Re-entrant Lattice Research Articles

A comprehensive study on numerical homogenization of re-entrant honeycomb lattice and analytical model assessment

A comprehensive study on numerical homogenization of re-entrant honeycomb lattice and analytical model assessment

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.

Flexural behaviors of asymmetric Re-entrant auxetic honeycombs

A family of negative Poisson's ratio honeycombs with asymmetric base units and potential applications in civil and marine industries are introduced by introducing asymmetricities to the geometry of regular re-entrant unit cell. These structures, namely the single symmetry-broken re-entrant (SSR), double symmetry-broken re-entrant (DSR), and hybrid symmetry-broken re-entrant (HSR) honeycomb lattices, are fabricated through fused filament fabrication and subjected to experimental three-point bending (TPB) experiments and simulations. The novel designs showcase exceptional specific energy absorption (SEA) attributes compared to the regular metamaterial, with the SSR structure exhibiting a remarkable 147.2% higher SEA. The asymmetric metamaterials also demonstrate higher flexural modulus (Ef) compared to the benchmark design, with the SSR and DSR models boasting approximately 29% and 19% higher Ef, respectively. Studies on design parameters show that internal angle of unit cells that creates the asymmetricity affects the flexural performance of the unique auxetic honeycombs, significantly. Finally, parametric investigation on out-of-plane bending of the honeycombs showed the dominance of all asymmetric-unit honeycombs over the benchmark due to having organized self-contact regions. The SSR and DSR structures own about 51% and 39% higher SEA than the benchmark honeycomb under out-of-plane TPB, respectively.

Design and numerical investigation of the 3D reinforced re-entrant auxetic and hexagonal lattice structures for energy absorption properties

Lattice Structures (LS) are widely recognized for their light weight and excellent mechanical properties. In this study, strut reinforcement technique is applied to enhance the energy absorption properties of 3D re-entrant auxetic (Aux) and hexagonal (Hex) LS. Numerical investigation using finite element analysis (FEA) was carried out to understand the mechanical and energy absorption properties of the novel designs through quasi-static compression test. The uniaxial loading results of the reinforced designs were compared to the traditional 3D hexagonal and re-entrant auxetic LS. In order to numerically simulate the mechanical behaviour of 3D printed LS, mechanical properties of experimentally studied PA2200 matrix material (manufactured via additive manufacturing) was used. Study of the mechanical behaviour of the 3D printed LS parts is essential because additive manufacturing has the capacity to fabricate the complex LS compared to conventional manufacturing processes, and innovative LS design can provide high specific strength of parts. The stress–strain and energy absorption curve results indicate that the purposed new designs are excellent choice for energy absorption properties at large strain. The findings of this study contribute towards the novel design of 3D hexagonal and re-entrant auxetic LS for superior mechanical strength and specific energy absorption properties.

Design of 2D re-entrant auxetic lattice structures with extreme elastic mechanical properties

Auxetics have emerged recently as an exciting class of mechanical metamaterials that are identified by negative Poisson’s ratio. The mechanical properties of rationally designed auxetic metamaterials are mostly influenced by their topological characteristics rather than the material properties of their constituent material. The goal of this paper is to use optimization algorithms to achieve the desired target mechanical properties for reentrant auxetic metamaterials. The GEKKO optimization package in Python is used for optimizing the geometry of auxetic lattice structures. Several mechanical properties including Poisson’s ratio, relative Young’s modulus, relative yield stress, and relative energy absorption capability, as well as the noted properties normalized with respect to relative density and relative Young’s modulus are chosen for being minimized or maximized. ANSYS finite element package is also implemented for validation of the results obtained from the optimization algorithm. According to the results, to achieve the maximum magnitude of Poisson’s ratio (positive and negative), the size of the unit cell in the lateral direction must be selected to be maximum and the thickness must become minimum. Moreover, to achieve the maximum value of the relative Young’s modulus or energy absorption in any direction, the size of the unit cell in that direction must be maximized. Also, to achieve the maximum amount of relative yield stress in both directions, the unit cell must have a maximum thickness and an internal angle close to zero.

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.

Impact resistance analysis of a dual-constituent negative Poisson’s ratio lattice metamaterial with tailorable coefficient of thermal expansion

The combination of parameters such as negative Poisson’s ratio (NPR) and negative coefficient of thermal expansion (NCTE) can result in novel materials with specialized functions to meet the requirements of multifunctional and multipurpose devices. However, the additional structural design to meet NCTE requires a re-examination of its impact resistance. In this paper, the deformation modes and impact resistance of dual-constituent re-entrant hexagonal lattice metamaterial (DRHLM) with both NPR and NCTE are investigated under different impact velocities, and the nominal stress–strain curves and energy absorption characteristics of DRHLM structures with different arch heights, chord lengths, and angles of the bottom edge ribs and the diagonal ribs are discussed. The results show that the bi-material bending ribs affect the energy absorption of the structure, and in most cases the DRHLM structure has a better energy absorption capacity than the traditional concave hexagonal structure, but too much or too little curvature of the bi-material portion reduces the energy absorption characteristics of the DRHLM structure. In addition, the increase in the pinch angle of the ribs leads to the weakening of the negative Poisson’s ratio effect, which is also detrimental to the energy absorption of the structure. The work done can provide guidance for finding the dimensional parameters of DRHLM structures with optimal energy absorption performance.

