Geometrical models for weft-knitted spacer fabrics

Abstract

Spacer fabrics are very attractive nowadays for use as technical fabrics. Our interest in this study is to give geometrical models for weft-knitted spacer fabrics which can be used in related engineering software. Models of two commonly used weft-knitted spacer fabrics are created here based on Kurbak’s 1998 plain knit model and are drawn to scale using the 3DS-MAX computer graphical program. It is observed that similar shapes to the real fabrics are obtained by the models.

Keywords

spacer fabric weft-knitted geometrical model

Spacer fabrics, in general, are obtained by having two layers of single fabrics, and these two layers are connected by some yarns or some single fabrics. There are mainly two groups of spacer fabrics; namely weft-knitted and warp-knitted spacer fabrics. Warp-knitted spacer fabrics are out of the scope of this paper; therefore, they are not mentioned hereafter. Traditional weft-knitted spacer fabrics are made on double-bed, flat or circular knitting machines by knitting two single-jersey (plain) fabrics at the same time with at least two different yarns on either bed at the back bed (or dial) and in the front bed (or cylinder) and connecting them by a third yarn which makes only tuck stitches at the back bed and in the front bed. Knitting instructions of the two most widely used weft-knitted spacer fabrics are given in Figure 1.

Figure 1.

Knitting instructions for weft-knitted spacer fabrics: (a) with rib gaiting and (b) with interlock gaiting.

A photograph of a weft-knitted spacer fabric is given in Figure 2.

Figure 2.

Photograph of a weft-knitted spacer fabric: (a) front view and (b) cross-sectional view.

Some other kinds of spacer fabrics, which are similar to the classical ones but instead of the yarns connecting the two fabric faces some single fabric parts connect the two surfaces, have also been developed to use as technical fabrics. These fabrics are also out of the scope of this work.

In the last decade, the technical potential of spacer fabrics has been distinguished and the properties have been improved for various technical applications. Some of the applications are outlined below.

Because of their higher recovery properties when a compression is applied on their surfaces and because of their cushioning effect, they are used inside the heel of sports shoes.

Because the gaps between the two layers (of the fabrics) make them highly water-absorbent and softer, they are used as pads, nappies, surgery pads, etc.

The formability, softness, elastic recovery, mechanical, noise reduction and thermal insulation properties of the spacer fabrics allow them to be use inside vehicles (cars, busses, etc.)

Because the two layers of the fabrics are connected by some connecting yarns, higher lamination properties are obtained when they are used as the reinforcing fabric materials in textile composites.

The two layers of spacer fabrics can be different; therefore, they can be used for items such as sportswear because of their transfer wicking properties and also for their higher microclimatic properties between the skin and the fabric.

They can be used as electromagnetic shielding fabrics.

They can be used as liquid cooling garment by inserting “tubes” between the layers.

They can be used as heating garments by inserting conductive yarns or wires inside the layers.

They can be used for the reinforcing materials, for items such as lightweight cement composites.

The above mentioned physical properties of spacer fabrics have been investigated experimentally in accordance with their application areas. At the same time some software engineering computer programs have being created for the estimation of the physical properties of textile fabrics. These software programs need full definitions of the geometrical properties of the fabrics used. As far as we know, there is no such properly-defined geometrical model created for weft-knitted spacer fabrics. Therefore, this work is aimed at creating geometrical models for at least the very basic weft-knitted spacer fabrics, of which the knitting instructions are given in Figure 1.

There are, however, some geometrical models that have been created for the plain-knitted fabrics and for structures that have tuck stitches. In this context, the models of Chamberlain,¹ Pierce,² Leaf and Glaskin,³ Leaf,⁴ Munden,⁵ Postle,⁶ Kurbak,⁷ Kurbak and Ekmen,⁸ Kurbak and Amreeva,⁹ Alpyıldız and Kurbak¹⁰ and Kurbak and Alpyıldız^11–14 can be counted. These models were investigated to see if some of them could be used for the basis of the work for the present models. The models of Kurbak and his colleagues^7–14 have been seen to be suitable for the present purpose; therefore, the present models will be based on the models created by Kurbak and his colleagues.^7–14

Creations of the models

Two models were created. Schematic pictures of the models are provided in Figure 3.

