Theoretical modeling of spacer-yarn arrangement for warp-knitted spacer fabrics and experimental verification

Abstract

Different surface structure of spacer fabric implies different spacer yarn arrangement and therefore the compression resistance of fabric. The spacer yarns' arrangement modeling of warp knitted spacer fabrics having identical parameters (thickness, density, material etc.) but different surface structures on double-needle-bar Raschel knitting machine is presented in the study. In a sample, there are several spacer yarns' inclination directions, which differ according to surface structures despite the same lapping movement of spacer guide bars. To simplify the arrangement modeling, it was considered that there was only one inclination angle in a sample, and that was weighted average angle η. The theoretical compression curves were obtained by substituting η into derived equation then simulating by Matlab. To verify proposed arrangement modeling, the compression experiment was done to gain experimental compression curves. The curves of theoretical simulation and experiment of each sample were compared in a figure and the curves exhibited agreement with each other while there were still some deviations derived from the partial approximate substitutions of equations and arrangement angles, as well as the differences between theoretical and actual fabric parameters after removing from knitting machine such as thickness, density, and so on.

Keywords

arrangement modeling compression resistance spacer yarn warp-knitted spacer fabric

Introduction

The engineering development of modern textile products is strongly connected with a large set of calculations.¹ The application engineers need to know fabric properties which are closely related to their structures. Warp-knitted spacer fabrics are widely used in areas such as mobile textiles (cushions² or car seats³), industrial textiles (composites), medical textiles (anti-decubitus blankets), sports textiles⁴ and foundation garments (bra cups, pads for swimwear), etc.⁵ due to their interesting three-dimensional (3D) structures.

A general model for warp-knitted fabric with 3D forms was presented in a paper by Renkens and Kyosev.¹ This paper concentrated on a computation of geometry of warp-knitted structures that had complex 3D forms or that were draped over similar ones produced on a single-bed Raschel knitting machine with three guide bars and focused on the creating geometry method by transferring the basic geometry to a 3D deformed state of the relaxed fabric.

The spacer-yarn arrangement angle was the one that could be calculated according to θ = tan⁻¹ t/w (where θ is the inclination angle, t is the thickness of the spacer fabric and w is the segment width).⁵ The angle of spacer yarn could be defined as the angle between the spacer yarn and the fabric surface; there were two arrangement angles in a spacer fabric with the surface structure of small mesh in the experimental samples – one was at an acute angle and the other was at a right angle³ – while the arrangement angle was not investigated in their researches.^3,5 Spacer-yarn material, pattern and threading on the compressions have been investigated by Armakan and Roye.⁶ It was observed that the compression behavior of spacer fabric could be influenced by location angles and the amount of spacer yarns; moreover, the compression resistance decreased as the spacer-yarn location angle decreased. The effect of the angle of the spacer yarn on the compression behavior was investigated by Miao and Ge.⁷ Thirteen samples were involved in this study, in which eight samples had a single spacer-yarn arrangement angle and the others had a mix of 45° and 90°. It was concluded that the more perpendicular the angle of the spacer yarn, the larger the compression pressure of the fabric. That three angles existed in a spacer fabric was involved in papers by Shen and Qian⁸ and Xia.⁹ The work showed that fabric with a smaller spacer-yarn inclination angle had a better cushioning effect. Compression properties of warp-knitted spacer fabric were investigated by Chen,¹⁰ who set up a mechanical and stress–strain model of the single spacer yarn without concerning the whole arrangement of the inner layer (a similar research has also been published¹¹). The compression properties were also studied by Liu et al.,¹² who concluded that the fabrics with lower spacer-yarn inclination angle and larger size mesh of the outer layers could be used to absorb lower energy with higher efficiency. None of the researches^13–18 directly focused on the arrangement of spacer yarns and its verification. Establishment of the modeling could contribute to the research on the compression behavior of spacer fabrics from the perspective of a full cycle of spacer yarns, not only a separated piece. Moreover, the influence of surface structures on compression properties could be investigated in detail based on the modeling.

