Geometrical and mechanical modelings of dry relaxed slack plain-knitted fabrics for the benefit of technical textile applications Part I: a geometrical model

Abstract

In the current literature, there is a lack of knowledge concerning which parameters are effective on relaxed slack plain-knitted conventional fabrics, although slack fabrics (i.e. fabrics with no occurrence of jamming) are commonly used for technical applications. Thus, the present series of works are conducted as further investigation on relaxed slack plain-knitted fabrics.

In the present part, Part I, a geometrical model is created based on Kurbak's 1998 model. The created model is then applied to conventional dry, wet and wash relaxed wool plain-knitted slack fabrics. The model is also applied to E-glass dry relaxed technical plain-knitted slack fabric. The applied models are then drawn to scale by using 3DS-Max computer graphical software. At first glance, the loop shapes obtained through the three-dimensional modeling of E-glass fabrics are observed to be similar to the E-glass fabric loops that were recorded in the photographs of the knitted samples.

An investigation on the physical conditions for obtaining such a special loop shape is going to be the subject of the following part, Part II.

Keywords

plain knit dry relaxed geometrical model slack fabrics Kurbak's model

Introduction

Technical and practical definitions of plain knitted slack fabrics

Technical definition

In the early literature,^1–3 plain-knitted fabrics are technically defined as tight and slack according to their jamming conditions (see Figure 1).

Figure 1.

Chamberlain's¹ plain knit model with jamming.

There are two kinds of jamming that occur in plain-knitted fabrics, namely course-jamming (see points P in Figure 1) and wale-jamming (see points S in Figure 1). Using his two-dimensional geometrical model, Chamberlain¹ calculated tightness ℓ/d at the jamming point as ℓ/d = 16.6688. Hepworth and Leaf³ calculated tightness ℓ/d at the jamming point in their three-dimensional (3D) mechanical model as ℓ/d = 16.75. In these works, ℓ stands for the loop length and d stands for the idealized yarn diameter, the cross-section of which is circular and is assumed not to change under compression or swelling.

In the above early theoretical works, plain-knitted fabrics with tightness (ℓ/d) smaller than 16.75 are referred to as tight fabrics and fabrics with tightness larger than 16.75 are referred to as slack fabrics. The difference between tight and slack fabrics from a technical point of view is that jamming forces occur in tight fabrics and thus must be considered in their mechanical modeling, whereas there are no jamming forces applied in slack fabrics, which makes it easier to obtain solutions in those cases.

From a practical point of view, however, since knitting yarns are mostly compressible and/or swellable by nature, defining an effective yarn diameter is necessary, which was defined by Shinn⁴ as d = 0.044 √Tex (mm). The reasons for using Shinn's formula for effective yarn diameter of knitting yarns were discussed elsewhere.⁵

In 1998, Kurbak^6,7 distinguished an additional region of medium tightness between the tight and slack tightness regions for swellable yarns, such as wool, acrylic, cotton, etc., as explained below. Kurbak completed a series of experiments in which 2/28 Nm wool yarn was used. Plain-knitted samples were prepared at different loop lengths from very tight to very slack ones. When the curves of course-spacing (c) versus loop length (ℓ) and wale spacing (w) versus loop length (ℓ) were drawn, three regions with different curve fittings could be determined, as shown in Figure 2. When tightness points, ℓ/d, were calculated (by measuring loop length (ℓ) and predicting effective yarn diameter, d, using Shinn's formula) at separation points of the curves in Figure 2, ℓ/d was about 17.29 at the point between tight and medium tightness regions and about 20 at the point between medium and slack regions.

Figure 2.

Curve fitting to the experimental results of plain-knitted wool fabrics (Kurbak⁷).

Since the theoretical jamming tightness point was calculated as about ℓ/d = 16.75 and it was obtained reasonably close to the theoretical one as 17.29 in practice, the tight fabric region extended until this point. In tight fabrics, yarn compressions occur between loops, whereas yarn swellings do not occur except in the direction of thickness. It was assumed that, in the medium tightness region in Figure 2, asymmetric yarn swellings occurred outside of the curvatures together with compression at interlocking places; therefore, the characteristic of the curve was changed. It was believed that fabric jamming continued to occur in the medium tightness region; thus, this region could not be referred to as slack fabric (see Figure 3).

Figure 3.

Yarn swellings that occur outside of yarn curvatures in plain knit loop at medium tightness region (ℓ/d = 20), as proposed by Kurbak.⁷

The starting tightness point of the slack fabric region for swellable yarns, therefore, was moved from ℓ/d = 16.75 to ℓ/d = 20. In the slack tightness region for swellable yarns in Figure 2, there should be no occurrence of jamming of fabric.

As a result of the discussions above, the technical definition of slack plain-knitted fabric can be stated as plain-knitted fabric produced from unswellable yarns with tightness greater than ℓ/d = 16.75 and plain-knitted fabric produced from swellable yarns with tightness greater than ℓ/d = 20.

Definition of the practical tightness factor

The practical tightness factor, or cover factor, was found necessary for checking the fabric tightness that was suitable to wear, hence a practical tightness factor K was defined by Munden⁸ as

K = \sqrt Tex / ℓ

If the K value was smaller than 15, the fabric was referred to as slack and if the K value was greater than 15, the fabric was referred to as tight. The relation between the practical cover factor K and the technical tightness factor ℓ/d can be obtained by using Shinn's formula as

K = 1 / (0, 0044 (ℓ / d))

The present work, however, uses the technical definition of slack fabric.