Figure 3.

Schematic pictures of the weft-knitted spacer fabric: (a) with rib gaiting and (b) with interlock gaiting.

Modeling of the fabric faces

In these models, the front and back faces are taken as the same, using the same yarn counts. Fabric faces are modeled based on the 1 × 1 rib models of Kurbak¹⁵ and Kurbak and Alpyıldız.¹⁴ That is, loop heads and loop arms are the same as in the 1 × 1 rib model in which the loop heads are elliptical curves and the loop arms are modeled by having vertical circular cylinders and wrapping the yarns axis on these cylinders using the parabolic equation below

z = a' S_{1}^{2} + b' S_{1} + c'

(1)

where S₁ is the peripheral length of the cross-section of the cylinder, z is the length taken on the cylinder axis and a′, b′ and c′ are constants.

During applications of the parabolic curves on cylinders, the helix angles at two ends of the cylinders are different as in the 1 × 1 rib model.

The models of plain-knitted fabrics on the faces of the spacer fabric are given in Figure 4.

Figure 4.

Model of the plain-knitted fabric which is used at the faces of weft-knitted spacer fabrics: (a) with rib gaiting and (b) with interlock gaiting.

In Figure 4, for the helix angle at the upper end, $α_{1} = 60 \circ$ is used, while at the lower end, $α'_{1} = 30 \circ$ is used. These values are used because the lower ends of the plain loop also accommodate the tuck stitch of the connected yarn. Since the calculations of the upper part and the arms of these loops were given in Kurbak and Alpyıldız¹¹ and Kurbak,¹⁵ they are not repeated here. The original Kurbak plain-knitted model^7,8 did not allow for the accommodation of the tuck stitch of the connected yarn; therefore, some other considerations were added at the feet of the plain loops as explained below.

Considerations for the faces of the weft-knitted spacer fabric with rib gaiting

First of all, the feet of the loops are wider than the loop heads. For this situation, a wale spacing w is defined, and then at the position of two feet of loops, $ξ d_{1}$ are added so that from the mid-points of the loop heads half of the new wale spaces (w_f) become

\frac{w_{f}}{2} = \frac{w}{2} + ξ d_{1}

(2)

while the major radius of the loop head is

α = \frac{w}{4} + \frac{d_{1}}{2}

(3)

The major radius of the loop feet is

α ″ = \frac{w}{4} + \frac{d_{1}}{2} + ξ d_{1}

(4)

where d₁ is the diameter of yarn used to knit the front face.

In addition to the above widening for accommodating the tuck stitch of the connected yarn in the fabric thickness direction, an elliptical cone given in Figure 5 is used at the legs of the plain loops.

Figure 5.

Elliptical cone defined for obtaining the lower ends of the plain loops.

The equations of the intersecting curve of the defined cone and DE plane in Figure 5 is obtained as

x = - a ″ \sin ϕ + \frac{w}{2} + ξ d_{1}

(5)

y = b \cos ϕ

(6)

z = b \cos ϕ \tan a'_{1}

(7)

where

b = \frac{b_{1}}{1 - (1 - \frac{b_{1}}{b_{2}}) \cos ϕ}

(8)

This curve given by equations (5)–(8) is used as the plain loop feet. In equations (5)–(8), b₁ is found from the curvature equalities at point D as

\frac{d_{1}}{2} ≅ \frac{b_{1}^{2}}{a ″}

(9)

From equation (9), b₁ is obtained as

b_{1} = \sqrt{\frac{a ″ d_{1}}{2}}

(10)

The parameter b₂ can be changed in accordance with the diameter of the connecting yarn d₂. Therefore the present models are versatile ones and can be used for connecting yarn of any yarn diameter. In this work, a parameter η is defined to consider the diameter of the connecting yarn as

b_{2} = b_{1} + η d_{1}

(11)

The parameter $η = 1 / 2$ is used in this work and used in the equations in the following sections.

If the above considerations are not enough and there are still some crossing yarns in the model, then b₂ is increased while a″ is decreased. In this case, it is thought that yarns in these sections are stretched; thus, some straight horizontal yarn sections are added to the feet to complete the original wale spacing (w_f).

