Re-entrant Hexagonal Honeycomb Research Articles

Efficacy of auxetic lattice structured shoe sole in advancing footwear comfort-From the perspective of plantar pressure and contact area.

Designing footwear for comfort is vital for preventing foot injuries and promoting foot health. This study explores the impact of auxetic structured shoe soles on plantar biomechanics and comfort, motivated by the integration of 3D printing in footwear production and the superior mechanical properties of auxetic designs. The shoe sole designs proposed in this study are based on a three-dimensional re-entrant auxetic lattice structure, orthogonally composed of re-entrant hexagonal honeycombs with internal angles less than 90 degrees. Materials fabricated using this lattice structure exhibit the characteristic of a negative Poisson's ratio, displaying lateral expansion under tension and densification under compression. The study conducted a comparative experiment among three different lattice structured (auxetic 60°, auxetic 75° and non-auxetic 90°) thermoplastic polyurethane (TPU) shoe soles and conventional polyurethane (PU) shoe sole through pedobarographic measurements and comfort rating under walking and running conditions. The study obtained peak plantar pressures (PPPs) and contact area across seven plantar regions of each shoe sole and analyzed the correlation between these biomechanical parameters and subjective comfort. Compared to non-auxetic shoe soles, auxetic structured shoe soles reduced PPPs across various foot regions and increased contact area. The Auxetic 60°, which had the highest comfort ratings, significantly lowered peak pressures and increased contact area compared to PU shoe sole. Correlation analysis showed that peak pressures in specific foot regions (hallux, second metatarsal head, and hindfoot when walking; second metatarsal head, third to fifth metatarsal head, midfoot, and hindfoot when running) were related to comfort. Furthermore, the contact area in all foot regions was significantly associated with comfort, regardless of the motion states. The pressure-relief performance and conformability of the auxetic lattice structure in the shoe sole contribute to enhancing footwear comfort. The insights provided guide designers in developing footwear focused on foot health and comfort using auxetic structures.

Energy absorption characteristics and multi-objective optimization of 3D bionic negative Poisson’s ratio honeycomb

This paper introduces a new bionic cell element with a negative Poisson’s ratio, which is derived from the traditional re-entrant hexagonal cell element. By incorporating rotation into the two-dimensional (2D) cell element, a three-dimensional (3D) cell element is formed, which exhibits enhanced mechanical properties. The 3D honeycomb is obtained by arraying the 3D cell elements and is enhanced by adding longitudinal ribs to the honeycomb. The present study employs the ABAQUS/EXPLICIT software to investigate the correlation between various compression velocities and the crashworthiness and energy absorption capacity of a given system. The effects of different compression velocities on the 3D honeycomb deformation pattern and energy absorption performance are analyzed. The simulation results show that the new bionic negative Poisson’s ratio honeycomb has better crashworthiness and energy absorption capacity than the traditional re-entrant hexagonal honeycomb. The nominal stress-strain curves of different honeycombs and the energy absorbed per unit mass of honeycomb are compared and analyzed, and the NSGA-II genetic algorithm is used to perform a multi-objective optimization analysis of the peak stress and energy absorbed per unit mass of honeycomb. The results show that the optimized honeycomb can increase the absorbed energy per unit mass by 7.5 % when the peak stress is the same as the initial honeycomb, while the peak stress can be reduced by 9.6 % when the absorbed energy per unit mass is the same.

