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... Because of these desirable properties, 11 auxetics possess many potential applications in civil engineering [10,11], protective engineering 12 [12, 13], medical treatment , intelligent materials , and so on. To be more specific, they 13 could be used in automobile anti-collision devices [16,17], esophageal stents , nails , and 14 shock absorbers . When an auxetic structure is compressed, the internal periodic cellular 15 structure gradually compacts. ...
... This work is applied to the application of auxetic materials that not only need negative 16 Poisson's ratio but also need high stiffness, e.g., energy absorption devices. It is worth noting that, 17 by coating nickel layer on the surfaces using electroless plating method, Zheng et al.  found 18 that the stiffness of auxetic metamaterials were enhanced for several orders without sacrificing 19 the auxetic effect. Although the structures proposed in the aforementioned publications have 20 improved the stiffness of auxetics, a systematic method to design the auxetic unit cells with 21 enhanced stiffness is rare reported. ...
... foam-filled re-entrant honeycomb had higher stiffness and bearing capacity. Meena et al.  18 introduced the star-shaped re-entrant unit cells between two consecutive S-shaped unit cells, it 19 shows better tensile results and higher compressive strength, stiffness and Young's modulus . 20 The auxetic effect of the proposed structure was achieved by sacrificing the structural stiffness. ...
Auxetic materials are a class of mechanical metamaterials. They exhibit lateral expansion under tension, and lateral contraction under compression. Such metamaterials have attracted increasing attention due to their unusual mechanical behaviour and various potential applications. However, the stiffness of auxetic structures is much weaker than that of solid ones due to their porous structure. Therefore, many researchers try to enhance the stiffness of auxetic cellular structures to broaden their potential applications. In this study, re-entrant unit cells with different variable stiffness factors (VSF) were designed to achieve the tunability of stiffness from the aspect of tuning the densification strain. Experimental and numerical analyses were carried out to verify the accuracy between the designed and the actual VSF. It is found that the compaction points of the proposed re-entrant structures could be tuned quantitatively using the defined VSF, which provides a new method for the optimal design of negative Poisson's ratio unit cell.
... As a result, auxetic structures are gaining popularity in a wide range of engineering applications. Some of the applications of auxetic structures include lightweight protective structures against blast and impact loads , automotive crash safety , sensors , fasteners , etc. In protective engineering applications, various auxetic structures have successfully been used in the core of protective sandwich panels to resist blast and impact loads [10,11,. ...
... By equating Eqs. (17) and (25), is obtained and given in Eq. (26). ...
This paper aims to investigate the crushing behaviour and protective performance of an innovative auxetic structure, named the hourglass structure (HGS), under in-plane compressive loads. The HGS was fabricated with a 3D printing method through fused filament fabrication technique. The 3D printed HGS specimens were subjected to a displacement-controlled in-plane quasi-static compressive load. A 3D comprehensive numerical model was developed and calibrated with the experimental results. The calibrated numerical model was used for a parametric study to explore the effects of relative density, functional grading, and crushing velocity on the protective performance of the HGS. Furthermore, a theoretical model was proposed to correlate plateau stress of the HGS with geometric design parameters, effective Poisson’s ratio, and matrix material properties. Verified with a series of numerical analyses of different HGS designs, the proposed theoretical model predicted plateau stress with less than 10% relative error. In addition, experimental, numerical, and theoretical analyses uncovered load–deformation relationship, Poisson’s ratio, crushing mechanism, and energy absorption capacity of the novel auxetic. Overall, the HGS exhibited excellent features for protective engineering applications such as shock mitigation and blast energy absorption.
... Auxetic metamaterials [8,9] are promising in the fields of aerospace, marine, civil engineering [10,11], fastener , piezoelectric materials [13,14], strain sensors [15,16], morphing structures , composite materials , medical engineering [21,22], tubular structures [10,23], smart materials [24,25] and cardiovascular stents  benefiting from their higher shear modulus, increased indentation resistance, synclastic behavior, fracture toughness, good energy absorption [27,28], blast resistance [29,30], acoustic damping , and other excellent properties. ...
... In some applications, it is a practical requirement that the stiffness and Poisson's ratio vary according to selfdeformation or external load. One example is the auxetic nail , variable stiffness is a requirement of auxetic nails when pushing in. Fig. 2 (a) shows the process of pushing in and pulling out of an auxetic nail. ...
An auxetic metamaterial composed of novel re-entrant unit cells was proposed. The new re-entrant structure was constructed by adding wedge-shaped parts to the conventional re-entrant structure. Not only can the additional part regulate the structural stiffness during compression but it can also increase the stability of the structure by hindering lateral buckling of the structure, endowing the metamaterial with more significant and stable auxetic behavior in compression. The mechanical and deformation characteristics of the proposed metamaterial were investigated experimentally and numerically. A parametric study was carried out using the validated finite element model to analyze the influence of the size, angle and stiffness of the wedge-shaped part. Due to its improved stiffness and tunability, the proposed auxetic metamaterial has huge potential to be utilized in civil engineering and protection engineering in the form of two-dimensional, three-dimensional and tubular structures. Furthermore, the self-adjusting stiffness property, better stability and enhanced auxeticity make this metamaterial useful for smart materials and intelligent sensors.
... In an auxetic structured model, the auxetic laminate enlarged strain energy release rate (thickness-through direction) and improved fracture toughness (in-plane) as tested by the compliance method (Wang et al., 2016). However, when applied to the wall structure of 3D printed nails, the auxetic effect was less impactful than the surface roughness in pull-in/out resistance against timber (Ren et al., 2018b). Ren et al. (2018b) stated that the auxetic effect may not be the governing factor in pull-out resistance of a specified debonding case. ...
... However, when applied to the wall structure of 3D printed nails, the auxetic effect was less impactful than the surface roughness in pull-in/out resistance against timber (Ren et al., 2018b). Ren et al. (2018b) stated that the auxetic effect may not be the governing factor in pull-out resistance of a specified debonding case. Therefore, it is suggested that the superiorities of applying auxetic insertion using the variant composition method should be examined to prevent overestimation. ...
Auxetic materials, possessing negative Poisson’s ratios (NPRs), have the ability to shrink (or expand) in the lateral direction under an axial compressive (or tensile) force respectively. Due to this unique feature, an auxetic material is found to sustain high energy absorption capacity, fracture toughness and shear resistance and thus regarded as one of the future materials in the field of impact protection. However, civil engineering applications of auxetic structures or materials are minimal due to miscellaneous restrictions on NPR effects. Accumulative developments in auxetics have facilitated their applications in cementitious materials in recent years. This paper presents an overview of recent advances in the development of auxetic cementitious composites and analyses and summarises their mechanical properties under different loading conditions. Prior to extensive finite element simulations, more attention has been given to the limited experimental results. Particular attention is paid to the expansionary feasibility of the parent material to introduce auxetic behaviour, with precise identification of the limitations, innovative composition methods and facilitation of auxetic features. Finally, the paper outlines the limitations of the current research and envisages few future research opportunities in auxetic cementitious composites.
... The Poisson's ratio and stiffness of generated auxetic metamaterials could be simply tuned by PSF. According to these publications, auxetic metamaterials usually possess a densification stage where the compressed stiffness increases sharply [40,41]. Thus, from the aspect of tuning densification strain, a novel methodology was firstly reported to design auxetic perforated structures to achieve tunable stiffness indirectly . ...
... The methodology of designing the auxetic metamaterial with tunable stiffness can be divided into four steps. Firstly, generating the original buckling-induced auxetic unit cell using the PSF method . The design regions and the variation range of VSF would be reduced when the original ellipse becomes narrow and oblate. ...
Auxetic metamaterials have attracted increasing attention due to their exceptional mechanical properties. However, the critical parameters of mechanical response and Poisson's ratio would be changed simultaneously when a geometrical parameter is tuned, which is adverse to achieving the quantitative design of energy absorption by tuning a single geometrical parameter. Thus, the methodology based on tuning densification strain is proposed to design auxetic unit cells with tunable stiffness. In this study, the static performance of 2D metallic auxetic metamaterials designed by the variable stiffness factor (VSF) method is examined experimentally and numerically. To further achieve tunable energy absorption under crushing load, the concept of VSF is extended to variable energy factor (VEF). The dynamic response of verified numerical models is investigated subjected to low-, medium-, and high-velocity crushing. Finally, a functionally graded auxetic structure with different design layers is proposed to effectively solve the issues of high stiffness ratio and initial peak force. These results show that the designed structure has the actual VSF and VEF percentages close to the designed value under low- and medium-velocity crushing. The findings from this study are useful for wider applications of auxetics in protective engineering.
... Auxetic materials [2,3,4] exhibit negative PR, i.e. they increase lateral dimensions in processes of stretching and decrease lateral dimensions under compression. This unique and counterintuitive property makes them attractive in the scope of basic research [5,6,7] and from the perspective of possible applications [8,9,10]. A search for auxetic properties is currently underway by considering various system structures [11,12,13,14,15], among metamaterials [16,17,18,19], as well as in heterogeneous systems such as hybrid materials and composites [20,21,22]. ...