Technical textile applications of slack plain-knitted fabrics with unswellable yarns

Most technical yarns, such as glass, carbon, aramid, wire, etc., are unswellable by nature. A plain-knitted fabric that is produced from such unswellable yarn with tightness greater than ℓ/d = 16.75 can be considered as slack fabric. Technical yarns are mostly incompressible and have higher bending rigidities; therefore, knitting problems increase when attempts are made to knit them as tight fabrics.

Since most plain-knitted fabrics that are produced from unswellable yarns, as discussed above, are slack, any work that is conducted on slack plain-knitted fabrics is eventually also beneficial for technical textile applications.

Some of the technical applications of slack plain-knitted fabrics produced from unswellable yarns are provided below.

Technical slack plain-knitted fabrics with E-glass, aramid, carbon, etc., can be used as composite reinforcements. Some research work^9,10 has shown that plain-knitted fabric reinforced composites, compared to other knitted fabric reinforced composites, give higher breaking strengths in their wale-wise direction. Moreover, researchers such as Khondker et al.⁹ and Bini et al.¹¹ found that the wale-wise breaking strengths were even higher when the plain-knitted reinforcing fabrics that were used in composites were slacker.

Aramid plain-knitted slack fabrics are used as reinforcement in radiator hoses of vehicles, as shown in Figure 4 (see MOD. RHU¹²).

Figure 4.

Slack plain-knitted fabric for reinforcing radiator hoses of vehicles.

Tubular plain-knitted fabrics with very small diameters, mainly made by using polyester yarn, are used as artificial textile-based blood vessels¹³ (see Figure 5).

Figure 5.

(a) Harry Lucas's miniature circular knitting machine and (b) polyester artificial blood vessels.

Electronically active “stretch sensors”^14,15 are being developed by using conductive knitted fabrics, including plain-knitted slack fabrics (see Figure 6).

Figure 6.

Stretch sensor made of slack plain-knitted fabric (see Satomi and Perner-Wilson¹⁴).

One emerging technology is “soft robotics,”^16,17 which is obtained by the controllable changing of the shape of objects made of elastic materials, pneumatically or by other means. Such elastic materials may benefit from elastic structures such as slack plain-knitted fabrics as reinforcing fabrics and as electronic sensors that sense the current form of the object and give feedback to the control system.

Technical applications and research interests for slack plain-knitted fabrics with swellable yarns

As explained above, the tightness region of slack plain-knitted fabrics with swellable yarns is narrower (ℓ/d > 20) than the tightness region of slack plain-knitted fabrics with unswellable yarns (ℓ/d ≥ 16.75). Most of the usable fabrics remain outside of the slack tightness area. A few technical applications of swellable slack plain-knitted fabrics can be found. In this context, the traditionally used slack plain-knitted cotton fabric backings of artificial leathers can be counted as fabrics for technical applications, since they are usually utilized in furniture coverings (chairs, etc.; see Figure 7).

Figure 7.

Artificial leather with cotton slack plain-knitted fabric backing: (a) front view; (b) back view.

The interest of researchers, from an academic point of view, in obtaining geometrical and physical properties of fabrics, including swellable slack plain-knitted fabrics, is also important and requires more work to be done.

Dry relaxed fabric

Excessive forces are applied to a fabric during the knitting process; therefore, the shapes of loops in the fabric become distorted. The loops change their shapes after the knitting process as well, as the external knitting forces reduce, resulting in a change of shape in the fabric that is called relaxation. Munden⁸ pointed out that after knitting the shape changes in a fabric occur in such a way that, at the end, the fabric reaches its minimum energy condition. It is believed that once a fabric reaches its minimum energy condition, its dimensions do not change anymore, thus a dimensionally stable fabric is obtained. Relaxation treatments such as dry, wet and full relaxations are defined and are applied to fabrics mainly for obtaining stable fabric dimensions. In this context, dry relaxed fabric was defined by Munden⁸ as a fabric that is laid on a smooth surface without being disturbed for a sufficient period of time. Munden observed that a period of 48 hours was enough to obtain stable dimensional properties in plain-knitted fabrics.

It is believed that, in the dry relaxed state, there are no external forces applied and a loop keeps its stable shape at its minimum energy condition through the reaction forces and moments that are applied from the neighboring loops. In further relaxation states, however, relaxation treatments affect fabric dimensions in different ways beside relaxation.

Some of these effects are given below.

Yarn setting occurs in water or in hot drying and the yarn bending energy reduces. Reaction forces and moments, which keep the loop shape in dry relaxed state, reduce as well.

Yarn swellings may occur further in water or in hot drying.

Characteristics of friction conditions on yarn surfaces may also change in water or in hot drying.

Friction forces, which hold the loop shape in a dry relaxed state, may be overcome in cases such as when some agitation is applied onto the fabric, etc.

For technical use of fabrics, on the other hand, the field of application dictates in which condition the fabric should be prior to the application. For all technical applications, however, dry relaxation can be defined as “the technical fabric, which is laid on a smooth surface after knitting without being disturbed until the related application condition starts”.

Since knitted fabric easily changes its shape, if it is subject to an investigation, the search can start with the defined dry relaxed state, but then the application conditions should be known to be able to investigate the fabric further for the related application.

The present studies deal with the above-defined dry relaxed conventional or technical fabrics.

Aims of the Present Studies

The objective of this series of papers is to create geometrical and mechanical models for the above-defined dry relaxed technical slack plain-knitted fabrics. Models should be beneficial and useful for technical textile applications, some of which are exemplified above. Models, at the same time, should be beneficial for understanding the fabric properties further from an academic point of view.