Samples

Warp-knitting spacer fabric consists of two surface layers and a layer of the yarns called spacer yarns. The spacer yarns connect two surface layers to form a special 3D structure. The spacer yarns play a supporting role and avoid the 3D structure being crushed under body pressure.

All the samples in the study were produced on a double-needle-bar Raschel knitting machine of gauge 18: a model of the machine is shown in Figure 1.⁴ The front guide bars (GB 1 and 2) knit a base fabric on the front needle bar only, while the back bars (GB 5 and 6) knit the other separate base fabric on the back needle bar. The middle bars (GB 3 and 4) carry the spacer yarns and knit on both needle bars in succession.⁸

Figure 1.

Model of double-needle-bar Raschel knitting machine.

Various means of forming the surface of spacer fabric exist. The surface structures can be plain or meshes with different shapes, and two surface stitches (top and bottom layers) can be identical or different. Five spacer fabrics with different surface structures were investigated in the study. The chain notations of two separate bases and threading methods are shown in Table 1. Lapping movements of guide bars 3 and 4 are given as below:

Table 1.

Chain notations for different surface structures of spacer fabrics

Code	Sample	Description	Chain notation
1	P-P	Both sides: pillar+weft–insertion	GB 1: 0-0 0-0/5-5 5-5// fully threaded
			GB 2: 1-0 0-0/1-0 0-0// fully threaded
			GB 5: 0-0 1-0/0-0 1-0// fully threaded
			GB 6: 0-0 0-0/5-5 5-5// fully threaded
2	R-R	Both sides: rhombic mesh	GB 1: 1-0 0-0/1-2 2-2/2-3 3-3/2-1 1-1// half threaded
			GB 2: 2-3 3-3/2-1 1-1/1-0 0-0/1-2 2-2// half threaded
			GB 5: 1-1 1-0/0-0 1-2/2-2 2-3/3-3 2-1// half threaded
			GB 6: 2-2 2-3/3-3 2-1/1-1 1-0/0-0 1-2// half threaded
3	H-H	Both sides: hexagonal mesh	GB 1: (1-0 3-3/3-2 1-1) × 2/1-0 3-3/3-2 4-4/(5-4 3-3 / 3-2 4-4) × 2/5-4 3-3/3-2 2-2// 2 threaded 2 empty
			GB 2: (5-4 3-3/3-2 4-4) × 2/5-4 3-3/3-2 1-1/(1-0 3-3/3-2 1-1) × 2/1-0 3-3/3-2 4-4// 1 empty 2 threaded 1 empty
			GB 5: (1-1 1-0/3-3 3-2) × 3/(4-4 5-4/3-3 3-2) × 3// 2 threaded 2 empty
			GB 6: (4-4 5-4/3-3 3-2) × 3/(1-1 1-0/3-3 3-2) × 3 // 1 empty 2 threaded 1 empty
4	P-R	One side: pillar+weft-insertion	GB 1,GB 2: the same as GB 1 and GB 2 for P-P
4	P-R	The other: rhombic mesh	GB 5,GB 6: the same as GB 1 and GB 2 for R-R
5	P-H	One side: pillar+weft–insertion	GB 1,GB 2: the same as GB 1 and GB 2 for P-P
5	P-H	The other: hexagonal mesh	GB 5,GB 6: the same as GB 1 and GB 2 for H-H

'P', 'R' and 'H' in the second column represent 'pillar + weft–insertion', 'rhombic mesh' and 'hexagonal mesh', respectively.

