Composites
ELSEVIER
Printed
PII:
SO266-3538(96)00098-X
Science and Technology
57 (1997) 1-22
0 1997 Elsevier Science Limited
in Northern Ireland. All rights reserved
0266-3S38/97/$17.00
CHARACTERIZATION
AND MODELING OF THE TENSILE
PROPERTIES OF PLAIN WEFT-KNIT FABRIC-REINFORCED
COMPOSITES
S. Ramakrishna”
Department of Polymer Science & Engineering, Faculty of Textile Science, Kyoto Institute of Technology, Matsugasaki,
Sakyo-ku, Kyoto 606, Japan
(Received 15 January 1996; accepted 11 July 1996)
Abstract
This paper describes analytical models for predicting
tensile properties of knitted fabric-reinforced
composites. Initially, tensile properties of plain weft-knit
glass-fiber fabric-reinforced
epoxy composites were
determined experimentally in the wale and course
directions. Elastic properties were predicted by using a
‘cross-over model’ and laminated plate theory. The
analytical model expresses the crossing over of looped
yarns of knitted fabric, and fiber- and resin-rich regions
of composite. Elastic properties of the composite were
determined by combining the effective elastic properties
of looped yarns and resin-rich regions. Study of tensile
failure mechanisms indicated that ultimate failure of a
knitted-fabric composite occurs upon the fracture of
.varns bridging the fracture plane. Tensile strengths
were predicted by estimating the fracture strength of
bridging yarns. Tensile properties of knitted-fabric
composites with different volume fractions of fibers
were predicted.
Analytical procedures
have been
validated by comparing predictions
with the experimental results. The applicability and limitation of
these models have been discussed. 0 1997 Elsevier
Science Limited
Keywords: knitted-fabric
composites, weft-knit fabrics,
analytical model, cross-over model, elastic properties,
tensile strength, fiber orientation, geometric model,
fiber volume fraction
NOTATION
Constant for expressing the radius of knit
loop
Planar area of the composite over which N is
measured (4 cm”)
* Present address: Department of Mechanical and Production Engineering, The National University of Singapore, 10
Kent Ridge Crescent, Singapore 119260.
Mb
4
A,
Pwlb
4
B
C
cd
d
4
E,
Ef
E,
EW
-&
[J%
&(~I
4
LE,lb
464
&I
-52
f(a)
g((+b)
Area fraction of yarns bridging the course
fracture plane
Area fraction of fibers in the yarn
Stiffness matrix of yarns in cross-over model
Area fraction of yarns bridging the wale
fracture plane
Cross-sectional area of yarn
Width of tensile specimen
Course density of knitted fabric (number of
loops/2 cm in wale direction)
Constant (9 X 105)
Diameter of resin impregnated yarn
of unimpregnated
yarn
Linear
density
(Denier)
Elastic modulus of knitted fabric composite
in the course direction
Elastic modulus of reinforcing fibers
Elastic modulus of matrix resin
Elastic modulus of knitted-fabric composite
in the wale direction
Average elastic modulus of yarn in x
direction
Elastic modulus of yarns of cross-over model
in the x axis direction
X direction elastic modulus of short segment
of yarn at ff orientation
Average
elastic modulus of yarn in y
direction
Elastic modulus of yarns of cross-over model
in the y axis direction
Y direction elastic modulus of short segment
of yarn at LYorientation
Longitudinal
elastic modulus of unidirectional lamina
Transverse elastic modulus of unidirectional
lamina
Orientation
distribution function of yarns
bridging the fracture plane
Yarn strength distribution function
2
G&)
G(ah)
G12
h
K
L
LS
n
[%lb
ilk
[db
N
P
Q
R
s
S
t
vh
v,
v,f
W
X
Y
Z
CY
ii
P
Y
6s
66
0
Yf
yrn
S. Ramakrishna
In-plane shear modulus of knitted-fabric
composite
Average shear modulus of yarn in xy plane
In-plane shear modulus of yarns in cross-over
model
XY plane shear modulus of short segment of
yarn at ff orientation
Yarns fractured due to the applied stress (Tb
In-plane shear modulus of unidirectional
lamina.