In recent years, the investigation of auxetic materials is receiving more and more attention due to their wide range of applications which follow enhancing indentation resistance, toughness, shear resistance, and other advantages of such materials. This work reports results of studies of models of auxetic metamaterials with nanoinclusions. Yukawa crystals with nanoinclusions in the form of nanochannels in the  crystallographic direction, filled by hard spheres, were simulated by Monte Carlo in a wide range of pressures to determine their elastic properties. Particular attention has been devoted to the Poisson’s ratio. It has been found that depending on the nanochannels’ type and pressure, the value of Poisson’s ratio can vary from -0.302(12) to 1.083(14). The microscopic structures of the crystals were also examined in detail. A solid-solid phase transition in a host-guest system (the Yukawa crystal with hard spheres) was observed. Interestingly, this phase transition generates a unit cell doubling along the nanochannels. To localize this phase transition, apart from studies of the structure, the Poisson’s ratio as a sensitive indicator of the phase transition was applied. In addition, it was found that the studied Yukawa systems with nanoinclusions for certain pressure values are completely non-auxetic, despite both the Yukawa and hard sphere crystals without inclusions are partially auxetic at the same conditions. This indicates that the presence of  nanochannels in the system not only can enhance auxeticity in comparison to the system without nanochannels but also, at some thermodynamic conditions, can lead to a completely non-auxetic behavior of the system which is partially auxetic without the nanochannels. Hence, one can use nanochannels to tune auxetic properties of crystals.
... However, no clear comparisons have been made that provides a more direct indication on which structures would be more suited to each specific practical application. There are a few papers that mention the implications of exploiting auxetic configurations, stressing the fact that auxetic solutions require design considerations that may not be present in more conventional manufactures, like for example auxetic nails by Ren et al. . Although some of the problems encountered by Ren et al. in creating an auxetic might not be as prominent for cases that already use cellular structures, it is worth to remember that most of the studies published on the topic have been mainly theoretical. ...
The research field of auxetics, materials or structures exhibiting a negative Poisson's ratio, has received attention because of the unusually advantageous material properties that can be achieved with it, such as high indentation resistance and high shear resistance. In the past decades, the theoretical understanding of different factors that can lead to an auxetic behaviour has advanced greatly, resulting in a rapid increase in the number and type of the structures designed to exhibit this behaviour. These now exploit a number of different mechanisms, providing a large selection of properties which can be tailored for the specific needs. This review aims to describes the auxetic structures that have currently been identified and designed, describing the different approaches utilised to define their mechanical behaviour and analysing their structural properties, limitations, and potential field of application. In particular, the focus lies on the major works within the field, discussing their limitations and addressing works done to complement them.
... In addition, ultrasonic attenuation , tendency for synclastic surfaces [138,139], variable permeability , and vibration control  are some of the other beneficial characteristics of auxetic metamaterials. By tailoring these properties through different internal architectures, auxetic metamaterials have been exploited in various sectors including protective devices , sensors , textiles , fasteners , and biomedical devices . ...
Auxetics are a class of structural metamaterials with a negative Poisson’s ratio. In comparison to conventional structures, auxetic structures are proven to exhibit several superior properties including higher energy absorption, enhanced indentation resistance, and improved mechanical properties. As a result, auxetic structures are gaining popularity as lightweight, high-performance protective structures to resist blast and impact loads. Although several past reviews on auxetics partially covered different aspects of auxetics, such as classification of architectures, mechanisms, mechanical properties, energy absorption behaviours, and applications, this field still requires a comprehensive overview of the anti-blast and -impact performances of the different auxetic structures. Therefore, this paper aims to review recent advances in auxetic-based lightweight, high-energy absorbing protective structures. Different design factors affecting the performance of the protective structures are critically discussed. Moreover, a classification of modern auxetic topologies, the status of large-scale auxetic fabrication, and anti-ballistic performance evaluation methods are presented in this paper. Overall, auxetic core sandwich panels offer superior protective performance than equivalent conventional protective armours. However, several limitations and challenges exist with the design, fabrication, and implementation of auxetic-based protective structures.
... Auxetic material behavior: structures: solids that exhibit a negative poisson's ratio are called auxetic materials , a metamaterial engineered through special microstructure design, where the mechanical properties can be adjusted over a wide range and have significant auxetic behavior - , the property of expanding contracting transversely while being tensioned compressed in the longitudinal direction , - . Improves various yields due to its unusual property -  isotropic , has been the subject of intensive research, where several different auxetic systems have been commonly based on designing special geometry of the material microstructure -  show better indentation resistance, impact shielding capability, and enhanced toughness , , as well as various design techniques based on finite elements have been developed to achieve auxetic materials with specific effective properties, mainly in a linear deformation regime . ...
span lang="EN-US">This study reviews and analyzes the different auxetic materials that have been developed in recent years. The search for research articles was carried out through one of the largest databases such as ScienceDirect, where 845 articles were collected, of which several filters were carried out to have a base of 386 articles. There are a variety of materials depending on their structure, composition, and industrial application, highlighting biomedical applications from tissue engineering, cell proliferation, skeletal muscle regeneration, transportation, bio-prosthesis to biomaterial. The present paper provides an overview of auxetic materials and its applications, providing a guide for designers and manufacturers of devices and accessories in any industry.</span
... According to the geometrical micro-structure, auxetics can be divided into the following categories: foam , re-entrant honeycomb , chiral and anti-chiral structures , etc. Different structural forms of auxetics can be applied in many fields according to their characteristics, such as cushion pads , nails , intelligent filters [27,28] and sensors [23,30], acoustic engineering [31,32] and civil engineering . Although auxetics could be used in many fields, the disadvantages of low early stiffness are often exposed. ...
A novel gradient auxetic tube is proposed, which is characterized by a gradual change in cross-section from top to bottom. Based on a verified numerical model, the mechanical properties of gradient auxetic tubular structure and traditional auxetic tubular structure under axial and inclined loads are analyzed through the finite element method. The results show that gradient auxetic tubular structures have higher overall stability, higher energy absorption and higher specific energy absorption under inclined loads. When the difference of wall thickness between the upper and lower sections of the gradient auxetic tubular structure is larger, the advantages of high energy absorption and specific energy absorption would be more prominent. Due to their excellent energy absorption properties, thickness gradient auxetic tubular structures have great potential for applications in civil engineering, vehicle crashworthiness and protective infrastructure.
... Auxetic materials have a counter-intuitive deformation characteristic of becoming thinner (fatter) when compressed (stretched). The properties of auxetic structures are beneficial to designing and fabricating honeycomb structures , sandwich panels [15,16], composite structures , and tubular structures , etc. Due to the various structural forms and mechanical properties of auxetic structures, they are promising in protective equipment , medical facilities , civil engineering  and aerospace engineering , etc. ...
Compared to traditional materials, auxetic materials possess superior bending resistance and energy absorption properties due to their special deformation mechanism. To take advantage of these desirable properties, a novel composite tube with auxetic honeycomb filler was proposed in this work. The honeycomb filler of the composite tube was manufactured by 3D printing technology. The bending properties of two kinds of tubes were studied, namely aluminum tube with re-entrant honeycomb filler and aluminum tube with hexagonal honeycomb filler, respectively. The mechanical responses, deformational characteristics, and failure mechanism of the composite tube under three-point bending were studied by experimental and numerical methods. The study showed that the experiment and numerical results match well with each other. In addition, parametric analysis was carried out to examine the effects of filler and aluminum alloy tubes on the bending properties of the composite tube. Finally, kriging model and the non-dominated sorting genetic algorithm (NSGA-Ⅱ) are used to optimize the design of composite tubes, and the parameter solution with better energy absorption capacity is obtained.
... One of these solutions can be changing or modifying the 2D constructing geometry of these structures. The important characteristic of auxetic metamaterials, which is negative Poisson's ratio, has given these metamaterials unique properties . As shown in Fig. 1, the auxetic metamaterials undergo expansion (contraction) under uniaxial tensile (compressive) loading [14,. ...
Selection of appropriate topology to use in the core of the sandwich structures is one of the serious challenges ahead to design of them. This study aims to investigate the influence of auxetic core topologies on the mechanical characteristics of fully integrated 3D printed polymeric sandwich structures. Specifically, three types of auxetic cores including square node anti-tetra chiral, re-entrant, and arrowhead were investigated and compared with the conventional honeycomb in terms of energy absorption, compressive strength, and Young’s modulus. The specimens were fabricated using FDM 3D printing method and quasi-static compression and low velocity impact loading tests were performed on the printed specimens. Moreover, finite element simulations were conducted to compare with experimental results and deformation patterns as well as for parametric study. Results indicate that the core topology is a critical parameter impressing the mechanical properties of sandwich structures, and using the auxetic cores improved the desirable properties of sandwich structures in both types of loading. Auxetic cores cause the sandwich structure to have more resistance to imported compression, while conventional honeycomb doesn’t have this superiority.
... Auxetic tubular structures exhibit excellent energy absorption properties  and can be used as energyabsorber. Studies have also shown that stretched tubular structures can be used as fasteners and nails [56,57]. ...
