Potential and applications of auxetic tubular: a review

Mohammad Javad Ramezani; Omid Rahmani

doi:10.1088/2631-6331/ad1dc6

1. Introduction

Materials with a negative Poisson's ratio (NPR), also known as auxetic materials, possess distinctive characteristics that make them appealing for diverse applications. The history of auxetic materials as one of the sub-branches of mechanical metamaterials traces back to the early 1980s when a group of researchers made a groundbreaking discovery, revealing that specific materials exhibited a remarkable mechanical property called NPR [1–4]. Before this revelation, it was widely held that all materials had a positive Poisson's ratio (PPR), indicating that they would contract in one direction when subjected to stretching [5].

The term 'auxetic' was first used in the 1990s to describe these materials with a NPR. The term is derived from the Greek word 'auxesis,' which means growth or expansion [6, 7]. Auxetic materials are materials that exhibit a NPR, meaning that they expand in lateral dimensions when stretched in the axial direction [8]. This is in contrast to conventional materials that have a PPR and contract in lateral dimensions when stretched in the axial direction.

Auxetic materials have unique mechanical properties that distinguish them from conventional materials. One of the most notable properties of auxetic materials is their NPR [9, 10]. This property results in several interesting mechanical properties, such as increasing toughness [11, 12], improving resistance to buckling [13, 14], increasing shear modulus [15–17], improving damping properties [18, 19]. By the way, auxetic materials have unique properties that make them attractive for use in various applications. For example, they can provide improved impact resistance [20–22] and energy absorption (EA) [23] properties compared to conventional materials. They can also be used in filtration applications where their ability to expand in lateral dimensions can improve their performance. There are several types of auxetic materials, including auxetic foams [24, 25], polymers [26, 27], textiles [28–30], and composites [31], metals [32, 33]. Researchers continue to explore the potential of auxetic materials for new applications, such as soft [34], acoustic insulation [35, 36], and energy harvesting [37].

The history of auxetic tubular materials finds its roots in the pioneering discovery of auxetic substances during the early 1980s. These unique structures have undergone thorough investigation by researchers, with a notable focus on the field of material science, culminating in substantial progress in the advancement and application of auxetic tubular materials. The first reported application of auxetic tubular materials was in the field of biomedical engineering [38, 39]. In the early 2000s, researchers began exploring the use of auxetic tubular materials for stent applications, where the expansion of the material upon compression could improve the performance and safety of the stent [38]. Since then, auxetic tubular materials have been explored for various other applications, including excellent EA [40], favorable mechanical properties [41], and impact protection [42]. The unique properties of auxetic tubular materials make them attractive for use in applications where their ability to expand upon compression can improve performance and safety.

An auxetic tube is a type of tube or cylinder that possesses the unique property of expanding in diameter when it is stretched, rather than contracting like most materials. Auxetic tubes have potential applications in areas such as medical implants, aerospace engineering, and protective wear, where their ability to expand upon impact could provide increased protection [43–45].

NPR and PPR are two distinct mechanical properties, which exhibit unique deformation behavior compared to conventional materials. NPR and PPR materials have opposing behaviors when subjected to axial loads. According to the figure 1, a simulation of auxetic tubes illustrated, allowing for the observation of Poisson's ratio behavior before and after the application of tensile load to the auxetic tubular structure. This structure is composed of missing rib unit cells, in direct comparison to a non-auxetic tubular structure that consists of unit cells with a PPR.

**Figure 1.** Deformation behavior of auxetic tube under tensile load [6]. Reprinted from [6], © 2021 Elsevier Ltd. All rights reserved.
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In summary, the distinctive characteristics of auxetic tubes render them a promising material for a diverse array of applications where flexibility, impact resistance, and shock absorption are crucial factors.

This present review article offers an extensive analysis of the design parameters governing auxetic tubular structures. It delves into their multifaceted applications spanning various industries, providing an in-depth investigation into the intricacies of their mechanical properties. Furthermore, this review article distinguishes itself from prior reviews by providing a cutting-edge analysis aimed at updating the findings within each segment of the auxetic tubes field. Additionally, it delves into the meticulous examination of recommendations and future challenges inherent to auxetic materials and structures, shedding light on the upcoming obstacles that necessitate further exploration. The review seeks to offer a refined understanding of the intricacies and potential applications of auxetic systems within the realm of scientific research and engineering.

2. Designing and manufacturing of auxetic tubular structures

In the realm of auxetic tube design, various methods are available, including 2D design techniques. However, one of the most prevalent and widely embraced approaches is the coordinate transformation design (CTD) method [6, 41, 42, 46]. Furthermore, the fabrication techniques for unique tubular structures encompass cutting-edge processes such as 3D printing [47–50], laser cutting, and weaving. These advanced methods play a pivotal role in shaping the future of materials engineering and design, offering unprecedented possibilities for tailoring the properties and applications of auxetic tubular materials with precision and efficiency.