Effects of Process Parameters on the Microstructure and Properties of Selective Laser Melting 316L Negative Re-entrant Hexagonal Honeycomb Porous Bone Scaffolds

Effects of Process Parameters on the Microstructure and Properties of Selective Laser Melting 316L Negative Re-entrant Hexagonal Honeycomb Porous Bone Scaffolds

A novel metamaterial with instantaneously sign-switchable coefficient of thermal expansion and Poisson's ratio

Most materials expand when heated and contract when cooled. In contrast, negative thermal expansion materials exhibit the opposite behavior. In this paper, we present a metamaterial that expands both when heated and cooled. The unit cell configuration is crafted by embedding two isosceles triangular structures composed of high thermal expansion materials within a re-entrant hexagonal honeycomb framework. Initially, the bases of the two isosceles triangles are in contact. The metamaterial exhibits two deformation states under different external loads. Analytical formulas for the equivalent parameters were derived based on classical beam theory within the regime of small deformation. Finite element simulations were conducted to validate the accuracy of these analytical formulas. The influence of geometric parameters on the metamaterial is discussed in detail. The results demonstrate that the metamaterial can exhibit excellent multi-functional properties within a small range of deformation, including instantaneous sign-variable coefficients of thermal expansion and Poisson's ratios, along with significantly distinct tensile and compressive stiffnesses. This metamaterial consistently exhibits expansive characteristics regardless of variations in ambient temperature. Additionally, the material always expands laterally regardless of the sign of uniaxial loads. Based on these properties, the metamaterial can be applied to fasteners in the aerospace industry that require permanent locking, ensuring stability in various temperature and mechanical load conditions.

On in-plane crushing behavior of a combined re-entrant double-arrow honeycomb

A novel re-entrant double-arrow honeycomb (RDAH) has been proposed and investigated by combining re-entrant hexagonal honeycomb and double-arrow honeycomb. Quasi-static compression experiments of 3D printed samples in different loading directions were carried out. Deformation modes and stress–strain response have been explored and identified. The corresponding finite element (FE) models were established and verified, and the dynamic crushing behaviors of RDAH were further investigated by FE simulations. The results show that RDAH has three typical deformation modes under different impact velocities. When subjected to quasi-static loading and low-velocity impact along the horizontal wall, two stress plateau regions of the stress–strain curves can be observed, while there is only one stress plateau when loaded perpendicular to the horizontal wall. Based on deformation modes, theoretical models of plateau stresses of RDAH under low-velocity and high-velocity impact were further established. The theoretical results of the plateau stresses agree well with the FE simulation results. The effects of impact velocity and geometric parameters on densification strain, Poisson's ratio and energy absorption were also discussed. The results show that the RDAH exhibits obvious negative Poisson's ratio effect in both directions. The energy absorption of RDAH is higher when loaded along the horizontal wall and can be 29.45 % higher than the traditional RH structure. The energy absorption of RDAH increases with the decrease of inclined wall l1, the decrease of inclined angle θ (perpendicular the horizontal wall), and the increase of inclined angle θ (along the horizontal wall).

In‐Plane Crushing Behaviors of Hexagonal Honeycombs with Hollow‐Circle Joint under Different Compressive Velocities

The in‐plane crushing behaviors of hierarchical hexagonal honeycombs, structured by replacing every three‐wall vertex of both the conventional and reentrant hexagonal honeycombs with a small hollow‐circle, are investigated herein. The finite element (FE) models are first verified by an empirical formula from the literature and then further used to simulate the in‐plane crushing behaviors of the honeycombs under different compressive velocities. The deformation mode, plateau stress, and specific energy absorption (SEA) are studied based on the FE simulations. With respect to the conventional hierarchical honeycombs, the reentrant hierarchical honeycombs, in most instances, are found to exhibit higher plateau stress but lower SEA. It is remarkable that the hierarchical hexagonal honeycombs exhibit higher plateau stress and SEA than the basic hexagonal ones, which indicates that the hierarchical design is effective in improving both the impact resistance and energy absorption capabilities of the honeycombs.

