Re-entrant Lattice Structure Research Articles

Additive manufactured 3D re-entrant auxetic structures for enhanced impact resistance

Abstract This study presents a novel exploration of the geometric parameters within a 3D re-entrant auxetic lattice structure, specifically focusing on their unique impact energy absorption properties, which were systematically evaluated through drop weight impactor testing. Each lattice configuration was additively manufactured using stereolithography, allowing for precise control over strut thickness (t), re-entrant angle (θ), and the aspect ratio (h/l) of unit cells during both low and high energy impact scenarios. This study found that the overall auxetic behavior is predominantly controlled by the aspect ratio of the cell ribs, while the modulus is governed by rib thickness. A finite element model was subsequently developed to simulate the experimental impact loading conditions and was used to examine a wider range of parameters that were not experimentally tested. The simulated dynamic test results displayed the deformation trends and changes to the Poisson's ratio. Among the studied parameters, experimental results highlighted that a lattice structure with t = 1.6 mm, θ = 65°, and a h/l ratio = 1.8 exhibited the highest specific energy absorption (SEA) under uniaxial impact deformation with 5 Joules of impact energy. Conversely, when employing 20 Joules of impact energy revealed the greatest SEA at t = 1.0 mm, θ = 65°, and an h/l ratio of 2.2. The results demonstrate unique deformation mechanism of auxetic structures under impact loading and the capacity to adapt the 3D re-entrant lattice structure for applications requiring tailored impact energy absorption.

Rounded corner thicken strut re-entrant auxetic honeycomb: Analytical and numerical modeling

An analytical model is formulated for 2-D periodic negative honeycomb thicken strut re-entrant lattice structures and a modified rounded corner negative honeycomb structure that shows negative Poisson’s ratios (NPR). Analytical modeling is done using Castigliano’s second theorem, where each beam is modeled using Timoshenko beam theory, considering bending, stretching, and transverse shearing. Elastic modulus and Poisson’s ratio have been formulated for both structures in the form of non-dimensional geometrical characteristics, such as length ratios, angles of re-entrant arms, shear correction factor, and the material’s Young’s modulus and Poisson’s ratios. Numerical simulations conducted in ABAQUS-CAE explicit solver validate the analytical model. The effect of the non-dimensional parameters on the qualities of the developed structure is demonstrated. It is observed that the structures with a low curvature ratio have a high fluctuation of Poisson’s ratio and Elastic constant when plotted against the other parameters. The slenderness ratio has little impact on Poisson’s ratio but significantly influences elastic modulus. It is shown that various needs can be satisfied by customizing the Poisson’s ratios and elastic constant of both forms of lattice construction over an extensive range by carefully choosing the geometrical parameters and material.

INVESTIGATION OF OVER OBSTACLE PERFORMANCE ANALYSIS OF AUXETIC AIRLESS TYRES

Advancing technologies are leading to the development of airless tire designs that can perform well on challenging road conditions. These designs include lattice structures, mesh structures, and periodic structures, among others. In this study, three different tire designs were analyzed using finite element analysis (FEA) to evaluate their strength and dynamic behavior. Dynamic analyses were conducted on two commercial designs and one original design with re-entrant lattice structures. The study found that these structures are versatile as they provide multiple load paths to resist deformation and failure, and they can be modified to produce different properties like stiffness and strength. The original design with re-entrant structures demonstrated mechanical properties that were twice as good as other commercial tires. Moreover, a spline-lined structure was developed, and it was discovered that a two-stage tire design could enhance strength. The analyses were conducted at specific and controlled speeds with a designated bump size. The new design demonstrated at least 66% higher impact absorption energy performance than other car tyres examined. In total, nine analyses were performed, making a significant contribution to the development of airless tire design.

