A finite element analysis of an auxetic warp-knitted spacer fabric structure

Abstract

This paper reports the tensile deformation behavior of an auxetic warp-knitted spacer fabric structure by using the finite element (FE) method. The precise geometry of the outer layer structure of the fabric was firstly obtained from a micro X-ray computerized tomography scan and then inputted to an ANSYS 13.0 software to generate geometrical models. The geometrical models were then meshed using the PLANE 183 element to build FE models. Based on the FE models built, deformation behaviors of the auxetic fabric in the course direction and wale direction were simulated and compared with experimental results. The FE analysis showed that the simulated fabric deformations at different tensile strains were very close to the real fabric deformations and the FE-generated Poisson’s ratio-strain curves agreed well with the experimental ones. Therefore, the FE models could be used for predicting the deformation behaviors of the auxetic warp-knitted spacer fabric stretched in different fabric plane directions.

Keywords

auxetic fabric warp-knitting spacer fabric finite element model deformation behavior

Fabrics that exhibit negative Poisson’s ratios (PR) are called auxetic fabrics.¹ They are different from conventional fabrics and laterally expand when stretched^2–4 or laterally shrink when compressed.^5,6 This unusual behavior provides them with a number of particular properties, among which it is possible to cite excellent formability under bending,⁴ indentation effect under compression,⁵ reduced garment pressure on human skin when extended, etc. These properties bring auxetic fabrics many application potentials in functional clothing, maternity dresses, sportswear, children’s wear, medical textiles, etc.⁷

Auxetic fabrics could be fabricated from different fabric structures using auxetic or non-auxetic yarns. Julian et al.⁸ designed auxetic woven fabrics with the use of auxetic helical yarns as warp yarns⁹ and found that the woven fabrics had both in-plane and out-of-plane negative PRs. Miller et al.¹⁰ reported a study on woven composites made from the same type of auxetic yarns and demonstrated that auxetic behavior could be realized in a woven composite structure. Although the woven fabrics and composites inherited the auxetic behavior of the auxetic yarns, the transversal expansion of the yarns under extension was restricted by the structures and matrix. As a result, the auxetic behavior of the fabrics and composites was lower than that of the auxetic yarns. Compared with auxetic woven fabrics, more auxetic fabrics were developed based on knitted structures using non-auxetic yarns. Ugbolue et al.^11,12 designed and fabricated different auxetic warp-knitted fabrics based on a re-entrant hexagonal structure and found that the highest negative PR value of the fabrics could reach about −0.5. Alderson et al.¹³ developed another kind of auxetic warp-knitted fabric based on a double-arrow structure and found that the auxetic effect of the fabric could only be obtained when stretched in a diagonal direction. In addition to warp-knitted structures, weft-knitted structures were also employed to produce fabrics with negative PRs. Liu et al.² and Hu et al.³ successfully developed a series of auxetic weft-knitted fabrics based on different auxetic geometries using flat knitting technology and found that the negative PR value of these fabrics could reach −0.6. Glazzard and Breedon¹⁴ designed another series of auxetic weft-knitted fabrics based on the double-arrow structure. Besides auxetic behavior, it was also found that auxetic weft-knitted fabrics could show an interesting effect for fashion design. In addition to the woven and knitted structures, other fabric structures were also adopted for producing auxetic fabrics. Ge and Hu⁵ and Ge et al.⁶ have recently developed a new kind of three-dimensional (3D) auxetic fabrics for composite reinforcement by combining non-woven and stitching structures. This kind of fabric showed auxetic behavior when compressed in the fabric thickness direction.

Although many auxetic fabrics have been developed, the auxetic effect of these fabrics is limited and most of auxetic fabrics are not very stable. Some auxetic fabrics even lose their auxetic effect after one stretch. To resolve these problems, a new kind of auxetic warp-knitted spacer fabric⁴ with higher and more stable auxetic effect was developed. The highest negative PR value of the fabric could reach −2.5. In our previous work,¹⁵ deformation behavior of this kind of auxetic fabric has been investigated based on the observations of the fabric structures at different tensile strains. Two semi-empirical equations for theoretically calculating PR values of the fabrics in the course direction and the wale direction were established based on a geometric analysis, in which the geometry of two outer layers of the fabric was simplified as repeated regular rectangles when stretched in the course direction, and the repeated hexagons when stretched in the wale direction. However, the real outer layer geometry of this kind of auxetic fabric is very complicated and irregular. It could not be accurately represented by the simplified regular rectangles or hexagons. Besides, the yarn material property could not be included in a geometry analysis. The more effective way to analyze the deformation behavior of this kind of auxetic fabric is to use the more precise geometry of the auxetic fabric with consideration of material properties. To do so, in this work, the auxetic fabric was first scanned by a micro X-ray computerized tomography (CT) machine to get the precise geometry of the fabric. Based on the precise geometry obtained, a finite element (FE) model was then established. Using the established FE model, the deformation behavior of the fabric in the uniaxial tensile condition was finally simulated and compared with the experiment results. Compared with the previous study, the advantage of this study is that the precise geometry of the auxetic warp-knitted spacer fabric was used. At the same time, the material property of the fabric could be included in the analysis, which makes the simulation more accurate and interesting. In addition, the deformation behavior of the fabric in any direction and stress distribution in the fabric structure could be simulated using FE analysis and the distribution of internal stresses in the fabric structure could be clearly demonstrated.