GB3: 1-0 3-2/3-2 1-0// half threaded

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

The materials of all the specimens were polyethylene terephthalate (PET) monofilament, 0.2 mm in diameter for spacer yarns and PET multifilament, 33.3 tex/96f, for the surface yarns. Although the distance between the needle beds was kept constant, 10.2 mm, and density was 10 cpc (courses per centimeter) for all spacer fabric types on the machine state, the specimens represented many different parameters when they were removed from the knitting machine. The resultant thickness and density after heat setting at 180℃ for 3 minutes are presented in Table 2. The front elevation and right view of P-P, which was chosen as the representative of five samples, are shown in Figure 2.

Figure 2.

The front elevation and right view of P-P.

Table 2.

Thickness and density of specimens

Code	Sample	Thickness (mm)	Density
Code	Sample	Thickness (mm)	Weft-wise (wale/cm)	Warp-wise (course/cm)	Area (loop /cm²)	Spacer yarns (spacer yarns/cm²)
1	P-P	7.38	7.03	5.42	38.10	76.20
2	R-R	6.60	7.07	5.84	41.22	82.44
3	H-H	7.26	7.08	5.79	40.96	81.92
4	P-R	7.06	P: 7.15	5.72	40.86	82.13
4	P-R	7.06	R: 7.24	5.70	41.27	82.13
5	P-H	7.29	P: 6.80	5.40	36.72	75.60
5	P-H	7.29	H: 6.93	5.61	38.88	75.60

'P', 'R' and 'H' in the fourth column mean the densities of pillar + weft-insertion, rhombic and hexagonal surface, respectively. The 'spacer yarns/cm²' represents the number of spacer yarns per square centimeter.

Figures 2(a) and (b) show the front elevation and right view of P-P, respectively. It can be seen from Figure 2(a) that there are several different spacer-yarn inclinations from the front side. Moreover, all spacer yarns bend slightly from the right-hand side shown in Figure 2(b).

Arrangement modeling of spacer yarns

Base modeling

The base modeling of spacer yarns is given in Figure 3, where the lapping movement of guide bar 3 (GB 3) is 1-0 3-2/3-2 1-0// and both top and bottom surface structures of the spacer fabric are pillar+weft–insertion (P-P).

Figure 3.

Base arrangement modeling (1-0 3-2 / 3-2 1-0 //).

In Figure 3(a), the X-, Y- and Z-axes indicate the directions of course, wale and thickness of the spacer fabric, respectively. All the points in the figure represent the junctions between monofilaments in the spacer layer and the surface layers. To distinguish the points in adjacent courses, three adjacent courses are put in three planes; the solid dots representing the first course are placed in the first plane, while the empty and patterned ones representing the second and third courses are in the subsequent two planes. The front elevation and right view of Figure 3(a) are shown in Figures 3(b) and (c). The solid pattern dots with black outline in Figure 3(b) mean that the points in the first plane are coincident with those in the two others.

Points A, B, C, D and A′ show the positions of 1-0, 3-2, 3-2, 1-0 and 1-0, respectively. Moreover, points A and B are in the first plane, points C and D are in the second plane and point A′, representing the beginning of next cycle process, is in the third plane. Areas AHH′A′ and FBB′F′ represent the top and bottom surfaces, respectively. Angles α, ζ, ϕ and λ in Figure 3(b) represent the acute or right angle between the fabric surface and spacer yarns AB, BC, CD and DA′ in the X–Z plane, where β and θ are the acute angles between the horizontal direction and BC, DA′ in the Y–Z plane, which can be calculated by Equation (1):

α = ϕ = \arctan \frac{h}{2 b} ζ = λ = \frac{π}{2} β = θ = \arctan \frac{h}{c}

(1)

where h is the thickness of the spacer fabric and b and c indicate distances between two adjacent wales and courses, respectively. Moreover, two spacer yarns in separate directions extend from each point (such as A), and the other points in Figure 3 follow the same rules in other cycles, then the global arrangement view is as described in Figure 4.

Figure 4.

Global arrangement view of spacer yarns.