Tensile Mechanical Behaviors of Re-entrant and Kelvin Cell Lattice Structures

Periodic lattice structures as lightweight and high-energy absorption materials have been widely used in various fields, among which re-entrant and Kelvin cell lattice structures have exhibited excellent mechanical behaviors under different loadings. Therefore, this study aims to numerically explore and compare the tensile mechanical responses of re-entrant and Kelvin cell lattice structures with the same relative density after validating with experimental tests. It has been found that the tensile behavior of the two stretching-dominated lattice structures resemble that of parent solid material but had smaller fracture stress and strain due to the lower ductility of the lattice structures. The re-entrant lattice structure displayed a better energy absorption capacity than the Kelvin cell lattice under tensile loading, i.e., the energy absorption and specific energy absorption of the re-entrant lattice were 3 times and 1.6 times, respectively, those of the Kelvin cell lattice. Meanwhile, the re-entrant lattice as expected exhibited auxetic behavior with a negative Poisson’s ratio during the whole stretching process, while the Kelvin cell had the mechanical behaviors of traditional materials with a relatively constant positive Poisson’s ratio. These results are expected to provide hints on mechanical references and guidance for their extensive applications in the future.

Effect of build direction on tension–tension low cycle fatigue behavior of polyamide 12 parts printed by Multi Jet fusion

The effects of build direction on the tensile properties, tension–tension low cycle fatigue behavior, and failure mechanism of polyamide 12 parts fabricated by Multi Jet Fusion have been investigated by addressing the roles of formed voids, sintering interfaces, and geometrical imperfections. Results indicate that the Young’s modulus of bulk material with vertical build direction is higher than that of bulk material with horizontal build direction due to the carbon black enhanced sintering interfaces. The tensile strength of the printed bulk material with horizontal and vertical build directions is comparable. It is found that the anisotropic tension–tension low cycle fatigue behavior for the bulk material is significant, while that for the re-entrant lattice structure is weak. The anisotropic fatigue strength of bulk material is derived from the anisotropic distribution of internal voids and sintering interfaces. The fatigue behavior of the re-entrant lattice structure is determined by the geometrical imperfections. The coalescence of the large voids or clusters of voids contributes to the crack initiation and propagation processes that lead to the failure of the bulk material. While the failure of re-entrant lattice structure is mainly determined by the intersection of strut junction rather than the void defect. This result provides the fatigue data that can be used for the structural design with Multi Jet Fusion polyamide 12 as well as reveals the effects of voids, sintering interfaces, and geometrical imperfections on the fatigue behavior of the printed parts.

Free Vibration Characteristics of Multi-Material Lattice Structures

This paper presents a modal analysis of honeycomb and re-entrant lattice structures to understand the change in natural frequencies when multi-material configuration is implemented. For this purpose, parallel nylon ligaments within re-entrant and honeycomb lattice structures are replaced with chopped and continuous carbon fibre to constitute multi-material lattice configurations. For each set, the first five natural frequencies were compared using detailed finite element models. For each configuration, three different boundary conditions were considered, which are free–free and clamping at the two sides that are parallel and perpendicular to the vertical parts of the structure. The comparison of the natural frequencies was based on mode-shape matching using modal assurance criteria to identify the correct modes of different configurations. The results showed that the natural frequency of the multi-material configurations increases from 4% to 18% depending on the configuration and material.

Influence of microstructure on stainless steel 316L lattice structures fabricated by electron beam and laser powder bed fusion

In this work, three-dimensional stainless steel 316L re-entrant lattice structures were fabricated by two mainstream powder bed fusion (PBF) techniques, namely electron beam PBF (EB-PBF) and laser PBF (L-PBF). Different grain morphology and crystallographic textures were found in the EB-PBF and L-PBF samples, which significantly influenced their mechanical properties through microscopic deformation. The EB-PBF and L-PBF samples achieved energy absorption capacities of 627.4 mJ/mm3 and 834.8 mJ/mm3, respectively, at a lattice relative density of ∼24%. The EB-PBF sample exhibited equiaxed and elongated grains, while elongated grains were primarily observed in the L-PBF sample. The dominant deformation mechanism of the EB-PBF sample was obtained through dislocation. In contrast, the dislocations trapped inside the solidification cellular walls and deformation-induced twinning were the dominant deformation mechanisms for the L-PBF sample, which contributed to its superior compressive strength and energy absorption capacities. This work provides insights into the enhancement of the mechanical properties of the additively manufactured metallic lattice structures through microstructural control.