Auxetic materials and structures have received extensive attention due to their unusual behavior. As one of the structures that can realize the auxetic effect, the elliptic perforated structure has been recently developed, but the previous research on the elliptic perforated is limited to the small deformation stage. At the end of the small deformation stage, the elliptic perforated structure becomes dense and loses the auxetic effect. The available compression stroke is very short and therefore the structure has relatively lower specific energy absorption (SEA). In order to make better use of material, through the buckling of lightweight perforated plates, a newly designed re-entrant elliptical perforated structure is proposed in this work. The re-entrant cells rotate and move inward (the first stage), and finally be compressed by neighbor ones (the second stage). Therefore, the structure undergoes rotational deformation in stage one same as previous perforated plates but specific re-entrant deformation in the second stage, thereby realizing the auxeticity in large deformation. Experimental and numerical results show that the stress–strain curve exhibits double platforms, and the novel structure possesses the auxetic effect throughout the whole compression process. Subsequently, a parametric analysis of the re-entrant distance and the ratio of horizontal and vertical wall thickness was carried out for obtaining greater auxetic effect and better stability of the novel re-entrant elliptic perforated structure. The material utilization rate of the re-entrant elliptic perforated plate was greatly improved, and the application prospect of the re-entrant elliptic perforated was also discussed.
... Metamaterials (cellular structures) represent a unique opportunity for adoption in lightweight design, due to their favourable characteristics related to different aspects of engineering design [5,6]. Generally, cellular structures are a relatively new class of metamaterials that present a unique opportunity for adoption in lightweight structures in modern engineering practice . The most important structural feature of the cellular structure is the relatively high stiffness with respect to the high porosity (low density) of the structure. ...
An investigation of the fatigue behaviour of the re-entrant auxetic structures made of the aluminium alloy AA 7075-T651 is presented in this study. The analysed auxetic structures represent a new class of cellular structures that show anomalous deformation responses such as negative Poisson’s ratio. In the proposed work, the influence of the unit cell orientation on the crack path and fatigue life was studied using experimental and computational approaches. The Low Cycle Fatigue (LCF) tests were performed at load control with the load ratio 0.1 in tension. In the LCF tests five loading levels were selected, and at least two tests were performed at each loading level. The same loading conditions were then applied in the computational model in the framework of the ANSYS software package, where a nonlinear kinematic material model was applied to obtain the stress–strain relationship. The strain-life approach, with consideration of the Morrow mean stress correction, was then used to obtain the fatigue life of the analysed auxetic structures. The experimental and computational results showed that the unit cell’s orientation has a minor influence on the fatigue life of both analysed auxetic structures, but impact on the direction of the fatigue failure path significantly.
... Auxetic structures have demonstrated good potentials to be used as auxetic nails , tubular  and energy absorption structures [54,12,55,56,57]. Moreover, recent studies show that the auxetic structures can play an important role for electrical-skin (e-skin) applications due to their extraordinary performance in controlling the electronic responses and conforming to human skins [58,59,60]. ...
Taking advantage of the powerful design potentials of isogeometric analysis, an integrated shape and size optimization framework for designing tetra chiral and anti-chiral auxetics is proposed to be capable of obtaining excellent designs without complicated implementation efforts. The framework utilizes a non-uniform rational basis spline (NURBS) based parametrization method that describes the chiral and anti-chiral structures with a small number of size and shape parameters. With this effective framework, systematic design studies considering both plane strain and stress conditions are performed to provide bounding graphs for the best achievable auxeticities under different stiffness requirements. Designs with tunable effective properties are also provided to demonstrate the capabilities of the proposed framework. The potential for electronic-skin applications is illustrated.
... So far, various auxetic structures have been systematically investigated, e.g., re-entrant structures , rigid rotation structures [31,32], chiral structures [33,34], bucklinginduced structures , etc. In particular, as a special part of auxetic structures, auxetic tubular structures require further research with the intention of capitalizing on the excellent auxetic behaviour for multi-functional applications . ...
Concrete-filled stainless steel tube (CFSST) members take advantage of the high strength and the outstanding corrosion resistance to act as an important role in civil engineering structures. However, the steel tube could not provide the perfect confinement effect for the core concrete during the initial elastic compression stage because Poisson’s ratio of the concrete is smaller than that of the stainless steel tube (SST). In this paper, a novel concrete-filled auxetic stainless steel tube (CFASST) composite structure was designed and manufactured to actively restrain the concrete, making the best use of the desirable deformation characteristics of auxetic tubular structures. The axial compressive performance of these CFASST members and their control factors were investigated experimentally and numerically. Test results were discussed in detail which included failure modes, load versus displacement curves and strain analysis. Finally, parametric analyses were conducted to further study the effects of different parameters (Poisson’s ratio, thickness of the stainless tube) on the CFSST composite structure under axial compression. It was found that CFASST composite structures possess an unusual deformation mode and an improved confinement effect.
... These structures shrink laterally under uniaxial compression. It is a typical mechanical metamaterial, which has advantages over traditional materials in terms of shear bearing capacity, fracture resistance, energy absorption, and indentation resistance . In 1991, Evans  first defined the term 'auxetic' to refer to materials with a negative Poisson's ratio. ...
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.
... Some studies have examined using Metamaterial structures in anchors 15,16 , composite structures, and reinforced concrete. Research on reinforced concrete seeks to offer improvements to composite structure strength via improved roughness or chemical adherence, or by adding ribs  . ...
Reinforced concrete beam (RCB) elements show low mechanical performance when interfacial bonding strength (IBS) is not well controlled. New tailorable material-structure arrangements - Metamaterials - offer solutions to the IBS problem. This paper analyzes the mechanical characteristics of IBS on RCBs for reinforced cement mortar containing Metamaterial bars (MMB) that were machined from SAE 1020 Carbon steel. Each MMB has a stepped geometrical shape, with a cylindrical bar divided into equal-length segments, along with a ’rise height’ (p) change. Four geometries were defined, i.e., R0− Smooth bar, R1− p = 0.1mm, R2− p =0.3mm and R3− p=0.5mm. Three-point flexural strength tests were performed on the RCBs to determine the maximum bond strength (ML) between the MMB and cement mortar. Images of interfacial regions were obtained using SEM and 3D Roughness Reconstruction software to calculate the average roughness (Ra) and the roughness height (Rz). The reinforcement MMB geometry had a significant impact on the ML results, particularly on the first crack strength and the failure mode. The R3 geometry ML values were higher than the other tested geometries (44.5%). The results of the scale models are encouraging and offer a novel and prospective direction for further experimental and even numerical Metamaterial research to improve interfacial bond strength.
... Metamaterials can have some extraordinary properties, such as being lightweight, 1 having excellent shock absorption and energy absorption, 2 and a negative Poisson's ratio. 3 The lattice metamaterial is an important member of mechanical metamaterials. Its mechanical properties have been extensively investigated during the last few decades. ...
Planar lattice metamaterials, such as periodic beam networks, are often considered as the micropolar continuum, where each material point has two translational degrees of freedom and one rotational degree of freedom. The joints through which bars are linked to one another are generally approximated as rigid. This study focuses on lattices with complex-structured deformable joints. The deformation field in each joint is obtained by conducting structural analyses. Once the “stiffness matrix” of the joint-centered unit cell is obtained by the finite element method, it can be used as the input for the standard procedure of calculating micropolar elastic moduli that are based on the equivalence of strain energy. As a result, effective moduli can be expressed in a semi-analytical form, meaning that only the cell structural stiffness is given numerically. The present model is validated by comparison to the FEM simulations. Particularly, the auxetic and anisotropic properties are discussed for various lattice metamaterials with deformable joints. We then take the obtained effective moduli as inputs to the in-house micropolar FEM code and obtain results agreeing well with the FEM structural simulations.
... Through the architectural design of the unit cell, various unusual mechanical properties-such as an NPR [6,7], high stiffness-to-weight ratio , vanishing shear modulus , negative stiffness [14,15], and negative compressibility -can be attained. Because of their extraordinary mechanical properties, mechanical metamaterials have been studied for a broad range of applications, ranging from auxetic textiles  and auxetic nails  to smart filters , vibration dampers [14,20,21], and protective devices . These properties, however, often require complex threedimensional (3D) topologies and small feature sizes, making it a challenge to be realized via conventional manufacturing techniques. ...
Mechanical metamaterials can be defined as a class of architected materials that exhibit unprecedented mechanical properties derived from designed artificial architectures rather than their constituent materials. While macroscale and simple layouts can be realized by conventional top-down manufacturing approaches, many of the sophisticated designs at various length scales remain elusive, due to the lack of adequate manufacturing methods. Recent progress in additive manufacturing (AM) has led to the realization of a myriad of novel metamaterial concepts. AM methods capable of fabricating microscale architectures with high resolution, arbitrary complexity, and high feature fidelity have enabled the rapid development of architected metamaterials and drastically reduced the design-computation and experimental-validation cycle. This paper first provides a detailed review of various topologies based on the desired mechanical properties, including stiff, strong, and auxetic (negative Poisson’s ratio) metamaterials, followed by a discussion of the AM technologies capable of fabricating these metamaterials. Finally, we discuss current challenges and recommend future directions for AM and mechanical metamaterials.
... Specifically, auxetic foam materials contract (expand) laterally under uniaxial compression (tension) . Along with the uncommon behavior, auxetic foam material has wide application in many fields  due to its superior shear resistance , indentation resistance , fracture resistance , energy absorption  and vibration isolation . ...