Constant for expressing the height of loop
above the plane of fabric
Packing fraction of yarn
Length of yarn under consideration
Length of yarn in one loop (stitch)
Number
Number of yarns bridging the course fracture
plane
Number of layers/plies of knitted fabric
Number of yarns bridging the wale fracture
plane
Stitch density of knitted fabric in the
composite (number of loops/4 cm’)
Parameter of exponential function of mh
Parameter of exponential function of (Th
Parameter of f(o) function
Distance between two points measured along
the loop
Parameter of f(a) function
Thickness of composite specimen
Volume fraction of impregnated yarns in
composite
Volume fraction of fibers in composite
Volume fraction of fibers in impregnated
yarn
Wale density of knitted fabric (number of
loops/2 cm in course direction)
X coordinate
Y coordinate
Z coordinate
Orientation of short segment of yarn with
respect to x axis
Average orientation of yarn of a knit loop
with respect to the loading direction
Maximum orientation of yarn with respect to
loading direction in the fracture plane
Orientation of short segment of yarn with
respect to y axis
Orientation of short segment of yarn with
respect to z axis
Length of a short segment of yarn
Angle of short segment of yarn at the center
of curved yarn
Angle of segment of yarn under consideration at the center of curved yarn
Poisson’s ratio of reinforcing fibers
Poison’s ratio of matrix resin
Poison’s ratio of knitted-fabric composite
Average Poison’s ratio of yarn in xy plane
with applied load in x direction
Poison’s ratio of yarns in cross-over model
Poison’s ratio of short segment of yarn at (Y
orientation
Average Poison’s ratio of yarn in xy plane
with applied load in y direction
Poison’s ratio of yarns in cross-over model
Poisson’s ratio of unidirectional lamina
Density of fiber (g/cm”)
Tensile strength of a yarn
Mean tensile strength of a set of yarns
bridging the fracture plane
Yarn strength corresponding to the orientation ayk
Maximum yarn stress
Tensile strength of knitted-fabric composite
in the course direction
Tensile strength of reinforcement fibers
Tensile strength of matrix resin
Tensile strength of knitted-fabric composite
in the wale direction
Longitudinal
tensile strength of unidirectional lamina
Transverse tensile strength of unidirectional
lamina
Shear strength of unidirectional lamina
Angle HCB
Angle OCQ, total angle of the portion of the
loop under consideration
Angle OCB
1 INTRODUCTION
Recently, composites reinforced
with textile fiber
preforms have been receiving greater attention owing
to the need for improvements in interlaminar shear
strength, damage tolerance and through-thickness
properties of composite materials. Textile composites
offer adequate
structural
integrity
as well as
shapeability for near-net-shape
manufacturing.
By
using conventional textile techniques such as weaving,
braiding, knitting and stitching, it is possible to
produce a wide range of two- and three-dimensional
fiber preforms (Fig. 1). Woven and braided fabricreinforced composites have been investigated extensively. However, so far only limited attention has been
given to knitted fabrics in the composites industry.
This is mainly due to the opinion that composites
reinforced with knitted fabrics possess low mechanical
properties owing to their looped-fiber architecture.
Recent research suggests that by selecting proper
knitted-fabric structure, it is possible to obtain the
desired mechanical properties.‘-7
Knitted fabrics are made by the interlocking of
loops of yarn. They are basically categorized into two
Tensile properties of fabric-reinforced
3
composites
Biaxial Weaving
Woven
-I
Triaxial Weaving
Flat Braiding
Braid
i
F
Circular Braiding
Warp Knitting
--I
Weft Knitting
Mechanical Process
-I Chemical Process
Knitting + Weaving
LCombination
Knitting + Nonwoven
Lock Stitching
-
Stitched
--I
Chain Stitching
Biaxial Weaving
-
Triaxial Weaving
Woven
-t
Multiaxial Weaving
2 Step Braiding
3D Preforms
-
Braid
4 Step Braiding
Solid Braiding
i
Warp Knitting
-
Knit
-I
-
Combination
Weft Knitting
Knitting + Weaving
-1 Knitting + Stitching
Fig. 1. Various
techniques
of manufacturing
types namely warp-knit fabrics and weft-knit fabrics
based on the yarn feeding and knitting direction (Fig.