Bending analysis of sandwich panel composite with a re-entrant lattice core using zig-zag theory

The sandwich panel structures have been widely used in many industrial applications because of their high mechanical properties. The middle layer of these structures is very important factor in controlling and enhancing their mechanical performance under various loading scenarios. The re-entrant lattice configurations, are prominent candidates that can be used as the middle layer in such sandwich structures because of several reasons namely the simplicity in tuning their elastic (e.g., values of Poisson’s ratio and elastic stiffness) and plastic (e.g., high strength-to-weight ratio) properties by only adjusting the geometrical features of the constituting unit cells. Here, we investigated the response of a three-layered sandwich plate with a re-entrant core lattice under flexural bending using analytical (i.e., zig-zag theory), computational (i.e., finite element) and experimental tests. We also analyzed the effects of different geometrical parameters (e.g., angle, thicknesses, and length to the height ratio of unit cells) of re-entrant lattice structures on the overall mechanical behavior of sandwich structures. We found that the core structures with auxetic behavior (i.e., negative Poisson’s ratio) resulted in a higher bending strength and a minimum out-of-plane shear stress as compared to those with conventional lattices. Our results can pave way in designing advanced engineered sandwich structures with architected core lattices for aerospace and biomedical applications.

Design and mechanical property studies of 3D re-entrant lattice auxetic structure

A new Ti-6Al-4V 3D re-entrant lattice auxetic structure is designed and manufactured by electron beam melting (EBM) in the present study. 2D structural components are combined into a 3D re-entrant lattice auxetic structure by a new connection and topological method. Under uniaxial loading, all 2D structural components show negative Poisson's ratio and bear loads. Four different configurations were fabricated and tested under uniaxial compression. The structure's deformation mechanisms during the compression process are analyzed. The relationship between the geometrical design parameters and the structure's mechanical properties is deduced by the beam theory. Furthermore, the compression test of the structure is simulated by the finite element method (FEM). It is shown that the theoretical and simulated results are consistent with experiments. Compared with classical re-entrant lattice structure, the new 3D re-entrant lattice structure has better mechanical properties, stronger energy absorption capacity and great larger design scope. The results show guiding significance for the study of 3D re-entrant lattice auxetic structures.

Investigation of the equivalent material properties and failure stress of the re-entrant composite lattice structures using an analytical model

In the present study, a novel theoretical model is developed, based on classical laminate theory, to predict the equivalent mechanical properties of the re-entrant lattice structures, which composed of continuous fiber reinforced composite struts. Three main mechanism of stretching, flexing and hinging are considered and a general closed-form formulation is derived to estimate the auxetic honeycomb’s elastic and shear modulus as well as Poisson’s ratios. In spite of previous studies in which the response of honeycomb structures is modeled using beam theory, here, each strut of unit cell is expressed as a composite laminate with orthotropic mechanical properties and classical laminate theory is implemented to calculate the mechanical constants. In addition, using the available relations, the elastic buckling stress and failure load of the auxetic unit cell is evaluated. The proposed model is validated using experimental results, which are available from the previous studies as well as FE simulations. A parametric study is also conducted on the model to study the effects of cell angle and honeycomb’s density on the mechanical properties and failure stresses of the re-entrant structure. The results show that, the proposed analytical model can finely predict the equivalent and failure properties of the auxetic honeycomb structures. Furthermore, it is observed that, for the considered re-entrant composite structures with the cell angles between −25 to 25, the vertical strut fails due to the damage evolution and for the unit cells with other cell angles, the buckling failure is determinative.

Correction to: Modeling of uniaxial compression in a 3D periodic re-entrant lattice structure

In the original article, there is a typographical error in the in-line equation on page 1418 following Eq. (21). The sentence should read as follows.

Micro and nanolattice fabrication using projection micro litho stereo exposure additive manufacturing techniques and synchrotron X-ray 3D imaging-based defect characterization

Synchrotron radiation X-ray micro-computed tomography (SR-µCT) is a 3D imaging technique that is widely employed for the characterization of defects in advanced materials and structures. In this study, we characterize several typical defects in octettruss and re-entrant 3D lattice structures by using SR-µCT. The 3D micro-lattice structures are manufactured using projection micro litho stereo exposure (PµLSE) additive manufacturing technology. The as-fabricated 3D lattice samples are characterized using optical microscopy, and subsequently, by SR-µCT. Further more, a statistical analysis is performed to characterize the surface roughness and internal defects qualitatively, whereby the statistical geometrical parameters of struts along different directions and strut joints are analyzed and classified. Consequently, several typical defects are identified: (1) holes at the joints of the strut and irregular diameter deviations of the strut in the octet-truss lattice structure; (2) irregular diameter variations, bulges, dislocations, grooves, accumulations, and torsion in the re-entrant lattice structure. All of these defects are related to the building direction, the weight of the structure, bubbles, dust, and impurities during the PµLSE additive manufacturing process.