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.

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.

Composites with Re-Entrant Lattice: Effect of Filler on Auxetic Behaviour.

This study is focused on the deformation behaviour of composites formed by auxetic lattice structures acting as a matrix based on the re-entrant unit-cell geometry with a soft filler, motivated by biomedical applications. Three-dimensional models of two types of the auxetic-lattice structures were manufactured using filament deposition modelling. Numerical finite-element models were developed for computational analysis of the effect of the filler with different mechanical properties on the effective Poisson's ratio and mechanical behaviour of such composites. Tensile tests of 3D-printed auxetic samples were performed with strain measurements using digital image correlation. The use of the filler phase with various elastic moduli resulted in positive, negative, and close-to-zero effective Poisson's ratios. Two approaches for numerical measurement of the Poisson's ratio were used. The failure probability of the two-phase composites with auxetic structure depending on the filler stiffness was investigated by assessing statistical distributions of stresses in the finite-elements models.

Deformation behaviors and energy absorption characteristics of a hollow re-entrant auxetic lattice metamaterial

Designing innovative architected metamaterials with enhanced energy absorption is a long-term pursuit to provide impact protection capacity for human beings and crucial structural components. Numerous natural materials prove that hollow structures present unique combinations of mechanical properties and functionalities with minimal weight cost. Therefore, a novel 3D hollow re-entrant auxetic (HRA) lattice metamaterial is proposed to improve the energy absorption capacity. To investigate the mechanical properties of the HRA lattices, an analytical model based on plastica theory is established to predict the crushing force. The energy absorption behaviors of the HRA lattices under quasi-static crushing load are studied based on finite element method, and the analytical predictions are in good agreement with the numerical simulation results. The analytical model can be used to optimize the appropriate design parameters of the HRA lattice and explain the simulation results in the parametric studies. Compared with the solid re-entrant auxetic (SRA) lattice with the same density, specific energy absorption of the HRA lattice can be improved by up to 27.43%. Finally, a parametric investigation is carried out to comprehend the influences of the length-to-height ratio, the re-entrant angle, the width-to-height ratio, and the wall thickness-to-height ratio on the energy absorption capacity of the HRA lattices. According to various energy absorption demands, the HRA lattices with different energy absorption properties can be designed by tuning these design parameters. This concept of building hollow auxetic lattice metamaterials can open up a new solution to design metamaterials with negative Poisson's ratio effects and enhanced energy absorption for potential engineering applications.

Geometric design and performance of single and dual-printed lattice-reinforced cementitious composite

Recent advances in additive manufacturing have enabled the design and fabrication of auxetic materials with programmable geometries and tailorable properties, but ongoing challenges remain in integrating them for composite design. In this work, we exploit a composite design strategy on cementitious material units embedded with three-dimensional re-entrant (3DR) lattices fabricated by single and dual material printing. Guided by experimental tests, the effect of additive manufacturing parameters on the mechanical properties of rigid and flexible 3DR lattices is confirmed. Then, we characterize the compressive behavior of lattice-reinforced cementitious composite (LRCC) units embedded with 3DR lattices printed using rigid, flexible, and dual materials with various spatial distributions. We found comparable ultimate strengths and higher energy-absorbing capabilities despite weaker 3D-printed materials embedded in LRCC units. A similar observation was found on LRCC columns with various geometric patterns. We envision that our proposed composite design strategy can produce next-generation low-carbon material systems with high performance and production efficiency toward a sustainable built environment.

Digital Light Processing of 2D Lattice Composites for Tunable Self‐Sensing and Mechanical Performance

This study investigates the mechanical and piezoresistive self‐sensing performance of additive manufacturing‐enabled 2D nanocomposite lattices under monotonic and cyclic tensile loading. Lattice structures comprising hexagonal, chiral, triangular, and reentrant unit cell topologies are realized via digital light processing using an acrylic photocurable resin filled with carbon nanotubes (CNTs). The results reveal that the piezoresistive sensitivity of reentrant and triangular lattices is nearly insensitive to changes in relative density. In contrast, the gauge factors of the hexagonal and chiral lattices rise by 300% and 500%, respectively, with an increase in relative density from 20 to 40%, which can be ascribed to their bend‐dominated behavior, causing an increase in surface strains in the lattice struts with increasing relative density for an imposed macroscopic strain. The measured stress versus strain responses compare well with nonlinear finite element results. Under strain‐controlled cyclic loading, the electrical resistance of the 2D lattices is found to decline over time due to reorientation of the CNTs in the surrounding viscoelastic polymer matrix. The findings provide valuable insights into the interrelations between sensing performance, cell architecture, and relative density of the lattices, and offer guidelines for the design of architected strain sensors and self‐sensing lightweight structures.

