Three-dimensional auxetic woven fabric structure with in-plane negative Poisson’s ratio for composite reinforcement: A finite-element analysis

Abstract

Numerical modeling of textile structures at the yarn level is challenging, yet provides information not obtainable through experimental studies alone. This study is focused on the development and finite-element analysis (FEA) of a three-dimensional (3D) woven structure with an in-plane negative Poisson’s ratio (NPR) effect, with the aim of exploring its potential applications in polymer composite reinforcement. While experimental and geometrical studies of 3D auxetic woven structures provide primary information, they have limitations in explaining the auxetic behavior of the structure at the yarn level. Additionally, prediction of the auxetic behavior of the 3D woven structure by varying material properties is only possible through FEA. Therefore, to overcome these limitations, a 3D FEA model of the structure was developed using commercially available software (Abaqus CAE-2020) to simulate the auxetic behavior. The structure was then studied for different binding yarn properties once a good agreement had been found between simulated and experimental results. The FEA provides new insights and provokes fresh discussion on the auxetic behavior of the 3D woven structure. One significant finding from the FEA is the strong influence of the axial ( $E_{x}$ ) and radial ( $E_{y}$ ) moduli of the binding yarn on the Poisson’s ratio of the 3D woven structure; $E_{x}$ has an inverse relationship, while $E_{y}$ demonstrates a direct relationship with the auxetic behavior of the structure. The developed FEA model is expected to provide a better foundation for the future development of 3D auxetic structures, particularly concerning their application in polymer composite reinforcement.

Keywords

Finite-element analysis three-dimensional woven fabric auxetic structure auxetic reinforcement composite structure

Auxetic textiles are a group of metamaterials that exhibit an unusual dimensional change in response to applied load.¹ Specifically, the transverse dimension of the material increases under tensile loading and decreases under compressive loading.² When this unique behavior is integrated in a woven textile, the textile is categorized as an auxetic woven fabric. Since 2009,³ auxetic fabrics have been utilized as reinforcements in the manufacture of polymer composites to produce, as an end product, an auxetic composite. The unique dimensional change in an auxetic textile gives rise to exceptional features, such as high surface area under tension, high indentation resistance owing to material compaction under compression,⁴ and easy shape adoption owing to synclastic deformation.^5
–7 As a result, the applications of auxetic textiles are diverse and advanced, ranging from impact protection⁸ and medical textiles to sportswear, composite reinforcement,^9,10 and auxetic metamaterials.^11
–13

Over the last two decades, researchers have gained considerable achievements by producing polymers,^5,14,15 fibers,^16
–18 yarns,^3,19,20 fabrics,^21
–25 and composites^26
–29 with auxetic properties, using various techniques. However, many researchers are still focusing on auxetic woven fabrics, owing to their potential applications and the challenges involved in their development. One of the two main methods of developing auxetic woven fabric is simple and elementary, and involves using auxetic yarns in the warp or weft of the fabric. Typically, helical auxetic yarn, developed by Sloan et al.,¹⁹ and double helix yarn, developed by Miller et al.,³ are utilized in this method. However, these fabrics have drawbacks, including reduced durability in terms of auxetic behavior, the special arrangement of auxetic yarns required in the warp or weft, incomplete transfer of auxeticity from yarn to fabric, and limited design variation. The second and more alluring method for developing auxetic woven fabric involves forming an auxetic geometry in the fabric structure by using conventional yarns in the warp and weft. This method was first proposed by Zulifqar et al.,³⁰ who reported a uni-stretch woven fabric with an in-plane negative Poisson’s ratio (NPR). The principle of differential shrinkage of different types of yarn (elastic and inelastic) in combination with different tight (plain) and tight (twill or satin) weaves was used to create auxetic patterns, such as foldable structures, rotating rectangles, or re-entrant hexagons. Zulifqar et al.³⁰ achieved a maximum of −0.17 NPR, which is considered insignificant. Therefore, another study³¹ was conducted on bi-stretch auxetic woven fabrics to improve the NPR. In that study,³¹ elastic yarn was used in both the warp and the weft, along with nonelastic yarns, resulting in an improved NPR of −0.36. However, there are some drawbacks associated with auxetic fabrics developed using this method, such as a low auxetic effect, high longitudinal deformation, and poor mechanical properties, owing to the use of elastic yarn. Nevertheless, this research gained a lot of attention, owing to the potential for improvement, such as the exploration of new auxetic structures and yarns that can offer robust properties for composite reinforcement.

