International Journal of Clothing Science and Technology

Shrinkage prediction of plain-knitted fabric based on deformable curve

Zhixun Su Xiaojie Zhou Guohui Zhao Xiuping Liu Ka-Fai Choi
Article information:
To cite this document:
Zhixun Su Xiaojie Zhou Guohui Zhao Xiuping Liu Ka-Fai Choi, (2008),"Shrinkage prediction of plain-knitted
fabric based on deformable curve", International Journal of Clothing Science and Technology, Vol. 20 Iss 4
pp. 222 - 230
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IJCST
20,4 Shrinkage prediction
of plain-knitted fabric based
on deformable curve
222
Zhixun Su, Xiaojie Zhou, Guohui Zhao and Xiuping Liu
Department of Applied Mathematics, Dalian University of Technology,
Received 8 April 2007
Revised 7 June 2007 Dalian, People’s Republic of China, and
Accepted 7 June 2007 Ka-Fai Choi
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Institute of Textiles & Clothing,

The Hong Kong Polytechnic University, Kowloon, Hong Kong

Abstract
Purpose – The aim of this paper is to develop a new method to predict the potential shrinkage of
plain-knitted fabric.
Design/methodology/approach – The presented method is based on deformable curve. The
delivered plain-knitted fabric is represented as a deformable parametric curve, and the relaxed fabric
can be reached by minimizing the energy of the curve. Compared to the delivered-knitted fabric, the
length and width shrinkage percentages can be calculated accordingly.
Findings – The new method is more convenient than the traditional trial and error method, and need
less-input parameters than the STARFISH technique. Experimental results show that this method is
feasible.
Originality/value – This paper presents a new method for shrinkage prediction of plain-knitted
fabric based on deformable curve and energy minimum. The work can be linked with shrinkage
control in textile industry.
Keywords Textiles, Predictive process, Fabric testing
Paper type Research paper

1. Introduction
Knitted fabric is a very popular product in textile industry, and it is faced with
ever-rising demands for better quality and reliability. One of the key demands is
dimensional stability, i.e. low levels of potential shrinkage. Potential shrinkage
is expressed as a percentage of the dimensions of the reference fabric:
ðcpcr 2 cpcd Þ ðwpcr 2 wpcd Þ
SL ¼ · 100% SW ¼ · 100% ð1Þ
cpcr wpcr
where SL, SW are length and width shrinkages, respectively, cpcr (courses per cm)
and wpcr (wales per cm) are the course and wale densities in the reference fabric
(after relaxation), cpcd and wpcd are corresponding values of the delivered fabric
International Journal of Clothing (before relaxation). The reference sample is the fabric after the standard relaxation
Science and Technology
Vol. 20 No. 4, 2008 procedure as the STARFISH project, i.e. five cycles of washing and tumble drying
pp. 222-230 under closely prescribed conditions (Heap et al., 1983).
q Emerald Group Publishing Limited
0955-6222
DOI 10.1108/09556220810878847 Project supported by New Century Excellent Talents in University in China (No. NCET-05-0275).
 Since traditional trial and error methods are too costly and uncertain for shrinkage Shrinkage of
control, many researchers studied this problem from different point of views (Postle,
1968; Postle and Munden, 1967; Shanahan and Postle, 1970; Knapton et al., 1975; Lo,
plain-knitted
1981; Doyle, 1953; Munden, 1959). Postle et al. (Postle, 1968; Postle and Munden, 1967) fabric
and Shanahan and Postle (1970) treated the yarn as an elastic object and proposed the
theoretical models of knitted fabrics. Experimental observation made by Doyle (1953)
confirmed the dependence of the fabric area on the loop length of plain-knit fabrics. 223
The dimensions of a knitted fabric are believed to be predominately determined by the
loop length and independent of other parameters such as yarn and knitting parameters.
k-values are designed to describe the relationship between the dimensions and loop
length. However, the k-values are affected quite significantly by several factors
including especially certain aspects of the yarn specification and wet processing. As a
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result, the k-values have limitations for the prediction of fabric dimensions due to a
poor precision. Instead of k-values, The STARFISH project is founded on database, in
which the effects of many factors, such as yarn type, yarn count, and fibre type, are
taken into account. It is based on statistical analysis of fabrics, and yarn and knitting
parameters have to be known as input. What is more, material properties of yarn,
which have significant effects on fabric dimensions, are not considered in their project.
In this paper, a new method for shrinkage prediction of plain-knitted fabric is proposed
based on deformable curve. The plain-knitted fabric is represented as a deformable
parametric curve. The reference state of the fabric is obtained by energy minimization,
and the shrinkage percentages can be achieved by the analysis of the delivered and the
reference fabrics. We need less-input parameters to make our method work.