In a literature review, it can be seen that valuable plain-knitted fabric models were created by Chamberlain,¹ Pierce,² Leaf and Glaskin,¹⁸ Leaf,¹⁹ Munden,²⁰ Hepworth and Leaf,³ Postle and Munden,^21,22 Postle,²³ Shanahan and Postle,²⁴ de Jong and Postle,²⁵ Choi and Ko,²⁶ Kurbak,⁷ Kurbak and Ekmen,²⁷ Popper,²⁸ MacRory et al.,²⁹ Hong et al.³⁰ and Vassiliadis et al.³¹ The present studies are based on Kurbak's^7,27 plain-knitted fabric model (shown in Figure 8).

Figure 8.

(a) Plain-knitted loop model proposed by Kurbak.^7,27 (b) 3DS-Max drawing of Kurbak's plain knit model with an effective yarn diameter.^7,27

In the first part of the present series of papers, a geometric model will be created. In the second part, the necessary physical conditions to obtain the geometrical model will be discussed.

When some parameters of the present model are modified or fixed to certain values, Kurbak's model can be obtained. Therefore, the present model will be given first and the differences between the two models will follow the creation of the present model.

Construction of the model

First of all, it can be said that the loop heads of the present series of models are assumed to be circular arches in two dimensions for reasons of simplicity.

The present model is constructed by leaning the arm cylinders of Kurbak's 1998 plain knit model^7,27 by an angle η, as illustrated in Figure 9.

Figure 9.

The slack plain knitted loop model proposed in the present work.

As can be seen in Figure 9, the plain-knitted loop is symmetrical in shape; hence, modeling a quarter part of it would be enough to obtain the whole loop shape. Therefore, the ABC quarter of the loop in Figure 9 is modeled as given below.

Loop head (part AB in Figure 9)

The following route is taken for obtaining the loop head curve.

The loop head is taken as a two-dimensional elliptical curve in general, as this was assumed in Kurbak's model extending from A to B (from θ = 0 to θ*) and making an angle α₁ with the horizontal y-axis, as shown in Figure 10.

The eccentricity value e = b / a of the elliptical loop head curve is equalized to one, for having the circular loop head curve to satisfy the assumption made for this series of papers, where b is the minor and a is the major axes of the elliptical curve. Calculations for the elliptical loop head can be given as follows.

Figure 10.

Slack plain-knitted loop head.

The equations of the loop head curve are as below

x = a \sin θ

(1)

y = b \cos α_{1} \cos θ

(2)

z = b \sin α_{1} \cos θ

(3)

The tangent of the xz component of the loop head curve at point B should be equal to the tangent of inclination angle η of the arm curve for continuity reasons. Therefore

| \frac{dz}{dx} |_{B} = - \tan η

(4)

can be written, thus

| \frac{dz}{dx} |_{B} = \frac{- b \sin α_{1} \sin θ d θ}{a \cos θ d θ} = - \frac{b}{a} \sin α_{1} \tan θ * = - \tan η

hence the equation

\frac{b}{a} \sin α_{1} \tan θ * = \tan η

(5)

is obtained.

If yarns of adjacent loops in a wale are assumed to be touching at B on the zx plane, BF in Figure 9 must be equal to the yarn radius. Thus, $x_{B}$ values would give another equation as

x_{B} = a \sin θ * = \frac{w}{4} + \frac{d}{2} \cos η

(6)

The third equation is obtained by writing the curvature equalities at B as explained below. The radius of the curvature for the loop head curve at B can be written as

ρ_{B} = \frac{a 2}{b} [1 - (1 - \frac{b 2}{a 2}) \sin 2 θ *] 3 / 322

(7)

while the radius of the curvature of the arm curve at B can be written as

ρ'_{B} = \frac{t_{1}^{2}}{2 e_{1} \cos 2 α_{1}}

(8)

By equalizing both radiuses of the curvatures $(ρ_{B} = ρ'_{B})$ , the equation

\frac{a 2}{b} [1 - (1 - \frac{b 2}{a 2}) \sin 2 θ *] 3 / 322 = \frac{t_{1}^{2}}{2 e_{1} \cos 2 α_{1}}

(9)

is obtained.

From equations (5), (6) and (9), the loop head parameters a, b and $θ *$ are obtained by knowing the parameters η, $t_{1}$ , $e_{1}$ and α₁. The η, $t_{1}$ , $e_{1}$ and $α_{1}$ parameters, in turn, are deducted from the loop arm equations together with the parameter $α_{2}$ , as given in the following section.

Loop arm (part BC in Figure 9)

Loop arm equations are obtained by wrapping the yarn axis on an elliptical cylinder, as given in Figure 11.

Figure 11.

Slack plain-knitted loop arms at the right-hand side.

It is obvious from Figure 11 that

h = c \cos η

(10)

and also

e_{1} = c \sin η + d

(11)

If the crossing of yarn axis in the thickness direction is assumed to occur on the line $B_{0} H$ in Figure 9, and also in Figure 11, the distance $F B_{0}$ in Figure 12 can be assumed to be half of the yarn diameter. The following can be written based on this assumption.

Figure 12.

The cross-section of an imaginary elliptical cylinder on which the loop arm is wrapped, according to the present model.