Over the last few years, there has been increasing interest in the development of three-dimensional (3D) auxetic structures to address high-profile applications that require high strength or energy absorption, such as protective gear, safety belts, and fiber-reinforced composite structures. In this regard, Ge and Hu³² developed what is thought to be the first ever 3D auxetic textile structure by modifying a conventional orthogonally through-the-thickness multilayered structure. A combination of knitting and nonweaving techniques was used to fabricate the structure, using a specially designed machine. This unique structure, when used as a reinforcement with a soft polyurethane foam as a matrix, exhibited an out-of-plane NPR under a compressive load. This means that the width (the perpendicular dimension of the composite) decreased under compression. Owing to this behavior, under a uniaxial compression test, the auxetic composite structure showed low specific stress at the initial strain, which increased later at a higher strain. Thus, the composite structure showed a higher energy absorption ability, yet with improved mechanical performance. Owing to the extraordinary response to compressive loading of the auxetic composites, the study was extended to analyze the low-velocity impact response of the composite reinforced with a 3D auxetic fabric structure.³³ Auxetic and nonauxetic composites were subjected to a low-velocity test with different energy levels (from 12.7 to 25.5 J). As expected, it was found that the auxetic composite structure absorbed a larger amount of impact energy than the nonauxetic composite structure when tested at an impact energy of 19.1 J. The auxetic composite absorbed 14.3 J, while the nonauxetic composite absorbed 12.8 J. The improved impact and indentation response of the auxetic composite corresponded to the material concentration phenomenon under compression or impact load, which was absent in conventional nonauxetic composites. In addition to experimental studies on the 3D auxetic structure, the authors also developed a finite-element model for the structure to validate its Poisson’s ratio behavior.³⁴ These auxetic composites, however, showed only an out-of-plane NPR effect, limiting their application to impact energy absorption and indentation resistance.

More recently, a novel 3D woven structure with an in-plane NPR effect was developed by modifying a conventional orthogonally through-the-thickness multilayered structure.³⁵ This structure offers a significantly higher NPR of −1.6 at a tensile strain of 8%. Notably, this structure exhibits NPR, even at higher tensile strains. Furthermore, the structure can be woven from high-performance yarns to achieve better mechanical properties, making it suitable for applications in automobile safety belts and composite reinforcement. Following the experimental study, a geometric model of the unit cell of the structure was developed to explore the fundamental properties of the auxetic geometry.³⁶ An equation was derived, based on the geometrical parameters of the yarns forming the auxetic geometry, and was used to predict the auxetic behavior of the 3D auxetic woven fabric at different longitudinal strains. Although the geometrical model provides further information about the structure, it is limited to a single geometric unit, which cannot provide detailed information about the structure under tensile deformation. Additionally, the geometrical model can only be applied based on geometrical parameters and does not consider material properties, limiting its applicability. Therefore, further study of the structure using the finite-element method is needed.^37
–39

This paper presents a finite-element analysis of the 3D woven structure with in-plane auxetic behavior for composite reinforcement. A 3D finite-element model of the fabric structure was developed and verified by comparing the calculated results with the experimental data. After verification, the model was used to identify the reasons for the auxetic behavior of the 3D structure at different longitudinal strains and the behavior of the warp and binding yarns. Furthermore, the model was used to evaluate the effect of the yarn properties on the auxetic behavior of the 3D structure. This study provides new insights into the factors that affect the auxetic behavior of a 3D structure under tensile extension. The results demonstrate that the finite-element model is a useful tool for predicting and designing the required properties of the 3D auxetic structure for specific applications, including composite reinforcement.

Methodology

Structure design and development

A conventional orthogonally through-the-thickness multilayered 3D structure, originally nonauxetic, was modified to create an auxetic property. Typically, this type of structure consists of three yarn systems: warp, weft, and binding yarns, in the X, Y, and Z directions, respectively, as shown in Figure 1. The warp and weft yarns are straight, while the binding yarn is woven through the thickness of the structure in an orthogonal manner along the warp direction. In most cases, the weave pattern of the binding yarn is a 1/1 plain weave to enhance the stability and compactness of the structure.

To convert this structure into an auxetic one, the three yarn systems were arranged in a special configuration with the desired diameters and properties of the yarns. In this study, a relatively thicker binding yarn was used, replacing a 1/1 plain weave in the warp direction with a 2/2 twill weave in the weft direction, as shown in Figure 2(a). This arrangement creates alternating spaces throughout the structure. In a subsequent modification stage, the weft yarn was replaced by a fine elastic yarn, as shown in Figure 2(b), causing the structure to shrink and fill the spaces by crimping the warp yarns around the orthogonal binding yarn in the lateral direction, as shown in Figure 2(c). With this modified arrangement, the 3D woven structure becomes auxetic, expanding in the lateral direction when a tensile deformation is applied in the warp direction. Therefore, the structure will switch from position (c) to position (b) under tensile deformation and from position (b) to position (c) in a relaxed state.

It is important to note that, during the fabrication process, the elastic weft yarns were inserted in a stretched form, as shown in Figure 2(b). It was intended to make the elastic yarn work as a restoring spring that could shrink the fabric structure when cut off from the weaving machine, as illustrated in Figure 2(c). At this stage, the 3D woven structure is referred to as being in an initial state or a relaxed state, because no external force or deformation is acting on it. Although the 3D auxetic woven structure is in a relaxed state, the elastic weft yarns are still in a stretched form, as they did not fully return to their initial length, owing to closed ends (the fabric selvage). Therefore, in the initial state, the elastic weft yarns are in a prestretched state.