Quasi-static and dynamic properties studies of a metamaterial with enhanced auxeticity and tunable stiffness

Auxetic metamaterials as a cutting-edge field have attracted extensive attention from researchers because of their peculiar deformation mode and excellent mechanical properties. However, the unstable deformation and low stiffness of auxetic structure limit their potential development. To address these limitations, a novel auxetic honeycomb (NAH) has been proposed by combining an original re-entrant hexagonal honeycomb (RH) with cross-chiral honeycomb (CH). The proposed structure has higher stiffness than RH, and more stability than CH. Additionally, the stiffness, densification point, and auxetic behavior can be adjusted by the angle of cell ribs (θ). In this study, the static mechanical properties and deformation behavior of NAH are investigated experimentally and numerically. Furthermore, the energy absorption and deformation behavior of NAH under different impact loadings (low-, medium- and high-velocity) are investigated numerically. Finally, a series of functionally graded NAHs with varying thicknesses are studied to improve the impact resistance of NAH. The results show that NAH has better specific energy absorption than RH and CH under static compression. Moreover, NAH with θ<90° exhibits greater energy absorption capacity compared to NAH with “θ > 90°” at high-velocity impacts.

Bending performance of 3D re-entrant and hexagonal metamaterials

Auxetic structures have theoretically superior bending resistance that was previously inaccessible. Compared with the planar two-dimensional (2D) auxetic structures, the three-dimensional (3D) counterparts may exhibit auxetic behavior in multiple directions. Recently, increasing studies have been devoted to studying the mechanical properties of 3D auxetic structures. In this paper, two 3D honeycomb structures were proposed, namely 3D re-entrant honeycomb and 3D hexagonal honeycomb. Subsequently, the mechanical properties, deformation modes, and deformation mechanisms of these two honeycomb structures under three-point bending were investigated experimentally and numerically. Compared with the 3D non-auxetic hexagonal honeycomb, the 3D auxetic re-entrant honeycomb exhibits higher ductility and fracture resistance. In addition, unlike the conventional honeycomb structure, the mid-span section of the 3D auxetic honeycomb exhibits a trapezoidal deformation mode under bending. Finally, the influence of the number of unit cells and geometric parameters on the mechanical properties of the 3D honeycomb structure were systematically studied by parametric analysis. The excellent bending performance provides a basis for the application of auxetic honeycomb in the fields of biomedicine, soft robots, and buffer devices.

Analytical relationships for 2D Re-entrant auxetic metamaterials: An application to 3D printing flexible implants

Both 2D and 3D re-entrant designs are among the well-known prevalent auxetic structures exhibiting negative Poisson's ratio. The present study introduces novel analytical relationships for 2D re-entrant hexagonal honeycombs for both negative and positive ranges of the cell interior angle θ (θ<0 showing a negative Poisson's ratio). The derived analytical solutions are validated against finite element method (FEM) and experimental results. The results show that, compared to the analytical solutions available in the literature, the analytical relationships presented in this study provide the most accurate results for elastic modulus, Poisson's ratio, and yield stress. The analytical/computational tools are then implemented for designing Kinesio taping (KT) structures applicable to treatment of Achilles tendon injuries. One of the main features of the Achilles tendon is a natural auxetic behavior. Poisson's ratio distribution of an Achilles tendon is obtained using longitudinal and transverse strains and are then used to design and 3D print thermoplastic polyurethane (TPU) KT structures with non-uniform distribution of auxetic unit cells. The presented novel KT shows that it is capable of replicating the deformation and global and local Poisson's ratio distributions, similar to those of the Achilles tendon. Due to the absence of similar formulations and procedures in the literature, the results are expected to be instrumental for designing and 3D printing of flexible implants with unusual auxeticity.

In-plane shear stiffness of diversified hexagonal honeycombs: A simple energy-based approach

The conventional hexagonal honeycomb is widely used in various practical applications. The re-entrant and semi re-entrant hexagonal forms derived from the conventional one have gained large attention in recent years due to their unusual Poisson's ratio and superior mechanical performance. However, the in-plane shear behavior of the semi re-entrant hexagonal honeycomb has not been reported yet. Especially, a comparative study on the shear stiffness of the three hexagonal honeycombs is lacking. A simple energy-based approach was used in this paper to establish theoretical models for predicting the in-plane shear modulus of the hexagonal honeycombs, which were later verified by the numerical simulations. Good agreement between the theoretical and numerical results is observed. The effect of the geometrical parameters on the shear modulus and the specific shear modulus were further discussed. It is found that the conventional hexagonal honeycomb displays higher shear modulus and specific shear modulus than both the re-entrant and semi re-entrant ones, while the re-entrant hexagonal honeycomb demonstrates the lowest values of shear modulus and specific shear modulus.