Considerations for the faces of the weft-knitted spacer fabric with interlock gaiting

The same considerations used for the weft-knitted spacer fabric with rib gaiting are used for the case with interlock gaiting, and a horizontal straight yarn path, which is equal to w_f, is added in the middle of the sinker loop (at the loop foot) as seen in Figure 4(b).

Modeling of the connecting tuck stitches

The connecting tuck stitches between the front and the back faces of the present spacer fabrics are modeled as given in Figure 6.

Figure 6.

The present model for the tuck stitch and the connection yarn sections of the weft-knitted spacer fabrics: (a) with rib gaiting and (b) with interlock gaiting.

The imaginary yarn wrapping cylinder diameters are taken as d₁.

The model is explained in parts as given in the following section.

Tuck stitch head for rib and interlock gaiting

The tuck stitch head (AB in Figure 6) is modeled by having an elliptical curve leaning toward the inner side of the fabric, in the fabric thickness direction, as seen in Figure 7.

Figure 7.

The tuck stitch head of the weft-knitted spacer fabric.

The equations of the tuck stitch head are given by

x = a \sin ϕ

(12)

y = b \cos α_{1} \cos ϕ

(13)

z = b \sin α_{1} \cos ϕ

(14)

The parameter $α_{1}$ in equations (13) and (14) can be assumed to be taken as equal to the inclination angle of a loop since the tuck stitch is situated just in front of a loop. The other parameters a, b and $ϕ *$ will be considered in the following section.

Tuck stitch arm for interlock gaiting

The tuck stitch arm (BCD in Figure 6(b)) is modeled by wrapping the yarn axis on a circular cylinder which is situated in the fabric plane and by making an angle γ with horizontal x-axis (Figures 6(b) and 8).

Figure 8.

The right-hand side arm of the tuck stitch of the weft-knitted spacer fabric with interlock gaiting.

At point B, the tangent of the upper curve and the arm curve should be equal to each other to obtain a continuous curve; therefore, the following equation can be written

\frac{dz}{dx} = \frac{b}{a} \sin α_{1} \tan ϕ * = + \tan γ

(15)

Again, at point B, the radii of the curvatures of the two intersecting curves should be equal; therefore, the following equation is applied

\frac{a 2}{b} [1 - (\frac{b 2}{a 2}) \sin 2 ϕ *] 3 / 2 = \frac{r}{\cos 2 a_{1}}

(16)

The third equation is the equality of the X_B coordinate at B which can be written as

a \sin ϕ * = \frac{w}{4} + r \cos γ

(17)

In this model, two adjacent tuck stitches in the same face are assumed to be touching each other for the rib gaiting. To satisfy this statement, the following equation is written for the wale spacing of the tuck stitch $(w_{t})$

\frac{w_{t}}{2} = \frac{w}{2} + ξ d_{1} - η_{1} d_{2}

(18)

where d₂ is the diameter of the yarn used to knit the connecting part.

Equation (18) is assumed to apply for interlock gaiting as well, even though the touching condition is not the case for interlock gaiting.

Instead of using lengthy calculations, the parameter $η_{1}$ will be estimated during the drawing of the tuck stitches to satisfy the touching condition.

From the above discussion, the $X_{D'} - X_{B'}$ distance in Figure 8 can be expressed as

X_{D'} - X_{B'} = \frac{w}{4} + ξ d_{1} - η_{1} d_{2}

(19)

Thus, if the height of the tuck stitch is denoted by c_t, then an expression for γ can be written as

\tan γ = \frac{c_{t}}{\frac{w}{4} + ξ d_{1} - η_{1} d_{2}}

(20)

For dry relaxed fabric, the following equation is assumed to apply

r = \frac{d_{1}}{2}

(21)

By using equations (15)–(17) and equation (20), the parameters a, b, $ϕ *$ and γ are obtained.