An Enhanced Three-Dimensional Auxetic Lattice Structure with Improved Property.

In order to enhance the mechanical property of auxetic lattice structures, a new enhanced auxetic lattice structure was designed by embedding narrow struts into a three-dimensional (3D) re-entrant lattice structure. A series of enhanced lattice structures with varied parameters were fabricated by 3D printing combined with the molten metal infiltration technique. Based on the method, parameter studies were performed. The enhanced auxetic lattice structure was found to exhibit superior mechanical behaviors compared to the 3D re-entrant lattice structure. An interesting phenomenon showed that increasing the diameter of connecting struts led to less auxetic and non-auxetic structures. Moreover, the compressive property of the enhanced structure also exhibited obvious dependence on the base material and compression directions. The present study can provide useful information for the design, fabrication and application of new auxetic structures with enhanced properties.

Damage characterizations and simulation of selective laser melting fabricated 3D re-entrant lattices based on in-situ CT testing and geometric reconstruction

In recent years, metal additive manufacturing (AM) are widely employed for industrial applications, such as: biomedical, aerospace, automotive, marine and offshore sections. AM demonstrated superior manufacturing efficiencies and economic advantages for advanced lightweight industrial components with unlimited arbitrary topological layouts and complex internal microstructures, and are also employed for fabrication of auxetic materials and structures. In this paper, damage characterizations and mechanical behaviors of selective laser melting (SLM) fabricated 3D re-entrant lattices are investigated based on in-situ interrupted micro-CT test, and simulation based on geometric reconstructed models are performed for exploring the underlying failure mechanisms. Firstly, theoretical models for predicting the mechanical properties of 3D re-entrant lattice are developed, such as stiffness, Poisson's ratio and strength, etc. Secondly, the geometrical errors and fabrication defects of 3D re-entrant lattices are analyzed based on 3D micro-CT scanning, in-situ micro-CT interrupted compression tests are performed for studying the deformation process and failure mechanisms. Finally, image finite element models with the detailed information of the shape, position and distribution of defects of the 3D reentrant lattices are constructed from 3D tomographic images, and numerical simulations are performed for studying the effects of the defects on the mechanical performances of the SLM additive manufactured 3D re-entrant lattice structures. It is shown that the failure behavior of the reentrant lattice is governed not only by its topology, but also by the geometric defects and surface defects. Moreover, the proposed interrupted in-situ micro-CT mechanical loading experiments and image finite element approaches can also shed lights on the relations between fracture failure around the edge and the powder adhesion. The damage evolution process is compared with the numerical simulation results to verify the materials failure modes.

Mechanical properties of 2D hierarchical re-entrant cellular structures with Voronoi sub-structures

In this study, 2D hierarchical re-entrant cellular structures with Voronoi sub-structures were proposed. Their Poisson's ratios and energy absorption capacities (in-plane) were studied using the finite-element method. The effects of geometric parameters (Voronoi sub-structure irregularity, k, concave angle, θ, ratio of cell wall thickness to cell height, w) on the lattice mechanical properties were investigated. Results showed that Poisson's ratio first decreased and then increased with an increasing θ when k is less than 0.5. A minimum Poisson's ratio of approximately −0.315 was achieved when and . The 2D hierarchical re-entrant lattice structure exhibited a maximum energy absorption capacity when and , which is 3.9 times higher than that of the traditional re-entrant structure with solid ribs and of identical relative density.

An analytical model for star-shaped re-entrant lattice structures with the orthotropic symmetry and negative Poisson's ratios