Fabric geometry acquiring

Micro X-ray CT scanning

Auxetic warp-knitted spacer fabrics are a special kind of spacer fabrics formed with outer layer geometrical structures having negative PRs. As shown in Figure 1(a), the auxetic warp-knitted spacer fabric used in this study was fabricated through a heat-setting of a base spacer fabric knitted on a warp-knitted machine equipped with six yarn guide bars. While guide bars 1, 2 and 5, 6 were respectively used to knit the two outer layers using 400D/96 F DTY polyester, guide bars 3, 4 were used to knit the spacer layer that connected the two outer layers together using 0.12 mm polyester monofilament. Below is the chain notation for each yarn guide bar. The thickness of the fabric produced was 7.9 mm.

Figure 1.

Auxetic fabric and scanned pictures: (a) fabric picture; (b) reconstruction of outer layer; (c) overview; (d) top view.

GB1: (1-0-0-0/1-2-2-2)*3/(2-3-3-3/2-1-1-1)*3//

GB2: (2-3-3-3/2-1-1-1)*3/(1-0-0-0/1-2-2-2)*3//

GB3: 4-5-4-3/4-5-4-3/4-5-4-3/4-5-4-3/4-5-4-3/4-5-3-2/3- 4-3-2/3-4-3-2/3-4-3-2/3-4-3-2/3-4-3-2/4-5-4-3//

GB4: 1-0-1-2/1-0-1-2/1-0-1-2/1-0-1-2/1-0-1-2/1-0-2-3/2- 1-2-3/2-1-2-3/2-1-2-3/2-1-2-3/2-1-2-3/1-0-1-2//

GB5: (2-3-3-3/2-1-1-1)*3/(1-0-0-0/1-2-2-2)*3//

GB6: (1-0-0-0/1-2-2-2)*3/(2-3-3-3/2-1-1-1)*3//

In our previous study, the deformation behaviors of the fabric were already investigated based on the observation of the fabric pictures taken at different tensile strains. However, the precise fabric geometry could not be obtained by these pictures due to very complicated yarn arrangement in the fabric structure. So, in this study, a Micro X-ray CT scan was firstly conducted to obtain the precise geometry of the auxetic spacer fabric structure. Since the middle layer yarns are perpendicular to the outer layer plane, their effect on auxetic behavior of the auxetic warp-knitted spacer fabric should be very small. This can be confirmed by the PR-tensile strain curves shown in Figure 2, from which it can be seen that the curve of the single outer layer cut from the 3D auxetic warp-knitted spacer fabric is not significantly different from that of the auxetic warp-knitted spacer fabric. For this reason, only one outer layer of the auxetic fabric was scanned to get the precise geometry of the fabric in this study. The scanning machine used was a Scanco medical VivaCT 40 micro-CT machine (SCANCO medical AG, Fabrikweg 2, CH-8306 Bruttisellen, Switzerland) and the scanning condition was 5 µm spot size, 50–70 kVp and 160 µA. The sample size used for scanning was 20 mm × 20 mm. The scan was performed along the thickness direction of the auxetic fabric and each scanned slice was parallel to the outer layer plane direction.

Figure 2.

Comparison of Poisson’s ratio-strain curves of single outer layer and three-dimensional (3D) auxetic spacer fabric.