Dynamic Compressive and Flexural Behaviour of Re-Entrant Auxetics: A Numerical Study.

Re-entrant auxetics offer the potential to address lightweight challenges while exhibiting superior impact resistance, energy absorption capacity, and a synclastic curvature deformation mechanism for a wide range of engineering applications. This paper presents a systematic numerical study on the compressive and flexural behaviour of re-entrant honeycomb and 3D re-entrant lattice using the finite element method implemented with ABAQUS/Explicit, in comparison with that of regular hexagonal honeycomb. The finite element model was validated with experimental data obtained from the literature, followed by a mesh size sensitivity analysis performed to determine the optimal element size. A series of simulations was then conducted to investigate the failure mechanisms and effects of different factors including strain rate, relative density, unit cell number, and material property on the dynamic response of re-entrant auxetics subjected to axial and flexural loading. The simulation results indicate that 3D re-entrant lattice is superior to hexagonal honeycomb and re-entrant honeycomb in energy dissipation, which is insensitive to unit cell number. Replacing re-entrant honeycomb with 3D re-entrant lattice leads to an 884% increase in plastic energy dissipation and a 694% rise in initial peak stress. Under flexural loading, the re-entrant honeycomb shows a small flexural modulus, but maintains the elastic deformation regime over a large range of strain. In all cases, the compressive and flexural dynamic response of re-entrant auxetics exhibits a strong dependence on strain rate, relative density, and material property. This study provides intuitive insight into the compressive and flexural performance of re-entrant auxetics, which can facilitate the optimal design of auxetic composites.

Porosity Effects on Static Performance of Carbon Nanotube-Reinforced Meta-Nanocomposite Structures.

A mixture of outstanding merits of polymer nanocomposites (PNCs) and metamaterials can lead to the development of ultra-light meta-nanomaterials whose high sensitivity can be efficiently used in wearable strain sensors. Thus, reliable data about the performance of structural elements manufactured from such meta-nanomaterials are needed before implementing their design. Motivated by this issue, the negative impacts of pores in the microstructure and carbon nanotubes' (CNTs') wavy configuration on the nonlinear bending features of thick beams consisted of auxetic CNT-reinforced (CNTR) polymers are probed for the first time. The impacts of distinct porosity distributions on the mechanical reaction of the system are covered in this article. In addition, a very low computationally cost homogenization is implemented herein to consider the waviness' influence on the reinforcement mechanism in the auxetic PNC material. Moreover, higher-order shear deformation theory (HSDT) is followed and merged with non-linear definition of strain tensor with the aid of von Kármán's theory to gather the equations describing the problem. Thereafter, the famous Navier's exact solution is employed towards solving the problem for thick beams with simple supports at both ends. A comparison of our data with those existing in the literature certifies the accuracy of the presented modeling. The outcomes indicate on the remarkable rise in the flexural deformation of the auxetic PNC beam while the coefficient of porosity is raised. It is also shown that utilization of thick-walled cells in the re-entrant lattice can help to control the system's total deflection. In addition, if the non-ideal shape of the nanofillers is ignored, the deflection of the meta-nanomaterial beam will be much larger than that of ideal calculations.

Zero-thermal-expansion metamaterial with broadband vibration suppression

Although conventional metamaterials possess extraordinary properties, they cannot meet the practical requirements of engineering structures subjected to both temperature fluctuations and vibration excitations. To bridge this research gap, this paper proposes a novel dual-functional metamaterial with zero thermal expansion and broadband vibration suppression. Inspired by the thermal mismatch and band gap effects, we present an innovative design strategy that incorporates star-shaped re-entrant lattices and locally rudder-shaped struts with two-phase materials. To effectively guide the coupled design, a theoretical model accounting for stretching-bending deformations is established to predict the thermoelastic behavior of the metamaterial. Moreover, a parameterized dynamic model using the spectral element method is developed to study the vibration characteristics. Particularly, the spectral formulation of curved Timoshenko beams is derived, including in-plane and out-of-plane vibrations. The comparisons between theoretical predictions and finite element simulations validate the accuracy of analytical models in characterizing thermal deformations and vibration responses. Finally, the case studies shed light on the underlying formation mechanisms of zero thermal expansion and multiple band gaps, while also unveiling the effects of geometric parameters on the performance of such dual-functional metamaterials. Our innovative design strategy and analytical methodology not only offer appealing alternatives for engineering applications but also push the boundaries of metamaterial properties by enabling the transition from single- to dual-functionality.