2.1. Designing of auxetic tubular structures

2.1.1. CTD

CTD is a powerful method for designing and optimizing auxetic structures, including auxetic tubular structures [51–53]. The CTD method involves mapping a conventional structure onto an auxetic structure using a series of coordinate transformations.

In the process of designing auxetic tubular structures using the CTD method, which is based on conventional tube geometry, the following steps are involved: First, one must choose the desired auxetic behavior and unit cell pattern. Then, the next step is to determine the coordinate transformation sequence. Afterward, the chosen coordinate transformations are applied, and finally, the design is validated [54, 55]. The CTD method provides a highly flexible and powerful approach for crafting auxetic tubular structures, offering a broad spectrum of mechanical properties. By meticulously selecting the coordinate transformation sequence, it becomes feasible to fabricate structures with specific auxetic behaviors tailored to address the specific requirements of a particular application. According to the figure 2, the CTD method is a fundamental tool for designing auxetic tubular structures with enhanced flexibility and bending properties, making them valuable in diverse engineering and material science contexts.

**Figure 2.** Auxetic tube design by coordinate transformation design method [56]. Reprinted from [56], © 2022 Elsevier Ltd. All rights reserved.
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Han et al [41] investigated the mechanical properties of novel auxetic tubes made by the coordinate transformation method and compared them with the original auxetic tube. Lou et al [46] studied stainless steel auxetic tubular that has high strength and outstanding corrosion resistance, In the experimental method they investigated creating all tubes in the CTD method. Ren et al [57] investigated aluminum foam-filled auxetic circular tubular with elliptical cells that were created in the CTD method. Furthermore, there are a lot of studies that reported on the CTD method, such as Han et al [56], Teng et al [58], and Gomes et al [39].

2.1.2. Computer-aided design (CAD)

CAD initiates by formulating the conceptual design of the desired auxetic tubular structure, encompassing the definition of its shape, dimensions, and structural attributes, all tailored to the intended application. Subsequently, CAD software options like AutoCAD, SolidWorks, or CATIA are carefully selected for the design process. These software solutions offer an intuitive interface for the generation of both 2D and 3D models. The utilization of CAD software facilitates the creation of a comprehensive 3D model of the tubular structure, encompassing the precise specification of its inner and outer geometries, layering, and various other structural features.

Abbas et al [59] conducted an inquiry into the adjustable compressive characteristics of an innovative auxetic tubular material under low-stress conditions. Their approach encompassed the design of the auxetic tubular structure utilizing CAD methodologies. Subsequently, they used this method by employing 3D printing technology to fabricate the auxetic tubular structures. Francisco et al [60] investigated the sandwich composite tubes with auxetic core for enhanced structural properties and mechanical performance. They employed CAD software to develop precise structural configurations. Subsequently, this optimized design was used through a 3D printing process and printed with PLA material.

2.2. Manufacturing of auxetic tubular structures

2.2.1. 3D printing

3D printing proves invaluable in the manufacturing of auxetic tubular structures [61–64]. To craft these structures using 3D printing, the process commences with the creation of a 3D model through CAD software. Various types of auxetic structures, such as re-entrant honeycombs [65, 66], chiral structures [67], and Kirigami patterns [68], are available. Once the desired structure type is chosen, CAD software facilitates the design of the structure's geometry, including the specifications for individual cell size, shape, and orientation.

Following the design of the auxetic structure, one can utilize a 3D printer to produce a physical prototype. 3D printing provides a rapid and efficient means to fabricate intricate geometries, some of which might be challenging or even unfeasible to manufacture using traditional production techniques. It offers versatility in material choices for 3D printing auxetic structures, encompassing thermoplastics, metals, and composites.

Once you have a physical prototype of the auxetic tubular structure, you can test its mechanical properties to determine its suitability for your intended application. 3D printing allows for rapid iteration and refinement of the design, enabling you to quickly fine-tune the structure's properties to meet your specific requirements. Figure 3 illustrated auxetic tubes with elliptical pattern designed using the 3D printing method.

**Figure 3.** Auxetic tubular design by 3D printing [69]. Reprinted from [69], © 2021 Elsevier Ltd. All rights reserved.
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Jiang et al [70] investigated the auxetic tubular lattice (ATL) in the 3D printing method and compared it with conventional diamond tubular lattice (DTL). Ni et al [45] manufactured aluminum tubular with auxetic foam filled. Kim et al [71] studied auxetic tubular for tissue engineering scaffolds by selected laser melting in 3D printing. Furthermore, other researchers manufactured auxetic tubular in 3D printing such as Ren et al [72], Ren et al [73], Ruan et al [74], and Zhang et al [75].

2.2.2. Laser cutting

Laser cutting is a popular manufacturing method for creating intricate and precise designs in a variety of materials, including metals [76, 77], plastics [78], and composites [79, 80]. In the context of manufacturing auxetic tubular structures [39, 81, 82], laser cutting can be a highly effective technique for creating the required patterns and shapes.