The length of cylinder 1 (in Figure 8) can be given by

h_{1} = \sqrt{c_{t}^{2} + (\frac{w}{4} + ξ d_{1} - η_{1} d_{2}) 2}

(22)

In order to draw the tuck stitch arm, other parameters are needed, namely the wrapping angle $(δ_{max_{1}})$ and the helix angles $(a_{3})$ at point D in Figure 8. These parameters can be calculated together with the model of the connecting yarn region; thus, they are given in the following section.

Equations (15)–(22) are also used to calculate the tuck stitch arms of the spacer fabric with rib gaiting.

Connecting yarn section for rib gaiting

The connecting yarn section (DE in Figure 6(a)) is modeled in three steps. First of all, a circular cylinder is considered, and the yarn axis is wrapped on this cylinder using equation (1) (see Figure 9(a)). Secondly, this cylinder with the wrapped yarn is bent so that the final shape becomes a part of a torus with radius R (see Figure 9(b). Thirdly the torus is placed in between the front and the back fabric faces making an angle β with the normal direction of the fabric plane as given in Figure 9(c).

Figure 9.

Modeling of the connecting yarn parts of the weft-knitted spacer fabrics: (a) yarn wrap on a straight cylinder, (b) straight cylinder bent together with wrap yarns on ıt into a torus shape and (c) torus placed between the front and the back faces (r₂ = r = d₁/2 is used).

From Figure 9(c), we can calculate h₂ and β as

h_{2} = \sqrt{(\frac{w}{4} + ξ d_{1} - η_{1} d_{2}) 2 + (t - 2 d_{av}) 2}

(23)

Figure 9(c) does not show it, but there is $- η_{1} d_{2}$ amount of space at point F; therefore, during the calculation of $\tan β$ in equation (24), $- 2 η_{1} d_{2}$ is used.

t is the distance between the two faces of the spacer fabric, and $d_{av}$ is equal to

d_{av} = \frac{d_{1} + d_{3}}{4} + \frac{d_{2}}{2}

(25)

where d₃ is the diameter of the yarn used to knit the back face. That is, in this study, d₃ equal to d₁ is used. The model can still be applied if d₃ is different from d₁.

It can be shown that the unit tangent vector $\vec{u}$ of the torus central axis at D is

\vec{u} = - \cos γ * \sin β \vec{i} + \cos γ * \cos β \vec{j} - \sin γ * \vec{k}

(26)

The unit vector of the tuck stitch arm DB at D is

\vec{v} = - \cos γ \vec{i} + \sin γ \vec{k}

(27)

The scalar product of these two vectors $\vec{u}$ and $\vec{v}$ in equations (26) and (27) gives the cosine of angle φ between these two vectors; therefore, the following equation can be written

\cos φ = | \vec{u} . \vec{v} | = | \cos γ \cos γ * \sin β - \sin γ \sin γ * |

(28)

Since the radii of imaginary cylinders 1 and 2 (see Figure 13 in the Appendix) are equal to each other, from the curvature equality, the helix angle of the wrapped yarn at D can be given by the equation

α_{3} = \frac{φ}{2}

(29)

It should be noted here that if the sign of the scalar product $\vec{u} . \vec{v}$ is a negative value, then equation $α_{3} = \frac{(π - φ)}{2}$ is used to obtain the helix angle at D instead of equation (29).

The calculation of the maximum wrapping angle $δ_{max_{1}}$ on the tuck stitch arm is lengthy; therefore, the calculation of this angle is given in the Appendix and only the result of it is given here

δ_{max_{1}} = \frac{π}{2} + β (1)

(30)

where

β (1) = \frac{1}{\frac{\cos γ}{\cos β} \tan γ * + \tan β \sin γ}

(31)

We thought that if we assume a small angle $γ *$ which is about 30°, we could obtain the practical curvature of the connected yarn. On the other hand, angle $α_{1}$ of the plain face loops must be a maximum one while angle $α'_{1}$ must be minimum one, since the tuck stitches of the connecting yarns are held by the lower ends of the face loops. Therefore, the values $α_{1} = 60 \circ$ and $α'_{1} = 30 \circ$ are chosen for the plain face loops and used in this work. The angle $α_{1}$ of the tuck stitch is also taken as $α_{1} = 60 \circ$ , which is the same angle $α_{1}$ as the plain face loops. If the connecting part is modeled as having a straight circular cylinder instead of it being bent in the torus shape, the helix angle $α_{3}$ of the tuck stitch at point D would be a maximum of 45°, and then φ would be $φ = 2 α_{3} = 90 \circ$ .