Auxetic foam materials contract (expand) laterally under uniaxial compressive (tensile) load. Due to superior characteristics of auxetic foam, e.g., shear resistance and in-plane indentation resistance, studies of auxetic foam composites have been increasing in recent years. In this paper, a novel cement-based auxetic foam composite is designed, fabricated and experimentally investigated. The influence of foam hole density, mass fraction and age on the flexural and compressive strength of the composite is analyzed. The failure modes and crack development of the specimen are examined. It is found that the flexural and compressive strength of composite are improved at the curing age of 7 and 14 days, and reduced at the curing age of 28 days with the incorporation of auxetic foam. And the flexural compression ratio of the composite is greater than that of matrix material. The integrity of the specimen is preserved during the compression failure process of cement-based auxetic foam composites. It is indicated that the incorporation of auxetic foam improves the toughness and deformation behavior of composites.
... AM, thanks to its flexibility, can be used to produce any kind of topology allowing the exploitation of lightened structures such as lattices and reticular. These latter structures have typically very small dimensions and, under operation, local plastic deformations could often not be avoided . ...
... Generally, cellular structures are a relatively new class of meta-materials which present a unique opportunity for adoption in light-weight structures, which are useful in modern engineering practice (X. Ren (2018), Hou (2015), Grima (2006), Meena (2019) and Zhao (20018)). Auxetic structures are a new type of porous material that exhibit a negative Poisson's ratio. The negative Poisson's ratio is a consequence of rotating cells in the geometry of the auxetic structure when an external load is applied ). The advanced geometrical possibilities of au ...
The investigation of fatigue behaviour of the re-entrant and rotated re-entrant auxetic specimens made of Al-alloy 7075-T651 is presented in this study. In the proposed work, the influence of the orientation of the unit cell on the crack path and fatigue life were studied by using both experimental and computational approaches. In the experimental and computational approach, two geometric layouts of the base unit cell were analysed (re-entrant and rotated re-entrant structure). The Low Cycle Fatigue (LCF) tests were performed at load control with the load ratio 0.1 in tensile. In the LCF tests, five loading levels were selected and at least two tests were performed at each loading level. The same loading conditions were then applied in the computational model in the framework of the ANSYS software package. For the fatigue life calculation, the strain life approach was used, based on the Coffin-Manson model with a Morrow mean stress correction. The comparison between computational and experimental results regarding Fa − N fatigue-life curves and observed fatigue failure path showed a reasonable agreement.
... The re-entrant unit cells were mapped on the tube's surface and studied using the finite element simulations in . The rotating rigid units were also considered for the auxetic tubular structure , as well as auxetic chiral unit cells [22e24], perforated sheets [25,26], and W geometry . At that point, it should be noted that by extending the auxetic 2D geometry in the radial direction, the extension should be relatively limited to retain the auxetic effect and prevent progressive and non-controlled buckling. ...
Novel three-dimensional (3D) axisymmetric chiral structures with negative and zero Poisson's ratios are presented based on the existing 3D conventional chiral unit cell. The conventional tetra-chiral unit cell is mapped to the axisymmetric space to form the new 3D axisymmetric chiral structure. Two different structure designs are characterised depending on the period delay of the sine curve representing the horizontal struts of the structure. The structures are fabricated using additive manufacturing technology and experimentally tested under compression loading conditions. The digital image correlation methodology is used to determine the Poisson's ratio dependence on the axial strain. The computational model of axisymmetric chiral structures is developed and validated using the experimental data. The computational model is then used to evaluate the new virtual axisymmetric chiral structures with graded cell structures. The newly developed axisymmetric structures show enhanced mechanical properties when compared to the existing 3D chiral structures.
... The ever growing interest in auxetics was sparked by the early theoretical  and experimental  studies performed in the 1980s. This interest is motivated by the vast potential applications  of materials that expand their transverse dimensions when stretched longitudinally (to point only one highly characteristic feature ). Since their discovery, auxetics have been extensively studied both theoretically  by computer simulations  and experimentally . ...
Negative Poisson’s ratio materials (called auxetics) reshape our centuries-long understanding of the elastic properties of materials. Their vast set of potential applications drives us to search for auxetic properties in real systems and to create new materials with those properties. One of the ways to achieve the latter is to modify the elastic properties of existing materials. Studying the impact of inclusions in a crystalline lattice on macroscopic elastic properties is one of such possibilities. This article presents computer studies of elastic properties of f.c.c. hard sphere crystals with structural modifications. The studies were performed with numerical methods, using Monte Carlo simulations. Inclusions take the form of periodic arrays of nanochannels filled by hard spheres of another diameter. The resulting system is made up of two types of particles that differ in size. Two different layouts of mutually orthogonal nanochannels are considered. It is shown that with careful choice of inclusions, not only can one impact elastic properties by eliminating auxetic properties while maintaining the effective cubic symmetry, but also one can control the anisotropy of the cubic system.
... Despite of its ability to generate unexpected material properties, it poses so strict manufacturing requirement due to its intricate interior hollow surfaces that makes it impossible for the conventional manufacturing process to handle . Compared to subtractive manufacturing and casting manufacturing, the development of additive manufacturing makes it possible to fabricate lattice structures of complex repetitive unit cells with high controllability . ...
In this work, we developed a Generalized Bayesian Regularization Network (GBRN) approach that can quantitatively identify the defect shapes and locations by mapping the distorted lattice structure to its original designed configuration, making registration between manufactured parts with defects and the perfect design models in the preliminary design stage of 3D printing.. On the one hand, it shows the proposed GBRN method has quantitatively comparable accuracy to the Coherent Point Drift (CPD) method in 2D boundary points registration problems. On the other hand, we have shown that the proposed GBRN method can find the possible geometric defects in the 3D printed lattice structure model and identify inherent defect-prone lattice structure parameters with obvious advantages over those two-dimensional point registration methods, i.e., coherent point drift (CPD) method, in registration of interior points of 3D lattice structures.
... Auxetic effect of honeycomb structures can be achieved by introducing specific cellular structures and periodically arranging these cells. Ren et al.  proposed a method to regulate the mechanical properties of auxetic structures and designed a tube with auxetic effect, and then manufactured the first nail with auxetic effect. Zhang et al.  proposed a novel type of tubular structure, which is the first tubular structure with auxeticity both in the wall thickness and in the radial direction. ...
Auxetic materials exhibit desirable mechanical properties, e.g., fracture resistance, shear resistance and energy dissipation due to their unique deformation characteristics. Honeycomb sandwich structures also have superior properties, e.g., energy absorption performance and low density. These desirable characteristics enable them as promising alternatives for building construction to meet the modern construction requirement which a higher safety standard is needed. In this study, a novel concrete composite with auxetic layered honeycomb was designed, manufactured and investigated for exploring the load-bearing performance of auxetic metamaterials under quasi-static compression. The layered honeycomb sandwich structure was analyzed by quasi-static compression test and its energy absorption performance was evaluated. Numerical models, validated by experimental results, were employed in parametric studies to further examine their performance. The results show that the layered structure has higher initial peak stress and stable platform stress. Moreover, it could improve the shear resistance and overall stability of the structure. These findings are beneficial to the applications of auxetic metamaterials in building construction.
... This class of mechanical metamaterial has been shown to possess superior properties compared to conventional ones, including, amongst others, larger shear stiffness [2,3], enhanced indentation resistance , energy absorption , and vibration control [2,. As a result they have been proposed for a wide range of applications . ...
The deformation behavior of intersecting ligaments forming variants of the square and rectangular grids under mechanical compression was investigated. It was shown that such systems are able to exhibit a negative incremental Poisson’s ratio at relatively large axial compressive strains. Numerical simulations and experimental studies indicated that the extent of auxeticity depends on the relative offset of successive ligaments, the relative lengths of the ligaments as well as on their thickness. It was also shown that there are two distinct modes of deformation, one resembling that of the reentrant hexagonal honeycomb and the other that of the meta-tetrachiral system.
... Due to this unusual deformation mechanism, auxetic structures have various excellent mechanical properties, such as indentation resistance , shear resistance , fracture resistance [7,8], and energy absorption capability . These superior properties were utilized in many applications for medical equipment , protective devices , sensors , civil engineering , etc. ...
Auxetic materials and structures have many potential applications due to their counter-intuitive deformation behavior and desirable mechanical properties. However, auxetic structures have some drawbacks, such as relatively low stiffness and stability. To improve their mechanical performance, one common method is to fill soft materials in auxetic frames, i.e., auxetic two-phase composites. In this paper, a novel method to further enhance the mechanical properties of auxetic composites has been proposed, which is implemented by designing joints of frames. Mechanical properties and deformation characteristics of these novel composites and their conventional counterparts are investigated experimentally and numerically. The results of finite element analysis and experiments exhibit a good agreement. The energy absorption capacity and auxeticity of the proposed composites could be enhanced by optimizing the design of joints. Subsequently, parametrical studies are conducted to quantify the effects of the geometrical parameters and the volume fraction of the frame.
... The new type of cellular structures are the auxetic cellular structures  which demonstrate the negative Poisson's ratio due to rotating of united cells when an external load is applied [12,13]. Different groups of the auxetic materials such as Auxetic honeycombs , Auxetic microporous polymers  and Auxetic composites [24,25] are using increasingly in the Aerospace and Automobile Industry . Because of dynamic loading of many engineering components integrated into the different engineering structures in the Aerospace and Automobile Industry, many researchers were focused on the experimental and/or computational study regarding the fatigue behaviour of cellular structures . ...