2). Warp-knit fabrics are produced by simultaneous
yarn feeding and loop forming at every needle of the
Y
I
textile fiber preforms.
needle bed during the same knitting cycle. Warp
knitting takes place in the wale direction (lengthwise
direction) of the knitted fabric. The warp knitting
direction is shown as a solid line in Fig. 2(b).
d
COURSE
c
(b)
(4
Fig. 2. Schematic
diagrams
of (a) weft- and (b) warp-knit
fabrics.
S. Ramakrishna
4
Weft-knit fabrics are produced by the same yarn
which forms into loops successively at each needle of
the needle bed during the same knitting cycle. The
weft-knitting action occurs in the course direction
(widthwise direction) of the knitted fabric (solid line
in Fig. 2(a)). A wide range of knitted fabrics, both
planar (two-dimensional, 2D) and three-dimensional
(3D), can be produced by selecting the type of
knitting machine, the number of needle beds, the
number of guide bars, etc. Some of the warp- and
weft-knit structures that can be mass produced in
conventional knitting machines are summarized in
Figs 3 and 4, respectively. Even though knitting
technology is well established in the textile industry, it
is relatively new to the advanced composites industry.
Several types of knitting machines are available
commercially. To give a comprehensive idea, efforts
have been made to classify the various knitting
machines. Figures 5 and 6 show the classification of
various warp and weft knitting machines, respectively.
From the large number of knitting machines and
fabric structures it is evident that composite properties
can be tailored to meet various end use requirements.
Warp-knit fabrics are rigid compared to weft-knit
fabrics. A composite material made from a weft-knit
fabric and a flexible resin matrix is highly deformable
and suitable for fabricating complex shaped and
deep-drawn components. On the other hand, rigid-
- 1 Single Dembigh (1X1 Tricot)
-
2 Single Vandyke (Single Atlas)
3 Single Cord (Plain Cord)
4 English Leather
5 Double Fabric (Two Needle Fabric)
6 Shell Fabric
1 Double Dembigh (Tricot, Plain Tricot)
2 Double Vandyke (Atlas, Diamond)
3 Double Cord
4 Half Tricot (Carmeuse)
5 Satin (Satin Tricot)
6 Sharkskin
7 Queen’s Cord (American Sharskin)
8 Idle Swing
9 Pile (Plush)
10 Milanese
11 Net (Open Work, Mesh)
12 Tulle
_13 Mesh
-l Tuck
2 Fleecy (Lined Warp Knits)
3 Jacquard
4 Inlaid Stitch
5 Inlaid Net
- 6 Jacquard Lace
Fig. 3. Broad
classification
of warp-knit
fabrics.
Tensile properties of fabric-reinforced
7
composites
5
Weft Lock Knit (Float Stitch)
2 Accordian (Single Jersey)
3 Hopsack (Inlaid Plain, Jersey)
4 Fleecy Rib (Plated Plain)
5 Lace Stitch
6 Half Point Transfer Stitch
7 Eyelet (Pereline)
8 Deflected Stitch
9 Sinker Wheel Fishnet (Expanded Loops)
10 Float Plated Fishnet
11 Coil Stitch
12 Intrasia
13 Binding Off
_lj Plush (Pile)
1 Double Jersey (l/l Rib)
2 Swiss Rib (212 Rib)
3 Derby Rib (6/3 Rib)
4 Interlock (Double l/l Rib,
Double Jersey Interlock)
5 Tuck-float-rib
6 Inlaid Rib
7 Rib Transfer
Knitted
Fabrics
Weft
8 Eyelet (Pereline)
9 Jacguard (with selected backing)
10 Roll welt (English welt)
11 Ripple Fabric
12 French or Tubular Welt
13 Rib Plating
14 Racked Rib
15 Royal Rib (Half Cardigan)
16 Polka Rib (Full Cardigan)
17 Binding Off
18 Sinker Loop Transfer
-19 Reverse Loop Plating
Fig. 4. Broad classification of weft-knit fabrics.
matrix composites reinforced with warp-knit fabric are
suitable for primary and secondary load-bearing
structural applications which require good stiffness
and strength properties.