To create an auxetic tubular structure using laser cutting, a pattern of slits or cuts is typically made in the material along the length of the tube. The specific design of the pattern will depend on the desired mechanical properties of the final structure. Laser cutting allows for precise control of the size, shape, and spacing of the cuts, which can be optimized to achieve the desired auxetic behavior [26, 57, 83, 84].

Once the pattern has been cut, the tubular structure can be formed by rolling or bending the material into a tube shape. The resulting structure will exhibit auxetic behavior when subjected to forces perpendicular to the axis of the tube. Overall, laser cutting is a valuable tool in the design and manufacture of auxetic tubular structures, allowing for precise control over the pattern of cuts and the resulting mechanical properties of the material. Figure 4 illustrated 2D planes of auxetic structure produced through laser cutting.

**Figure 4.** 2D planes of auxetic structure produced via laser cutting [85]. Reprinted from [85], © 2019 Elsevier Ltd. All rights reserved.
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Chen et al [86] investigated paper tubes of sandwich panels with the auxetic re-entrant core. They used the laser cutting method to cut the face skins of samples. Balan et al [87] prepared a review about metamaterials and explained about auxetic tube designing. Cai et al [88] studied tunable auxeticity in carbon nanotubes and they used this method to create cylindrical tubes. There is a lot of research in this way as Huang et al [89], Suh et al [90] and Zhu et al [91].

2.2.3. Knitting and braiding

Knitting and braiding [92] proves to be a valuable technique for crafting auxetic tubular structures, as it grants precise control over both the structure's geometry and material properties [39, 92, 93]. Through knitting, a diverse array of patterns and structures, including ribbed and mesh-like configurations, can be created, all of which exhibit auxetic behavior. A common strategy for manufacturing auxetic tubular structures via knitting involves forming a structure with a specific stitch pattern that allows for lateral expansion when the tube is longitudinally stretched. For instance, one can knit a tube with a ribbed pattern alternating between rows of knit and purl stitches. When the tube undergoes longitudinal stretching, these ribs expand laterally, resulting in an auxetic response [94].

In contrast to woven or braided textiles, knitted fabrics exhibit elevated productivity and exceptional adaptability to intricate geometries owing to the remarkable deformable nature of their distinctive knitted structure. The utilization of knitting technology represents a compelling avenue within the realm of textile engineering for crafting NPR materials, primarily owing to its extensive structural diversity. Flat knitting technology, in particular, stands as a widely employed method in fabric production, distinguishing itself from warp and circular knitting by virtue of its heightened process versatility and broader spectrum of fabric structures [92].

Figure 5 illustrated the intricate manufacturing process of knitted auxetic tubular structures, showcasing the advanced techniques used in their creation. This unique property, characterized by its intricate design, holds significant value in applications such as tissue engineering scaffolds [71] and multifunctional foldable structures [95].

An alternative approach involves utilizing a mesh-like structure capable of deformation in multiple directions. This can be accomplished by employing a combination of knit and purl stitches to form a mesh-like configuration that exhibits multi-directional expansion when subjected to stretching. In summary, knitting emerges as a potent tool for the design of auxetic tubular structures. It enables precise control over material properties and geometry and facilitates the creation of a diverse range of patterns and structures that demonstrate auxetic behavior [75, 98]. Ma et al [93] and Boakye et al [92, 99] conducted comprehensive studies focusing on the design and fabrication of knitted tubular fabric to investigate the auxetic effect. Their research delved into the intricate principles and methods underpinning the creation of these specialized tubular structures. Chen et al [96] investigated into the mechanical modeling of auxetic tubular braided structures through a combination of experimental and numerical analyses. Their groundbreaking study revealed that these braided structures possess the capacity to exhibit a robust auxetic behavior. Importantly, their research emphasized that key mechanical properties, including Poisson's ratio and stiffness, are highly sensitive to the initial micro-geometric parameters, with a particular emphasis on the initial braiding angle and the diameter of the component yarns. Table 1 displayed examples of auxetic tubes produced with various patterns that are utilized in scientific research.

Table 1. Common auxetic tubes in recent research.

Geometry	Cell	Tubular model	References
		Elliptical auxetic	Reprinted from [57], © 2023 Elsevier Ltd. All rights reserved.
		Rotating square	Reprinted from [58], © 2023 Elsevier Ltd. All rights reserved.
		Re-entrant	Reprinted from [48], © 2022 Elsevier Ltd. All rights reserved.
		Curved	Reprinted with permission from [100]. Copyright (2019) American Chemical Society.
		Chiral	Reprinted from [101], © 2022 The Authors. Published by Elsevier Ltd.
		Anti-tetra chiral	Reprinted from [102], © 2022 Elsevier Ltd. All rights reserved.
		Arrowhead	Reprinted from [39], © 2023 Elsevier Ltd. All rights reserved.
		Orthogonal peanut-shaped	Reprinted from [47], © 2021 Elsevier Ltd. All rights reserved.