This study presents the experimental and computational analysis for determining the fatigue life of the auxetic cellular structures made of aluminium alloys. For the fatigue life calculation, the strain life approach was applied in the framework of the ANSYS software. The obtained experimental results were evaluated using the comprehensive statistical analysis, and presented in the form of different fatigue-life curves.
Based on the reasonable agreement between computational and experimental results the proposed computational model was validated and can be used further for the fatigue studies of various auxetic structures made of Al-alloys.
... Auxetic metamaterials exhibit good mechanical properties  such as high indentation resistance, large shear modulus, large fracture toughness, natural tendency to adopt synclastic curvature when bent along the out-of-plane direction, variable permeability, fatigue properties and energy absorption, good acoustic and thermal absorption. Since the first report of auxetics in 1985 , auxetic metamaterials have attracted an increasing interest and been proposed to be used in various applications such as cores for sandwich panels [11,12], medical stents , smart filters , sound absorbers , vibration dampers , textiles , seat cushions , human protection equipment , nails and screws [20,21], etc. ...
This paper reports a structural modification of an auxetic metamaterial with a combination of representative re-entrant and chiral topologies, namely, a re-entrant chiral auxetic (RCA). The main driving force for the structural modification was to overcome the undesirable properties of the RCA metamaterial such as anisotropic mechanical response under uniaxial compression. Additively manufactured polyamide 12 specimens via Multi Jet Fusion (MJF) were quasi-statically compressed along the two in-plane directions. The experimental results confirmed that the modified structure was less sensitive to the loading direction and the deformation was more uniform. Moreover, similar energy absorptions were obtained when the modified metamaterial was crushed along the two in-plane directions. The energy absorptions were improved from 390 to 950 kJ/m³ and from 500 to 1000 kJ/m³ compared with the RCA when they were crushed along the X and Y directions, respectively. The absorbed energy per unit mass (SEA) also improved from 1.4 to 2.9 J/g and from 1.78 to 3.1 J/g compared with that of the RCA under the axial compression along the X and Y directions. Furthermore, parametric studies were performed and the effects of geometric parameters of the modified metamaterial were numerically investigated. Tuneable auxetic feature was obtained. The energy absorption and Poisson’s ratio of the modified metamaterial offer it a good alternative for a wide range of potential applications in the areas such as aerospace, automotive, and human protective equipment.
... Along with these research productions, auxetic materials have been proved to process excellent impact resistance, fracture resistance, shear resistance and energy dissipation properties, etc. Therefore, auxetic metamaterials have great potential in the field of buffer energy absorption, medical instruments, artificial intelligence, e.g., buffer cushion , esophageal stent , nail , intelligent sensor . ...
Auxetic materials have attracted a considerable attention due to their excellent properties, e.g., fracture resistance, shear resistance, energy dissipation, etc. However, the stiffness of auxetics tends to be much weaker than solid structure because of the existence of internal holes. Inspired by tuning the compacted point of auxetic structures to enhance their stiffness, a systematic methodology for defining a single parametric of variable stiffness scale factor (VSF) to generate auxetic unit cell with variable stiffness has been proposed and verified in this study. Two models with different VSF proportions were investigated experimentally. Different centres of rotation, heights of deformation area, and VSF percentages were analyzed to prove the effectiveness of the method numerically. The results indicate that the compacted strain can be tuned effectively using the designed VSF proportions, and the difference between the designed VSF and real VSF could be reduced by slightly changing the height of deformation area. These desirable characteristics provide a new idea for the optimal design of auxetics, with potential application in protective structures.
... According to the uncommon feature of auxetics, Ren et al.  designed and manufactured the first auxetic nail for easier push-in and harder pull-out. Likewise, because of the higher energy absorption properties which is equipped by auxetics, Qi et al.  invented a novel re-entrant circular auxetic honeycombs. ...
Auxetic materials could exhibit desirable mechanical properties, e.g., fracture resistance, shear resistance and energy dissipation due to their unique deformation characteristics. However, the seismic performance of auxetics in disaster prevention and mitigation of building structures is rarely studied. In this study, a novel perforated negative Poisson’s ratio core buckling-restrained brace (NP-BRB) was designed and manufactured for exploring the hysteretic performance of auxetic metamaterials under cyclic load. Experiments and verified numerical simulations were conducted to investigate the effects of porosity and section weakening rate on the seismic performance. The results show that the NP-BRB has stable hysteretic curves and low compression strength adjustment factor. In addition, the parametric analysis indicated that the energy dissipation capacity of NP-BRB with relatively large section weakening rate could be improved due to auxetic behavior when the average strain exceeds 1%. These findings are beneficial to the applications of auxetic metamaterials in damping devices for mitigating seismic effects.
... It also possesses superior bending performance [30,31] and interesting extension twist deformation response [32,33]. Utilizing its special deformation characteristics, auxetic tubular structure could be used in fastener and nail [34,35]. Besides, it could be used for the production of angioplasty and oesophageal stents in the medical field [36,37]. ...
Auxetic materials exhibit interesting deformation characteristics and excellent mechanical properties. A novel combined tubular structure with tunable stiffness is proposed in this work, aiming to improve the bearing capacity and stability by length design of the central column. Specimens were fabricated via 3D printing technique. Experimental test was performed to study their mechanical property and deformation characteristics under uniaxial compression. The validity of the finite element model was proved by comparing the experimental result with simulation prediction. The compression process and stress-strain curve of the tubular structure with tunable
stiffness exhibited four distinct stages (elastic, stiffness change, densification and buckling). Subsequently, a parametrical analysis was conducted to investigate the influences of the central connecting column on the stressstrain response, Poisson’s ratio and stability of the structure. By properly choosing the length of the central connecting column, the tubular structure could possess tunable stiffness, higher stability and compressive capacity. Furthermore, this design concept could be of benefit to the development of adaptive structures, smart devices and applications for civil engineering and protective engineering.
Auxetics are mechanical metamaterials with the unique properties of expanding their transversal section upon longitudinal positive strain, decoupling the deformations in normal and transversal directions. Such property can be exploited to develop soft sensors that can provide feedback to different mechanical stimuli, e.g. pressure and shear force. In this work, we propose for the first time a mathematical model to analytically simulate and design the auxetic behavior in a capacitive strain gauge, and show that, for a Polyurethane (PU) auxetic foam, Poisson Ratio’s values can satisfy the negative gauge factor condition. We develop an innovative thermo-compressive process to obtain anisotropic auxetic polyurethane sponges both in normal and normal/radial directions, and their mechanical properties are in agreement with the theoretical calculations validating our model. Then, we develop a capacitive strain gauge by integrating a normal auxetic PU foam with PDMS/CNTs electrodes. Results show that the capacitive change caused by an external force, is proportional to the induced deformation, but importantly it is also dependent on the direction of the applied force. A negative gauge factor of GF = -2.8 is obtained for a longitudinal strain range up to 10%. This auxetic foam structure guarantees flexibility and paves the way for an improved design freedom for multimodal mechanical soft sensors providing new opportunities towards smart wearables and perceptive soft robots.
Auxetic materials or structures possess a negative Poisson's ratio in contrast to conventional materials, and they shrink or expand transversely under uniaxial compression or tension, respectively. These unique deformation features leads to enhance the mechanical properties compared with the conventional materials. Auxetic tubular structures are of significant interest in the literature because of their superior mechanical qualities, applicability and extensive application. Various auxetic tubular structures with different geometries have been proposed and examined before including conventional peanut-shaped tubular structures. However, application of the peanut-shaped structures is limited due to their low stiffness. In this study, it is aimed to enhance the stiffness of the peanut-shaped tubular auxetic by either adding stiffener to the conventional structure or rotating the unit cell of the structure by a certain angle. Also, the effect of the above-mentioned modifications on the Poisson’s ratio of the structure is investigated. A total of twelve different peanut-shaped auxetics are modelled and the elastic behaviour of these structures under uniaxial compression is compared numerically using finite element simulation. As a result of this analysis, it is observed that both the Poisson’s ratio and stiffness values obtained from the models utilising stiffener were higher than the values obtained from their conventional counterparts. Besides, it is seen that the stiffness values increased while the Poisson’s ratios decreased with the rotation of the unit cell in all of the peanut-shaped tubular auxetics.
Carbon dioxide has a rich phase diagram boasting a number of different solid polymorphs. One such solid is CO2‐II, a high‐pressure polymorph of carbon dioxide. In this work the structural, mechanical and vibrational properties of CO2‐II have been studied by employing first principles DFT simulations. In particular, it has been shown that the CO2‐II single crystal has the potential to exhibit off‐axis auxetic behaviour (negative Poisson’s ratio) in the (010) and (001) plane. Moreover, the auxetic behaviour is pressure dependent in the pressure range of 15 GPa to 20 GPa, with an increase in pressure resulting in an increase auxetic behaviour, for loading in certain directions, of this system. This anomalous mechanical property was also studied through the use of Raman and IR spectroscopy. The results obtained suggest that the auxetic behaviour of CO2‐II may be the result of the interplay between rotations and distortions of the system. This article is protected by copyright. All rights reserved.