Preliminary
experimental
studies have been made on the mechanical properties
of warp-knit7-12 and weft-knit4-6,13-22 fabric composites. The effects of variables such as stitch density,
knitted-fabric structure, number of plies of knitted
fabrics, percentage pre-stretching of knitted fabrics,
inlay fibers, tow size of yarn bundles, etc., on the
mechanical properties of a composite material have
been identified. However, only a limited amount of
attention has been given to the modeling of the
mechanical properties of knitted-fabric composites.
Ko et aL8 proposed a fabric geometry model based
on the unit cell concept and laminate theory for
predicting the tensile properties of warp-knit fabric
composites. They investigated composites reinforced
with multi-directional warp-knit fabric with five basic
yarn components: 0” (weft), 90” (warp), 45” (bias),
-4.5” (bias) and the stitching yarn (through-thickness).
A good match was reported between the experimental
and predicted tensile properties. The fabric structure
investigated by Ko et al. is a special case. In general
knitted fabrics, the yarns are curved and their
orientation
changes continuously
along the loop.
Recently,
Gommers et a1.12 extended
the same
concept to composites reinforced with conventional
warp-knit fabrics, although this study was limited to
predicting only the elastic properties.
The yarn
orientation
was determined
mainly from the dry
knitted fabric. This analysis was based on the
S. Ramakrishna
Milanese Single Stroke
1 Double Stroke
I
Double Guide Bar
Triple Guide Bar
Multiple Guide Bar
Pentihose
Cut Presser
Chain Raschel
Tricot
Single Stroke
-1 Double Stroke
Jacquard
Bearded
Needle
Special Type
L Double
Needle Bed
Knock-off Lap Tricot
Chain Automat Tricot
Swan Warp Tricot
Guide Bar Transfer
Sinker Loop
Simplex Machine
Circular Milanese
Single
Needle Bed
I
Raschel
Latch
Needle
Multi-target
Jacquard
Circular
Needleless
Lace Type
Less Guide Bars
i
Multiple Guide Bars
i
Tulle Machine
Fish Net Machine
Power Net Machine
Net Type
Crochet
Special Type
Double
Needle Bed
Raschel -
Weft Insert
Co-We-Nit
i Double Stroke
Multi-target
Jacquard
Carnet Tyne
Thermal ?loth
Tubular Fabric Raschel
Special Type
Narrow Fabric Raschel
Single Needle Bed Raschel
Double Needle Bed Raschel
Tricot
Malimo
Stitch Bonding Loom
Maliwatt
Malipol
Compound
Needle
Single Needle Bed Raschel
Self-Forced
Beard Needle
i
.Multi-target
Fine Net
Crochet
1
Fig. 5. Broad classification
assumption that the yarn orientation in dry knitted
fabric is exactly same as that in the composite form.
that the
Mundenz3 and Postlez4 demonstrated
knitted-fabric geometry is influenced by the medium
in a
in which it is kept. The yarn orientation
knitted-fabric composite can be determined accurately
by following the methods used for determining fiber
orientation in random short-fiber-reinforced
composites. In these methods, often a number of sections
CarpetType
Jacquard Crochet
of warp knitting
machines.
through the composite are cut, polished and image
analyzed for determining
fiber orientation.
This
process is laborious and time consuming. Ideally, a
geometric model that can express orientation of yarns
in the knitted-fabric composite is needed, so that with
such a model the yarn orientation can be predicted for
different knitting variables such as stitch density of
knitted fabric, linear density of yarn, etc.