The reconstructed outer layer structure of the auxetic fabric at different views is shown in Figures 1(b)–1(d). From Figure 1(b), it can be seen that the outer layer structure of the auxetic fabric is constructed by short ribs and long ribs that are formed with yarns. Although the profile of each rib in the outer layer geometry of the auxetic fabric was clearly and precisely reconstructed, the geometrical configuration of each yarn could not be clearly observed in the reconstructed figures due to the limitation of the CT machine resolutions. Therefore, it is not suitable to model the fabric structure at yarn level using the scanned geometry. Since the rib structure is very clear in the scanned geometry, the FE analysis could be done at rib level. In this case, ribs could be considered as a homogenous material and their properties could be determined from the testing result of the real rib structure. On the other hand, since the rib geometry has no obvious difference in the thickness direction, the geometry of ribs could be obtained from a scanned slice that most clearly shows the geometry of ribs. Under this condition, the outer layer structure of the auxetic fabric could be modeled as a two-dimensional (2D) structure.

CT image processing

The CT scanned slice, which could more clearly show the view of the rib geometry of the outer layer structure of the auxetic fabric, is shown in Figure 3(a). Since the original CT images were very dark and it was not easy to capture the rib geometry, the brightness and contrast of the figure were first adjusted to make the profile of the figure clearer (Figure 3(b)). Then the figure was processed through Matlab software to acquire the outlines of the ribs, as shown in Figure 3(c). The outline dots and their coordinates could be obtained from the Matlab software. Figure 3(d) shows the outline dots of one repeating unit acquired. Finally, the coordinates of outline dots were inputted to ANSYS 13.0 to build the geometrical model of the auxetic fabric for further analysis. From Figure 3(b), it can be seen that the structures of the repeating units in a real fabric are not very uniform. In order to facilitate the building of the geometrical model, only the outline dots of one repeating unit acquired was used to build the geometrical model with the uniform repeating units.

Figure 3.

Computerized tomography slice image of auxetic fabric: (a) original slice; (b) after adjusting brightness and contrast; (c) after Matlab treatment; (d) acquired outline dots.

Finite element analysis

Geometrical models

Geometric models were built by inputting the coordinates of the outline dots above obtained to the ANSYS 13.0 software. The dots obtained embrace four hole areas in each repeating unit, as shown in Figure 4(a). To obtain the geometry of the outer layer structure of the fabric, the hole areas were repeated and subtracted from a rectangle plane having the same size of the tested fabric sample. Figures 4(b) and (c) show the geometrical models of the outer layer structure built for simulating stretch in the course direction and wale direction, respectively.

Figure 4.

Geometrical models of auxetic fabric: (a) geometry of hole areas in a repeating unit; (b) geometrical model in the course direction; (c) geometrical model in the wale direction.

Material properties

For facilitating the FE analysis, the ribs were considered as continuous homogenous material. To determine the rib material properties, a tensile test of the ribs cut from the auxetic fabric was carried out by using a SANSI CMT-1000 machine equipped with a 100 N unit cell. Due to the restriction of the rib length, the effective length for test is limited for 4 mm. So the gauge length of the machine was set at 4 mm. The testing speed used was 5 mm/min. The stress–strain curve of the rib from the test is shown in Figure 5. It can be seen that the curve is very close to a linear curve. Therefore, the rib material was modeled as an isotropic linear elastic material that has two elastic constants. The first constant is the Young’s modulus. To determine the Young’s modulus, a fitting line was plotted according to the testing data. The R-squared value of the fitting line was 0.99794. The slope of the fitting line gave the modulus of the material, which was 234 MPa. The second elastic constant is the PR. Since the fabric was made of polyester yarns with a PR of 0.3, the same value was selected as the PR of the modeled material for FE analysis.

Figure 5.

Stress–strain curve of ribs.

FE models and boundary conditions

Based on the fabric outer layer geometrical models and material properties obtained above, FE models for simulating the deformations of the fabric in the course direction and wale direction could be built, respectively. Since the fabric outer layer structure was assumed as a 2D structure and the fabric was only stretched in the fabric plane directions, the PLANE 183 element was employed for meshing the geometry of the out layer structure to build FE models. PLANE183 is a 2D, eight-node or six-node element, and each node has X and Y translational degrees of freedom (DOFs). It is very competent for modeling irregular geometries, which is very suitable for the outer layer structure of the auxetic spacer fabric. The boundary conditions were completely set according to the real testing conditions. Since there were no rotational DOFs in the tensile condition, only translational DOFs were applied on each FE model. As the fabric was simulated as a plane form, there are only 2D DOFs. In this study, the wale direction of the fabric was defined as the X direction, and the course direction was defined as the Y direction. To simulate the real tensile testing process, a tensile displacement was applied onto the fabric instead of a load, and the maximum displacement applied was equal to the original length of the fabric, which corresponded to 100% of tensile strain. To do that, one end of the fabric structure was fixed, that is, UX = UY = 0. On the other end, a displacement was applied in the tensile direction. On this end, UX = 0, and UY is equal to the tensile displacement when stretched in the course direction, and UY = 0, and UX is equal to the tensile displacement when stretched in the wale direction. The boundary conditions applied on the FE models in the course direction and the wale direction are shown in Figures 6(a) and (b), respectively.