In this case, if we assume that $α_{3}$ of the tuck stitch at point D is equal to $α_{1}$ at point B, the angle $γ *$ will be about 30° since the equation $2 α_{3} = φ + γ * ≅ 90 \circ + γ *$ is applied here.

Therefore, in this study, for relaxed spacer fabrics, the helix angle $α_{3}$ of the tuck stitch at point D is assumed to be equal to the helix angle $α_{1}$ at point B; thus, the angle $γ *$ is found using this assumption and also using equations (24), (28) and (29).

For the above assumed case ( $α_{1} = α_{3}$ ), the course-spacing equation for the plain-knitted fabric model which was given in Kurbak⁷ and Kurbak and Ekmen⁸ can be adopted here to obtain the helix angle $α_{2}$ at point C in Figure 8. The equation is

c = \frac{E_{1} t_{1}}{2} (u_{2} + u_{1})

(32)

where

c = h_{1}

$E_{1} = \frac{δ_{max_{1}}}{2} and t_{1} = d_{1}$ can be written. Therefore, from equation (32), an equation for u₂ is obtained as

u_{2} = \frac{4 h_{1}}{d_{1} δ_{max_{1}}} - u_{1}

(33)

where

α_{2} = \tan - 1 (u_{2})

and

α_{1} = \tan - 1 (u_{1})

can be written.

Since the unknown parameters $δ_{max_{1}}$ , $α_{2}$ and $α_{3}$ are calculated in this section, the tuck stitch arm curve can now be drawn.

If the helix angle $α_{3}$ is not equal to $α_{1}$ , then $α_{2}$ can be calculated again by using the procedure given in Kurbak.¹⁵ This case may occur while applying pressure to the surface of the spacer fabric.

The parameter $h'_{2}$ can be calculated as the arc length BEF in Figure 9(b) as given below

R = \frac{h_{2}}{2 \sin γ *}

(34)

and thus

h'_{2} = 2 γ * R

(35)

The maximum wrapping angle δ_max2 of the connected yarn between B′E′F′ in Figure 9(a) can be given by

δ_{max_{2}} = 2 \tan - 1 [\frac{\tan γ}{\cos β} \cos γ * + \tan β \sin γ *]

(36)

The calculation of equation (36) is also given in the Appendix together with the parameter $β (1)$ of $δ_{max_{1}}$ (see equation (31)).

Correction factor for α₃

While bending the straight rod into the torus shape, the length of the central axis of the rod does not change, but the outer side extends and the inner side contracts. This situation creates a problem if the starting point of the parabolic curve on the rod (at D and F in Figure 9(b)) is not at the central axis of the rod. Instead of having $α_{3}$ as shown in the parabolic curve on a straight rod in Figure 9, helix angle $α'_{3}$ should be taken and an additional parameter, f, (the correction factor) should be calculated so that when the rod is bent (as in Figure 9(b)) the final helix angle should be $α_{3}$ .

From Figure 9(c), the following equation can be written

u_{3} = \tan α_{3} = \frac{[R + r \sin (\frac{π}{2} - ϕ)] d γ *}{d S_{1}}

(37)

From the geometry, $d γ *$ will be

d γ * = \frac{dy'}{R}

(38)

and thus the following is obtained

u_{3} = \tan α_{3} = [1 + \frac{r}{R} \sin (\frac{π}{2} - ϕ)] \frac{dy'}{d S_{1}}

(39)

We already know that

\frac{dy'}{d S_{1}} = \tan α'_{3} = u'_{3}

(40)

Therefore, the following equation is obtained

u_{3} = \tan α_{3} = [1 + \frac{r}{R} \sin (\frac{π}{2} - ϕ)] u'_{3}

(41)

In equation (41), the following can be written for point D

ϕ = ϕ_{B} = \frac{δ_{max_{2}}}{2}

Therefore, the correction factor at point D can be written as

\frac{1}{f} = 1 + \frac{r}{R} \sin (\frac{π}{2} - \frac{δ_{max_{2}}}{2})