To develop the modified 3D auxetic structure, the first step is to create the weave design, also known as the interlacing pattern, for the warp and weft yarns. The widely used technique of coding the warp and weft yarns of different layers was applied for this purpose, as shown in Figure 3(a). The corresponding weave of the fabric structure is represented in Figure 3(b). This weave design was then input into a semi-automatic machine equipped with automatic dobby shedding and manual weft insertion mechanisms for sample production, as depicted in Figure 3(c). Additionally, the machine was loaded with 16 harnesses, of which 8 harnesses were used based on weave design requirements. To save time during weaving preparation and production, a narrow sample, 25 mm wide, consisting of two layers was produced. Subsequently, this sample was utilized to evaluate its auxetic behavior through uniaxial tensile testing.

Characterization of auxetic behavior

The developed 3D woven fabric was tested using a universal tensile testing machine (Instron 5982) to assess its deformation behavior. The fabric sample was clamped between the two jaws of the machine, with the lower jaw being stationary and the upper jaw moving at a constant speed of 30 mm/min. Additionally, a high-resolution camera (Canon EOS 800D) was positioned in front of the sample at a reasonable distance to capture a video of the fabric’s lateral and longitudinal deformation, as shown in Figure 3(d). To facilitate analysis, the sample was premarked with horizontal and vertical dots. The recorded video was further processed by extracting images at regular intervals, corresponding to a specific percentage of longitudinal strain. The distance between the marked dots was measured using screen ruler software (open access) to calculate the lateral and longitudinal strains. Finally, these two strains were utilized to calculate the Poisson’s ratio of the fabric as

υ = - \frac{ε_{y}}{ε_{x}} = - \frac{y - y_{0} / y_{0}}{x - x_{0} / x_{0}}

(1)

where ε_x is the longitudinal strain and ε_y is the transverse strain; y₀ and x₀ are the initial width and length, respectively; and y and x are the width and length, respectively, after deformation.

Numerical modeling

Modeling the textile structure at the mesoscale (yarn level) is considered a challenging task, owing to architectural complexity, material anisotropy, and meshing challenges. Before establishing the finite-element model, the following assumptions were made, based on the findings of the experimental analysis conducted on the textile structure.

In the finite-element model, the initial state will be referred to as the postdeveloped state of the fabric structure under no external force, while the deformed state will be referred to as the fabric structure under tensile deformation.

In the initial state, two adjacent warp yarns within the same layer are in contact with each other.

The warp yarns are in contact with the binding yarns and the elastic weft yarns.

At the initial state, the elastic weft yarn is under a prestretch of 14.8%, while the warp and binding yarns are under zero extension or compression.

During the tensile extension, the deformation is applied to one end of the structure in the warp direction, while the other end will remain fixed.

The ends of the structure in the weft direction (along the width) should act as fabric selvage (i.e., fixed ends of weft yarns).

These assumptions are expected to enhance the accuracy of the finite-element model and ensure that the simulated results obtained will closely match the experimental findings.

Establishing a model of the fabric structure

The process used to develop the model for the 3D auxetic textile structure is explained in detail, ensuring the reproducibility of the model. To reduce the processing time, a relatively small yet adequately representative geometric area was chosen for the finite-element model. This area consists of 4 × 4 geometric unit cells, in both the warp and weft directions, representing a single repeat of the weave pattern, specifically a 2/2 twill weave, as shown in Figure 4(a). The geometric arrangement of yarns forming a 3D textile assembly was initially generated using open-source TexGen software,⁴⁰ and the resulting geometry was extracted as a STEP file for further processing. The STEP file was then imported into commercially available Abaqus software for simulation. as shown in Figure 4(b). Geometric parameters of the fabric model, such as the diameter of yarns and the distance between yarns, are shown in Figure 4(b), while their respective values are given in Table 1.

Table 1.

Geometric parameters of fabric structure for finite-element model

Description	Symbol	Value (mm)
Diameter of binding yarn	$D_{b}$	2.28
Diameter of warp yarn	$D_{p}$	0.7
Diameter of weft yarn	$D_{f}$	0.25
Distance between axial lines of two consecutive binding yarns	$A_{o}$	3
Distance between surfaces of two consecutive elastic yarns or elastic yarn and binding yarn	$B_{o}$	0.073
Distance between axial lines of warp yarns of top and bottom layers in thickness direction	$T_{o}$	0.95
Length of representative model structure	$L_{o}$	12
Width of representative model structure	$W_{o}$	7.36