Figure 6.

Boundary conditions of FE models: (a) stretched in the course direction; (b) stretched in the wale direction.

After all the boundary conditions were set, the FE models could be solved. Different mesh sizes were tried during the meshing process. It was found that too large mesh size would cause meshing problem, and too small meshing size would extend the solving time. After several trials, the mesh size of 0.4 was used in this work, and 35,000 elements were generated under this mesh size.

Results and discussion

Fabric deformations simulated at different tensile strains

The simulated deformations of the fabric at different tensile strains in the course direction are shown in Figure 7. For comparison, the deformations of a real fabric stretched with the same strains are also shown in Figure 7. It can be seen that the simulated deformations of the fabric are very close to the real deformations of the fabric. Both the simulation and deformed fabric pictures show that the fabric outer layer structure opened up gradually with the increase of the tensile strain. As a result, a high auxetic effect was obtained when stretched in the course direction. However, the simulation could only reach 30% of the tensile strain, which was much lower than the maximal tensile strain of the fabric. After this tensile strain, the simulation process was stopped and no calculated results could be obtained. The reason is that in the FE model, the ribs were considered as continuous homogenous material and their deformations were limited. However, in a real fabric, the ribs are formed with yarns, and the yarn slippage effect could considerably increase the deformation rate of ribs under tension. Therefore, a higher tensile strain could be reached in a real fabric.

Figure 7.

Deformations of fabric structure at different tensile strains when stretched in the course direction: (a)(c)(e) finite element simulation; (b)(d)(f) real fabric.

The simulated and tested deformations of the fabric when stretched in the wale direction are shown in Figure 8. It can be found that the simulated deformations of the fabric structure in this direction are also very close to the real deformations of the fabric. However, the transverse size has no obvious change in a large tensile strain. This implicates that the fabric has low auxetic effect when stretched in the wale direction. The same as stretch in the course direction, the maximal tensile strain reached in the FE simulation is also lower than that of a real fabric. However, in the wale direction, the FE simulation could reach about 75% of the tensile strain as ribs are easier to rotate to the wale direction when stretched.

Figure 8.

Deformations of fabric structure at different tensile strains when stretched in the wale direction: (a)(c)(e) finite element model simulation; (b)(d)(f) real fabric.

Von Mises Stress distribution

The Von Mises Stress (VMS) distribution pictures are shown in Figure 9. It can be found that the stress mainly distributes on the short ribs towards the tensile direction when stretched in the course direction. As the long ribs bear very small stress, they could rotate more easily in the transverse direction, causing the fabric to open up more easily under tension and resulting in a high auxetic effect of the fabric.

Figure 9.

Von Mises Stress distributions at different tensile strains: (a)(b) stretched in the course direction; (c)(d) stretched in the wale direction.

When stretched in the wale direction, the stress is lower than that when stretched in the course direction for the same tensile strains. This means the fabric is easier to deform when stretched in the wale direction. Different from extension in the course direction, the VMS is higher in the short ribs towards the transverse direction. The stress on the long ribs is small, which enables the long ribs to very easily rotate to the tensile direction. That is why the tensile strain could reach a very high value when stretched in the wale direction.

Poisson’s ratio

After the simulation finished, the simulated tensile and transverse displacements of the fabric could be obtained from the ANSYS 13.0 software to calculate PR values of the fabric. In order to compare the calculated PR values with those from experiment, a tensile test of the real auxetic fabric was conducted using an Instron 5566 machine. The gauge length of the machine was set as 150 mm and the tensile speed was set as 30 mm/min. The fabric sample size used for the testing was 200 mm × 50 mm. Before testing, as shown in Figure 10(a), nine dots were first marked on the fabric sample for recording the size change of the fabric in both the tensile and transversal directions during test. The fabric structures before stretch and stretched at different strains were photographed by a camera. Then the size changes of the fabric could be obtained by measuring the distances of the points on the pictures via a screen ruler. The measured values were finally used to calculate the tensile and transverse strains through Equations (1) and (2):

ɛ_{a} = - \frac{X - X_{0}}{X_{0}}

(1)

ɛ_{t} = - \frac{Y - Y_{0}}{Y_{0}}

(2)

where x₀, x, y₀ and y are the distances between two marked dots in the transverse direction and tensile direction in the initial state and deformed state, respectively;

ɛ_{a}

and

ɛ_{t}

are the tensile strain and the transverse strain. After

ɛ_{a}

and

ɛ_{t}

were known, PR ν could be obtained through Equation (3):

ν = - \frac{ɛ_{t}}{ɛ_{a}}

(3)

Figure 10.