From Figure 12,

x_{F} = \frac{e_{1}}{2} \sin γ * = \frac{c}{2} \sin η + \frac{d}{2} - \frac{d}{2} = \frac{c}{2} \sin η

(12)

and also

y_{F} = \frac{t_{1}}{2} \cos γ * = \frac{d}{2}

(13)

are found. From equation (12), the parameter

γ *

can be found as

γ * = \sin - 1 [\frac{c}{e_{1}} \sin η]

(14)

and from equation (13),

t_{1}

is obtained as

t_{1} = \frac{k_{2} d}{\cos γ *}

(15)

where, in spite of this assumption, the

k_{2}

parameter is also put in equation (15), which can be necessary for extreme cases. In this work k₂ = 1 is taken.

A parabolic curve is wrapped on the defined cylinder in order to obtain the yarn arm curve. The parabolic curve is the same as the curve used in Kurbak's (1998) model^7,27 given as

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

(16)

where z is measured on the cylinder length direction and

S_{1}

is the peripheral elliptical length on the cross-section of the cylinder. The parameters

a'

b'

and

c'

are constants. The differentiation of equation (16) in respect to

S_{1}

gives the tangent of the inclination angle α of the curve on the cylinder as

\frac{dz}{d S_{1}} = 2 a' S_{1} + b' = \tan α = u

(17)

Starting from point C, the peripheral length can be defined as

x = \frac{e_{1}}{2} \sin ϕ'

(18)

y' = \frac{t_{1}}{2} \cos ϕ'

(19)

and the differentiations of equations (18) and (19) as

dx = \frac{e_{1}}{2} \cos ϕ' d ϕ'

(20)

dy' = - \frac{t_{1}}{2} \cos ϕ' d ϕ'

(21)

From equations (20) and (21), the elementary length $d S'_{1}$ on the cross-section of the cylinder can be obtained as follows:

d S'_{1} = \sqrt{d x 2 + (dy') 2} = \sqrt{\frac{e_{1}^{2}}{4} \cos 2 ϕ' + \frac{t_{1}^{2}}{4} \sin 2} ϕ' d ϕ'

d S'_{1} = \frac{e_{1}}{2} \sqrt{\cos 2 ϕ' + \frac{t_{1}^{2}}{e_{1}^{2}} \sin 2} ϕ' d ϕ'

d S'_{1} = \frac{e_{1}}{2} \sqrt{1 - (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin 2} ϕ' d ϕ'

(22)

Integrating equation (22) gives the peripheral length $S'_{1}$ as

S'_{1} = \frac{e_{1}}{2} \int_{ϕ' = 0}^{ϕ'} \sqrt{1 - (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin 2} ϕ' d ϕ' = \frac{e_{1}}{2} F'

(23)

If the parameter $ϕ' = π / π 22$ , then $F' = E'_{1}$ is taken.

If the starting point is the point B, which is illustrated in Figure 13, the equations

ϕ = \frac{π}{2} - ϕ'

(24)

F_{1} = E'_{1} - F'_{1}

(25)

S_{1} = \frac{e_{1}}{2} ({E'}_{1} - {F'}_{1}) = \frac{e_{1}}{2} F_{1}

(26)

can be used. It should be noted here that

E'_{1}

E_{1}

can be taken.

Figure 13.

The cross-section of the imaginary elliptical cylinder showing the relations of the peripheral length $S_{1} ‘$ and the angle $ϕ_{1} ’$ .

The application of equation (16) was given by Kurbak⁷ and Kurbak and Ekmen,²⁷ and also provided below.

Starting from point B in Figure 11, a local coordinate system x y′ z′ is defined in which x and y′ coordinates are the same as the x and y′ coordinates in Figure 13. The z′ coordinate, however, starts from point B and extends in the direction opposite to the z-axis given in Figure 11 as

z' = (z - \frac{h}{2})

(27)

According to this coordinate system, the right arm curve can be obtained as below. When equations (16) and (17) are applied to the points B (S₁ = 0, z′ = 0, ϕ = 0) and C (S₁ = (e₁/2)E₁; z′ = h/2; ϕ = π/2), the equalities below are derived

c' = 0

(28)

b' = u_{1}

(29)

u_{2} = \tan α_{2} = (2 h / (e_{1} E_{1})) - u_{1}

(30)

a' = \frac{u_{2} - u_{1}}{e_{1} E_{1}}

(31)

Using equations (28)–(31), the BC curve can be drawn by the equations

x = \frac{e_{1}}{2} \sin ϕ y' = - \frac{t_{1}}{2} \cos ϕ z' = a' S_{1}^{2} + b' S_{1} + c'

(32)

where S₁ is a variable that was given in equation (26). Using equation (17), the yarn arm curve between B and C, according to the x y z coordinate system provided in Figure 11, can be given as

x = \frac{e_{1}}{2} \sin ϕ y = \frac{t_{1}}{- 2} \cos ϕ z = \frac{h}{2} - z'

(33)

The yarn arm curve between C and D can be given easily because of the symmetry about the point C as

x = \frac{e_{1}}{2} \sin ϕ y = - \frac{t_{1}}{2} \cos ϕ z = - (h / 2 - z')

(34)

Calculation of loop length (l)

The yarn length of the elliptical curve between point A and point B in Figure 10 can be calculated as

l_{AB} = a \int_{θ = 0}^{θ *} \sqrt{1 - (1 - e 2) \sin 2 θ} d θ

(35)

where

e = \frac{b}{a}

is the eccentricity value of the loop head ellipse.

The yarn length between point B and point C in the arm curve, on the other hand, can be obtained as explained below.

