial stage of the impact, before any significant local cell collapse or buckling occurred. The evaluation was thus performed at small strain levels (below approximately 1%), where the response could still be approximated as quasi-linear.

It should be noted that due to the transient and highly dynamic nature of impact loading, the calculated Poisson’s ratio represents an approximate instantaneous value rather than a constant material property, so this value corresponds to the early response phase of the impact and does not reflect the material behavior after progressive collapse mechanisms dominate.

Results of sensitivity analysis

Sensitivity analysis was performed on the results obtained from the impact tests. In this analysis, a Cubic model was employed. After ensuring the adequacy of the model, a sensitivity analysis was conducted. The results of this analysis for the maximum acceleration during the impact test are presented in Table 10. According to the table, the model is significant.

Table 10.

ANOVA results.

Source	Sum of Squares	df	Mean Square	F-value	p-value
Model	35217.78	8	4402.22	1148.41	< 0.0001
A - Poisson’s ratio	12545.33	1	12545.33	3272.70	< 0.0001
B - Height (h)	14628.57	1	14628.57	3816.15	< 0.0001
C - Cell	4671.41	2	2335.71	609.31	< 0.0001
AB (Interaction)	4641.33	1	4641.33	1210.78	< 0.0001
AC	0.0000	0
BC	1653.41	2	826.71	215.66	< 0.0001
A²	0.0000	0
B²	44.08	1	44.08	11.50	0.0080
Pure Error	34.50	9	3.83
Total	35252.28	17

The p-values for all three input factors (cell type, Poisson’s ratio, and impact height) are less than 0.05, indicating that all three parameters significantly influence the maximum acceleration. The F-values reveal the relative importance of these factors, showing that the impact height has the most significant effect, followed by the Poisson’s ratio, and then the cell type. These findings confirm that all three variables contribute meaningfully to the observed variation in acceleration, with height being the dominant factor in the model’s response behavior.

Conclusion

The present study introduces a new class of auxetic lattice metamaterials obtained by optimally combining two- and three-cell geometries, and demonstrates their mechanical performance under dynamic impact loading through systematic experiments and sensitivity analyses. The main innovations lie in the integrated design approach—merging distinct unit-cell topologies to tailor auxetic behavior and energy-absorption capacity—and in the combined experimental–analytical framework used to evaluate the effects of geometric parameters, Poisson’s ratio, and impact conditions on performance.

Based on the experimental and analytical results, the key findings can be summarized as follows:

• The hybrid HTC (honeycomb–tetra-chiral–cubic) structure exhibited the highest specific energy absorption among the investigated configurations. At the maximum impact height considered in this study, the SEA of the HTC structure reached approximately 18% higher than that of the TC, HT, and HC structures under identical test conditions.

• Across all configurations, increasing the impact height from the lowest to the highest level resulted in a nonlinear increase in absorbed energy, with the total absorbed energy rising by approximately 600%, indicating a response governed by plastic deformation and local buckling rather than elastic behavior.

• Parametric analysis showed that impact height had the dominant influence on the peak acceleration, leading to an increase of approximately 33%, while variations in unit-cell topology contributed changes of approximately 30%.

These findings highlight the practical significance of the designed metamaterials. By combining tailored cell geometries and controlled porosity, the proposed structures offer a lightweight yet efficient solution for energy absorption. Such properties make them strong candidates for applications in personal protective equipment, automotive crash components, and aerospace energy-absorbing systems, where minimizing weight while maximizing impact resistance is critical. Future work will focus on multiscale optimization, material diversification, and prototype integration to further enhance their functional performance in specific industrial applications.

The results underscore the potential of geometrically complex, multi-cell auxetic metamaterials in applications requiring high energy absorption and impact resistance.

Author contributions

A. S.: Conceptualization, Methodology, Data curation, Visualization, Investigation, Validation, Writing – original draft. R. H.: Supervision, Conceptualization, Methodology, Writing – review & editing. M. R.: Supervision, Conceptualization, Methodology, Writing – review & editing. All authors reviewed the manuscript.

Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

Declarations

Competing interests

The authors declare no competing interests.