In Figure 4, the solid lines represent lapping movement of GB 3 (1-0 3-2/3-2 1-0//) and the dotted lines represent that of GB 4 (3-2 1-0/1-0 3-2//). Two surface layers are connected with two systems of the symmetrical inclined spacer yarns to enhance the stability of the structure. However, it should be pointed out that the base modeling is for the monofilaments' inclination way of spacer fabric whose surface structures are P-P, the method varies according the surface structures, which is researched in the next section.

Modeling variations

The front elevations of the spacer yarns arrangement of five different samples are shown in Figure 5, according to the base modeling above; their right-hand views, which are the same as each other, have been given in Figure 3(c).

Figure 5.

Front elevation of spacer-yarn arrangement of five different samples.

For P-P, the top and bottom surfaces are pillar+weft–insertion; moreover, they are completely identical and symmetrical: the drawing of one side is shown in Figure 5(a₁) and the front elevation is shown in Figure 5(a₂). For R-R, both top and bottom surfaces, are rhombic mesh and they are also identical and symmetrical; a cycle process needs two courses, the drawing of one layer is given in Figure 5(b₁), while the front elevation is shown in Figure 5(b₂). For H-H, the top and bottom surfaces are given in Figures 5(c₁) and (c₂) respectively, the two layers are hexagonal mesh but not symmetrical, and a cycle process needs 12 courses. There are three different arrangements for the structure shown in Figures 5(c₃)–(c₅); the first arrangement (c₃) occurs a total of six times in a cycle, and they are the courses numbered 1, 2, 3, 7, 8 and 9. The second arrangement (c₄) occurs four times in a cycle process, on courses numbered 4, 6, 10 and 12. The third arrangement (c₅) occurs twice on courses numbered 5 and 11. Therefore, the weight of the three arrangements is 1/2, 1/3 and 1/6. For P-R, the top and bottom surfaces shown in Figures 5(a₁) and (b₁) are different; the front elevation is shown in Figure 5(d). Finally, for P-H, the top and bottom surfaces are the same as shown in Figures 5(a₁) and (c₁); it also has 12 courses in a cycle process and three arrangement methods (e₁, e₂ and e₃). The times, courses occurred and weight for each method are the same as those of H-H.

In the model, b is equal to 2a. The weft-wise density wpc (wales per centimeter) and thickness h are theoretical values according to the processing parameters on the knitting machine. h is 10.2 mm and there are 18 needles per inch, so a can be calculated as 0.7056 mm. Then, the angle of each spacer yarn and weighted average angle in a cycle process can be calculated as shown in Table 3.

Table 3.

Angles of spacer yarns in a cycle process

Code	Sample	Angle of spacer yarn
Code	Sample	α(rad)	ζ(rad)	ϕ(rad)	λ(rad)	Weighted average η (rad)
1	P-P	1.3009	1.5708	1.3009	1.5708	1.4358
2	R-R	1.3009	1.5017	1.3009	1.5017	1.4240
3	H-H	F.M. 1.2423	1.4020	1.2423	1.4020	1.3455
		S.M. 1.2423	1.4577	1.2423	1.4577
		T.M. 1.2423	1.5708	1.2423	1.5708
4	P-R	1.3333	1.5362	1.3333	1.5362	1.4353
5	P-H	F.M. 1.2810	1.4437	1.3609	1.5282	1.3984
		S.M. 1.2810	1.4999	1.3072	1.5282
		T.M. 1.1801	1.4999	1.3072	1.4999

F.M., S.M. and T.M. represent the first, second and third arrangement method mentioned above.

It can be seen from Table 2 that the sequence of average angle η for each sample is P-P > P-R > R-R > P-H > H-H. The warp-wise densities cpc are also set to be theoretical values according to the theoretical parameters; they are all 10 courses per centimeter, so c in Figure 3(a) is 1.00 mm, and β and θ can be calculated according to Equation (1), which shows that they are both equal to 1.4731 rad (84.40 degrees). To further investigate the arrangement of spacer yarns, in the next section, the theoretical angle values will be substituted into derived compression equations to obtain the theoretical data, and they will be compared with experimental compression data to verify the model above.