Water is known to have a complex phase diagram with numerous solid polymorphs, one of which is ice VIII, a proton ordered high pressure form of ice. In this work the mechanical properties of ice VIII were studied through the use of first principles density functional theory (DFT) simulations. In particular, it was shown that ice VIII has the potential of exhibiting a negative Poisson's ratio in the (110) plane on application of an on-axis stress. The auxetic behaviour of ice VIII was found to increase with an increase in hydrostatic pressure. This auxetic potential together with its pressure dependence was rationalised through the nanoscale deformation of this system when subjected to a series of stresses as well as through spectroscopic techniques. Our simulations suggest that the auxetic behaviour of ice VIII is due to the geometry of the oxygen sublattice in conjunction with an interplay between two deformation mechanisms namely stretching and hinging. As the hydrostatic pressure is increased the stretching mechanism becomes less significant resulting in the hinging mechanism becoming the predominate deformation mechanism.
An auxetic metamaterial consisting of a re-entrant honeycomb structure with hierarchical characteristics is proposed. The new structure is constructed by attaching small re-entrant structural unit cells to the nodes of the traditional re-entrant structures. Not only can the overall stiffness and stability of the proposed structure be tuned during compression and tension, but better acoustic performance is also obtained compared with traditional re-entrant honeycomb structures (T-RHS). Firstly, the deformation mechanism of the bandgap is numerically explored by analyzing the dispersion curve of the microstructure as well as the upper and lower bounds of the bandgap vibrational modes. Secondly, the bandgap tunability of the designed structure under uniaxial compression or tension is discussed. Finally, the transmittance of finite period size is calculated to verify the numerical results of the bandgap. Numerical simulation results show that the proposed novel re-entrant honeycomb structure with hierarchical characteristics (RHS-H) has attenuation characteristics of a tunable low-frequency plane wave through a reasonable selection of compressive strain, tensile strain and geometric parameters. The vibration damping strength of the bandgap increases under tensile strain. When the auxetic effect is enhanced, the first and second bandgaps become lower and wider. The novel metamaterial has potential applications in vibration and noise reduction and the design of acoustic devices in dynamic environments, while providing new ideas and a methodology for the real-time adjustment of bandgaps.
4D printing is an innovative manufacturing approach that combines 3D printing and stimuli- responsive abilities to produce objects with complex geometry and capable of shapeshifting over time (the fourth dimension). To pursue such an approach this paper proposes to develop re-entrant honeycomb auxetic grids with tunable shape reconfigurable behavior. Particularly, the work combines 3D printing and a photopolymer exhibiting the so-called temperature memory effect (TME), a peculiar shape memory behavior expressing the capability of the material to remember not only the original shape but also the deformation temperature. A thorough experimental activity was carried out on single auxetic unit cells, chosen as representative of the whole auxetic grid, to properly highlight and assess their response upon heating after single-step and multiple-step deformation histories and to describe the recovery process as a function of time and temperature. Results demonstrate the possibility to achieve an easily controlled TME and to successfully exploit it for autonomous, complex hierarchical transformations over a large range of temperatures. As a proof-of-concept, the study of the sequential recovery of an entire auxetic grid subjected to double-step programming allowed highlighting a decoupled in-plane elongation and out-of-plane bending. The behavior of the 4D-printed auxetic structures was simulated by means of finite element (FE) analysis, using a thermoviscoelastic model of the photopolymer and viscoelastic experimental data obtained by time-temperature superposition analysis applied to multifrequency dynamic mechanical tests and to isothermal recovery tests. A good correspondence between experiments and simulations was obtained for all shape memory tests, demonstrating that the proposed FE approach is a suitable tool to support the design of these structures. The combination of 3D printing and TME opens new perspectives to achieve dynamic tunability in mechanical metamaterials, that is a key ingredient in several application fields.
In the past decades, auxetic metamaterials have attracted extensive attention due to their desirable properties. In order to further improve the energy absorption properties of auxetic metamaterials, this work investigates the design of lightweight auxetic metamaterials based on the elliptic perforated plate. The material in the large deformation region is retained and the material in the small deformation region is removed as much as possible. The experimental and numerical results indicate that the mechanical properties of the novel metamaterials are almost unchanged, but the specific energy absorption is substantially enhanced owing to the reduced material overall and the distribution of the optimized materials to the most needed area. In this paper, the influence of the amount and mode of the material removal on the mechanical properties of the auxetic metamaterial is explored through the numerical method. The proposed auxetic metamaterials with lightweight characteristics have potential applications in many fields, e.g., civil engineering, aerospace and vehicle engineering.
Additive manufacturing technologies such as fused filament fabrication (FFF) allow the production of metastructures with global properties that can be tailored to their specific application. This study simulated and optimized an auxetic re-entrant structure with a stiffness gradient for enhanced energy absorption with low acceleration peaks under different low-velocity impact conditions. The finite-element method (FEM) was used, and appropriate constitutive models were fitted to static and dynamic tensile and compressive data of acrylonitrile butadiene styrene (ABS) tested under various strain rates. A Johnson–Cook plasticity model demonstrated the best compromise between accuracy and computational efficiency. A simulation strategy using explicit FEM was developed to simulate additively manufactured auxetic metastructures under impact conditions. There was good agreement between the model prediction and the experimentally observed structural response. A parametric optimization was implemented to enhance the energy absorption capability with low acceleration peaks of a graded auxetic re-entrant structure for different impact velocities.
Novel 3D printed square auxetic tubular lattice (SATL) structures were designed, fabricated and investigated. Their mechanical properties were examined by the finite element method and experiments. The height and wall thickness show different effects on the mechanical properties of SATL structures. Compared with the circular auxetic tubular (CATL) structures, the SATL structure has a lower peak force under axial load. Under lateral load, the SATL structure has higher stiffness and specific energy absorption. Moreover, the auxetic effect of the proposed SATL structure is also obvious under lateral load. Then, numerical investigations of several improved SATL structures were carried out, the results show that the improved square auxetic tubular lattice (ISATL) structures have stronger energy absorption capacity under axial and lateral loads. Due to their unique structural design and excellent mechanical properties, the SATL structures and ISATL structures have great potential for applications in civil engineering, vehicle crashworthiness and protective infrastructure.
Auxetic foams counter-intuitively expand (shrink) under stretching (compression). These foams can exhibit superior mechanical properties such as resistance to shear and indentation, improved toughness and energy absorption (EA) under several types of loadings. Their unique deformation mechanism and manufacturing process lead to special multiphysics properties such as variable permeability, synclastic curvature and shape memory. Except for traditional energy absorber stuff, the potential applications of auxetic foams have involved biomedicine, aerospace, smart sensing, etc. However, most of the potential applications are restrained in the theoretical stage due to complicated fabrication and a deficiency of stability. For removing the barrier for practical application, a series of issues remain to be resolved, though the explorations of the manufacture methodologies and potential applications are fruitful in the past decades. We present here a review article discussing the state-of-the-art for manufacturing, characterization and applications of auxetic foams. We also provide a view of the existing challenges and possible future research directions, aiming to state the perspective and inspire researchers to further develop the field of auxetic foams.
Two-dimensional star-shaped honeycombs (2D SSHs) exhibit an effective negative Poisson's ratio due to the abundant internal space and re-entrant angle, while the strength and the deformation of the three-dimensional (3D) SSHs are still limited owing to the finite tuneability of the tip angle. By adding different tip re-entrant angles into the SSH, multiple improved star-shaped honeycombs (ISSHs) with tunable Poisson's ratio are proposed. The in-plane elastic properties, including the effective Young's moduli and effective Poisson's ratio, are both derived by 2D analytical model using the energy method. The finite element simulation and compression experiment are used to verify the correctness of theoretical results. Based on the work, the deformation mechanism of the 3D ISSHs is discussed by quasi-static compression experiment and numerical simulation. The simulation results and experimental results show a great agreement with theoretical results. Different tip angles heighten the normalized Young's modulus and make the Poisson's ratio more tunable, respectively. In addition, the 3D ISSHs show an enhanced effect among higher strength and stability while bearing the compression load. This work provides a good reference for constructing 3D symmetrical multicellular structures, especially honeycombs.
In this study, the deformation and energy absorption characteristics of a novel hybrid tube is investigated both experimentally and numerically. The architecture of the proposed hybrid tube composed of an inner conventional tube and a co-axially arranged outer auxetic tube. Quasi-static compressive tests were conducted on the auxetic tubes, the conventional tubes and the hybrid tubes. A three-dimensional digital image correlation technique was used to monitor the evolution of radial contraction of the tubes. Measured performance of the tubes were compared in terms of force, energy absorption, and specific energy absorption. Finite element (FE) models were also developed for the three types of tubes, analyzed using ANSYS/LS-DYNA, and validated by experimental measurements. Both experimental and numerical results showed that the presence of the auxetic tube in the hybrid tube alters the deformation mode of the conventional tube by exerting a radial force. Parametric studies were performed to investigate the effects of the outer auxetic tube, wall thickness of the inner conventional tube, the failure strain, and yield strength of the outer tube’s material on mechanical performance of the hybrid tube. Increasing the yield strength of the outer auxetic tube improved both the energy absorption and specific energy absorption characteristics of the hybrid tube significantly.