When the differentiating of equation (17) in respect to S is taken, an equation is obtained as

2 a' \frac{d S_{1}}{d S} = \frac{du}{d S}

(36)

In equation (36), dS₁/dS is equal to the cosine of the defined helix angle α. If cosine α is written in tangent form, equation (36) becomes

2 a' \frac{1}{\sqrt{1 + \tan 2 α}} = \frac{du}{d S}

(37)

In place of tanα in equation (37), the parameter u can be written as defined above, thus equation (37) becomes

dS = \frac{1}{2 a' l} \sqrt{1 + u 2} du

(38)

By integrating equation (38) between u₂ and u₁, the yarn length between point B and point C on the arm curve is obtained as

l_{BC} = \frac{1}{4 a'} [u_{2} \sqrt{1 + u_{2}^{2}} - u_{1} \sqrt{1 + u_{1}^{2}} + \ln \frac{u_{2} + \sqrt{1 + u_{2}^{2}}}{u_{1} + \sqrt{1 + u_{1}^{2}}}]

(39)

Because of symmetry, the total loop length (ℓ) can be obtained as

ℓ = 4 ℓ_{AB} + 4 ℓ_{BC}

(40)

The flowchart of the computer program is given in Figure 14.

Figure 14.

Flowchart of the computer program to obtain the slack plain-knitted loop.

It should be noted here that the present model that is given above can easily be changed into Kurbak's model by setting the parameters $η = 0$ and $e \leq 1$ . There is also a parameter $θ *$ that transforms into $θ * = \frac{π}{2}$ , when the parameter η is fixed to zero.

As seen from the flowchart given in Figure 14, for any given parameter η, a plain loop model can be drawn. The parameter η can be obtained from the minimum yarn strain energy.

It is a well-known fact that the strain energy of yarn in loop form has three components, namely extension energy $E_{e}$ , bending energy $E_{b}$ and torsional energy $E_{c}$ . For relaxed fabric, the yarn extension energy $E_{e}$ can be omitted. The total energy of the relaxed loop for the present model can be formulated as

E_{T} = E_{b} + E_{c}

These energy terms for the present model are calculated as follows.

There is no torsional energy at the loop head or feet. The total bending energy for those parts of the loop can be given as

\frac{E_{u}}{B} = \frac{4}{2} \int_{ϕ = 0}^{ϕ *} K 2 d S = \frac{4}{2} \frac{b 2}{a 3} \int_{ϕ = 0}^{ϕ *} \frac{d ϕ}{[1 - (1 - \frac{b 2}{a 2}) \sin 2 ϕ] 5 / 522}

(41)

In the loop arms there are bending and torsional energies. The total energy of the loop arms can be given as

\frac{E_{a}}{B} = \frac{4}{2} \int_{ϕ' = 0}^{π / π 22} K 2 d S + \frac{C}{B} \frac{4}{2} \int_{ϕ' = 0}^{π / π 22} τ 2 d S

(42)

where

K = \frac{\cos 2 α \sqrt{(1 + P 2)}}{L} τ = \frac{\sin α \cos α}{L (1 + P 2)} [1 - 3 \frac{K_{p} P}{\sin α}]

d S = \frac{e_{1}}{2 \cos α} \sqrt{1 - (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin 2 (\frac{π}{2} + ϕ')} d ϕ' L = \frac{e_{1}^{2}}{2 t_{1}} [1 - (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin 2 (\frac{π}{2} + ϕ')] 3 / 322 P = 2 a' L \cos α K_{p} = \frac{e_{1}}{t_{1}} (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin (\frac{π}{2} + ϕ') \cos (\frac{π}{2} + ϕ') α = \tan - 1 [2 a' \frac{e_{1}}{2} F_{1} + u_{1}] a' = \frac{u_{2} - u_{1}}{e_{1} E_{1}} u_{1} = \tan α_{1} u_{2} = \tan α_{2} F_{1} = \int_{ϕ' = 0}^{ϕ'} \sqrt{1 - (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin 2 (\frac{π}{2} + ϕ')} d ϕ' E_{1} = \int_{ϕ' = 0}^{π / π 22} \sqrt{1 - (1 - \frac{t_{1}^{2}}{e_{1}^{2}}) \sin 2 (\frac{π}{2} + ϕ')} d ϕ'

The total energy of the loop will be

\frac{E_{B}}{B} = \frac{E_{u}}{B} + \frac{E_{a}}{B}

(43)

where B and C are the bending and the torsional rigidities, respectively. There has been no proper study done regarding obtaining torsional rigidities, C, for knitting yarns. Lomov and Verpoes,³² however, gave an empiric equation to be used for E-glass yarns as

C = \frac{B}{d / d 22}

(44)

In this work, this simple equation is used to estimate the torsional rigidity of the used E-glass yarn.

The calculation of the total energy of the loop is also included in the computer program written to calculate the loop parameters, as shown in the flowchart given in Figure 14. The minimum of the total energy functions are obtained by changing the parameter η. The results will be provided later in the section concerning the application of the model to E-glass technical slack plain-knitted fabrics.

Resultant drawings of the present model for two specific applications

Application to the slack wool plain-knitted fabrics

The model created by Kurbak was applied to a series of experimental results. In those applications, 2/28 Nm wool yarn was knitted at eight different tightness points between tight to very slack on an E = 10 gauge V-Bed knitting machine. Dry relaxation was applied to the samples by laying them on a smooth surface without disturbance for one week. Wet relaxation was applied to the samples by soaking them in water for 12 hours without agitation and drying them on a smooth surface for one week. During wet relaxation, the initial temperature of the water was 50℃. Sufficient wetting agent was added to the water.