Verification of arrangement modeling

Theoretical simulation

According to the weighted average angle calculated in Table 3, it can be considered that all spacer yarns in a sample incline toward the same direction. Figure 3(a) can be simplified by Figure 6(a), while the front elevation and right-hand view are shown in Figures 6(b) and (c), respectively.

Figure 6.

Simplified modeling of spacer yarns in a cycle process.

The average angle η can be seen in Figure 6(b), and β (1.4731 rad) is taken as the right angle to simplify the modeling, since it is near to so, then, the length of spacer yarn l, FC in Figure 6a, is approximate to DC (Figure 6(b)), which can be found out according to Equation (2):

l = \frac{h}{\sin η}

(2)

The compression schematic of single vertical spacer yarn is shown in Figure 7.

Figure 7.

Compression schematic of single vertical spacer yarn.

In Figure 7, there was an assumption that the bending form of the spacer yarn on complete relaxation and small deformation conditions was sinusoid proximately, which could be described as Equation (3):¹⁹

y = a \sin \frac{π x}{L}

(3)

where a (cm) is the mid-point displacement of the spacer yarn and L (cm) is the vertical height of the spacer yarn.

The curvature k is related to the bending moment M(x):²⁰

k = - \frac{M (x)}{EI}

(4)

where E (cN/cm²) is the elastic modulus of the spacer yarn and I (cm⁴) is rotary inertia.

In addition:

k = \frac{y ″}{(1 + y'^{^{2}})^{3 / 2}} M (x) = Py

(5)

According to Equations (3)–(5), the pressure P and height of the spacer yarn satisfy Equation (6):

P = \frac{π^{2} EI}{L^{2}}

(6)

If the spacer yarn inclines with an angle η, Equation (6) can be written as Equation (7):

P = \frac{π^{2} EI}{h^{2}} \sin^{2} η

(7)

where h (cm) is the height of the spacer yarn (thickness of the spacer fabric).

Supposing there are n spacer yarns under the pressure area, Equation (7) can be written as Equation (8):

P = \frac{n π^{2} EI}{h^{2}} \sin^{2} η

(8)

According to Equation (8), the curve of pressure P followed the spacer fabrics' thickness h during the compression process, which can be simulated by Matlab. Here, E can be obtained from a tensile property test of the monofilament, and I can be calculated by Equation (9):

I = \frac{1}{4} π r^{4}

(9)

where r (cm) is the radius of the monofilament.

In Equation (8), h was as listed in Table 2. The number of spacer yarns in 20 cm² 'n' can be calculated according to the spacer yarns' amount per square centimeter, shown in Table 2. The bending rigidity EI was 1.1422 (cN cm²).

The theoretical pressure–compressive strain curves according to the arrangement modeling are shown in Figure 8.

Figure 8.

Simulation compression curves.

It can be revealed from Figure 8 that, under the same compressive strain, the pressure sequence of the five samples is P-P > P-R > R-R > P-H > H-H; in addition, with regard to the pressure values, for which the horizontal dashed and dotted lines referred to are those of P-P and P-R under different strains, it can be seen that as the pressure of P-P is a little greater than that of P-R, the sequence is identical with the weighted average angles of five samples. The more perpendicular the the spacer yarn, the larger the compression pressure of the fabric and, therefore, the fabric has better compression resistance as well.⁷

Compression experiment

The compression tests were carried out by TexLab Precision Instruments CT250, and the compression curves were obtained for all fabrics after conditioning at 20℃ and 65% relative humidity (RH) based on Chinese standard FZ/T01051.2-1998. The instrument was composed of a presser foot with a contact area of 20 cm², which moved vertically at a speed of 12 mm/min. As soon as the presser foot touched the sample, the sensor at the top of the presser foot started measuring the value of the pressure applied, which was recorded. At the same time, another sensor measured the position of the presser foot, that is, the thickness of the sample under a particular pressure, which was also recorded. The maximum compression rate was 20%. All the samples for testing were cut into circular form of 50 cm² in area. The compression curves of the five different spacer fabrics are shown in Figure 9.

