Fabrication and property of auxetic warp-knitted spacer structures with mesh

Abstract

For the purpose of reconstructing a three-dimensional auxetic structure adopting spacer warp-knitting techniques, a simple mathematical model based on the structure of auxetic warp-knitted spacer hexagonal meshes was established and several samples were produced using an E22 RD7/2-12EN warp-knitting machine. It is predicted from synthesizing the results of both mathematical modeling and experimental sampling that auxetic performance of this structure mainly depends on the contraction distortion before stretching in three directions, especially in the y-axis. When it comes to knitting parameters, yarn fineness, machine gauge, fabric density, movement of guide bars, and inclination angle of wales all matter for the auxetic performance, which is also affected by the stiffness of materials and the lodging of spacer yarns. It is also concluded that successive inlaying in the same direction of the spacer guide bar favors contraction distortion in the y-axis, which means auxetic performances can be more easily achieved in this way.

Keywords

negative Poisson’s ratio warp-knitted spacer fabric mathematical modeling performance

Auxetic fabrics, namely fabrics with negative Poisson’s ratio, are gaining greater attention, having been studied and developed rapidly over the past 20 years. Compared to conventional materials, many properties of auxetic fabrics – such as mechanical properties, indentation resistance, breaking tenacity,¹ and energy absorbance² – are enhanced due to the correspondence of deformation in the directions along and perpendicular to loading, which means these fabrics could expand or shrink in both directions at the same time. Warp-knitted spacer fabrics, namely sandwich fabrics, are composed of two independent face layers and spacer yarns or filaments connecting face layers and supporting the space in between. Double-bed raschel knitting machines are usually used to produce spacer fabrics, during which guide bars are assigned to knit the front and back pieces separately, and one or two bars in the middle are used to knit on two beds alternately to connect and support two face pieces. Warp-knitted spacer fabrics possess many superior properties, such as good air and moisture permeability, cushioning performance, rebound resilience as well as excellent shock resistance, filtration, and sound insulation.

A series of structural models exhibiting auxetic properties including re-entrant models,^3,4 rotating models,⁵ nodule and fibril models, chiral models, liquid crystalline models, helical models, and rod models have been constructed so far; the most commonly adopted are re-entrant models and rotating models. Ugbolue and colleagues⁶ developed a kind of warp-knitted structure composed of pillar open stitches, which are knitted from thicker and softer filaments, and inlay yarns that are stiffer. The problem is that the wales are still not able to be connected with each other to form a complete sheet. Ugbolue and colleagues also developed an auxetic warp-knitted inlay structure adopting two guide bars, one of which is fully threaded to knit pillar chain and the other one of which is partially threaded to inlay yarns. Based on the structure of warp-knitted hexagonal meshes, Ugbolue and colleagues⁷ developed another auxetic structure of re-entrant hexagonal meshes adopting two guide bars threaded with elastic yarns wrapped by polyester inlaid between wales along the knitting direction. Alderson and colleagues⁸ reconstructed the double-headed arrow structure using warp-knitting techniques and designed a series of plane warp-knitted fabrics that are mainly formed with deforming and stable parts using a tricot machine with four or fewer guide bars. Wang and colleagues⁹ developed a novel warp-knitted spacer fabric whose face layers have a geometric structure composed of two parallelograms aligned in a V-shape. Hu and colleagues¹⁰ also developed two kinds of plane weft-knitted re-entrant hexagonal structures using different knitting techniques. Making use of flat knitting machines, Hu and colleagues proposed another folded structure combining face and back loops in the shape of zig-zags, rectangles, or strips. Ma and colleagues¹¹ studied auxetic properties of two-dimensional warp-knitted rotating hexagonal structures composed of basic hexagonal meshes and deforming partial miss-lapping chain stitches.

Based on the deforming structure of rotating hexagonal meshes, which may be the most feasible warp-knitted structure, geometrical models were established in this paper for detailed mathematical analysis, and spacer fabric samples were knitted on the RD7/2-12EN warp-knitting machine with different knitting parameters to help evaluate the influence of parameters including material fineness, chain notations, let-off values, and fabric density on auxetic performance.