Anti-blast performance of 3D-printed sandwich panels with auxetic hexagonal and regular hexagonal honeycomb cores

3D-printed auxetic honeycomb sandwich panels (HSPs) have considerable potential for blast resistance enhancement. This study aims to bridge the gap with regard to the experimental data on the effect of 3D-printed auxetic hexagonal honeycomb cores on the blast response of HSPs. To this end, the blast resistance of HSPs developed using auxetic re-entrant and regular hexagonal honeycomb cores is investigated experimentally and numerically. First, six HSPs are tested under close-in blast loadings. The HSPs comprise Q345 steel top and bottom sheets as well as a honeycomb aluminum alloy core with two configurations (regular hexagonal and auxetic re-entrant). According to the results, the former configuration enhances the HSP blast resistance by not only decreasing the deformation of the HSPs but also improving their damage tolerance. In addition, the responses of the HSPs are numerically studied via finite element analysis. The numerical method using the finite element program LS-DYNA achieves reasonable accuracy. Finally, a parametric study is conducted to investigate the effects of the honeycomb type, cell angle, debonding effects, and core material on the blast resistance of the HSPs. The results demonstrate that employing auxetic hexagonal honeycomb core, smaller cell angle, and ductile core material can improve the blast resistance of HSPs.

A novel series of mechanical metamaterials with sign-changing coefficient of thermal expansion and their parameter analysis

Mechanical metamaterials show great application prospects in many fields such as aerospace, biomedical engineering, energy and transportation. The development of mechanical metamaterials with multiple peculiar properties is an important prerequisite for the development of intelligent sensors with excellent performance and the realization of multifunctional integration of structures. In this paper, a series of novel mechanical metamaterials are designed by adding rhomboid negative thermal expansion units into re-entrant hexagonal honeycombs, which have different signs of the coefficient of thermal expansion when the temperature rises or falls, and the analytical formula of their equivalent thermal expansion coefficient are derived. Parameter analysis and numerical simulations show that the metamaterial can realize the sign-changing properties of thermal expansion coefficient without entering the range of large geometric deformation under specific constituting materials and geometric parameters. In addition, this design idea is extended to three-dimensional form.

Energy absorption characteristics and stability of novel bionic negative Poisson’s ratio honeycomb under oblique compression

Due to the excellent energy absorption performance of re-entrant hexagonal honeycomb (RHH) with negative Poisson’s ratio (NPR) property, an improved Diabolo shaped honeycomb (DSH) based on RHH is proposed in this paper. With the application of bionic principle, three DSHs with adding ribs are further extended in design. A comparative analysis between RHH and DSHs under axial compression is performed by the finite element calculation software ABAQUS/ EXPLICIT, and subsequently the energy absorption performance and energy absorption stability of DSHs under oblique compression are investigated. The relationship between the cell wall thickness of the honeycomb with the best energy absorption effect in DSHs and the compression crashworthiness criteria is also predicted. The results show that the plateau stress and specific energy absorption of DSHs under axial compression are better than those of RHHs, and the energy absorption capacity of RHH is improved by a novel design. In addition, the energy absorption capacity and stability of the honeycomb under oblique compression are influenced by the number position of the ribs and the cell wall thickness. Therefore, the improved design of NPR metamaterials with oblique compression can provide a reference for the practical application of honeycombs with good energy absorption capacity and stability.