Experimental Investigations on the Mechanical Performances of Auxetic Metal-Ceramic Hybrid Lattice under Quasi-Static Compression and Dynamic Ballistic Loading

In recent years, there have been increasing research interests in investigating the compression and ballistic responses of metal-ceramic hybrid structures, mainly making use of the synergistic effects of conventional metal honeycomb structures and infilled ceramic matrix materials. In this paper, a novel hybrid auxetic re-entrant metal-ceramic lattice is designed and manufactured to overcome the intrinsic conflicts between the strength and toughness of architected mechanical metamaterials, synergistic effects of auxetic re-entrant metal honeycombs and infilled ceramic materials are experimentally and numerically studied, and auxetic deformation features and failure modes are characterized with the digital image correlation (DIC) technique as well. It was found that (1) the infilled ceramic matrix of conventional honeycomb frames only endure longitudinal compression or impact loading along the external loading direction, while auxetic metal re-entrant honeycomb components endure both longitudinal and transverse loading due to the negative Poisson′s ratio effect and (2) the collaborative effects of infilled auxetics and the constraint frames’ hybrid structure dramatically moderate the stress concentration state and improve the impact resistance of single-phase ceramic materials. Our results indicate that the auxetic hybrid design exhibits promising industrial application potentials for blast protection engineering.

Thermal expansion regulation and bandgap analysis of a novel dual-constituent negative Poisson’s ratio lattice metamaterial

Negative Poisson's ratio lattice metamaterials (NPR-LM) have broad application prospects in explosion-proof and impact resistance due to their excellent energy absorption and cushioning properties. The development of science and technology has put forward higher requirements for the versatility of materials, but the currently existing lattice metamaterials are often limited to a single function. In this paper, a novel dual-constituent re-entrant hexagonal lattice metamaterial (DRHLM) with tunable thermal expansion behavior, negative Poisson's ratio (NPR) effect, and outstanding band-gap characteristics are designed based on traditional re-entrant hexagonal metamaterial and bi-layer curved rib. The tunable thermal expansion characteristics and NPR effect of DRHLM are verified by analytic and finite element numerical simulation methods. Finally, the bandgap of DRHLM is studied by using the finite element method, and the influence of the bending angle of the ribs on the bandgap is analyzed for DRHLM configurations with different coefficients of thermal expansion (CTEs) and NPRs. The results show that the designed DRHLM can simultaneously achieve different positive/negative/zero CTE and NPR with outstanding band gap characteristics by reasonably selecting material parameters and geometric parameters. This paper provides a new idea for the design of mechanical metamaterials with various properties and functions.

Light-powered self-oscillation in liquid crystal elastomer auxetic metamaterials with large volume change

Self-oscillation can harvest energy directly from the environment to maintain its motion and is characterized by its autonomy, with various promising applications in the fields of actuators, soft robots and medical devices. Limited to conventional materials, challenges remain in designing novel self-oscillating systems. Auxetic metamaterials can exhibit exceptional properties of negative Poisson's ratio and volume expansion, and the integration of metamaterial structures into self-oscillation can provide additional functionality and improve existing properties and its application. This paper proposes a liquid crystal elastomer (LCE) auxetic metamaterial, and theoretically investigates the self-oscillation of the LCE auxetic metamaterial-mass system under steady illumination. The system can undergo a supercritical Hopf bifurcation between the static pattern and self-oscillation pattern, which arises from the light-driven contraction of LCE fibers and is maintained by light energy input from the environment to compensate for the damping dissipation. This paper also investigates the Hopf bifurcation conditions, as well as the vital systematic parameters affecting frequency and amplitude of the self-oscillation pattern in detail. Different from the existing self-oscillating systems, this LCE metamaterial structure can provide additional functionality and potentially improve existing properties such as large volume change. More generally, the idea of integrating LCE fibers into the re-entrant honeycomb lattices can be applied to other auxetic structures such as arrows and chirality, and it is expected to offer more diverse design ideas of metamaterial and motion control for artificial hearts, hydraulic pumps, sensors, and energy harvesters.

In-plane elastic properties of meta-chiral star-shaped lattice structure

A combination of chirality, anti-chirality, and re-entrant lattice pattern is proposed to form a new two-dimensional meta-structure, and a comprehensive parametric study is conducted by using experimental and numerical analysis. The proposed structure can undergo larger deformations without significant stress concentrations. Based on the verified finite element model, the effects of various geometrical parameters on the in-plane mechanical response of the structure are investigated. Results show that the thickness of the structure is the most dominant factor in affecting the Young modulus while other parameters can alter Poisson’s ratio. In addition, the proposed structure exhibits tunable negative/near-zero Poisson’s ratios.