A new analytical model is developed for three types of 2-D periodic star-shaped re-entrant lattice structures that possess the orthotropic symmetry and exhibit negative Poisson's ratios. Contributions from both the re-entrant and connection struts are considered using an energy method based on Castigliano's second theorem. Each re-entrant strut is treated as a Timoshenko beam, and stretching, transverse shearing and bending deformations are all incorporated in the formulation. Unlike existing studies, the overlapping of struts at joints is included in determining the relative density, which is analytically expressed for each lattice type. Closed-form formulas are derived for the effective Young's moduli and Poisson's ratios of each type of lattice structure, which contain three non-dimensional length ratios, two re-entrant angles, one shear correction factor, and Young's modulus and Poisson's ratio of the strut material. The new analytical model is validated against finite element simulations conducted in the current study and two existing analytical models for simpler square lattice structures without re-entrant struts. To illustrate the newly developed analytical model, a parametric study is conducted to quantitatively show the effects of the five geometrical parameters on the effective properties of each type of lattice structure. It is found that for the effective Poisson's ratios the key controlling parameters are the two re-entrant angles, while for the effective Young's moduli all of the geometrical parameters can have significant effects except for the external connection length ratio. It is demonstrated that through a proper selection of the geometrical parameters, vertex connections and strut material, it is possible to tailor the effective Poisson's ratios and Young's moduli of each type of lattice structure over wide ranges to satisfy different needs in various applications.

Micromechanical modeling of 3D printable interpenetrating phase composites with tailorable effective elastic properties including negative Poisson's ratio

3D printable two-phase interpenetrating phase composites (IPCs) are designed by embedding a 3D periodic re-entrant lattice structure (as the reinforcing phase) in a matrix phase. These IPCs display the cubic or tetragonal symmetry. A micromechanical model is developed to evaluate effective elastic properties of the IPCs. Effective Young's moduli, shear moduli and Poisson's ratios (PRs) of each IPC are determined from the effective stiffness and compliance matrices of the composite, which are obtained through a homogenization analysis using a unit cell-based finite element (FE) model incorporating periodic boundary conditions. The FE simulation results are also compared with those based on various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. The effective properties of the IPC can be tailored by adjusting five geometrical parameters, including two strut lengths, two re-entrant angles and one strut diameter, and elastic properties of the two constituent materials. The numerical results reveal that IPCs with a negative PR can be generated by using a compliant matrix material and large re-entrant angles. In addition, it is found that the two re-entrant angles can greatly affect other effective elastic properties of the IPC: the effective shear modulus can be enhanced, while the effective Young's modulus can be enhanced or compromised with the increase of the re-entrant angles. Furthermore, it is seen that by adjusting one of the two re-entrant angles or one of the two strut lengths, the material symmetry exhibited by the IPC can be changed from cubic to tetragonal.

Wave propagation in hexagonal and re-entrant lattice structures with cell walls of non-uniform thickness

On the basis of the Bloch’s theorem, the in-plane wave propagation in hexagonal and re-entrant lattice structures with cell walls of non-uniform thickness is investigated using the dynamic stiffness matrix in conjunction with the Wittrick–Williams algorithm. Special attention is devoted to the effects of the internal angle, the slenderness ratio and the material distribution on the directional and band gap behaviors. Results show that the three considered parameters can significantly influence the band gap characteristics. For the wave propagation directionality, however, the internal angle is more prominent than the other two factors. The work expects to serve as a guide for the optimal design of directional mechanical filters.

Modeling of uniaxial compression in a 3D periodic re-entrant lattice structure

In this study, the behavior of a parametric 3D re-entrant dodecahedron lattice structure with negative Poisson’s ratio was studied. Four geometrical configurations for the re-entrant dodecahedron were designed, and the relationship between the mechanical properties and the design parameters was determined through beam theory. Samples were fabricated successfully via electron beam melting. Compressive tests as well as finite element analysis (FEA) were performed, and the results were compared with theoretical predictions. The modeling yielded explicit analytical equations of various mechanical properties including Poisson’s ratios, modulus and strength, and the compressive strength and the modulus from the prediction match well with the experiments, as well as the FEA results. The methodology used by this study also demonstrated a feasible approach to design 3D auxetic cellular structure for various applications.

Compressive properties of Ti–6Al–4V auxetic mesh structures made by electron beam melting

In this current work, a Ti–6Al–4V 3-D re-entrant lattice auxetic structure is manufactured by the electron beam melting (EBM) process. Four different design configurations (two negative Poisson’s ratio values×two relative densities) were manufactured and tested under compressive stress. Two failure modes were observed whose occurrence appeared to be dependent on the ratio of vertical strut length to re-entrant strut length. A small deflection analytical model is presented that predicts yield strength and modulus for one type of design with good accuracy. Results also show that the re-entrant lattice structure possesses superior mechanical properties compared to regular foam structures. Limitations of the analytical model are also discussed.