Theoretical analysis

Mathematical analysis on the geometric model of auxetic spacer warp-knitted hexagonal meshes were conducted defining the weft direction as the x-direction or horizontal direction, the warp direction as the y-direction or longitudinal direction, and the vertical spacer direction as the z-direction. As shown in the geometric model in Figure 1, the structure consists of red ribs and blue ribs that denote different deformations of the ribs under stretch in the x–y plane, and in the z-direction the structure is formed with two face layers and a spacer layer in between. Before stretch, the hexagon structure is inclined and shrunk; after stretch it shows a certain degree of expansion in at least two directions. Deformations on both the x –y plane and the x–z plane were modeled and analyzed, considering the whole structure as perfectly rigid in order to simplify the calculations.

Figure 1.

Three-dimensional geometric model.

In the x–y plane the hexagonal mesh structure formed by loops in the fabric was simulated as a rigid hexagon taking a repeating unit as the object. As shown in Figure 2, wales of loops are simplified as red and blue ribs, the shaded areas represent loops, and the blank space inside represents the mesh area of the structure. The y-direction length of a repeating unit in its natural state and after deformation are represented by L₁ and L₂_, respectively, given by:

L_{1} = 2 (h_{0} + a \sin ν + b \sin α) \approx 2 (h_{0} + a \sin α + b \sin α)

(1)

L_{2} = 2 (h_{0} + a \sin β + b)

(2)

where h₀ is the intrinsic width of the ribs, a is the length of shorter ribs named AB of the normal hexagon, b is the length of longer ribs named BC of the normal hexagon, α is the acute angle between ribs of BC and x-direction naturally, ν is the acute angle between ribs of AB and x-direction naturally, which approximates angle α when the structure is fully shrunk, and β is the acute angle between ribs of AB and x-direction after deformation. The x-direction length of a repeating unit at natural state and after deformation are represented by H₁ and H₂, respectively, given by:

H_{1} = h_{0} + x (x \to 0)

(3)

H_{2} = h_{0} + 2 a \cos β

(4)

where x is the natural width of the space between the ribs of BC and B₁C₁, which is infinitesimally close to zero when it is fully shrunk in the x-direction.

Figure 2.

Distortion model in the x–y plane.

This structure deforms in two processes when tensile force is exerted in the x-direction. Step 1: the angle α increases to 90° and the zig-zag shape becomes straight. Step 2: the angle β starts to decrease from 90° gradually. In the process of step 1, extension in the y-direction dominates, with the height increasing to its maximum of 2(a + b). In the process of step 2, extension in the x-direction dominates, with the width increasing from its minimum of h₀ and the height decreasing slightly at the same time. The y-direction strain and x-direction strain of the structure after experiencing the aforementioned two processes could be represented by ɛ_L and ɛ_H respectively, given by:

ɛ_{L} = \frac{L_{2} - L_{1}}{L_{1}} = \frac{a (\sin β - \sin α) + b (1 - \sin α)}{h_{0} + (a + b) \sin α}

(5)

ɛ_{H} = \frac{H_{2} - H_{1}}{H_{1}} = \frac{2 a \cos β}{h_{0}}

(6)

The Poisson’s ratio of this structure when stretched in the x-direction is represented by μ, given by:

μ = - \frac{ɛ_{L}}{ɛ_{H}} = - \frac{a (\sin β - \sin α) + b (1 - \sin α)}{h_{0} + (a + b) \sin α} \cdot \frac{h_{0}}{2 a \cos β}

(7)

Defining a : b = 1 : n, then Poisson’s ratio can be represented by:

μ = - \frac{h_{0}}{2} \cdot \frac{\sin β - \sin α + n (1 - \sin α)}{[h_{0} + a (n + 1) \sin α] • \cos β}

(8)

Considering that the values of h₀, a, and n are all positive, we are going to discuss the effects of angle α and angle β, and find out under what conditions Poisson’s ratio could be negative. To simplify the analysis, specific angles are selected to discuss the auxetic performance under specific conditions.

If α = 90°, then μ can be represented by:

μ = \frac{h_{0}}{2 [h_{0} + a (n + 1)]} \cdot \frac{1 - \sin β}{\cos β}

(9)

Since 0 ≤ β<90°, the value of μ is constantly positive under this condition.