Determining material properties for the model

The material composition and properties of the three yarn systems forming the textile assembly are given in Table 2. In the finite-element model, the warp yarn was defined as an isotropic elastic material, considering its braided structure, in which component yarns are oriented in various directions to form a quasi-isotropic structure. The elastic modulus of the warp yarn was calculated from region A (elastic deformation) of the stress–strain curve, as illustrated in Figure 5. Owing to its braided structure, the individual filaments of the yarn adjust to the applied tensile force, resulting in an initial low modulus at a negligibly small strain of 0.11%. As the filaments align, the yarn starts to resist the deformation, leading to an increase in modulus, as highlighted in yellow (region A) in Figure 5. This modulus of the warp yarn represents the elastic modulus; therefore, this is considered for the finite-element model. Subsequently, the yarn enters the plastic deformation stage at a tensile strain of 2.34%, highlighted in green (region B) in Figure 5. Hence, region A was defined with an isotropic property by inputting the axial modulus and Poisson’s ratio, while region B, which reflects the plastic deformation of the warp yarn, was defined with a plastic property by inputting the plastic engineering stress–strain data points of the curve. In contrast to the warp yarn, the binding yarn was defined with a transversely isotropic material property, owing to the presence of polyurethane in its core, which imparts transverse isotropy to the yarn. In addition, the radial modulus of the binding yarn is crucial in the structure, as the yarn undergoes compressive loads during deformation. A discrete local orientation of the binding yarn was defined to specify its axial modulus ( $E_{x}$ ) and radial modulus ( $E_{y}$ ). It is important to note that $E_{x}$ was calculated from the tensile stress–strain data within the elastic limit, while $E_{y}$ was obtained from a compression test performed on the binding yarn. The elastic weft yarn demonstrates substantially significant deformation, with a strain of over 400%, and it was defined with the hyperelastic property. The deformation of the elastic weft yarn remained within an elastic limit during the tensile process of the 3D woven fabric. Hence, assigning the hyperelastic property to the elastic weft yarn is appropriate for the finite-element model. The properties of the warp and binding yarns used in the finite-element model are given in Table 2. To assign the property of the weft yarn, the uniaxial test data (nominal stress–strain data) were initially input for the hyperelastic mechanical property. Subsequently, the material was evaluated to calculate the strain energy potential constants, based on the Ogden model of elasticity (for incompressible materials):⁴¹

W = \sum_{i = 1}^{N} \frac{µ_{i}}{α_{i}} (λ_{1}^{- α_{i}} + λ_{2}^{- α_{i}} + λ_{3}^{- α_{i}} - 3) + W_{volume}

(2)

where N is an order of the model that ranges from 1 to 3,

µ_{i}

and

α_{i}

are material constants,

λ_{1}

λ_{2}

and

λ_{3}

are variables of principal stretches in the three directions of strain, and

W_{volume}

indicates the volume change of the material that can be eliminated, in the case of incompressible materials.

Table 2.

Material properties of warp yarn and binding yarn

Yarn type	Material and structure	Young’s modulus (MPa)		Poisson’s ratio (–)
Yarn type	Material and structure	Axial ( $E_{x}$ )	Radial ( $E_{y}$ )	Poisson’s ratio (–)
Warp yarn	Polyester-based braid	1645.8	1645.8	0.35
Binding yarn	Polyester-wrapped polyurethane-based braid	39.7	9.9	0.35

The calculated values of the strain energy potential constants, μ and α, are given in Table 3. Furthermore, based on the nominal stress–strain data of the weft yarn, the strain energy potential constants remained stable for the Ogden model of order 2. The material models used in the Abaqus interface for all yarns are given in Table 4.

Table 3.

Material law parameters for elastic weft yarn

Order	µ	α
1	2.555 × 10⁻¹⁰	5.981
2	0.561	−3.401

Table 4.

Summary of material models used for each yarn

Yarn	Material behavior
Warp yarn	Elastic isotropic
Warp yarn	Plastic isotropic
Binding yarn	Elastic transversely isotropic
Elastic weft yarn	Elastic hyperelastic

Setting up the testing environment and executing the model

The three yarn components that make up the assembly were relatively straightforward for the finite-element modeling. However, the prestretch condition of elastic weft yarn (according to Assumption 4) complicates the model. Specifically, the elastic weft yarns are under a strain of 14.8% at the initial state. To carry this argument accurately, a two-step simulation method using a dynamic explicit solver was chosen. More specifically, the prestretch condition of the elastic weft yarn was achieved in the first step (Step 1) of the analysis by applying a displacement of 1.09 mm (equivalent to a strain of 14.8%) to the ends of the elastic weft yarns in the positive and negative y-directions, as shown in Figure 6(a). In addition, other yarns were temporarily fixed in this step to prevent unintended displacements. In Step 2, a tensile deformation of 4.2 mm (equivalent to a tensile strain of 35%) was applied to one end of the model in the positive x-direction, while the other end remained fixed, as shown in Figure 6(b). Moreover, the end columns of the binding yarns were fixed in the z-direction to prevent undesired rotation.

The interaction between yarns in the textile assembly is both crucial and challenging. To address this, a general contact was established for Step 1 with a global property of frictionless behavior to smoothly model the prestretch condition of the elastic weft yarn. However, the general contact was modified for Step 2 by enabling a penalty friction formulation of 0.4 between the surfaces of the binding and elastic weft yarns and the surfaces of the warp yarns. Additionally, to satisfy Assumption 6, a rough interaction was defined between the ends of the elastic weft yarns and the warp yarns, effectively acting as fabric selvage, with specific conditions of no slip and no separation after contact.