Wash relaxation was applied to the samples after wet relaxation, by washing them in a domestic washing machine for 45 minutes at 30℃ temperature. Loop lengths (ℓ), course-spacings (c) and wale spacings (w) of the samples were measured in the known manner. The thicknesses of samples (t) were also measured by a Zweigle DM 100 T thickness tester under 10 g pressure. The yarn diameter d was calculated by using Shinn's formula as d = 0.044 √Tex = 0.3719 mm.

The results of the above-described experiments are provided in Table 1 as well as in Figure 2. Kurbak's model was applied to the experimental results in order to obtain the loop head eccentricity values e = b/a, which are also provided in Table 1.

Table 1.

The eccentricity, e, values of plain-knitted loop heads obtained according to Kurbak's plain knit model.⁷

Rel.	ℓ/d	w/d	c/d	t/d	e
Dry	15.7569	3.9222	2.9826	3.9954	1.0699
	16.7249	4.5982	3.3672	3.7255	0.9000
	18.0021	5.0915	3.8019	3.6991	0.8168
	19.5267	5.3027	3.9596	3.8526	0.9161
	20.2904	5.8005	4.1574	3.8228	0.8428
	21.4466	6.4390	4.4201	3.8044	0.7782
	22.7615	6.9434	4.6486	3.8374	0.7705
	24.3721	7.4571	4.8938	3.8989	0.7872
Wet	15.7569	3.8515	2.8532	4.1755	1.1020
	16.7249	4.2567	3.1430	3.9323	1.0370
	18.0021	4.7852	3.3970	4.0797	0.9810
	19.5267	5.1701	3.5380	4.2489	1.0263
	20.2904	5.4598	3.8432	4.1861	0.9698
	21.4426	5.8403	4.2352	4.1291	0.9190
	22.7615	6.1665	4.5499	4.1400	0.9170
	24.3721	6.5115	4.8713	4.1839	0.9376
Wash	15.7569	3.8529	2.8428	4.2446	1.0994
	16.7249	4.2073	3.1110	3.9957	1.0564
	18.0021	4.6689	3.3567	4.1455	1.0180
	19.5267	5.0375	3.4960	4.3174	1.0651
	20.2904	5.2796	3.8091	4.2570	1.0155
	21.4466	5.6018	4.2115	4.2032	0.9719
	22.7615	5.8843	4.5312	4.2172	0.9745
	24.3721	6.1885	4.8579	4.2645	0.9990

As explained in the introduction, Kurbak's^7,27 model gave $e = \frac{b}{a}$ values (eccentricity values of the loop head) that were smaller for dry relaxed slack fabrics and greater for tight fabrics, which seems unrealistic; that is, Munden,²⁰ and Postle and Munden^21,22 pointed out that, for a two-dimensional model, the shape of loop heads in a relaxed fabric must be circular, otherwise external forces would be applied in the course-wise direction. Moreover, as known from the circular ring bending rigidity tests of Pierce,³³ the minimum energy condition of an elastic ring is circular. However, Table 1 illustrates that loop heads of dry relaxed wool fabrics are not circular, but might be elliptical with higher eccentricity values, which are, in general, higher than 0.85 (e = b/a ≥ 0.85). This might be occurring because of frictional restraints and/or the 3D nature of loop shapes, and conceivably, can be tolerable. However, the eccentricity values of slack fabrics are much lower than the eccentricity values of tight fabrics and are not considered to be tolerable. In other words, it can be said that the dry relaxed slack plain-knitted fabric region could not be explained by Kurbak's model^7,27 and thus a new model is required. For this reason, the present model was applied to the slack region of wool plain-knitted fabrics, as described below.

For the first approximation $e = 1$ was taken and the present model was applied to the $\frac{l}{d} = 24$ tightness case in Table 1 for dry relaxed, wet relaxed and wash relaxed plain-knitted wool fabrics. The computer program gave $η = 11 \circ$ , $η = 4 \circ$ and $η = 0 \circ$ for dry, wet and wash relaxed wool fabrics, respectively. It was evident from $η = 0 \circ$ for wash relaxed fabrics that Kurbak's original model was still applicable for fully relaxed wool slack fabrics. The models were drawn to scale by using the 3DS-Max computer graphical program, as provided in Figure 15(a)–(c). It was interesting to see that the eccentricity value e could be increased towards 1 by the present model; however, the loop length was about 3% bigger than the measured one. It was also interesting to see that the yarn length contraction with relaxation was about 3%, as had been found by Smirfitt³⁴ as well as by Kurbak.³⁵

Figure 15.

Relaxed plain-knitted slack conventional wool fabric at ℓ/d = 24 tightness point drawn to scale according to the present model using the 3DS-Max computer graphical program: (a) dry relaxed; (b) wet relaxed; (c) wash relaxed.

It is believed that the loop head eccentricity values of wool fabric would be higher than e = 0.85, and the η values would be lower than $η = 11 o$ .

It is evident from the above application and discussion that the present model is a useful tool for a further understanding of fabric properties from an academic point of view.

In addition, by using the present model, the experimentally obtained situations given in Table 1 and Figure 2 can fully be explained.

Application of the model to dry relaxed plain-knitted E-Glass technical fabrics

Some research works, such as Khondker et al.⁹ and Stolyarov,¹⁰ have shown that, compared to other knitted fabric reinforced composites, plain-knitted fabric reinforced composites give higher breaking strength values in the wale-wise direction. Furthermore, Khondker et al.⁹ and Bini et al.¹¹ obtained even higher wale-wise breaking strength values when they used slacker plain-knitted fabrics as reinforcement material in their composites. As the second application, the present model is applied to this specific case as described below.