3. Mechanical properties of auxetic tubular

3.1. Tensile properties

Engineering materials or structures commonly possess positive poisson ratios, meaning they contract laterally under axial tensile loads and expand laterally when exposed to axial compressive forces. In contrast, auxetic structures exhibit a different behavior than materials with positive poisson ratios [103]. For that reason, the tensile properties of an auxetic tube, a unique structure, are a subject of interest in materials science and engineering [56].

Numerous research investigations have provided compelling evidence that auxetic materials exhibit outstanding resistance to indentation, exceptional shear stiffness, remarkable fracture toughness, and enhanced EA [104]. For instance, research studies [41, 105–108] investigating the mechanical characteristics of auxetic re-entrant foam tubular structures subjected to tensile loading and uniaxial compression revealed that auxetic foam displayed notably higher tensile strength compared to conventional foam of equivalent density.

Chen et al [96] investigated various factors, including the geometric characteristics of a braided structure, the impact of yarn component diameter, the effects of initial braiding angle, and the influence of the inherent Poisson's ratio within the context of an auxetic tubular braided structure undergoing tensile testing. Their research, illustrated in figure 6(a), demonstrated that an increase in strain and the diameter of stiff yarn components corresponded with a notable decrease in Poisson's ratio. However, as depicted in figure 6(b), an increase in the diameter of stiff yarn components coincided with elevated stress levels, indicating enhanced strength in samples with larger diameters. The stiffness ratio of the yarn components, which exhibits a close relationship with diameter, emerged as a significant factor influencing the overall structural performance. Moreover, the initial braiding angle exhibited a substantial impact on the effective tensile stress. As the initial braiding angle decreases, there was a proportional increase in tensile stress.

**Figure 6.** (a) The impact of the stiff yarn diameter on the Poisson's ratio of the braid, (b) the relationship between stiff yarn diameter and nominal stress [96]. Reprinted from [96], © 2019 Elsevier Ltd. All rights reserved.
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The advent of auxetic materials has prompted extensive research into the assessment of auxetic composites for potential engineering applications. In pursuit of this goal, Strek and Jopek [109] investigated using the finite element method (FEM) to determine young's modulus and Poisson's ratio of auxetic composites comprising concentric cylindrical inclusions composed of various combinations of auxetic and conventional materials. Their findings showed the mechanical properties of distinct composite phases impact the overall mechanical properties of the composite.

Strain hardening, as a fundamental feature of metal plasticity, has a significant effect on the mechanical performance of cellular structures and materials. Ren et al [110] studied on mechanical properties of auxetic tubular. According to figure 7, their results showed that increasing the strain hardening modulus (Ep)/ the elastic modulus (Es) ratio of displacement diagram-Poisson's ratio increases.

**Figure 7.** Poisson's ratio-displacement diagram with EP /ES ratios [110]. Reproduced from [110]. © IOP Publishing Ltd. All rights reserved.
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Wu et al [111] conducted an investigation into the mechanical properties of arterial stents, specifically focusing on the mechanical integrity and biomechanical performance reliability within the context of the stent-plaque-artery system. Moreover, they explored the influence of stent geometrical parameters on the tensile mechanical behaviors of these stents using finite element analysis. Leveraging auxetic deformation characteristics inherent in chiral structures and the mechanical advantages associated with structural hierarchy, the study introduced two novel types of chiral stents with auxetic properties. These innovations include an anti-tetrachiral and hierarchical anti-tetrachiral stents featuring circular and elliptical nodes. As depicted in figure 8, the tensile test results revealed a clear relationship between the increment in the number of structs nodes along the circumferential direction and a subsequent reduction in the NPR. Similarly, when there is an augmentation in the number of structs nodes along the axial direction, particularly in the case of stents featuring elliptical and circular nodes, the NPR demonstrated an increase. Furthermore, as the elliptical radius ratio (r/rz θ) experienced growth, it is associated with a corresponding decrease in the NPR.

**Figure 8.** The interplay between the geometric characteristics of anti-tetrachiral stents featuring circular and elliptical nodes and the Poisson ratio under the tensile test: (a) the repercussions of varying the quantity of structural elements, denoted as structs along the circumferential direction (Nθ) and (b) the impact of structural modifications, once again quantified as the number of structs along the axial direction, represented as (Nz) [111]. Reprinted from [111], © 2017 Elsevier Ltd. All rights reserved.
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Tabacu and Stanescu [112] investigated into anti-tetra-chiral auxetic tubes exposed to quasi-static tensile loads. This study involved the development of an analytical solution to estimate the reaction force resulting from tensile loads, utilizing the plastic hinge mechanism. The findings revealed that when the reaction force surpasses the axial force necessary to yield the vertical ligaments, a significant alteration occurs in the deformation mechanism. Table 2 illustrated Poisson's ratio of some auxetic tubes with different geometry and materials.

Table 2. Investigate Poisson's ratio of auxetic structures.