2. Parametric model of plain-knitted fabric

Many researchers discussed the geometric models of knitted fabric (Postle, 1968; Postle
and Munden, 1967; Peirce, 1947; Leaf and Glaskin, 1955; Choi and Lo, 2003; Demiroz
and Dias, 2000). These models lay emphasis on the visualization of fabric. We need
a parametric model suitable for shrinkage prediction, which can describe the
plain-knitted fabric accurately with several variables. Choi and Lo (2003) presented a
parametric model. In their model, the fabric loop is treated as symmetrical parts, and
the central axis of the left part is represented as:

xðtÞ ¼ a · t 3 2 1:5 · t 2 þ 0:5 · ða þ wÞ · t ð2:1Þ

yðtÞ ¼ 0:5 · ðc þ 2eÞ · ð1 2 cos p tÞ ð2:2Þ
zðtÞ ¼ 0:5 · ðth 2 2rÞ · ð1 2 cos 2p tÞ ð2:3Þ
where w is the loop width, (c þ 2e) is the loop height, e is the adjacent loop overlapping
distance, tk is the fabric thickness, r is the radius of the yarn, a is a free parameter,
t [ [0, 1] is the parameter of the space curve (shown as Figure 1(a)).
Suppose A is the extremum point of x(t) in the interval t [ [0.5, 1], and B is the
extremum point of x(t) in the interval t [ [0, 0.5] of the adjacent loop (Figure 1). The
above-parametric model assumes that y(t) component of A is the same as B. However,
we observe that vertical shift may occur due to relaxation procedure. So, a new variable
s is introduced to describe the vertical shift, that is:
yðtÞ ¼ 0:5 · ðc þ 2e þ sÞ · ð1 2 cos ptÞ ð2:20 Þ
 IJCST e
20,4 e

c c

224 A B
e s
w e
w
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Figure 1.
Parametric model of fabric (a) Geometric model (b) The modified geometric model (c) Knitted fabric
loop
Source: Choi (2003)

equations (2.1), (2.20 ) and (2.3) give the modified parametric model (Figure 1). Similar to
the original parametric model, e can be denoted by a, w, c, s, i.e.


ðc þ sÞ 1
e¼ 21 ;
2 cos pb
where:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 a 2 2w
b¼ 2 :
2 2 3a
In the modified geometric model, the five independent variables a, w, c, s, tk determine
the plain-knitted fabric.

3. Energy model
Kass et al. (1987) introduced deformable curve model, named active contour models or
“snakes,” for solving various problems in computer vision and image analysis. In the
active contour models, curves are used to deform or define edges or contours or to track
motion in a moving image driven by energy. The energy of a curve is composed of
internal energy, including stretching and bending energy ensuring its smoothness, and
the external energy pulling it to the edges. In our context, fabric loop is represented as a
spatial curve, besides the stretching and bending energy, twisting energy should be
considered. The internal energy of the loop is:
U int ¼ U t þ U t þ U k ; ð3Þ
where Ut, Uk, and Ut are stretching, bending and twisting energy, respectively. We call
the yarn unraveled from a finished fabric the natural state of a loop, or natural loop.
And we observed that it is very rare that the loop of natural state is straight, it looks
wavy. So, the energy of the loop should be relative to natural state. Stretching energy
due to elongation can be expressed as:
Z
E Ly 2
Ut ¼ e ds;
2 0 y
 where E is the tensile modulus, L, is the length of loop, ey is the yarn strain. By Shrinkage of
assuming that the yarn extension is constant along the yarn and the loop length does
not change significantly, the stretching energy can be rewritten as:
plain-knitted
fabric
Ee2y L2y EðLy 2 Ly0 Þ2
Ut ¼ < ; ð4Þ
2 2
225
where Ly0 is the length of natural loop. Bending energy illustrates the bending degree
of the yarn deviating from the natural state:
Z Ly
B
Uk ¼ ðk 2 k0 Þ2 ds; ð5Þ
2
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where B is the bending modulus, k and k0 are the curvatures of the loop and the natural
loop, respectively. Twisting energy describes its twisting degree deviating from the
natural state:
Z Ly
C
Ut ¼ ðt 2 t0 Þ2 ds; ð6Þ
2 0

where C is torsional modulus of the loop, t and t0 are the torsions of the loop and the
natural loop.
For the shrinkage prediction of fabric, the reference state is the fabric after
relaxation, and no additional load on the fabric. The external energy is due to yarn
jamming in the fabric. Larger jamming volume brings larger energy, so the external
energy can be expressed as:

X
N
U ext ¼ P Vj ð7Þ
j¼1

where N is the number of loops which contact with the current loop, Vj is the jamming
volume with loop j. Although the jamming volume can be accurately calculated by
intersecting volume of two polyhedrons, it is too costly and not necessary. By
assuming that the yarn near the contact point can be treated as a cylinder, the jamming
volume can be approximately calculated by the intersecting volume of cylinders. First,
we have to find the contact point, i.e. the point where the distance of the central axis of
two cylinders attains minimum. We define the corresponding distance function of two
loops as:

ds : ½0; Ly
£ ½0; Ly
! R d s ðs1 ; s2 Þ ¼ dðX 1 ðs1 Þ; X 2 ðs2 ÞÞ ð8Þ

where d (· , · ) compute the distance of two points in the space, si, X i(si), i ¼ 1, 2 are the
arc length and the spatial coordinates at si respectively. The contact point is where
ds ðs1 ; s2 Þ , 2r and ds ðs1 ; s2 Þ attain extremum. In practice, possible contact region can
be estimated beforehand, and local distance function is calculated to determine the
contact point (Figure 2). According to calculus, the intersecting volume of two
cylinders with radius r is:
 IJCST Z
20,4
X Y

p
226 2

1.5

1
0
3
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1
4
Figure 2. s2 2 5 s1
Sketch map of local 3
distance function
(a) Possible contact region (b) The distance of possible contact region

Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
r ð1=sin uÞ r 2 2ðx2d Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ¼4 r 2 2 x 2 dy dx
d2r 0
Z ð9Þ
4 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ r 2 2 x 2 r 2 2 ðx 2 d Þ2 dx
sin u d2r

where d is the distance of the central axis of the two cylinders at contact point, u is the
angle between the two cylinders. The algorithm of computing the jamming volume of
loops is as follows:
(1) Initialize N, and the numbers Ml of possible contact regions for loop
lðl ¼ 1; L N Þ, and jamming volume V ¼ 0, i ¼ 0, and j ¼ 0.
(2) If i $ N, stop; else goto (3).
(3) For one possible contact region Aij, compute the distance function, and get the
contact point by searching for the extremum of the distance function.
(4) Compute the angle between two loops at the contact point.
(5) Compute intersecting volume Vij from equation (9), and V ¼ V þ Vij. If j $ Mi,
i ¼ i þ 1, goto (2), else j ¼ j þ 1, goto (3).
The jamming volume can be calculated by the above method, however, the integral
may cause computation complexity, an alternative method is to take d 3 as the
approximation of the jamming volume. It is not as accurate as the above method.
According to the principle of minimum potential energy, the fabric is in stable
equilibrium, i.e. the reference state, while the total energy is at a minimum. So, the
reference fabric can be achieved by minimizing the total energy. Taking the geometric
meanings of the variables in the parametric model of the loop, we have w . 0, c . 0,
and tk $ 2r. Considering the trend of x component along t, we have:


dx 1
,0 ð10Þ
dt 2
 From equations (2.1)-(10), we get: Shrinkage of
3a · 0:52 2 3a · 0:5 þ 0:5ða þ wÞ , 0 ð11Þ plain-knitted
fabric
that is a . 2w. So, a constrained optimization:
(
min U
ð12Þ
w; c . 0; th $ 2r; a . 2w 227
is utilized to compute the reference fabric, where U ¼ Uint þ Uext is the total energy.
The constrained optimization can be converted to an optimization problem without
constraints through punish function. And gradient descent method or quasi-Newton
method can be used to solve it. After the variables a, w, c, s, tk of the reference fabric are
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obtained, the shrinkage percentages can be calculated by equation (1).

4. Determination of k0 and t0
The determination of k0 and t0 is important to the energy computation. In practice, it is
difficult to get the shape of natural loop. Choi and Lo (2003) defined the degree of set c
to describe the shape of natural loop relative to the loop of the delivered fabric. c ¼ 0
illustrates the natural loop is a straight line, and c ¼ 1 shows that the natural loop is
the same as the loop of the delivered fabric. By assuming that k0 =ks ¼ t0 =ts ¼ c,
k0, t0 can be obtained by:
k0 ¼ ks · c; t0 ¼ ts · c ð13Þ
where ks and ts are the curvature and torsion of a standard loop. According to the
curve theory in differential geometry, the shape of a spatial curve, the natural loop
here, is determined by the curvature and torsion. While the shape obtained through the
above k0, t0 is not the natural loop obviously (shown as Figure 3). So, we need a new
method to compute k0 and t0. Instead of determine them by degree of set, we compute
them by estimating the natural loop directly.
Suppose that the natural loop can also be represented by the parametric model in
Section 2, and denote h ¼ c þ 2e þ s. The natural loops are put between two glasses
(shown as Figure 4), then we can measure the width w
0 and height h
0 at this state,
assume that the projection of the natural loop onto the yOz plane is an arc of a circle
(shown as Figure 5), then we have:

Z Z

X Y X Y

Figure 3.
The natural loop
(a) The natural loop of Choi's method (b) The natural loop of the new method
 IJCST h
0
Ru ¼ R cos u þ t h0 2 d ¼ R h0 ¼ 2R sin u ð14Þ
20,4 2
Since the width of the loop almost maintain constant before and after put into glasses,
we assume w0 ¼ w
0 . And the thickness of natural loop is between the yarn diameter
and the thickness of original loop, and can be approximated by:
228 2r w
0
th0 ¼ þ 2r: ð15Þ
Ly
From equations (14) and (15), we can get w0 ; th0 ; h0 , and, a0 can be computed by solving
the non-linear equation:
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Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ly0 ¼ ðx0 ðtÞÞ2 þ ð y 0 ðtÞÞ2 þ ðz 0 ðtÞÞ2 dt:
0

5. Data acquisition and experiments

We need only measure some properties of the delivered fabric and the yarn unraveled
from it to make our method work. First, it is easy to measure the yarn count,
pffiffiffiffifficpc,
ffi wpc,
and loop length, then the yarn diameter is approximated by D ¼ 2:54=23 Ne, and the
width and the course of a loop are w ¼ 1/wpc and c ¼ 1/cpc. w0, h0 can be measured as
discussed in Section 4. For simplicity, the corresponding data are divided by yarn
diameter.
The real length and width percentages can be obtained by measuring the fabric
before and after relaxation. The prediction results are compared to them to illustrate
the validity of the present method. We arbitrarily choose 14 plain-knitted fabrics, the
tensile, bending and torsional modules are set to be constants. The prediction results
are listed in Table I, the subscript d, r, p represent the values corresponding to the

Figure 4.
The natural loop
between glasses

h0
2
t h0 - d
h0
Figure 5. 2
The sketch map of the
q R
projection of natural loop
on yOz-plane
 Shrinkage of
No Ly Ne wpcd cpcd wpcr cpcr wpcp cpcp SW SL SWp SLp
plain-knitted
1 0.307 19.9 11.70 16.95 12.46 18.02 13.35 18.52 6.14 5.93 7.10 2.80 fabric
2 0.300 19.3 11.76 17.47 12.62 18.69 13.53 19.44 6.76 6.55 7.20 4.00
3 0.302 19.9 10.99 17.47 12.62 18.43 14.46 18.91 12.91 5.24 14.62 2.58
4 0.308 20.1 11.46 17.02 12.50 18.60 14.05 19.83 8.31 8.51 12.41 6.60
5 0.258 26.5 14.44 20.30 15.33 21.51 16.34 22.16 5.78 5.58 6.65 3.05 229
6 0.256 26.2 14.55 20.20 15.27 21.51 16.48 22.38 4.73 6.06 7.95 4.09
7 0.259 27.6 14.39 20.20 15.63 21.28 16.80 22.65 7.91 5.05 7.55 6.47
8 0.264 26.0 14.04 20.10 14.98 22.10 15.93 23.94 6.32 9.05 6.36 8.33
9 0.266 31.8 14.34 18.69 15.15 19.70 16.13 20.43 5.38 5.14 6.46 3.67
10 0.267 31.5 14.13 19.42 15.15 20.41 16.48 20.68 6.71 4.85 8.75 1.34
11 0.263 30.3 14.65 19.05 15.27 19.80 16.22 19.82 4.03 3.81 6.24 0.11
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12 0.271 31.7 14.49 18.52 15.38 19.61 16.47 19.94 5.80 5.56 7.08 1.67 Table I.
13 0.265 31.5 13.84 18.52 15.15 19.60 16.73 20.03 8.65 5.52 10.40 2.18 Fabric parameters and
14 0.304 20.0 11.05 17.70 12.31 18.35 13.34 18.58 10.22 3.54 8.36 1.28 the prediction results

delivered, reference and predicted fabric. There are three samples with error within
2 percent, 7 within 3 percent, and almost all of them within 4 percent.

6. Conclusion
In this paper, we present a shrinkage prediction method for plain-knitted fabric.
The reference fabric is obtained by minimizing the total energy to predict the
shrinkage. Experimental results show that this method is feasible, but the precision
need to be promoted. Further research focus on the measurement of the actual tensile,
bending and torsional modules of the yarn, and we believe that accurate measurement
of the material properties help to improve the prediction precision. The optimization in
Section 4 is a nonlinear constrained optimization, finding efficient algorithm to solve it
is also an important issue.

References
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Journal, Vol. 73 No. 8, pp. 739-48.
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structure, Part I: stitch model for the graphical representation of plain-knitted structures”,
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Institute, Vol. 44 No. 8, pp. 561-78.
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Corresponding author
Zhixun Su can be contacted at: zxsu@comgi.com

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