For the meshing, the element size was determined based on the yarn diameter. For example, the elastic weft yarn was seeded with an element size of 0.17 mm, the warp yarn with an element size of 0.2 mm, and the binding yarn with an element size of 0.25 mm. Owing to the nonlinearity in the yarn’s initial geometry, a tetrahedral element shape with a linear element type (C3D4) was assigned to the warp and binding yarns, while a hybrid element type (C3D4H) was chosen for the weft yarn. The selection of the hybrid element for the weft yarn was based on the strain energy potential constants, as weft yarn material was considered incompressible by the model, and could only be modeled using hybrid elements.

To calculate the finite-element simulated Poisson’s ratio of the auxetic 3D structure, pairs of nodes from a specific region (as specified in Figure 6(b) by red dots) were selected, in both the length (x) and width (y) directions. The data point at each deformation state was then extracted. The distance between the two points in the x-direction can be used to calculate the longitudinal deformation, while the distance in the y-direction can be used to calculate the transverse deformation. Once the two deformations are known, the Poisson’s ratio of the structure can be calculated using equation (1).

Results and discussion

Verification of finite-element model with experimental results

To verify the finite-element model, the simulated Poisson’s ratio and tensile stress of the 3D auxetic structure were compared with the experimental results calculated for real fabric in a previous study,³⁵ as shown in Figure 7(a) and (b), respectively. From Figure 7(a), it can be seen that both simulated and experimental Poisson’s ratios of the 3D textile structure exhibited a decrease at the initial tensile strain, with the smallest lowest Poisson’s ratios of −1.66 and −1.61 at tensile strains of 7.77% and 7.26%, respectively. After reaching the minimum Poisson’s ratio, each curve began to increase again, though at a slower pace. Since the simulated Poisson’s ratio followed the same trend as the experimental Poisson’s ratio, it can be concluded that the simulated and experimental results for the 3D auxetic structure are in good agreement, with a strong coefficient of determination (R²) of 0.882. However, it should be noted that the Poisson’s ratio at the initial tensile strain is smaller for the finite-element model than for the experimental Poisson’s ratio. This difference can be attributed to the ideal arrangement of the warp yarns in the finite-element model. Another reason for the difference is the nonlinear tensile modulus of the warp yarn; specifically, the modulus at initial strain from 0% to 0.9% is 10.1 MPa, followed by an increase in modulus to 27.18 MPa for a tensile strain of 0.9% to 3%. The modulus of 27.18 MPa is the elastic modulus of the yarn, while the initial lower modulus, of 10.1 MPa, is caused by the alignment of filaments with the tensile force. However, the modulus of 27.18 MPa is considered in the finite-element model, resulting in the initial difference in Poisson’s ratio.

From Figure 7(b), it can be observed that, although there are some differences in the simulated and experimental tensile stress–strain curves, they still show good agreement, with a strong R² value of 0.997. The slight discrepancies could be attributed to the slippage phenomenon occurring during experimental testing, whereas no slippage condition is assumed in finite-element analysis (FEA). It is challenging to eliminate microslippage of the fabric in the machine’s clamps during experimental tensile testing, especially for high-strength fabrics. This slippage leads to an increase in the longitudinal strain of the fabric, which adversely affects both the stiffness and the auxetic behavior. Based on the tensile stress–strain and Poisson’s ratio–tensile strain results, which exhibit significantly high values of R², the model is considered validated and can be used to predict and analyze the auxetic behavior of the 3D auxetic woven structure.

Deformation of structure and stress distribution analysis

In the geometry, the initial length of the elastic weft yarns was intentionally kept smaller than the total width of the model to attain the prestretching condition, as explained before. Consequently, the weft yarns were subjected to an initial tensile strain of 14.8% in Step 1, as shown in Figure 8(a) and (b). Once the initial strain conditions of the elastic weft yarns were met in Step 1, the structure was subjected to an axial tensile extension in the warp (x) direction. In Step 2, at zero strain, the warp yarns were perfectly crimped by the elastic yarns in the 3D structure, as shown in Figure 8(b). However, when a tensile displacement is applied, the warp yarns start to straighten in response to the applied force. This causes the binding yarns to be pushed laterally by the warp yarns, because of the unique alternating arrangement of these two types of yarn. This causes a lateral expansion of the whole structure, as shown in Figure 8(c) to (h), and leads to a NPR.

Owing to the application of an initial strain of 14.8% to the elastic weft yarns, they exhibit comparatively higher stresses than the warp yarns at the initial strains in Step 2. However, at higher longitudinal strains, as shown in Figure 8(h), the stresses in the warp yarns become dominant over those in the elastic weft yarns. Since the displacement in Step 2 was applied to the warp yarns, they play a central role in bearing the applied tensile load and exhibit a higher stress contribution. The binding yarns experience compression forces, owing to the pushing force exerted by the warp yarns, which contributes to the lateral expansion of the structure. Therefore, compressive stresses are generated in the binding yarns, as shown in Figure 8(i).