An experimental work was carried out regarding the dimensional properties of E-glass technical plain-knitted fabrics by having 10 samples knitted on an eight-gauge V-bed knitting machine at a suitable cam setting. Sample widths were set to 100 needles and input yarn tension and the fabric take-down tension were kept constant during the knitting. Samples were dry relaxed by being laid on a flat surface for more than one week. The averages of wale-spacing (w), course-spacing (c) and loop length (ℓ) were calculated. The average fabric parameters, together with the yarn properties (tex, the calculated yarn diameter), are provided in Table 2.

Table 2.

E-glass plain-knitted fabric parameters obtained from the present model.

Parameter	Yarn nr.	Yarn diameter d [mm]	Stitch length ℓ/d	Wale-spacing w/d	Course-spacing c/d
	200 Tex^a	0.36	31.51	11.28	5.51
Parameter	a/d	b/d	e = b / a	$t_{1} / / t_{1} d d$	$e_{1} / / e_{1} d d$
	3.33	3.33	1	1.048	1.43
Parameter	$α_{1}^{\circ}$	$α_{2}^{\circ}$	a′/d	$ϕ * (0)$	$η 0$
	70.14°	70.57°	0.034	85.7°	4.5°

EC9 200 untwisted E-glass yarn.

The present model was applied by using the average values c/d, w/d and ℓ/d that are given in Table 2. First of all, equation (43) was included in the flowchart of the program given in Figure 14 and the total energy of the loop was calculated by having e equal to 1 and having different values for the parameter η each time. It is observed that, when the parameter η is about η = 4.5 °, the total energy of loop is at its minimum. The other model parameters were then calculated by setting e = 1 and η = 4.5 ° and are also given in Table 2. The model is drawn to scale by using the 3DS-MAX program and given in Figure 16, together with a photograph taken from a sample.

Figure 16.

Dry relaxed plain-knitted slack technical glass fabric at ℓ/d = 31.5 tightness point drawn to scale according to the present model using the 3DS-Max computer graphical program: (a) face, side and upper views of the created model; (b) a comparison of the model and a photograph taken from the actual knitted sample.

It should be noted here that the diameter of untwisted E-glass 200 tex yarn is calculated by assuming the open fiber packaging model³⁶ in the yarn cross-section.

Figure 16 shows that, at first glance, the shapes of loops obtained from the model and the photograph are similar, except that the loop heads in the photograph might be elliptical rather than circular.

This similarity generates thoughts on the physical conditions for obtaining such a shape and this will be given in the Part II of this series of papers.

Conclusion

As a result of the above-given discussions on slack plain-knitted fabric and dry relaxation, the following points can be outlined.

(a) Technical slack fabric is defined as a fabric with no occurrence of jamming between courses and between wales.

(b) Jammings occur up to the tightness point ℓ/d = 16.75 in fabrics produced from unswellable yarns, whereas they occur up to the tightness point ℓ/d = 20 in fabrics produced from swellable yarns.

(c) Since technical yarns are mostly unswellable by nature, most technical fabrics made of those yarns can be considered to be slack fabrics. Some technical applications of plain-knitted slack fabrics are given in the introduction section.

(d) Relaxation treatments are mainly defined for conventional uses of knitted fabrics. For technical textile applications of knitted fabrics, however, there is no definition of relaxation, since every case needs its own specific fabric condition prior to the application. In this study, it is suggested that at least the dry relaxation of conventional uses of fabrics can be defined and used as a start to researching technical fabrics as well.

Secondly, a geometrical model is created based on Kurbak's 1998 plain-knitted fabric model⁷ for the above-defined slack dry relaxed plain-knitted fabrics for two reasons, namely (I) to be used in technical textile applications and, at the same time, (II) to understand the fabric properties further from an academic point of view.

Thirdly, resultant drawings of the model are given for two special applications as follows.

(i) The first special application

It is believed that there is geometrical lack of fit in Kurbak's model for explaining dry relaxed slack conventional plain-knitted fabrics. Kurbak's model gave lower elliptical loop head eccentricity values for slack fabrics and higher elliptical loop head eccentricity values for tight fabrics. The present model is applied to this case to see whether this lack of fit in geometry can be overcome.

For the above reason, the model is applied to the plain-knitted wool fabric samples at the tightness point ℓ/d = 24. It is seen that higher elliptical loop head eccentricity values can be obtained through the present model, although the very same dimensional parameters with Kurbak's model are used. In this respect, the present model becomes a useful tool for a further understanding of the fabric properties from an academic point of view.

The drawings of the model for dry, wet and wash relaxed conditions of related fabrics are carried out using the 3DS-Max computer graphical program and are given in Figure 15.

(ii) The second special application

Some research works^9,10 have shown that compared to other knitted fabric reinforced composites, plain-knitted fabric reinforced composites gave higher breaking strengths in the wale-wise direction. Moreover, researchers such as Khondker et al.⁹ and Bini et al.¹¹ discovered even higher wale-wise breaking strengths when they used slacker plain-knitted reinforcing fabrics in their composites. The present model is applied to this special case as follows.

The model is applied to a dry relaxed E-glass plain-knitted technical fabric at the tightness point ℓ/d = 31.5. In this application, the leaning angle η is obtained by minimizing the total energy of the loop.

This model is also drawn to scale by using the 3DS-Max graphical program. The created image is then compared with a photograph of the knitted E-glass sample. At first glance, it is observed that loop shapes obtained by the model and the photograph appear to be similar, except that the loop heads in the photograph might be elliptical rather than circular.

An investigation on the physical conditions for obtaining such a special loop shape is going to be the subject of the following part.