	Model	Cell	Material	Poisson's ratio	References
1	Circular tube	Elliptical	Stainless steel	−0.95 to −0.79	Reprinted from [42], © 2023 Elsevier Ltd. All rights reserved.
2	Square tube	Elliptical	Stainless steel	−0.83 to −0.84	Reprinted from [42], © 2023 Elsevier Ltd. All rights reserved.
3	Foam filled	Elliptical	Stainless steel+ aluminum	−0.71 to −0.69	Reprinted from [42], © 2023 Elsevier Ltd. All rights reserved.
4	Foam filled	Elliptical	Stainless steel+ aluminum	−0.8 to −0.78	Reprinted from [42], © 2023 Elsevier Ltd. All rights reserved.
5	Circular tube	Orthogonal peanut	VeroWhitePlus	−0.75	Reprinted from [47], © 2021 Elsevier Ltd. All rights reserved.
6	Circular tube	Elliptical	TPU	−0.27 to −0.49	Reprinted from [69], © 2021 Elsevier Ltd. All rights reserved.
7	Circular tube	Circular	TPU	0.05 to −0.06	Reprinted from [75], © 2021 Elsevier Ltd. All rights reserved
8	Square tube	Re-entrant	Aluminum+ epoxy resin	−1.01	Reprinted from [113], © 2021 The Author(s). Published by Elsevier B.V.
9	Circular tube	Re-entrant	nylon	−0.49 to −0.53	Reprinted from [107], © 2022 Elsevier Ltd. All rights reserved.
10	Circular tube	Anti-tetrachiral	Nylon powder	−1.35 to −0.4	Reprinted from [102], © 2022 Elsevier Ltd. All rights reserved.

3.2. Enhanced bending properties and flexibility

Enhanced flexibility and bending in auxetic tubular structures pertain to their capability for undergoing substantial bending deformations or demonstrating increased flexibility in comparison to conventional tubular structures. This unique mechanical attribute arises from the structural configuration and behavior of the employed auxetic pattern within the tubular structure [114]. The heightened flexibility observed in auxetic tubular structures can be attributed to their NPR, which allows them to expand in transverse directions when subjected to axial loading. They can bend and twist more readily without significant resistance. This expansion or widening effect in response to applied forces facilitates greater bending capabilities and improved flexibility.

Auxetic structures can resist shear forces more effectively. This means that they can better withstand torsional loads, making them suitable for applications requiring twisting or torque resistance. However, the auxetic behavior also results in improved EA capabilities. In bending or twisting situations, the material can absorb and distribute energy more efficiently, which can be advantageous in impact or vibration absorption applications.

Figures 9(a) and (b) illustrated auxetic tubular with rotating square and honeycomb pattern under bending test. Furthermore, the figure displayed the rotation of the square pattern mechanism during deformation in a bending test.

**Figure 9.** Auxetic tubular under bending (a) rotating square [4] (b) peanut-shaped [70, 115]. Reprinted from [4], © 2020 The Author(s). Published by Elsevier Ltd. Reproduced from [115]. © IOP Publishing Ltd. All rights reserved. Reprinted from [70], © 2021 Elsevier B.V. All rights reserved.
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Jiang et al [70] investigated and compared the bending behavior, including bending stiffness and plasticity of DTL with auxetic tubular lattice (ATL). The ATL sample exhibited an 85.4% increase in bending ductility compared to the DTL sample. Furthermore, among the differences discussed and examined in this study, the bending behavior of the ATL sample is dependent on the indenters, whereas the DTL sample demonstrates global bending behavior. The investigation also delved into the impact of circumferential unit cells. When comparing the simple tube, it was observed that the bending stiffness of the ATL increased as the circumferential number of unit cells. Furthermore, in the ATL, it was noted that the contribution of bending deflection decreased as the circumferential number of unit cells increased. Additionally, the thickness of the samples directly affects their bending stiffness. Figure 10(a) showed deformation patterns under the bending test for ATL, figure 10(b) deformation patterns for DTL.

**Figure 10.** (a) Deformation patterns under bending test for ATL. (b) Deformation patterns for DTL [70]. (c) Auxetic tubular under torsion [4]. Reprinted from [70], © 2021 Elsevier B.V. All rights reserved. Reprinted from [4], © 2020 The Author(s). Published by Elsevier Ltd.
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The unique structure and properties of auxetic materials cause them to expand when subjected to stretching forces. Consequently, stents with these distinct characteristics are highly sought after in tracheal applications. When the neck undergoes bending or movement, auxetic stents exhibit a wider diameter compared to traditional metal or simple stents, thereby ensuring excellent ventilation. For this reason, Liu et al [116] conducted a study on chiral auxetic stents for ciliated epithelium.

3.3. Enhanced impact resistance

Auxetic tubular structures are a unique class of materials known for their exceptional mechanical properties, including enhanced impact resistance [117–120]. Characteristic with NPR allows auxetic tubular structures to distribute and absorb impact energy more effectively compared to traditional materials [57, 82, 105, 121].