In this paper, rigid polyurethane (PU) foam was filled into hollow auxetic tube for enhancing the energy absorption. The rigid PU foams and stainless-steel auxetic tube with different geometrical parameters were manufactured, respectively. They were then assembled to form composite foam-filled auxetic tube (FFAT). Four auxetic tubes and four FFATs were respectively tested to examine the enhancement of energy absorption. Tubular types and the effects of parameters including wall thickness, and ellipticity, on FFATs were analyzed numerically by using the validated models. The results show that the overall absorbed energy of FFAT is larger than the sum of single foams and hollow auxetic tube under compression. The geometrical parameters of wall thickness, ellipticity, have a considerable effect on the structural deformation mode and energy absorption. These findings could promote the applications of auxetic composite structures in protective engineering.
Auxetic materials have excellent mechanical properties, e.g., indentation resistance, shear resistance, fracture toughness and energy absorption. However, the stiffness of auxetics is normally lower than that of solid structures due to the existence of voids. In this study, to improve the mechanical properties of re-entrant honeycombs, a buffer material called slow recovery foam is filled into re-entrant honeycombs. The mechanical properties and deformation patterns of slow recovery foam-filled re-entrant honeycombs are investigated numerically and experimentally. Parametric studies are conducted to investigate the effects of geometrical parameters on the Poisson’s ratio and energy absorption capacity. The results show that filling foam into re-entrant honeycombs will prevent lateral buckling of the structure. Compared with foam-filled hexagonal honeycombs, foam-filled re-entrant honeycombs have higher stiffness. With the increase of the strain rates, the stiffness and energy absorption capacity of slow recovery foam-filled re-entrant honeycombs will increase. With the increase of cell wall thickness and the decrease of cell angle, the energy absorption capacity of slow recovery foam-filled re-entrant honeycombs will also increase. The results indicate that slow recovery foam-filled re-entrant honeycombs are promising in the field of protective engineering.
A new auxetic structure was designed with rigid diamonds incorporated to enhance the strength of a star-shaped auxetic structure along with the symmetry in the present work. The new structure was fabricated through Multi-Jet Fusion process and subjected to quasi-static compression tests for static analysis. Finite element model was developed using ABAQUS and validated by experimental results in terms of deformation mode, nominal stress-strain curve, and Poisson’s ratio of the auxetic structure. Theoretical model was also developed to predict the Poisson’s ratio of a unit cell of the designed auxetic structure. The theoretically predicted Poisson’s ratio is in tandem with that of obtained from a finite element model, which utilized periodic boundary conditions. Parametric study using the theoretical model revealed that the angle of the inclined members has shown a significant effect on the Poisson’s ratio. The newly proposed structure exhibited 36% better specific plateau stress than a conventional 3D star-shaped auxetic structure of similar geometrical parameters.
Equivalent sandwich panels composed of auxetic and conventional honeycomb cores and metal facets are analysed and compared for their resistance performances against impulsive loadings. The dynamic behaviours of these structures are numerically investigated, taking into account the rate-dependent effects. The Johnson-Cook model is employed to describe the dynamic responses of the composite sandwiches subjected to high strain-rate loadings. Analytical models are derived correlating unit cell geometrical parameters and crushing strengths of the representative panels at different impact velocities. Parametric studies are conducted to evaluate the performances of different sandwich panel designs under impulsive loadings. In particular, transmitted reaction forces and maximum stresses on the protected structure are quantified for various design parameters including the geometrical factors and the effective Poisson’s ratios. A quarter of the panel is symmetrically modelled with shell elements and the CONWEP model is used to simulate the blast loading. Auxetic panels demonstrate interesting crushing behaviour, effectively adapting to the dynamic loading by progressively drawing material into the locally loaded zone to thereby enhance the impact resistance. Meanwhile, conventional honeycomb panels deform plastically without localised stiffness enhancement.
Auxetic materials exhibit uncommon behaviour, i.e. they will shrink (expand) laterally
under compression (tension). This novel feature has attracted intense research interest. However,
most of previous works focus on auxetic behaviour in either compression or tension. Most of the
auxetic materials are not symmetric in tension and compression under large deformation. Studies on
the auxetic performance of metamaterials both in compression and tension are important but rare.
As an extension of our previous research on compressive auxetic performance of 3D metallic
auxetic metamaterials, numerical simulations were carried out to investigate the auxetic and other
mechanical properties of the 3D metallic auxetic metamaterials in tension. The preliminary results
indicated that the designed 3D metallic auxetic metamaterials exhibited better auxetic performance
in compression than in tension. By increasing a pattern scale factor, auxetic performance of the 3D
metallic auxetic metamaterials under tension can be improved. With proper adjustment of the
pattern scale factor, an approximately symmetric auxetic performance could be achieved in
compression and tension.
Auxetic materials and structures are increasingly used in various fields because of their unusual
properties. Auxetic tubular structures have been fabricated and studied due to their potential to
be adopted as oesophageal stents where only tensile auxetic performance is required. However,
studies on compressive mechanical properties of auxetic tubular structures are limited in the
current literature. In this paper, we developed a simple tubular structure which exhibits auxetic
behaviour in both compression and tension. This was achieved by extending a design concept
recently proposed by the authors for generating 3D metallic auxetic metamaterials. Both
compressive and tensile mechanical properties of the auxetic tubular structure were investigated.
It was found that the methodology for generating 3D auxetic metamaterials could be effectively
used to create auxetic tubular structures as well. By properly adjusting certain parameters, the
mechanical properties of the designed auxetic tubular structure could be easily tuned.
Metallic auxetic metamaterials are of great potential to be used in many applications because of their superior mechanical performance to elastomer based auxetic materials. Due to the limited knowledge on this new type of materials under large plastic deformation, the implementation of such materials in practical applications remains elusive. In contrast to the elastomer based metamaterials, metallic ones possess new features as a result of the nonlinear deformation of their metallic microstructures under large deformation. The loss of auxetic behaviour in metallic metamaterials led us to carry out a numerical and experimental study to investigate the mechanism of the observed phenomenon. A general approach was proposed to tune the performance of auxetic metallic metamaterials undergoing large plastic deformation using buckling behaviour and the plasticity of base material. Both experiments and finite element simulations were used to verify the effectiveness of the developed approach. By employing this approach, a 2D auxetic metamaterial was derived from a regular square lattice. Then by altering the initial geometry of microstructure with the desired buckling pattern, the metallic metamaterials exhibit auxetic behaviour with tuneable mechanical properties. A systematic parametric study using the validated finite element models was conducted to reveal the novel features of metallic auxetic metamaterials undergoing large plastic deformation. The results of this study provide a useful guideline for the design of 2D metallic auxetic metamaterials for various applications.
Sandwich panels with auxetic lattice cores confined between metallic facets are proposed for localised impact resistance applications. Their performance under localised impact is numerically studied, considering the rate-dependent effects. The behaviour of the composite structure material at high strain rates is modelled with the Johnson-Cook model. Parametric analyses are conducted to assess the performance of different designs of the hybrid composite structures. The results are compared with monolithic panels of equivalent areal mass and material in terms of deformations and plastic energy dissipation. Design parameters considered for the parametric analyses include the auxetic unit cell effective Poisson’s ratio, thickness of the facet, material properties and radius of the unit cell’s struts. Significant reduction in computational time is achieved by modelling a quarter of the panel, with shell elements for facets and beam elements for the auxetic core. With projectile impacts up to 200 m/s, the auxetic composite panels are found to be able to absorb a similar amount of energy through plastic deformation, while the maximum back facet displacements are reduced up to 56% due to localised densification and plastic deformation of the auxetic core.
Auxetic materials are endowed with a behavior that contradicts common sense, when subjected to an axial tensile load they increase their transverse dimension. In case of a compression load, they reduce their transverse dimension. Consequently, these materials have a negative Poisson’s ratio in such direction. This paper reviews research related to these materials. It presents the theories that explain their deformation behavior and reveals the important role represented by the internal structure. Their mechanical properties are explored and some potential applications for these materials are shown.
Describes development work to combine the basic ESO with the additive evolutionary structural optimisation (AESO) to produce bidirectional ESO whereby material can be added and can be removed. It will be shown that this provides the same results as the traditional ESO. This has two benefits, it validates the whole ESO concept and there is a significant time saving since the structure grows from a small initial one rather than contracting from a sometimes huge initial one where 90 per cent of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Presents a brief background to the current state of Structural Optimisation research. This is followed by a discussion of the strategies for the bidirectional ESO (BESO) algorithm and two examples are presented.
A constant thermodynamic tension Monte Carlo method is introduced and applied studies of the elastic properties of a two-dimensional system of hard cyclic hexamers. Elastic compliances and elastic constants are determined at a number of different values of the pressure. The existence of the phase transition between a tilted and a straight phase is confirmed. The results obtained strongly suggest that the Poisson modulus can be negative in the tilted phase.
Auxetics exhibit the anomalous property of getting fatter rather than thinner when uniaxially stretched [negative Poisson's ratio (NPR)]. This paper presents an analytical model of a structure constructed from umbrella type sub-units which is predicted to exhibit this property together with the results of mechanical tests of a structure having this geometry. It is shown that such systems can exhibit negative on-axis Poisson's ratio of −1 which means that they maintain their aspect ratio, while Poisson's ratio becomes less negative for loading off-axis. It is also shown that this model can explain the sign and anisotropy of Poisson's ratio demonstrated by a molecular level system constructed from calix(4)arenes having the same geometry.