Dots marked on the fabric sample: (a) before stretch; (b) under stretch.

For the FE analysis, two pairs of nodes located in the central part of the fabric structure in both tensile and transverse directions were selected to determine the size changes of the simulated fabric, respectively. The original distances of each pair of nodes were 40 mm, which equaled the same value used in the experiment. As the original position and the displacement of the nodes could be directly obtained from the ANSYS 13.0 software, both the tensile strain and transverse strain could be calculated. After the tensile strain and transverse strain were known, the PR could be finally calculated.

The PR-strain curves from the FE analysis and experiment in both the course and wale directions are shown in Figure 11. It can be seen that the curves from the FE analysis have the same variation trends as those from the experiment. Although some differences exist between the FE analysis and experiment, the curves from the FE analysis could well predict the deformation behaviors of the auxetic fabric. The differences between the FE analysis and experiment mainly come from the simplification of the FE models and material properties. In particular, ribs were simplified as isotropic continuous homogenous material and the yarn slippage effect was ignored in the FE analysis. To get better understanding of deformation behavior of the auxetic fabric, a FE model at yarn level seems to be required. However, that will considerably increase the number of elements and calculating time and cost.

Figure 11.

Comparison of finite element (FE) simulated and experimental Poisson’s ratio-strain curves of auxetic fabric.

Another interesting phenomenon between the FE analysis and experimental results is that the simulated auxetic effect of the fabric is lower than that from the test when stretched in the course direction, and higher than that from the test when stretched in the wale direction. As shown in Figure 11, when stretched in the course direction, the tested PR value of the auxetic fabric gradually increases from −2.0 to −0.9 when the tensile strain increases from 5% to 30%. In the same tensile strain range, the simulated PR value increases from −1.8 to −0.8, which means the simulated auxetic effect is always lower than that of the real fabric. Conversely, when stretched in the wale direction, the simulated auxetic effect is higher. The simulated PR value gradually increases from −0.25 to 0 when the tensile strain increases from 5% to 70%. The tested PR value of the auxetic fabric gradually increases from −0.15 to 0.5 at the same tensile strain range. The main reason for the difference is that the vertexes of ribs in the real fabric are more flexible than those of the simulated ribs. When a real fabric is stretched in the course direction, the long ribs are more sensitive and easier to rotate to the transverse direction, resulting in more open up effect. Consequently, the real auxetic fabric has higher auxetic effect than the simulated structure when stretched in the course direction. However, when stretched in the wale direction, the flexible vertexes of the real fabric enable the short ribs to rotate more easily to the tensile direction, making the transverse strain of the real fabric decrease more quickly than that of the simulated fabric structure. As a result, the auxetic effect of the real fabric is lower than that of the simulated fabric structure.

Conclusions

This study is focused on the simulation of the deformation behavior of auxetic warp-knitted spacer fabric using the FE method. The precise geometry of the outer layer of the fabric was firstly obtained from a micro X-ray CT scan and then inputted to an ANSYS 13.0 software to generate geometrical models. The geometrical models were then meshed by the PLANE183 element to build FE models for simulating deformation behaviors of the auxetic fabric in the course and wale directions, respectively. Some conclusions can be drawn from this study as follows.

The FE simulation could well predict deformation behaviors of the auxetic fabric. The simulated fabric deformations at different tensile strains are very close to those obtained from experiments. High auxetic behavior of the fabric is obtained when stretched in the course direction.

The low stress is distributed on the long ribs when the fabric is stretched in a uniaxial direction. The long ribs are easy to rotate to the transverse direction when stretch is in the course direction, and are easy to rotate to the tensile direction when stretch is in the wale direction. This makes the fabric have high auxetic effect when stretched in the course direction and high tensile strain when stretched in the wale direction.

Although the FE simulation can only be performed in a relatively low tensile strain range, the simulated PR values of the auxetic fabric agree well with the experimental PR values. The differences between the FE analysis and experimental results mainly come from the simplification of the FE models and material properties. To get a better understanding of deformation behavior of the auxetic fabric, a FE model at yarn level should be conducted in a future study.

Footnotes

Funding

This work was supported by the Research Grants Council of Hong Kong Special Administrative Region Government (grant no. 518109).

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