Rudd et al.13 and Ramakrishna and HullI made
Tensile properties of fabric-reinforced
composites
7
Latch Needle - Traverse Knitting Machine
Flat Bed
Milanorib Knitting Machine
Jacquard Knitting Machine
Beard Needle
Semi-Jacquard Knitting Machine
r
i
Rohben Knitting Machine
Single
Needle Bed
Latch Needle_
i
Circular Bed
1
Sinker Wheel with Jacquard
Knitting Machine
1Sinker Top Knitting Machine
L Beard Needle
Loop Wheel Knitting Machine
Tompkin Knitting Machine
1 Silver Knitting Machine
Fraise Knitting Machine
Dial & Cylinder -
Latch Needle
i
Double Side Knitting Machine
Seal Knitting Machine
Double Jersey Knitting Machine
Double
Needle Bed V-bed Knitting Machine
Double Cylinder -
Beard or
Double Needle
Cowper Knitting Machine
Jersey Knitting Machine
Hosiery Knitting Machine
Fig. 6. Broad classification
efforts to predict the elastic moduli of weft-knit
fabric-reinforced
composites. They predicted elastic
moduli by using a combination of the rule-of-mixtures
and a reinforcement efficiency factor. The agreement
with the experimental results appeared to be good.
This method is limited to the prediction of the elastic
modulus of the composite material, and these authors
considered only the yarn orientation in the planar
direction of the knitted fabric. However, in knittedfabric composites the yarns are oriented threedimensionally.
Recently, Ramakrishna”
proposed a ‘cross-over
model’ based on a geometric model of a plain
weft-knit fabric to determine the elastic properties of
knitted glass-fiber fabric-reinforced epoxy composites.
This analytical model considers the three-dimensional
orientation of yarn in the knitted-fabric composite.
This model was applied to a composite with a specific
fiber volume fraction. In the present paper, the same
analytical model has been applied for predicting the
elastic properties of knitted-fabric composites with
different fiber volume fractions. Attempts have been
made to identify
the necessary
equations
for
predicting the volume fraction of fibers in knittedfabric composites. Efforts were also made to develop
analytical procedures for predicting tensile strength of
knitted-fabric composites.
of weft knitting
machines.
2 EXPERIMENTAL
2.1 Knitted fabrics
The work described in the present paper is concerned
with a plain weft-knit fabric. A schematic diagram of
this knitted fabric is shown in Fig. 7. Knitted fabrics
were produced on a flat bed weft knitting machine
with a single set of needles and glass-fiber yarn (ECD
Fig. 7. Schematic
diagram
of
a plain weft-knit fabric.
S. Ramakrishna
8
450 l/2 4-45Y-23, Nippon Electric Glass Co., Japan).
The yarn runs widthwise, and the knit loops are
formed by a single yarn. The linear density of the
glass-fiber yarn is D, = 1600 Denier. The row of loops
in the width direction is called the ‘course’ and the
row of loops in the longitudinal direction of the fabric
is called the ‘wale’. Knitted fabrics are often specified
by the terms ‘wale density’ and ‘course density’. W is
defined as the number of wales per unit length in the
course direction, and C is the number of courses per
unit length in the wale direction of the fabric. C and
W can be measured experimentally from the knitted
fabric in the composite form. Sometimes knitted
fabrics are specified by the ‘stitch density’, N, given by
the product of W and C. In other words, N is defined
as the number of knit loops per unit area of the fabric.
Knitted fabrics with N = 20 loops/4 cm2 (W = 4 loops/
2 cm and C = 5 loops/2 cm) were made.
2.2 Composite fabrication
Knitted-fabric-reinforced
composites were made by
the hand lay-up method with a mixture of epoxy resin
(Epikote
828) and triethylenetetramine
hardener
(11% by weight of epoxy resin). The composite was
cured at 100°C for 1 h. The value of V, of the
composite was estimated by the combustion method.