Tsang and Raza [122] investigated the impact of EA capability on hierarchical tubular. Their results demonstrated that these tubes exhibit exceptional impact EA when compared to standard tubes. Furthermore, the stress levels within the samples were somewhat alleviated. The crash box is a widely utilized component in the automotive industry, typically located at the front of the car. Its primary purpose is to absorb impact energy and mitigate damage. In light of this, Yuan et al [123] studied on quasi-static impact on origami crash boxes. Their findings revealed a significant enhancement of up to 107% in EA, coupled with a remarkable reduction of 68.3% in the initial peak force compared to the conventional crash box. Table 3 illustrated the specific EA (SEA) values using various materials and unit cells in auxetic tubular structures.

Table 3. Specific energy absorption with different material and unit cell in auxetic tubular.

Number	Material	Cell	SEA (kJ kg⁻¹)	Year	References
1	Stainless steel 304;	Elliptical	14.20	2022	Reprinted from [40], © 2022 Elsevier Ltd. All rights reserved.
2	Aluminum	Double-arrowed	4.062	2018	Reprinted from [124], © 2018 Elsevier Ltd. All rights reserved.
3	Stainless steel 316L	Circular	25.20	2022	Reprinted from [40], © 2022 Elsevier Ltd. All rights reserved.
4	Stainless steel	Re-entrant	2.67	2019	Reprinted from [19], © 2018 Elsevier Ltd. All rights reserved.
5	TPU 95A	Helically oriented tubular	0.71	2022	Reproduced from [125].CC BY 4.0.
6	Aluminum	Elliptical	3.95	2022	Reprinted from [56], © 2022 Elsevier Ltd. All rights reserved.
7	Stainless-stee	Elliptical	1.37	2022	Reprinted from [126], © 2022 Elsevier B.V. All rights reserved.
8	Aluminum foam filled	Elliptical	5.84	2023	Reprinted from [57], © 2023 Elsevier Ltd. All rights reserved.
9	Aluminum foam filled	Elliptical	1.35	2023	Reprinted from [42], © 2023 Elsevier Ltd. All rights reserved.
10	Aluminum	Elliptical	0.84	2023	Reprinted from [42], © 2023 Elsevier Ltd. All rights reserved.

3.4. Enhanced resistance to torsional deformation

The NPR plays a significant role in auxetic torsional behavior. Unlike conventional materials with a PPR, which tends to contract laterally under axial loading, auxetic materials expand laterally. This lateral expansion enables auxetic tubular structures to resist torsional forces more effectively [127, 128]. The unique geometric arrangement of auxetic tubular structures contributes to their torsional resistance. This arrangement facilitates rotational deformation under axial loading. Consequently, when torque is applied to an auxetic tubular structure, it undergoes twisting motion with minimal deformation or warping [129].

Auxetic tubular structures compared to simple tubes, demonstrate a negative twist-to-length relationship, whereby an elongation of the tube leads to a reduction in the twist angle. This phenomenon arises from the distinct geometric arrangement of the structure, enabling rotational deformation when subjected to axial loading [39, 130, 131]. Furthermore, the NPR in auxetic tubes allows them to expand laterally when subjected to twisting forces. This expansion reduces the risk of kinking, buckling, or deformation during twisting. Figure 8(c) illustrated an auxetic tubular under torsion and its behavior.

3.5. Compression behavior

The compression behavior of auxetic tubular structures is another important mechanical property to consider [132]. Auxetic materials typically exhibit unique behavior under compression, such as resistance to buckling or the ability to absorb energy through their lattice-like structures. Compression testing can help characterize the compressive strength and deformation response of auxetic tubular structures. Zhang et al [47] investigated the mechanical properties of auxetic tubes with a peanut-shaped hole pattern under uniaxial pressure and compared them with the orthogonal elliptical hole pattern using both experimental and FEM analysis. Ren et al [57, 106] studied the properties under the uniaxial pressure of hollow auxetic tubular made of stainless steel and aluminum with an elliptical pattern and then filled them with rigid polyurethane (PU) foams. Figure 11 showed auxetic behavior under uniaxial pressure, demonstrating that the material expanded in the direction of the applied force. This behavior is a characteristic of materials with a NPR, where they exhibit the opposite response to conventional materials, which typically contract laterally under such loading conditions.