If α = 60°, then μ can be represented by:

Since 0 ≤ β < 90°, then

when \sin β > \frac{(\sqrt{3} - 2) n + \sqrt{3}}{2}, μ < 0; when \sin β \leq \frac{(\sqrt{3} - 2) n + \sqrt{3}}{2}, μ \geq 0

At this time, when n > $\frac{\sqrt{3}}{2 - \sqrt{3}}$ , the inequality of $\frac{(\sqrt{3} - 2) n + \sqrt{3}}{2}$ < 0 is constant. Since 0 ≤ sinβ < 1, the inequality of sinβ > $\frac{(\sqrt{3} - 2) n + \sqrt{3}}{2}$ is constant and the value of μ is constantly negative.

If α = 45°, then μ can be represented by:

μ = - \frac{h_{0}}{2 [2 h_{0} + a (n + 1) \sqrt{2}]} \cdot \frac{2 \sin β + (2 - \sqrt{2}) n - \sqrt{2}}{\cos β}

(11)

Since 0 ≤ β < 90°, then

when \sin β > \frac{1 - (\sqrt{2} - 1) n}{\sqrt{2}}, μ < 0; when \sin β \leq \frac{1 - (\sqrt{2} - 1) n}{\sqrt{2}}, μ \geq 0

At this time, when n > $\frac{1}{\sqrt{2} - 1}$ , the inequality of $\frac{1 - (\sqrt{2} - 1) n}{\sqrt{2}}$ < 0 is constant. Since 0 ≤ sinβ < 1, the inequality of sinβ > $\frac{1 - (\sqrt{2} - 1) n}{\sqrt{2}}$ is constant and the value of μ is constantly negative.

If α = 30°, then μ can be represented by:

μ = - \frac{h_{0}}{2 [2 h_{0} + a (n + 1)]} \cdot \frac{2 \sin β + (n - 1)}{\cos β}

(12)

Since 0 ≤ β < 90°, then

when \sin β > \frac{1 - n}{2}, μ < 0; when \sin β \leq \frac{1 - n}{2}, μ \geq 0 .

At this time, when n > 1, the inequality of $\frac{1 - n}{2}$ < 0 is constant. Since 0 ≤ sinβ < 1, the inequality of sinβ > $\frac{1 - n}{2}$ is constant and the value of μ is constantly negative. If the value of α is infinitesimally close to zero, the value of μ tends to negative infinity.

It can be seen from the mathematical analysis mentioned above that auxetic performance of this structure in the x–y plane is significantly influenced by angle α and ratio n with constant intrinsic width of ribs and constant length of diagonals. If α = 90°, apparently there is no y-direction shrinkage in the natural state and thus no auxetic properties. If 0° < α < 90°, this structure is auxetic only when n > n_cri where n_cri is the critical ratio. The smaller α is, the smaller n_cri is, and the auxetic property is more easily obtained. If α tends to be infinitesimally close to zero, this structure is constantly auxetic, having nothing to do with ratio n, and its Poisson’s ratio tends to be infinitesimally negative, implying that its auxetic performance is optimal.

In the x–z plane (Figure 3), T represents the length of spacer yarns connecting two face layers in three-dimensional warp-knitted structures and θ represents its lodging degree. The thicknesses between two face layers in the natural state and after deformation are represented by T₁ and T₂, respectively, given by:

T_{1} = T \sin θ

(13)

T_{2} = T

(14)

where T is the length of rigid connections between two face layers and θ is the acute angle between rigid connections and the x-direction.

Figure 3.

Distortion model in the x–z plane.

When stretched in the z-direction the strain can be represent by ɛ_T, given by:

ɛ_{T} = \frac{1 - \sin θ}{\sin θ}

(15)

Apparently, strain in the z-direction is determined by the inclination of rigid connections, and the smaller angle θ is, the larger is strain ɛ_T.

Experimental details

Sample preparation

Samples with four different structures of GB4 were fabricated using the RD7/2-12EN type of raschel warp-knitting machine (German Karl Mayer, supplied by Huayu Knitting Co. Ltd. in Jinjiang, China). Material specifications are listed in Table 1, in which D stands for the common unit of denier used to characterize yarn fineness. Chain notations are listed in Table 2 and structural changes of GB4 are shown in Table 3. Chain notations represent the movement of yarn guide bars in front of needle hooks. Draw-off density of the samples is set as 11 cpc (courses per centimeter) and the interval between the front and back bed of the machine is set as 4 mm. Let-off parameters are listed in Table 4, in which sectional multivelocity letting-off is adopted in GB3 and GB5. The meaning of one rack is millimeter length of 480 courses and the sectional multivelocity letting-off of GB3 and GB5 in Table 4 is a segmentation of a 12-course unit into two courses with 860 racks, two courses with 290 racks, four courses with 860 racks, two courses with 290 racks, and two courses with 860 racks. Partial magnification of the fabric sample is shown in Figure 4.