(42)

When we put the correction factor into equation (41), we obtain

u'_{3} = {fu}_{3}

(43)

It became clear from the above discussion that, if we want helix angle $α_{3}$ at point D of the tuck stitch when the connecting rod is bent into the torus shape as in Figure 9(b), we should take the helix angle

α'_{3} = \tan - 1 (u'_{3})

at point

D'

on the straight rod given in Figure 9(a), and the correction factor f relates these two angles,

α_{3}

and

α'_{3}

Because of the symmetry the helix angles, $α'_{3}$ at point $D'$ and $α'_{5}$ at point $F'$ in Figure 9(a) are equal to each other. Therefore, a similar equation as equation (33) can be written to obtain $α'_{4}$ at point $E'$

u'_{4} = \frac{4 h'_{2}}{d_{1} δ_{max_{2}}} - u'_{3}

(44)

where

α'_{4} = \tan - 1 (u'_{4})

can also be written. Since

δ_{max_{2}}

α'_{3}

α'_{4}

and

h'_{2}

are known, the parabolic curve on the straight cylinder given in Figure 9(a) can be drawn. The equations for drawing the part

D' E'

are given below as an example. For

0 \leq ϕ \leq \frac{δ_{max_{2}}}{2}

x = - r \cos (\frac{π}{2} - \frac{δ_{max_{2}}}{2} + ϕ) y = α' (r_{2} ϕ) 2 + b' (r_{2} ϕ) + c' z = - r \sin (\frac{π}{2} - \frac{δ_{max_{2}}}{2} + ϕ)

(45)

where

r = \frac{d_{1}}{2}

a' = \frac{u'_{4} - u'_{3}}{r δ_{max_{2}}}

b' = u'_{3}

and

c' = 0

When the straight cylinder in Figure 9(a) is bent to the torus shape given in Figure 9(b), the equations for the parabolic curve for $0 \leq γ' \leq γ *$ become

x' = x y' = R \sin γ * - R \sin (γ * - γ') z' = R \cos γ * - (R - z) \cos (γ * - γ')

(46)

where

γ' = \frac{y}{R}

and x, y and z are given by equation (45).

Again, the equations of the curve between D and E in Figure 9(c) can be written in the $XYZ$ coordinate system as

X = - y' \sin β - x' \cos β Y = y' \cos β - x' \sin β Z = z'

(47)

where

x', y'

and

z'

are given by equation (46).

Connecting yarn section with interlock gaiting

The present model for the connecting yarn section (DE in Figure 6(b)) of the weft-knitted spacer fabric with interlock gaiting is given in Figure 6(b). To obtain this model, the very same equations for the model of the connecting yarn section (DE in Figure 6b) of the weft-knitted spacer fabric with rib gaiting are used except that the parameter β is taken as zero. Therefore, the calculations of the model are not given here. Equation (18) is assumed (and is used) to consider the widths of the tuck stitches of the weft-knitted spacer fabric with interlock gaiting as well, although the touching of the adjacent tuck stitches in the same face are not applied in this case.

Results

The three-dimensional models created for the weft-knitted spacer fabrics are drawn to scale using the 3DS-MAX computer graphical program and given in Figure 10. It is seen that the shapes obtained by the models are very similar with the shapes observed in the real fabrics. The parameters used to draw the models are as below:

The parameters used to draw the model given in Figure 10(a) for the weft-knitted spacer fabrics with rib gaiting are $w = 4 d_{1}, c = 3 d_{1}, t = 8 d_{1}, c_{t} = 0.85 c$ ; $d_{2} = \frac{d_{1}}{2}, η_{1} = - 0.56, ξ = 0.22, d_{3} = d_{1}$ ; and $α_{1} = 60 \circ, α'_{1} = 30 \circ, α_{3} = 60 \circ$ .

The parameters used to draw the model given in Figure 10(b) for the weft-knitted spacer fabric with interlock gaiting are $w = 4 d_{1}, c = 5 d_{1}, t = 8 d_{1}, c_{t} = 0.85 c$ ; $d_{2} = \frac{d_{1}}{2}, η_{1} = - 0.56, ξ = 0.22, d_{3} = d_{1}$ ; and $α_{1} = 60 \circ, α'_{1} = 30 \circ, α_{3} = 60 \circ$ .