A simple 3D re-entrant auxetic metamaterial with enhanced energy absorption

As a branch of auxetics, the re-entrant hexagonal honeycomb has many superior mechanical performances. However, most research focused on two-dimensional (2D) re-entrant structures or three-dimensional (3D) structures without high compressibility. The energy absorption capacity of such structures is deficient for practical applications. Therefore, it is important to investigate the 3D re-entrant honeycomb structures which can sustain large deformation in order to make the most use of the materials. In this work, a simple 3D re-entrant unit cell is designed, manufactured and examined. The influence of geometric parameters on the deformation mode and energy absorption capacity is investigated numerically. The experimental results are in good agreement with the finite element prediction. The proposed 3D re-entrant auxetic metamaterial not only possesses a greater bearing capacity but also presents a stable compression deformation. The structure exhibits a desirable energy absorption behavior, including obvious auxetic behavior and a long stress plateau. By adjusting the geometrical parameters of the structure, the performances of energy absorption and compression stiffness can be improved. These findings provide a new idea to design 3D auxetic metamaterials and promote their utilization in protective structures.

Crushing behaviors of novel Diabolo shaped honeycombs with enhanced energy absorption performance

This paper presents the design of a novel Diabolo shaped honeycomb (DSH) based on re-entrant hexagonal honeycomb (RHH), and four enhanced honeycombs are designed based on the DSH by adding horizontal or longitudinal ribs. The finite element software ABAQUS/EXPLICIT is applied to investigate the relationship between crashworthiness and energy absorption performance with different impact velocities and geometric parameters. The simulation results show that DSH has better impact resistance and energy absorption than RHH. DSH has improved plateau stress by 141.3%, 78.7%, and 22.7% of low-velocity, medium-velocity, and high-velocity compression, respectively, and SEA by 103.7%, 79.3%, and 7.3%, respectively. In addition, the impact resistance of the four enhanced honeycombs is further improved. The shrinkage effect of DSH after impact is not as good as that of RHH, and the horizontal ribs suppress the negative Poisson's ratio property of the honeycomb. Different impact velocities affect the deformation pattern of the honeycomb. With the increase of impact velocities, the honeycomb has a transition from overall deformation to local deformation with increasing velocity. The plateau stress and specific energy absorption of the honeycomb are significantly influenced by the geometric parameters. This research provides a reference for the optimal design and application of honeycombs as energy absorbing components.

Energy Absorption Characteristics and Functional Gradient Optimization of a Hybrid Honeycomb

Metastructures are extensively used in aerospace engineering. In the present work, a high-strength structure was designed based on a reentrant hexagonal honeycomb (RHH) by topology optimization. To generate a negative Poisson’s ratio (NPR) effect, topologically optimized cells were alternately arranged with RHH cells to obtain a hybrid hexagonal honeycomb (HHH). In comparison with other NPR structures, this structure had superior energy absorption characteristics. In addition, the mechanical properties and deformation behavior of this structure were analyzed. Finally, single, double, and triple functional gradients were introduced into the HHH. It was found that the introduction of functional gradients improved the energy absorption capacity of the structure, and that the energy absorption capacity increased with the increase in the number of functional gradients.

In-plane elasticity of the re-entrant auxetic hexagonal honeycomb with hollow-circle joint

This work proposed an improved modification on the common re-entrant hexagonal honeycomb (RHH) by replacing the solid junction of the cell walls with a small hollow circle, and thus achieving a novel kind of honeycomb. A simple energy-based approach was adopted to develop theoretical models for predicting the in-plane equivalent elastic moduli of the new honeycomb. Finite element analysis was employed to verify the theoretical models and then the effects of the micro geometrical parameters on the in-plane elastic behaviors were further discussed. The results showed that the new honeycomb possesses an obvious negative Poisson's ratio effect, meanwhile, can achieve higher specific Young's modulus and shear modulus than the RHH. Moreover, the new honeycomb was found to exhibit a more tailored anisotropy than the RHH from the off-axial properties analysis. As a result, the new honeycomb may be an effective substitute for the RHH in application and the new layout can be a good guide for the improved design of auxetic honeycombs.