Table 1.

Material specifications

Guide bar	GB1, GB2, GB6, GB7	GB3, GB5	GB4
Yarn material	Polyester	Polyester	PE monofilament
Specifications	200 D	100 D	0.09 mm

Table 2.

Chain notations of samples

Guide bar	Chain notations	Threading
GB1	(2-3-2-2/2-1-2-2) × 3/(1-0-1-1/1-2-1-1) × 3//	1 in 1 empty
GB2	(1-0-1-1/1-2-1-1) × 3/(2-3-2-2/2-1-2-2) × 3//	1 in 1 empty
GB3	1-0-0-0/0-1-0-0/0-0-0-0/0-0-0-0/0-1-1-1/1-0-0-0/ 0-1-1-1/1-0-1-1/1-1-1-1/1-1-1-1/1-0-0-0/0-1-1-1//	Full
GB4	Adjusting	Full
GB5	1-1-1-0/0-0-0-1/1-1-0-0/0-0-0-0/0-0-0-1/1-1-1-0/ 0-0-0-1/1-1-1-0/0-0-1-1/1-1-1-1/1-1-1-0/0-0-0-1//	Full
GB6	(1-1-1-0/1-1-1-2) × 3/(2-2-2-3/2-2-2-1) × 3//	1 in 1 empty
GB7	(2-2-2-3/2-2-2-1) × 3/(1-1-1-0/1-1-1-2) × 3//	1 in 1 empty

Table 3.

Changes of GB4 chain notations

Samples	Chain notations of GB4
1#	0-1-0-1/1-0-1-0//
2#	1-0-1-0/0-1-0-1//
3#	(0-1-0-1) × 6/(1-0-1-0) × 6//
4#	(1-0-1-0) × 6/(0-1-0-1) × 6//

Table 4.

Let-off parameters

Guide bar	Letting-off values/rack
GB1, GB2, GB6, GB7	970
GB3, GB5	2	860
	2	290
	4	860
	2	290
	2	860
GB4	3700

Figure 4.

Partial enlarged details of sample fabric

Sample testing

Samples with an overall size of width ×height = 50 × 180 mm were cut out for testing along the x-direction and y-direction. The extension test with fixed elongation was conducted using a HD026N + fabric strength tester. The value of fixed elongation was set as 10 mm, stretching velocity was set as 200 mm/min, and initial clamping distance was set as 100 mm. When starting the stretching test, three pictures per second were taken at the same time to record the deforming process of the samples under stretch. Every sample was recorded with nine pictures, after which picture processing was conducted to measure the length and medial width of the sample using Adobe Photoshop CS3 Extended in order to calculate the Poisson’s ratio. Photos of samples before and after stretch along the y-direction and x-direction are shown in Figure 5, where parts 1-1, 2-1, 3-1, and 4-1 show samples under y-direction stretch and parts 1-2, 2-2, 3-2, and 4-2 show samples under x-direction stretch.

Figure 5.

Actual fabrics under stretch. 1-1, 2-1, 3-1, 4-1: Samples under y-direction stretch; 1-2, 2-2, 3-2, 4-2: Samples under x-direction stretch. (a) At natural state; (b) under stretch.

Results and discussion

Experimental results

Poisson’s ratios of samples with different structures under x-direction and y-direction stretch were measured and calculated respectively. Statistical analyses were conducted using OriginPro8 software, and ratio-strain line charts are shown in Figure 6, among which black lines refer to y-direction stretch and red lines refer to x-direction stretch.

Figure 6.

Diagrams of Poisson’s ratio and uniaxial strain. (a) Structure 1#; (b) Structure 2#; (c) Structure 3#; (d) Structure 4#.

Analysis and discussion

It can be seen from the line charts in Figure 6 that samples with structure 1# and structure 2# cannot exhibit negative Poisson’s ratio under either x-direction stretch or y-direction stretch while samples with structure 3# and structure 4# can only be auxetic under x-direction stretch, and its auxetic performance gets worse as strain increases.