Composites reinforced with (a) single ply and (b) four
plies of knitted fabrics were produced. The thicknesses of single-ply and four-ply composites were O-6
and O-7 mm, respectively. Composite specimens of
200 mm length and 20mm width were prepared by
cutting parallel to the wale and course directions of
the knitted fabric. Glass/epoxy end tabs of 50 mm
length were glued to both ends of each specimen.
2.3 Tensile tests
Tensile tests were carried out in an Instron testing
machine (Type 4206) at a cross-head speed of
1 mm/min. Strains were measured with bi-axial strain
80.0
2.0
1.0
Strain,
%
Fig. 8. Tensile stress/strain curves for single-ply knittedfabric reinforced composite.
gauges. A minimum of five specimens were tested for
each case. Fracture surfaces were studied by optical
and scanning electron microscopy techniques.
3 EXPERIMENTAL
RESULTS
Typical stress/strain
curves obtained from tensile
testing of single-ply knitted-fabric
composite are
shown in Fig. 8. Stress increased linearly with
increasing strain up to a knee point which occurred at
approximately 0.45% strain. The stress corresponding
to the knee point was higher for wale-tested
specimens than course-tested
specimens. The nonlinearity of the stress/strain curves above the knee
point was associated with material deformation and
microfracture
processes in the specimen. At strain
levels immediately above the knee point, whitening of
knit loops and matrix cracking were observed. The
whitening of knit loops is due to debonding at the
yarn/resin interface. With further increase of applied
strain the cracks grew through the resin-rich regions in
the widthwise direction of the specimen. A number of
such cracks were observed across the width of the
specimen in the gauge length, and it could be seen
that there were yarns bridging the cracks. With further
increase of applied strain, peeling (further debonding)
of yarns from the fracture surface was also observed.
Finally, fracture of the bridging yarns occurred,
resulting in separation of the fracture surfaces. Optical
photographs
and related schematic diagrams of
fractured wale and course specimens are shown in Fig.
9. SEM study of fracture surfaces indicated that river
lines in the epoxy resin matrix originated from the
yarn/resin interface region. Initially, microcracking
occurs as a consequence of the debonding of yarns
oriented normal to the testing direction. The cracks
nucleated from the debonded sites, propagate into
resin-rich regions and coalesce into large transverse
cracks. The fracture plane is bridged by unfractured
yarns, and fracture of these bridging yarns resulted in
complete separation of fracture surfaces. In other
words, the tensile strength of knitted-fabric composite
is determined mainly by the fracture strength of
yarns bridging the fracture plane. Similar observations
were made in the case of the 4-ply knitted-fabric
composites.
Elastic modulus was calculated from the initial
linear portions of the stress/strain curves. Elastic
modulus and tensile strength values obtained from
tensile tests are summarized in Table 1, where
standard deviations are given in parentheses adjacent
to each result. The plain-knit fabric composites
possess better tensile properties in the wale direction
than in the course direction. Both the wale and course
tensile properties
increased with increasing fiber
content.
Tensile properties of fabric-reinforced
z.
7
7
3
9
composites
!-
COURSE
Fig. 9.
4 ANALYSIS
Optical photographs and schematic diagrams of tensile tested specimens: (a) wale; (b) course.
APPROACH
An outline of the analytical procedure for predicting
the elastic properties of knitted-fabric composites is
shown in Fig. 10. Initially, a geometric model was
identified for determining the orientation of yarn in a
knitted-fabric
composite
(Section 5). The input
parameters for this model are W, C and d: both C and
Table 1. Tensile properties
of plain knitted glass-fiber-fabric/epoxy
experiments
Number of plies Fiber content,
of knitted fabric,
vol. %
nk
1
4
9.5
32.35
W can be determined experimentally. The procedure
for estimation of d is outlined in Section 6.1. A
‘cross-over model’ based on the unit cell approach has
been proposed for expressing the crossing over of
looped yarns of knitted fabric (Section 7) and the
effective elastic properties of the yarns were estimated
from laminated plate theory. The elastic properties of
the composite were determined by combining the
composites
obtained
from
Elastic modulus, GPa
Tensile strength, MPa
Wale
Course
Wale
Course
5.35 (0.33)
10.28 (O-35)
4.37 (0.07)
8-49 (0.21)
62-83 (7.1)
152.7 (9.5)
35.5 (2.21)
75.4 (4.5)
S. Ramakrishna
10
N=C*W
Ls
ANALYTICAL
Vyf Ef, Y f -j
“k
I DY t
Pf
CROSS-OVER MODEL
\ UD Lamina Properties \
i E11,E2>G12 y12 ;
L~_~~.__.__________.._________,
E rn’ >,r
.--._--...______
4..__.__...________,
Yam Properties
[Exlb,
lEylb. [Gxylb, i
._..!_%i!~.T “.??...........~
,---.._.__.___
i ____._.._._____.