Figure 9.

Compression curves based on experiment.

Figure 10.

Comparison of simulation and experiment results on compression behavior.

It can be seen from Figure 9 that under the same compressive strain, the pressure sequence of the samples is P-P > P-R > R-R > P-H > H-H, resulting from the same reason, that is, the more perpendicular the angle of the spacer yarn, the larger the compression pressure of the fabric.⁷

Comparison of simulation and experiment

Comparison of the compression results of theoretical simulation and the experiment for five samples is shown in Figure 10.

It can be concluded from Figure 10 that the calculated results of theoretical simulation conform to the experimental results, while there are still some deviations between them. Several factors can account for the deviation, but the following may be the most critical ones. Firstly, β in Figure 6(c) was taken as a right angle to simplify the modeling, since it was near to so. Several different angles of spacer yarns existed in a sample; for instance, there were four different angles for H-H and seven different angles for P-H. To calculate conveniently, the arrangement model was simplified, and a weighted average angle was used in Equation (2) to obtain the pressure–compressive strain curve. The method definitely influences to some extent the precision of results. Secondly, in the study, same parameters were used for theoretical simulation, except the surface structure, in other words, except the arrangement of spacer yarns, while the samples used for the compression experiment represented different thicknesses and densities after being removed from the knitting machine. Thirdly, Equation (6) used in the theoretical simulation existed in the condition that the deformation curves of spacer yarns were sinusoid in the situation of small strain, ignoring the friction among them; in fact, the curves were not perfect sinusoid and the friction existed.

Conclusions

Theoretical modeling of spacer-yarn arrangement for warp-knitted spacer fabrics was investigated in this paper. To verify the proposed modeling, a compression experiment was conducted. Weighted average angles of five spacer fabrics with different surface structures could be found based on the modeling. The maximum simulative and experimental compression pressure (under the compressive strain of 20%) could also be obtained. The results are shown in Figure 11. According to comparison of theoretical and experimental results, the following conclusions can be drawn.

1. The arrangement angles of spacer yarns could be affected by different surface structures of spacer fabrics. The spacer yarns' weighted average angles of five samples are P-P (1.4358 rad) > P-R (1.4353 rad) > R-R (1.4240 rad) > P-H (1.3984 rad) > H-H (1.3455 rad). In other words, the weighted average angle of spacer yarns with a plain surface layer (pillar+weft–insertion) is a little greater than that with rhombic mesh, while the average angle of spacer yarns with a rather small rhombic mesh surface layer is larger than that with hexagonal mesh, and the average angle of spacer yarns with the plain-mesh surface is between those with plain-plain and mesh-mesh. The weighted average angle of spacer yarns is closely related to the surface structure and chain notations of it. With a minor change of chain notations of the surface structure, completely different simulated results may occur.

2. The maximum forces of theoretical and experimental compression approach each other. That means the calculated results of theoretical simulation conform to the experimental results, while there are still some deviations between them due to some simplification and approximation of modeling. The method demonstrates arrangement modeling of spacer yarns, without taking into account the friction among spacer yarns.

3. The order of maximum forces (both theoretical and experimental force) is in complete accord with that of weighted average angles (P-P > P-R > R-R > P-H > H-H), because the more perpendicular the angle of the spacer yarn, the larger the compression pressure of the fabric.

Figure 11.

Weighted average angles, the maximum simulative and experimental pressures.

Spacer fabrics having identical parameters (thickness, density, material, etc.) but different surface structures affect their arrangement of spacer yarns and then influence the compression resistance of fabrics. The generated arrangement model of spacer yarns is a required initial state. It can be referenced for further calculations of mechanical and other properties of spacer fabrics using computer simulation, such as the finite element method.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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