Insights into the deformation behavior of the 3D auxetic structure

In the experimental study, the results were analyzed based on overall physical assessments of the 3D auxetic structure under tensile deformation.³⁵ Based on those assessments, a geometrical model was developed to predict the auxetic behavior of the structure.³⁶ However, it is evident that general physical assessments cannot provide detailed information about the behavior of the individual yarns during structural deformation. Owing to this limitation, the Poisson’s ratio results obtained from the geometrical model²⁴ did not fit with the experimental results, as shown in Figure 9. Nonetheless, the finite-element model is based on the true material properties used in the experiment study, along with a significantly larger geometrical unit representing the whole 3D auxetic structure. Therefore, the finite-element model provides a more accurate and justifiable calculation of the Poisson’s ratio.

During the experimental and geometrical studies of the 3D auxetic structure, it was previously believed that the compression experienced by the binding yarns during deformation was caused solely by the restoring force of the elastic weft yarns. Although the analysis was correct, however, it was only one of two factors causing the compression of the binding yarn. The finite-element model revealed an additional factor that contributes to the compression of the binding yarn, which is the uniaxial stretching of neighboring pairs of warp yarns that attempt to move closer to each other during the decrimping process. To comprehend this phenomenon, the graphical representation of the structure was modified to display a top cross-sectional view with the elastic weft yarns omitted, as shown in Figure 10(a). If the binding yarns are assumed to be absent from the geometry, as shown in Figure 10(b), the neighboring pairs of warp yarns (referred to as Pair 1 and Pair 2) would certainly move closer to each other under the defined boundary conditions. However, owing to the presence of binding yarns in the actual scenario, the warp yarns experience a transverse displacement in the y-direction, as shown in Figure 10(c). As a result, the tendency of the warp yarns to approach each other generates an additional compressive force on the binding yarns, which acts as a second factor that influences the NPR of the structure.

Prediction of auxetic behavior with finite-element model

In science, FEA holds significant importance in predicting the properties and behavior of materials, products, or structures in specific conditions. Hence, the finite-element model developed in this study can be utilized to assess the auxetic behavior of the 3D auxetic woven fabric, considering different geometrical parameters and material properties. However, the previously developed geometrical model³⁶ of the 3D auxetic structure was primarily employed to predict and analyze its auxetic behavior based on various geometrical parameters, such as warp or binding yarn diameters. Therefore, the developed finite-element model will only be used to predict the auxetic behavior for varying material properties. Although the geometrical model successfully predicted the Poisson’s ratio for different geometrical parameters, it has limitations in predicting the auxetic behavior for varying material properties. Thus, the finite-element model is employed to predict the Poisson’s ratio of the 3D auxetic structure by modifying the most important factor, i.e., the stiffness of the binding yarn.

As the 3D auxetic woven fabric was formed on the weaving machine, it was observed that the bending stiffness, also known as the axial stiffness, of the binding yarn played a critical role in both the weaving process and the auxetic behavior of the structure. However, no physical tool was available to assess and quantify the effect of the bending stiffness of the binding yarn on the auxetic behavior of the 3D fabric. In addition, as explained in the previous section, the binding yarn undergoes a compression process during tensile deformation, which also affects the Poisson’s ratio of the 3D auxetic structure. Therefore, it can be concluded that both the axial and radial stiffnesses, which are correlated to the bending and compression of the binding yarn, respectively, play a significant role in the auxetic behavior of the 3D auxetic fabric structure. To investigate these two factors, the model was modified solely for binding yarn. The binding yarn was assigned different moduli for the axial ( $E_{x}$ ) and radial ( $E_{y}$ ) directions of the binding yarn. The values of $E_{x}$ and $E_{y}$ given in Table 5 represent the moduli used in this study to obtain conclusive results, explaining the effect of the binding yarn moduli on the auxetic behavior of the 3D woven structure. The 3D auxetic structure exhibited intriguing outcomes when the given values of $E_{x}$ and $E_{y}$ for the binding yarn were employed to predict the auxetic behavior.

Table 5.

Axial and radial moduli of binding yarn

Factor	Level
Axial modulus, $E_{x}$ (MPa)	40	40	40	40	50	60	70
Radial modulus, $E_{y}$ (MPa)	10	20	30	40	10	10	10

Figure 11(a) provides clear information about the relationship between the axial and radial moduli of the binding yarn and the NPR of the 3D auxetic structure. The auxetic behavior of the 3D structure becomes stronger when the radial modulus of binding yarn is increased while keeping the axial modulus constant. In contrast, the auxetic behavior becomes weaker when the axial modulus of the binding yarn is increased with a fixed radial modulus.

In the first scenario, increasing the radial modulus makes the binding yarn stiffer in the thickness direction. As a result, it becomes more difficult for the warp yarn to compress the binding yarn under tensile deformation. This reduced compression of the binding yarn leads to a higher NPR of the 3D auxetic structure. This claim is supported by Figure 11(b), which illustrates the compressive strain of the binding yarn as a function of the tensile strain of the 3D auxetic structure. It can be observed that the compressive strain decreases as the radial modulus increases. A smaller compressive strain indicates a smaller decrease in the diameter of the binding yarn. Additionally, the compressive strain remains the same for samples with similar out-of-plane moduli, regardless of the change in axial modulus.