**Figure 11.** Aluminum foam-filled auxetic tube deformation under compressive [57]. Reprinted from [57], © 2023 Elsevier Ltd. All rights reserved.
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4. Applications of auxetic tubular

Auxetic tubular structures, with their unique properties and behavior, have found applications in various fields. Here are some examples of applications where auxetic tubular structures have been explored:

4.1. Protective wear

A significant advantage of auxetic structures in the realms of apparel and sports industries is their capacity to undergo precise geometric transformations while assuming intricate configurations with accuracy. Auxetic tubular materials have been investigated for use in protective wear such as helmets, knee pads, and body armor [133]. Their ability to absorb impact energy and enhance cushioning properties makes them potentially useful in improving safety and reducing the risk of injuries. Specifically, these materials provide comfortable and effective body protection by mitigating impact forces, thereby reducing the risk of injuries, making them suitable for a variety of high-risk individuals, including the elderly, industrial workers and athletes [134]. Furthermore, auxetic tubular structures have been explored for use in smart textiles and wearable technologies [135]. These materials can provide enhanced comfort, flexibility, and stretchability, making them suitable for applications such as sportswear, compression garments, and stretchable electronics.

Figure 12 showed the wearable sleeve, fabricated from bilayer polymers, exhibits thermoresponsive behavior, expanding in response to ambient heat [133]. This unique characteristic enables the sleeve to dynamically adapt to environmental conditions, providing controllable breathability and enhanced thermal comfort. Additionally, the thermoresponsive properties of the bilayer polymers make them ideal for applications in soft robotics, allowing robots to efficiently respond to dynamic changes in temperature, similar to living organisms [136].

4.2. Biomedical devices

Auxetic tubular structures have shown promise in biomedical applications [137]. They can be utilized in implantable devices [111, 138–141], tissue engineering scaffolds [142, 143], drug delivery systems [26, 142, 144], biosensors, and diagnostics [145]. Their unique properties, such as improved flexibility, enhanced surface area, and compatibility with biological tissues, make them attractive for various biomedical applications. In the design of stents, it is essential to incorporate features such as bending, tensile strength, compressive strength, and torsional resistance [146]. As a result, auxetic stents are developed to possess these desired characteristics.

Failure, twisting, and migration of stents can occur in endovascular aortic repair due to the interplay of radial stiffness and bending flexibility characteristics. Vellaparambil et al investigated the mechanical behavior of auxetic endovascular stents and they found that chiral stents can be used in conditions that require greater radial stiffness [139]. Furthermore, various researchers have investigated auxetic stents, including those by Asadi et al [138], Liu et al [116], Shirdel et al [147], and Xue et al [148]. Figure 13 exhibit auxetic stents employed during the research stages.

**Figure 13.** Auxetic stents employed during the research stages [149, 150]. Reproduced from [149]. CC BY 4.0. Reprinted from [150], Copyright © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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4.3. EA systems

Auxetic tubular structures have been investigated for EA systems, such as crash protection in automobiles or impact mitigation in structural components. Their ability to dissipate energy efficiently under compression can enhance the safety and durability of such systems [151]. Lee et al [19] investigated the behavior of auxetic honeycomb and re-entrant tubes and compared them with simple tubes in terms of their response to low-speed impact and crash behavior. Their findings demonstrated that auxetic tubes exhibit significantly higher SEA compared to plain tubes under low-speed impact conditions. This enhanced EA capability can be attributed to the improved damping performance exhibited by these Auxetic tubes. Usta et al [117] studied the behavior and mechanical response of triggered and non-triggered tubes with auxetic foam under low-velocity impact to investigate EA (SEA) and crash force efficiency (CFE). Doudaran et al [119] conducted an investigation into the behavior of auxetic tubular structures with Re-entrant, arrow-head, and anti-tetrachiral patterns, comparing them with honeycomb tubes under quasi-static loads and impact using a drop weight apparatus. The results revealed that the auxetic tubular structures exhibited a notable increase in parameters such as CFE, SEA, and EA. However, among the tested samples, the honeycomb structure demonstrated the highest SEA. Figure 14(a) illustrated the deformations of auxetic and non-auxetic tubular materials under low-velocity impact, while figure 14(b) showed crash boxes containing auxetic tubular structures designed for EA.

**Figure 14.** (a) Deformations of auxetic under impact [56]. (b) Crash boxes containing auxetic tubular [57]. Reprinted from [56], © 2022 Elsevier Ltd. All rights reserved. Reprinted from [57], © 2023 Elsevier Ltd. All rights reserved.
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4.4. Auxetic fasteners and nails

Auxetic fasteners and nails are fastening devices that exhibit auxetic properties, characterized by their lateral expansion when subjected to longitudinal tension or compression. While auxetic materials have undergone extensive research and application in various fields, the development and application of auxetic fasteners and nails represent relatively new areas of study. The potential advantages of auxetic fasteners and nails include improved gripping strength, enhanced resistance to loosening or pullout forces, and potential applications in areas where dynamic loadings or vibrations are present. However, it is crucial to note that further research and development are required to optimize their design, material selection, and performance for specific applications. In contrast to ordinary nails, which commonly experience displacement and loosening over extended periods, auxetic materials demonstrate lateral expansion when subjected to uniaxial compression and tension. Building on this understanding, Ren et al [73] conducted an investigation on auxetic nails, comparing their behavior to that of conventional nails. A comprehensive study was performed by Kasal et al [152] on a significant quantity of auxetic dowels with varying diameters and sizes, fabricated from Polyamide using a 3D printer. The research findings revealed a positive correlation between the inclusion size and the diameter of the hole on the dowels and the withdrawal strength of the dowels. Specifically, an increase in the inclusion size and the diameter of the hole led to an enhancement in the withdrawal strength of the dowels. Several researchers have explored auxetic fasteners and nails, including Kuşkun et al [153, 154] and Harinarayana and Shin [155]. However, as this is an emerging field, specific scientific references or commercial examples of auxetic fasteners and nails may be limited. Figure 15 displayed several examples of auxetic nails.