Fracture toughness of re-entrant foam materials with a negative Poisson's ratio is explored experimentally as a function of permanent volumetric compression ratio, a processing variable. J
values of toughness of negative Poisson's ratio open cell copper foams are enhanced by 80 percent, 130 percent, and 160 percent for permanent volumetric compression ratio values of 2.0, 2.5, and 3.0, respectively, compared to the J
value of the conventional foam (with a positive Poisson's ratio). Analytical study based on idealized polyhedral cell structures, approximating the shape of the conventional and re-entrant cells, disclose for re-entrant foam, toughness increasing as Poisson's ratio becomes more negative. The increase in toughness is accompanied by an increase in compliance, a combination not seen in conventional foam, and which may be useful in some applications such as sponges.
Foam materials based on metal and several polymers were transformed so that their cellular architecture became re-entrant, i.e. with inwardly protruding cell ribs. Foams with re-entrant structures exhibited negative Poisson's ratios as well as greater resilience than conventional foams. Foams with negative Poisson's ratios were prepared using different techniques and materials and their mechanical behaviour and structure evaluated.
Although a negative Poisson's ratio (that is, a lateral extension in response to stretching) is not forbidden by thermodynamics, for almost all common materials the Poisson's ratio is positive. In 1987, Lakes first discovered negative Poisson's ratio effect in polyurethane (PU) foam with re-entrant structures, which was named anti-rubber, auxetic, and dilatational by later researchers. In this paper, the term 'auxetic' will be used. Since then, investigation on the auxetic materials has held major interest, focusing on finding more materials with negative Poisson's ratio, and on examining the mechanisms, properties and applications. Therefore, more materials were found to have the counter-intuitive effect of auxeticity due to different structural or microstructrual mechanisms. The present article reviews the latest advances in auxetic materials, their structural mechanisms, performance and applications.
Elastic instability has been increasingly used to design buckling-induced auxetic metamaterials. However, only limited patterns of existing three dimensional (3D) buckling-induced auxetic metamaterials exhibit a reliable 3D auxetic behaviour, i.e., the material will contract along two normal lateral directions under compression along the third direction. In this paper, we study a simple geometrical shape for achieving 3D auxetic behaviour. The unit cell of the proposed 3D design is composed of a solid sphere and three cuboids. Two representative models, one with slender connecting bars and the other with thick connecting bars, are investigated both numerically and experimentally. The results indicate that the designed material with thick connecting bars did not exhibit auxetic behaviour while the one with slender connecting bars did. However, the anticipated 3D auxetic behaviour degraded to a two dimensional (2D) auxetic behaviour, i.e., the material would contract only in one lateral direction and maintain nearly the same dimension in the other lateral direction. This finding was further confirmed by experiments. This research has demonstrated challenges in designing and manufacturing of buckling-induced auxetic metamaterials with 3D auxetic behaviour and highlighted the importance of optimising and fine-tuning the auxetic unit cell using the pattern scale factor.
The phenomenon of negative linear compressibility has attracted much interest because of its unusual deformation features with many potential applications. However, the design and fabrication of materials and structures with negative linear compressibility are limited. In this paper, we proposed two approaches to designing and fabricating new composite structures with negative linear compressibility. The effectiveness of the proposed design approaches was validated experimentally by applying uniformly distributed pressure to all surfaces of bulk specimens. The deformation features, strain history, and the effective area reduction of the specimens were analyzed from the experimental data. The results clearly demonstrated the feasibility of the proposed designing and manufacturing approaches for realizing composites with negative linear compressibility.
In this paper, a new kind of chiral three-dimensional honeycomb material was designed by orthogonal assembling based on chiral two-dimensional honeycomb with four ligaments. The analytical formulae of equivalent Young’s modulus and Poisson's ratio are deduced using the beam theory. The calculations of the analytical formulae can be well consistent with those of finite element method. The theoretical and numerical results show that the honeycomb material proposed in this paper is isotropic at the macroscopic scale and its Poisson's ratio is close to -1, which means the material have larger ratio of shear modulus to Young's modulus. Furthermore, the influence of geometries on the equivalent elastic parameters are also discussed.
Materials with negative Poisson's ratio attract considerable attention due to their underlying intriguing physical properties and numerous promising applications, particularly in stringent environments such as aerospace and defense areas, because of their unconventional mechanical enhancements. Recent progress in materials with a negative Poisson's ratio are reviewed here, with the current state of research regarding both theory and experiment. The inter-relationship between the underlying structure and a negative Poisson's ratio is discussed in functional materials, including macroscopic bulk, low-dimensional nanoscale particles, films, sheets, or tubes. The coexistence and correlations with other negative indexes (such as negative compressibility and negative thermal expansion) are also addressed. Finally, open questions and future research opportunities are proposed for functional materials with negative Poisson's ratios.
Monte Carlo simulations in the isobaric–isothermal ensemble with variable shape of the periodic box are used to study elastic properties of two‐dimensional (2D) model systems of hard discs with parallel layers of hard cyclic hexamers. Both the particles in their pure phases form elastically isotropic crystals. However, the crystal of discs shows positive and that of hexamers – negative Poisson's ratio (PR). The studied systems with layers are of low symmetry. Hence, a general analytic formula for the orientational dependence of PR in 2D systems is applied to such systems. It is shown that, by changing parameters of the particles and their relative concentration, as well as orientation of the layers, one can obtain systems which are non‐auxetic (their PR is positive), auxetic (their PR is negative), or partially auxetic ones (their PR is negative for some directions and positive for other ones). If the disks and hexamers are interpreted as small molecules, the obtained results indicate that by introducing auxetic nano‐layers one can strongly modify PR of crystals or even thin layers.
The periodic box with particles can be thought of as a model of a thin layer or as a model crystalline structure with parallel horizontal layers built of four hard cyclic hexamer rows in the hard disc system.
A new approach to search for materials with auxetic properties by modifying structures of solids at molecular level has been proposed. The analysis of elastic properties of the face-centered cubic Yukawa crystals with very narrow nanochannels in the  crystallographic direction using Monte Carlo simulations in the isothermal–isobaric ensemble has been done. An influence of the size of nanochannels on the value of Poisson's ratio in main crystallographic directions has been studied. It has been shown that the insertion of nanochannels in the system causes a decrease of the Poisson's ratio in the direction from –0.15(2) to –0.29(3). That means an amplification of auxetity in the studied system twice as compared to the system without nanochannels.
Composites with elliptic inclusions of long semi-axis a and short semi-axis b are studied by the Finite Element method. The centres of ellipses form a square lattice of the unit lattice constant. The neighbouring ellipses are perpendicular to each other and their axes are parallel to the lattice axes. The influence of geometry and material characteristics on the effective mechanical properties of these anisotropic composites is investigated for deformations applied along lattice axes. It is found that for anisotropic inclusions of low Young's modulus, when a + b → 1 the effective Poisson's ratio tends to -1, while the effective Young's modulus takes very low values. In this case the structure performs the rotating rigid body mechanism. In the limit of large values of Young's modulus of inclusions, both effective Poisson's ratio and effective Young's modulus saturate to values which do not depend on Poisson's ratio of inclusions but depend on geometry of the composite and the matrix Poisson's ratio. For highly anisotropic inclusions of very large Young's modulus, the effective Poisson's ratio of the composite can be negative for nonauxetic both matrix and inclusions. This is a very simple example of an auxetic structure being not only entirely continuous, but with very high Young's modulus. A severe qualitative change in the composite behaviour is observed as a/b reaches the limit of 1, i.e. inclusions are isotropic. The observed changes in both Poisson's ratio and Young's modulus are complex functions of parameters defining the composite. The latter allows one to tailor a material of practically arbitrary elastic parameters.
As promising metamaterials, 3D periodic auxetic cellular structures (PACSs) have attracted great interest. However, they usually consist of intricate geometries which make their fabrication a significant challenge. The present paper is focused on introducing the interlocking assembly concept into the fabrication of 3D PACSs. There are distinct advantages of the interlocking assembly method compared with the additive manufacturing methods mainly used before. Based on the interlocking assembly method, the dependences of mechanical properties mainly including the Poisson's ratio and the Young's modulus of the structure on the re-entrant angle were investigated through a combination of uniaxial compression experiments and numerical simulations, excellent qualitative and quantitative agreement was found. Using the experimentally verified numerical model, the effects of the strut thickness and the ratio of the vertical strut length to oblique strut length on the mechanical properties of the structure were investigated. Results show that the compression modulus of the structure will increase with the structure becomes more re-entrant, but there exists an extreme value for Poisson's ratio with the re-entrant angle around 45° which differs from former studies. With the thickening of the struts the compression modulus of the structure monotonously increases and the Poisson's ratio of the structure will gradually changes from negative to positive then gradually approaches to the Poisson's ratio of the parent material. The vertical strut length to oblique strut length ratio plays fewer roles on the mechanical properties compared with the re-entrant angle and the strut thickness.
Designing structures that dilate rapidly in both tension and compression would highly benefit devices such as smart filters, actuators or fasteners. This property however requires an unusual Poisson's ratio, or Poisson's function at finite strains, which has to vary with applied strain and exceed the familiar bounds: less than 0 in tension and above 1/2 in compression. Here, by combining mechanical tests and discrete element simulations, we show that a simple three-dimensional architected material, made of a self-entangled single long coiled wire, behaves in-between discrete and continuum media, with a large and reversible dilatancy in both tension