In the second scenario, a larger axial modulus indicates a larger bending stiffness of the binding yarn. This larger bending stiffness causes two problems for the 3D structure: the first problem is related to the practical weaving process of the 3D woven structure, where a binding yarn is more difficult to bend when it changes its movement direction alternately from the top face to the bottom face and from the bottom face to the top face of the structure. The second problem is relevant to the stress concentration, as shown in Figure 12(a) and (b), where significantly larger stresses are developed in a binding yarn with a larger axial modulus than in one with a lower axial modulus. From the aforementioned scenarios, it could be concluded that a better auxetic behavior can be achieved by using a binding yarn that possesses the unique property of being easy to bend but difficult to compress.

Conclusions and recommendations

This research has successfully demonstrated the potential of FEA in providing detailed insights into the auxetic behavior of 3D woven structures at the yarn level, which was previously a challenging task. A reliable 3D finite-element model has been established that simulates the behavior of the structure effectively, bridging the gap left by experimental and geometrical studies. The main findings from the study are stated next.

The finite-element model provides a significant advantage over the previously developed geometrical model in predicting the Poisson’s ratio of the 3D auxetic structure, considering the material’s properties. Thereby, results achieved through the developed finite-element model are more accurate.

The FEA highlighted the critical role of the uniaxial stretching of neighboring warp yarns that causes compression of the binding yarn during the decrimping process.

Based on the findings from the FEA, an important question regarding the disparity between the experimentally determined Poisson’s ratio and the geometrically calculated Poisson’s ratio of the 3D auxetic structures has been addressed. Specifically, it has been determined that the compression of binding yarns during the lateral expansion of the structure is not caused solely by the elastic weft yarns, but that the warp yarns themselves also contribute to this compression.

The analysis also revealed the significant impact of the axial and radial moduli of the binding yarn on the Poisson’s ratio of the 3D woven structure. The results show that a better auxetic effect can be achieved by using binding yarn with a lower axial modulus (easy to bend) and a higher radial modulus (difficult to compress).

The previously conducted experimental study³⁵ and the FEA of 3D auxetic woven fabrics provide concrete information on their fundamental properties and behavior. This information can be valuable for designing composite structures that utilize 3D auxetic woven fabric as reinforcement. To advance this research, future studies could focus on the development and simulation of polymeric composites reinforced with 3D auxetic woven fabrics.

Footnotes

Data availability statement

All data that support the findings of this study are included within the article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Hong Kong General Research Fund (project number 15607920) and the Research Institute for Intelligent Wearable Systems of The Hong Kong Polytechnic University, in the form of an internal project (number P0039471).

ORCID iDs

Muhammad Zeeshan

Li Li

Hong Hu

References

Luan

West

DenHartog

, et al. Auxetic deformation of the weft-knitted Miura-ori fold. Text Res J 2020; 90: 617–630.

Lakes

Foam structures with a negative Poisson’s ratio. Science 1987; 235: 1038–1040.

Miller

Hook

Smith

, et al. The manufacture and characterisation of a novel, low modulus, negative Poisson’s ratio composite. Compos Sci Technol 2009; 69: 651–655.

Parisi

Allen

Colonna

, et al. Indentation and impact response of conventional, auxetic, and shear thickening gel infused auxetic closed cell foam. Smart Mater Struct 2023; 32: 074004.

Duncan

Alderson

, and Allen

Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams. Smart Mater Struct 2021; 30: 035014.

Alderson

KL.

Auxetic materials. Proc Inst Mech Eng, Part G: J Aerosp Eng 2007; 221: 565–575.

Evans

Nkansah

Hutchinson

, et al. Molecular network design. Nature 1991; 353: 124–124.

Liu

Zhao

, et al. Auxetic meta-materials and their engineering applications: A review. Eng Res Express 2023; 5: 042003.

Ntintakis

Stavroulakis

, and Karathanasopoulos

Auxetics, theory and textile applications. In: Proceedings of the joint international conference: 10th textile conference and 4th conference on engineering and entrepreneurship (ed Guxho G, Spahiu TK, Prifti V, et al.), Tirana, Albania, 19–20 October 2023, pp. 509–517. Cham: Springer.

10.

Nguyễn

Fangueiro

Ferreira

, et al. Auxetic materials and structures for potential defense applications: An overview and recent developments. Text Res J 2023; 93: 5268–5306.

11.

Mizzi

Salvati

Spaggiari

, et al. Highly stretchable two-dimensional auxetic metamaterial sheets fabricated via direct-laser cutting. Int J Mech Sci 2020; 167: 105242.

12.

Mizzi

Salvati

Spaggiari

, et al. 2D auxetic metamaterials with tuneable micro-/nanoscale apertures. Appl Mater Today 2020; 20: 100780.