Figure 10.

Weft-knitted spacer fabric drawn to scale using the 3DS-MAX computer program according to the present model: (a) with rib gating and (b) with interlock gaiting.

Discussion on the widths and lengths of the tuck stitches

During the calculations of the tuck stitches, in equation (18), the wale spacing w_f is used instead of w; therefore, there are some openings occur between the adjacent face loops. The reason for using w_f was that if we use a normal wale spacing (w), the maximum width points of the tuck stitches would be at the upper side of them instead of at point D. In most of the spacer fabrics, however, the openings between the face loops are visible. Therefore, in the present models, the parameter $ξ d_{1}$ is added to the feet of the loops. If the face fabrics are very tight, during modeling the parameter $ξ d_{1}$ cannot be added to the face loops, and in this case, in our opinion, the tuck stitches are placed on top of each other at their maximum width points as given in Figure 11. This situation can also occur when a twist lively yarn is used for the knitting of the tuck stitches.

Figure 11.

The tuck stitches placed on top of each other.

At the first stage, the spacer yarn between the front and back fabric was modeled by wrapping the yarn on a straight cylinder as in Figure 9(a). The model of the spacer fabric with this assumption was drawn as in Figure 12(a). This model did not satisfy our observations of the real spacer fabrics; therefore, we bent the straight cylinder into a torus shape (as in Figure 9(b)) and obtained the shape given in Figure 12(b) for the present model. The curves of the spacer yarns, given in Figure 12(b), were the same as we observed on the real fabrics. Therefore, in the present model shapes given in Figure 12(b) are used for the modeling of the spacer yarn curves.

Figure 12.

Modeling of the spacer yarn: (a) modeling with wrapping the yarn on a straight cylinder and (b) modeling with wrapping the yarn on a torus.

Figure 13.

Crossing of two cylinders with equal radii in three-dimensional space to obtain a continuous wrapping curve.

Conclusion

Nowadays, weft-knitted spacer fabrics are used for creating various functional materials, such as materials with a cushioning effect, liquid absorbent materials, materials that have thermal insulation and/or noise reduction, conductive materials, sport-wear with different yarn properties for front and back layers of spacer fabrics, electromagnetic shielding materials, lightweight cement composite materials, liquid cooling materials with inserted tubes in between the layers, etc. Since these applications are costly, some software package programs have been developed to check whether the cultivated technical function of the material is suitable for the related applications. These software package programs, then, use geometrical models of the materials as input data.

In this study, geometrical models for two commonly used weft-knitted spacer fabrics are given based on Kurbak and his colleague’s models^7–15 to be used in the above given software package programs or in theoretical calculations of the above given functions. One of the created models is for spacer fabrics obtained by rib gaiting and the other created model is for spacer fabrics obtained by interlock gaiting of the machine. The models created were drawn to scale using the 3DS-MAX computer graphical program. It is seen that the shapes obtained by the models are very similar to the shapes observed on real fabrics.

Footnotes

Acknowledgement

The author would like to thank Tuba Alpyıldız, Mustafa Değirmencioğlu and Neyra Nam for their contributions to the initial drawings.

Declaration of conflicting interests

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

Funding

The author received no financial support for the research, authorship and/or publication of this article.

Appendix

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Geometrical models for weft-knitted spacer fabrics

Abstract

Keywords

Creations of the models

Modeling of the fabric faces

Considerations for the faces of the weft-knitted spacer fabric with rib gaiting

Considerations for the faces of the weft-knitted spacer fabric with interlock gaiting

Modeling of the connecting tuck stitches

Tuck stitch head for rib and interlock gaiting

Tuck stitch arm for interlock gaiting

Connecting yarn section for rib gaiting

Correction factor for α3

Connecting yarn section with interlock gaiting

Results

Discussion on the widths and lengths of the tuck stitches

Conclusion

Footnotes

Acknowledgement

Declaration of conflicting interests

Funding

Appendix

References

Correction factor for α₃