Footnotes

Acknowledgement

The author would like to thank Assoc. Prof. Dr Tuba Alpyıldız for her contributions in the initial drawings.

Declaration of conflicting interests

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

Funding

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

References

Chamberlain

. Hosiery yarns and fabrics 1949; Vol. II, Leicester: Leicester College of Technology and Commerce, pp. 107.

Pierce

. Geometrical principles applicable to the design of functional fabrics. Text Res J 1947; 17: 123–147.

Hepworth

Leaf

GAV

. The mechanics of an idealized weft knitted structure. J Text Inst 1976; 67: 241–248.

Shinn

. An engineering approach to jersey fabric construction. Text Res J 1955; 25.3: 270–277.

Amreeva

Kurbak

. Experimental studies on the dimensional properties of half Milano and Milano rib fabrics. Text Res J 2007; 77: 172–178.

Kurbak

. Plain-knitted fabric dimensions Part I. Text Asia 1998; 29: 34–36. 41–44.

Kurbak

. Plain-knitted fabric dimensions Part II. Text Asia 1998; 29: 34–36. 41–44.

Munden

. The geometry and dimensional property of plain knitted fabrics. J Text Inst 1959; 50: T448–T471.

Khondker

Herszberg

Leong

. An investigation of the structure-property relationship of knitted composites. J Compos Mater 2001; 35: 489–508.

10.

Stolyarov

. Mechanical properties of polymer composites reinforced with knit fabrics made of high-strength aramid fibers. Fiber Chem 2009; 41: 53–55.

11.

Bini

Ramakrishna

Huang

. structure-tensile property relationship of knitted fabric composites. Polym Compos 2001; 22: 11–21.

12.

MOD. RHU. http://www.lucas-elha.de/pdf-daten/ckm/sjm/RHU.pdf (accessed 18 November 2015).

13.

Soydan AS and Kurbak A. Küçük Çaplı Yuvarlak Tekstil Malzemeleri ve Üretim Metodları. Tekstil ve Mühendis Yıl 17, Sayı 80, 18 November 2015.

14.

Satomi M and Perner-Wilson H. KOBAKANT, http://www.kobakant.at/DIY/?p=1996 (accessed 18 November 2015).

15.

Qureshi

Guo

Peterson

. Knitted wearable stretch sensor for breathing monitoring application. ambience 2011 2011; Vol. 9, Sweden: Borås.

16.

Harvard Biodesign Lab. http://biodesign.seas.harvard.edu/soft-robotics (accessed 18 November 2015).

17.

Carmel

. Soft robotics: a perspective—current trends and prospects for the future. Soft Rob 2014; 1: 5–11.

18.

Leaf

GAV

Glaskin

. Geometry of plain-knitted loop. J Text Inst 1955; 46: T587–T605.

19.

Leaf

GAV

. Models of plain knitted loop. J Text Inst 1960; 51: T49–T58.

20.

Munden

. The geometry of a knitted fabric in its relaxed condition. Hosiery Times 1961, pp. 43. April.

21.

Postle

Munden

. Analysis of the dry-relaxed knitted loop configuration, Part I: two-dimensional analysis. J Text Inst 1967; 58: T329–T351.

22.

Postle

Munden

. Analysis of the dry-relaxed knitted loop configuration, Part II: three-dimensional analysis. J Text Inst 1967; 58: T352–T365.

23.

Postle

. Structure shape and dimensions of wool knitted fabrics. Appl Polym Symp 1971; 18: 1419.

24.

Shanahan

Postle

. A theoretical analysis of the plain knitted structure. Text Res J 1970; 40: 656–664.

25.

De Jong

Postle

. An energy analysis of the mechanics of weft knitted fabrics by means of optimal control theory, Part II: the plain knitted structure. J Text Inst 1977; 68: 307. 316.

26.

Choi

. An energy model of play knitted fabric. Text Res J 2003; 73: 739–748.

27.

Kurbak

Ekmen

. Basic studies for modelling of complex weft knitted fabric structures Part I: creation of a geometrical model for widthwise curlings of plain knitted fabrics. Text Res J 2008; 78: 198–200.

28.

Popper

. The theoretical behavior of a knitted fabric subjected to biaxial stresses. Text Res J 1966; 36: 148.

29.

MacRory

McCraith

McNamara

. The biaxial load-extension properties of plain, weft-knitted fabrics theoretical analysis. Text Res J 1975; 45: 746.

30.

Hong

de Araujo

Fangueiro

. Theoretical analysis of load-extension properties of plain weft knits made from high performance yarns for composite reinforcement. Text Res J 2002; 72: 991–995.

31.

Vassiliadis

Kallivretaki

Provatidis

. Mechanical simulation of plain weft knitted fabrics. Int J Clothing Sci Technol 2007; 19: 109–130.

32.

Lomov

Verpoest

. Model of shear of woven fabric and parametric description of shear resistance of glass woven reinforcements. Compos Sci Technol 2006; 66: 919–933.

33.

Pierce

. The handle of cloth as a measurable quantity. J Text Inst 1930, pp. T337–T415.

34.

Smirfitt

. Worsted 1 × 1 rib fabrics part i dimensional properties, Part II some physical properties. J Text Inst 1965; 56: T248–T259.

35.

Kurbak A. Some effects of substituting a presser foot for take down tension in weft knitting. PhD Thesis, Leeds University, Leeds, 1982.

36.

Hearle

JWS

Grosberg

Backer

. Structural mechanics of fibres, yarns and fabrics, New York, Sidney, Toronto: Wiley-Interscience, 1969.