All samples can shrink significantly in the x-direction, while samples with different structures vary in the y-direction. Samples with structure 1# and structure 2# can barely shrink in the y-direction, which means the value of α is close to 90°; samples with structure 3# and structure 4# could shrink to a certain extent. It is predicted that the lapping movement of GB4 has something to do with the longitudinal shrinkage. If the lapping movement of GB4 is homodromous in both front bed and back bed and remains homodromous for the first successive six courses of a repeating unit, hexagonal meshes of the fabric would deflect more easily, leading to naturally longitudinal shrinkage.

The GB4 structures of samples are one-needle chains, which means that the spacer yarns stay vertical theoretically, but lodging in practice due to the pressure of draw-off rollers and winding rollers. It is predicted that the irregular lodging of spacer yarns is to some extent responsible for the instability of auxetic performances of the fabric.

Combining geometrical models and knitted samples, the value of h₀ approximates the width of two wales, which is mainly determined by the fineness of yarn materials and the gauge of the warp-knitting machine; the value of l₀ approximates the height of a loop which is mainly determined by letting-off values; the value of a approximates the length of two courses which is also mainly determined by letting-off values; the value of n approximates 2 in this experiment, which is usually determined by the structure of face layers. With fixed variables of h₀, l₀, a, and n, the Poisson’s ratio of fabrics only relates to α and β. The β value decreases with expansion of the structure under tensile forces and thus the value of β is mainly determined by tensile force. Thus with fixed yarn materials, machine gauge, longitudinal density, and face layer structures, Poisson’s ratio of fabrics mainly relates to the initial value of α under compression, which signifies the degree of deflection of wales in the natural state. By comparison, fabrics with structure 3# and structure 4# can achieve better deflecting deformation, thus exhibiting better auxetic performances under x-direction tension. It is obvious that homodromous lapping movement of GB4 has a positive effect on the achievement of predicted structure, thus the lapping movement of other guide bars could be further researched in future studies.

The limitation of the work proposed here is that the geometric model is totally rigid, while fabric materials are flexible. Subtle deformation of flexible yarns in fabric is ignored in the simulations, while in practice its flexible deformation matters during the knitting process. For the purpose of weakening the effect of minor deformation and better fitting the geometric model, the stiffness of fabric structures could be improved by selecting stiffer yarn materials or increasing the density of the fabric.

Conclusion

The reconstruction of a three-dimensional auxetic structure using spacer warp-knitting techniques is one of the most important research directions in industrial fabrics. Warp-knitted spacer fabrics with negative Poisson’s ratio are endowed with fantastic shape-fitting abilities, energy-absorbing abilities, along with possessing lightweight and low-density properties. There is great potential in macroscopic materials such as reinforcement in composites with complex curves, coating materials, arthrosis or head-protecting equipment, smart filters and smart medical textiles, etc. We established a geometric model based on the warp-knitted spacer hexagonal meshes with auxetic properties for mathematical analysis and adopted the RD7/2-12EN warp-knitting machine for sample manufacturing. Essential points for this structure to be auxetic and its influential factors were thoroughly analyzed, combining theories and practice as follows:

Better auxetic performances can be achieved when hexagonal meshes shrink drastically in three axial directions, especially the y-axial direction. The auxetic performance is mainly determined by fineness of yarn materials, machine gauge, fabric density, face layer structure, as well as deflection angle of wales and stiffness of materials and fabric, among which the last two factors may play the most important role.

Lapping movement of spacer yarns can affect the shrinkage deformation in the y-direction. Homodromous lapping in the front and in the back, along with homodromous lapping in successive courses of a half-repeating unit can help the structure shrink in the y-direction and achieve better auxetic performances. The effect of other guide bar lapping movements could be considered in subsequent research.

The stability of the shrinkage deformation is influenced by the lodging of spacer yarns to some extent. The irregular lodging of spacer yarns could cause discrepant shrinkage in the y-direction, leading to unstable auxetic performance.

There is some error introduced between practical flexible fabrics and rigid geometric models. And it is feasible to use stiffer yarn materials or increase fabric density properly to reduce the difference between practice and theory.

Footnotes

Declaration of conflicts of interest

The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support from the China Postdoctoral Science Foundation (No. 2016M591767), the Fundamental Research Funds for the Central Universities (JUSRP51625B), the Applied Foundation Research Funds of China Textile Industry Association (J201604), and the Natural Science Foundation of Jiangsu Province (No. BK20151129).

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