:
Rule of Mixtures
j
Lack..____._____._____._______,
Composite E+ astic Properties
Ew,EC,Gwc,yWC
Fig.
10.Flow chart of the analysis method
for predicting
elastic properties of yarns and resin-rich regions. The
necessary equations for estimating the fiber volume
fraction of the composite theoretically are given in
Section 6.2.
Experimental
studies indicated that the ultimate
failure of knitted-fabric composite occurs upon the
fracture of yarns bridging the fracture plane. Hence,
tensile strengths were predicted by estimating the
fracture strength of bridging yarns (Section 8).
5 GEOMETRIC
5.1 Geometric
MODEL
model of plain weft-knit fabric
On the basis of the geometric model of Leaf and
Glaskin for plain-knit
fabricz6
a mathematical
description of yarn orientation can be obtained in
terms of the known parameters C, W and d. Figure 7
shows the schematic diagram of a projection of knit
loops on the plane of the fabric, and a schematic
diagram of an idealized unit cell of knitted fabric is
shown in Fig. 11. Basic assumptions are that (1) the
the elastic properties
of knitted-fabric
reinforced
composite.
yarns assume a circular cross-section and (2) the
projection of the central axis of the yarn on the plane
of the fabric is composed of circular arcs, i.e. the yarn
forming a course lies on the surface of a series of
circular cylinders whose generators are perpendicular
to the plane of the fabric. These assumptions are
reasonable
as the knit loops are formed during
knitting by bending the yarn round a series of equally
spaced knitting needles and sinkers. The composite is
made when the fabric is in a relaxed condition without
any pre-stretching.
Consider the rectangular axes Ox and Oy parallel to
the wale and course directions of the fabric. The OQ
portion of loop is assumed to have its center at C. The
total angle of the portion of the loop under
consideration
is OCQ = cp. ad is the radius of
projection of knit loop, CO in Fig. 11, where a is a
constant. Q is the point at which the central axis of
this loop joins the central axis of the loop with center
F. H and J are the points at which the yarns of
adjacent loops (loops with centers at C and B) cross
over. The angles OCB = $ and HCB = 4. If P is any
Tensile properties of fabric-reinforced
composites
11
parameters experimentally; they may be determined
from the simple geometric relationships of the loop.
From Fig. 11, l/W is given by:
l/W = 2EF = 4ad sin 9
a=
(4)
1
(5)
4Wd sin cp
and l/C is given by:
l/C=AC+CE=(2a-l)dcos$-2adcosrp
(6)
Fig. 11. Schematic representation of a unit cell of plain-knit
This equation contains two unknown parameters, $
and cp, for the determination
of which we need
another simultaneous equation.
If we assume that the loops are closely fitted at G,
then CB = 2GC = (2a - 1)d. Also, CF = 2CQ = 2ad,
AB = CB sin 1c,and EF = CF sin(l80” - cp). But AB =
EF, therefore
fabric.
(2a - 1)d sin I++
= 2ad sin p
point on the projection of the central axis, OCP = 8,
then the coordinates of P in the xy plane are:
x = ad(l - cos 19)
(1)
y = ad sin 0
(2)
The boundary conditions for the t coordinate
central axis in space are as follows:
From eqns (6) and (7):
l/C = (d(4a2 cos2 q - 4a + 1))d - 2ad cos cp (8)
From eqns (5) and (8):
of the
1. at 0 the height of the central axis above the
plane of the fabric is zero, i.e.
z=O
2.
when
3.