A lightweight rotationally arranged auxetic structure with excellent energy absorption performance

As one of the representatives of advanced lightweight cellular materials, the auxetic structure has a great potential for energy absorption due to its unique deformation mechanism, which makes a very promising application prospect in the fields of aerospace, vehicle and engineering protection. In order to further increase the energy absorption capacity of auxetic structures, this paper designed and characterized a novel two-dimensional (2D) lightweight rotationally arranged auxetic structure. The elastic properties, plastic collapse stress and specific energy absorption (SEA) of the novel auxetic structure were systematically investigated through theoretical calculations, numerical simulations and experimental method. Our results verify that the proposed theoretical model is accurate and applicable. The rotationally arranged shape and stable triangular design inside unit cells lead to an excellent energy absorption capacity, which is significantly higher than that of conventional auxetic structures (star-shaped honeycombs and re-entrant hexagonal honeycombs). Moreover, the SEA of novel structure has extremum conditions, by which the optimal relative density can be determined. According to different user requirements, the novel structure can be designed as the best energy absorption type or the most concise and lightweight type by adjusting geometrical parameters.

Large Torsion Deformation: Centrosymmetric Reentrant Honeycomb

There exist some problems in the crash box and anti-collision beam sandwich structure, such as monotone deformation pattern and uneconomical energy absorption performance. In order to raise the deformation capacity and energy absorption performance of sandwich structure, centrosymmetric reentrant honeycomb (CRH) and hexagonal centrosymmetric reentrant honeycomb (HCRH) are proposed based on auxetic reentrant honeycomb (ARH) in this work. Based on HCRH, four kinds of transverse combination structures and two kinds of longitudinal combination structures are obtained. The results of specific energy absorption show that the energy absorption capacity of the angular contact homodromous combination structure (ACOC) is about 3 times that of the other three transverse combination structures. Compared with longitudinal heterodromous combination structure (LHEC), the energy absorption capacity of longitudinal homodromous combination structure (LHOC) is improved by 72.7%.

Paper tube-guided blast response of sandwich panels with auxetic re-entrant and regular hexagonal honeycomb cores – An experimental study

Auxetic honeycomb structures have drawn increasing attention for impact/blast protection purposes owing to their superior indentation resistance and energy absorption capability. However, few comparative study on the blast performance among sandwich structures with different auxetic and non-auxetic cores has been performed to date. In this paper, the dynamic behaviors of metallic sandwich panels with auxetic re-entrant and regular hexagonal honeycomb cores (i.e. RHSPs and HHSPs) under the paper tube-guided explosions were investigated by a series of blast tests. The failure modes and underlying mechanisms of the target panels with different core relative densities under different impulse levels were classified and analyzed in detail. The panel blast resistance was evaluated by means of the back-face maximum permanent deflections, introducing the testing results of double arrowhead honeycomb sandwich panels (DAHSPs) in a separate paper. It indicates that the introduction of the paper tube achieved the concentration of shock energy, thus aggravating the panel central damage levels remarkably. For the RHSPs with NPR effect, the predominated failure modes of front faces transitioned essentially from the complete petal-like tearing to a local inner dome as the blast impulse decreased. The ones for the back faces ranged from partial tearing to pure global bending deformation, while the core failure modes tuned from central crushing and peripheral buckling to global compression deformation with central shrinkage and boundary adduction. Relative to the RHSPs, the equivalent HHSPs with a higher compressive strength were less insensitive to the blast intensity. Within the scope of present study, the DAHSPs could experience lower back face deflections than those of RHSPs regardless of core relative density. The auxetic honeycomb sandwich panels not always exhibited better over the non-auxetic HHSPs, which depended greatly on the blast impulse and core density level. This comparative research provides a useful reference for the auxetic sandwich honeycomb core design in protection applications.