**Figure 15.** Auxetic nails [73]. Reprinted from [73], © 2017 Elsevier Ltd. All rights reserved.
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5. Opportunities and challenges

Auxetic tubular structures pose both challenges and exciting opportunities across various disciplines. Overcoming the hurdles related to material selection, manufacturing techniques, stability, and standardization will unlock the full potential of these structures. The capacity of these structures to laterally expand when subjected to longitudinal forces presents a wide range of possibilities in various fields. Nevertheless, alongside these opportunities, several challenges need to be addressed to fully harness the potential of auxetic tubular structures. This article explores the challenges and opportunities associated with the development and application of auxetic tubular structures.

5.1. Opportunities

5.1.1. Impact and EA applications

Auxetic tubular structures demonstrate significant potential in applications related to impact and EA. Their unique ability to expand laterally allows for the efficient absorption and dissipation of impact energy. Consequently, these structures find suitability in various fields, including protective gear design, crash safety systems, and vibration-damping solutions.

5.1.2. Biomedical and tissue engineering

In the field of biomedical engineering, auxetic tubular structures offer exciting opportunities. Their unique properties can be utilized in tissue engineering scaffolds, implantable devices, and drug delivery systems. The ability of auxetic structures to adapt to biological environments and provide enhanced mechanical cues can greatly benefit tissue regeneration and therapeutic interventions.

5.1.3. Smart materials and textiles

Auxetic tubular structures can contribute to the development of smart materials and textiles. By integrating auxetic properties into fabrics, it is possible to create stretchable, flexible, and adaptive textiles for applications such as wearable electronics, smart clothing, and responsive sensors.

5.1.4. Aerospace and engineering applications

The aerospace and engineering sectors can also benefit from the use of auxetic tubular structures. These structures can be employed in lightweight structures, morphing components, and deployable systems. Their unique properties, such as improved stiffness-to-weight ratios and resistance to buckling, make them appealing for aerospace and engineering applications.

5.2. Challenges

5.2.1. Material selection and design optimization

One of the primary challenges in the field of auxetic tubular structures is the selection of appropriate materials and the optimization of their design. Identifying materials that exhibit the desired auxetic behavior, along with the necessary mechanical properties, is crucial. Additionally, designing the geometry and internal structure of the tubular forms to achieve the desired auxetic response presents a significant challenge.

5.2.2. Manufacturing and fabrication techniques

The fabrication of auxetic tubular structures with precise geometry and internal structure is another challenge. Existing manufacturing techniques often struggle to produce intricate auxetic designs consistently and at scale. The development of advanced manufacturing techniques such as additive manufacturing (3D printing) and advanced weaving methods holds promise in addressing this challenge.

5.2.3. Stability and durability

Ensuring the stability and long-term durability of auxetic tubular structures is essential for their practical applications. Challenges such as maintaining the auxetic behavior under cyclic loading, resistance to fatigue and degradation, and environmental factors need to be addressed to ensure reliable and long-lasting performance.

5.2.4. Characterization and standardization

Reliable characterization methods and standardized testing protocols for auxetic tubular structures are crucial for comparing and benchmarking their performance. Developing standardized procedures to quantify parameters such as Poisson's ratio, EA, and mechanical properties will facilitate accurate assessment and comparison among different materials and designs.

5.2.5. Material limitations

Many auxetic materials inherently possess low stiffness and strength, limiting their utility in structural applications. Researchers are exploring composite materials, which combine auxetic substances with stronger ones, to enhance stiffness and strength.

5.2.6. Design optimization

Achieving the desired mechanical properties while maintaining auxetic behavior requires meticulous design. Advanced modeling and simulation software enable engineers to optimize the tube's geometry to achieve both high strength and auxetic properties.

6. Conclusion

This review article presented a comprehensive review covering design methodologies, mechanical properties, including NPR and tensile behavior, comparative analysis of these properties across various materials, bending characteristics, torsional properties, impact resistance, and EA in auxetic tubular structures. The investigation also explored the applications of these materials across diverse sectors, including aerospace, automotive, and medical industries, with a particular emphasis on their significant potential for medical applications, especially in the domain of auxetic stents. Additionally, the article addressed existing challenges and offers suggestions for future research and development in this field. The information presented is grounded in recent and updated scholarly articles, ensuring a comprehensive and up-to-date comprehension of the subject matter.