13.

Zahra

Behaviour of 3D printed re-entrant chiral auxetic (RCA) geometries under in-plane and out-of-plane loadings. Smart Mater Struct 2021; 30: 115011.

14.

Alderson

Evans

The fabrication of microporous polyethylene having a negative Poisson’s ratio. Polymer 1992; 33: 4435–4438.

15.

Caddock

Evans

KE.

Microporous materials with negative Poisson’s ratios. I. Microstructure and mechanical properties. J Phys D: Appl Phys 1989; 22: 1877–1882.

16.

Ravirala

Alderson

, et al. Expanding the range of auxetic polymeric products using a novel melt-spinning route. Phys Status Solidi 2005; 242: 653–664.

17.

Ravirala

Alderson

Davies

, et al. Negative Poisson’s ratio polyester fibers. Text Res J 2006; 76: 540–546.

18.

Alderson

Smart

, et al. Auxetic polypropylene fibres: Part 1—Manufacture and characterisation. Plast Rubber Compos 2002; 31: 344–349.

19.

Sloan

Wright

, and Evans

KE.

The helical auxetic yarn—A novel structure for composites and textiles; geometry, manufacture and mechanical properties. Mech Mater 2011; 43: 476–486.

20.

Lim

T-C.

Semi-auxetic yarns. Phys Status Solidi 2014; 251: 273–280.

21.

Verma

Smith

Griffin

, et al. Wool nonwovens as candidates for commodity auxetic materials. Eng Res Express 2020; 2: 045034.

22.

Verma

Shofner

Lin

, et al. Inducing out-of-plane auxetic behavior in needle-punched nonwovens. Phys Status Solidi 2015; 252: 1455–1464.

23.

Verma

Wagner

Griffin

, et al. Reversibility of out‐of‐plane auxetic response in needle‐punched nonwovens. Phys Status Solidi 2022; 259: 2200387.

24.

Chang

, and Jiang

Design and fabrication of auxetic warp-knitted structures with a rotational hexagonal loop. Text Res J 2016; 86: 2151–2157.

25.

Zeeshan

Ali

Riaz

, et al. Optimizing the auxetic geometry parameters in few yarns based auxetic woven fabrics for enhanced mechanical properties using grey relational analysis. J Nat Fibers 2022; 19: 4594–4605.

26.

Gonçalves

Magalhães

Rana

, et al. Novel high performance auxetic fibrous structures for composite reinforcement. IOP Conf Ser Mater Sci Eng 2018; 406: 012046.

27.

Steffens

Oliveira

, and Fangueiro

Energy absorption from composite reinforced with high performance auxetic textile structure. J Compos Mater 2021; 55: 1003–1013.

28.

Sun

Lin

, et al. Preparation of soft composite reinforced with auxetic warp-knitted spacer fabric for stab resistance. Text Res J 2020; 90: 323–332.

29.

Liaqat

Samad

Hamdani

STA

, et al. The development of novel auxetic woven structure for impact applications. J Text Inst 2017; 108: 1264–1270.

30.

Zulifqar

Hua

, and Hu

Development of uni-stretch woven fabrics with zero and negative Poisson’s ratio. Text Res J 2018; 88: 2076–2092.

31.

Cao

Zulifqar

Hua

, et al. Bi-stretch auxetic woven fabrics based on foldable geometry. Text Res J 2019; 89: 2694–2712.

32.

Innovative three-dimensional fabric structure with negative Poisson’s ratio for composite reinforcement. Text Res J 2013; 83: 543–550.

33.

Zhou

Zeng

Jiang

, et al. Low-velocity impact properties of 3D auxetic textile composite. J Mater Sci 2018; 53: 3899–3914.

34.

, and Liu

A finite element analysis of a 3D auxetic textile structure for composite reinforcement. Smart Mater Struct 2013; 22: 084005.

35.

Zeeshan

, and Zulifqar

Three-dimensional narrow woven fabric with in-plane auxetic behavior. Text Res J 2022; 92: 4695–4708.

36.

Zeeshan

, and Etemadi

Geometric analysis of three-dimensional woven fabric with in-plane auxetic behavior. Polymers 2023; 15: 1326.

37.

Luan

Sun

, and Gu

Ballistic impact damages of 3-D angle-interlock woven composites based on high strain rate constitutive equation of fiber tows. Int J Impact Eng 2013; 57: 145–158.

38.

Luan

Energy absorption of three-dimensional angle-interlock woven composite under ballistic penetration based on a multi-scale finite element model. Int J Damage Mech 2015; 24: 3–20.

39.

Luan

West

McCord

, et al. Mosquito-textile physics: A mathematical roadmap to insecticide-free, bite-proof clothing for everyday life. Insects 2021; 12: 636.

40.

Brown

Long

AC.

Modeling the geometry of textile reinforcements for composites: TexGen. In: Boisse

(ed) Composite reinforcements for optimum performance. Duxford: Woodhead, 2021, pp. 237–265.

41.

Ogden

RW.

Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc London, Ser A 1972; 326: 565–584.