(9)
From eqn (7):
8=0
if hd is the maximum height (at Q) of the axis
above the plane of the fabric, we must have
z=hd
when
0’9
at 0 and Q the central axis must lie parallel to
the xy plane, and so we have
dz/ds=O
when
satisfies
z = y (1 - cos 7&/p)
2a
.
za - 1 lslIl q >
(
Cc,= sin-’
The parameter h can be determined
height of the central axis at H is:
(10)
as follows. The
z,,=F[l-cos+)]
e=Oorcp
where s is the distance OP measured
loop.
A simple function which
mentioned conditions is:
(7)
(11)
along the
all the
Similarly, the height at J is:
above
z,=h$
l-cos;(+++)
[
1
(12)
But
(3)
where h is a constant used for representing
the
maximum height hd (at Q) of the central axis above
the plane of the fabric. Equations (l)-(3) give the X,
y, and z coordinates of any point on the central axis of
the loop. The unknown parameters in eqns (l)-(3) are
a, h and rp. It is difficult to measure
these
ZJ
Rearranging
-
ZH
=
d
eqns (II)-(13)
gives:
h = [sin(T)
sin(y)l-l
To determine
(13)
(14)
h we need to know the angles 4 and
S. Ramakrishna
12
+. $t is given by eqn (10) and 4 can be determined as
follows. The yarns forming the loops with centers B
and C cross each other at H and J, where angle
HCB = angle JCB = 4. From Fig. 11, HC = HB = ad,
and CG = BG = (a - 1/2)d. Hence, the triangles HCG
and HBG are congruent and henec angle HGC = 90”.
Therefore
CG
2a-1
cos
+=HC=- 2a
2a - 1
__
2a
( >
-x,-1
=
sscosff
yn - y,_,
=
sscosp
2, -
=
Gscosy
xn
i&l
(15)
Hence, 4 is given by:
6 = coscl
vector, as, with respect to the x, y and z axes,
respectively. From the vector analysis principles, the
Cartesian components of the vector are:
(16)
where (A_,, Y,-~, z,_~) and (x,, y,, z,) are the
Cartesian coordinates of the start and end points of
each segment under consideration. (Y, p and y are
given by:
5.2 Determination of loop length
The length of yarn from 0 to Q, is given dy:
L-adcp
(17)
p = cos
-fyn
-r)
y=cos
-fZn
-Z-‘1
Considering the symmetry of the loop, the length of
yarn in one loop (MNOQR) is given by:
L, = 4adq
(18)
(20)
where L, is the length of yarn in one loop.
5.3 Determination of yam orientation
The orientation of yarn in the knit loop (MNOQR)
can be determined by knowing the orientation of yarn
in the portion OQ (Fig. 11). It can be assumed that
the OQ portion of loop is an assemblage of many
small pieces of straight lines. The length of each small
piece, SS, is given by:
8s = ad(80)
(19)
where 88 is the angle of segment 6s at the center of
the curved yarn.
The orientation
of each segment in a threedimensional space can be represented by the vector 6s
in Fig. 12. (Y,/3 and y represent the orientations of the
6 ESTIMATION OF YARN DIAMETER
FIBER VOLUME FRACTION
AND
6.1 Estimation of yarn diameter
Hearle et aiF proposed two basic idealized packings
of circular fibers in yarns: open packing and close
packing, in which the fibers are arranged in concentric
and hexagonal patterns, respectively. The packing
fraction of the yarn, K, is defined as:
K+f
Y
(21)
where A, and Af are the cross-sectional areas of the
yarn and the fibers in the yarn, respectively. Typical
values of K are 0.75 and 0.91 for open and closed
packing patterns, respectively. However, experimental
investigations
indicate”
that for weft-knit fabricreinforced composites K = 0.45, much smaller than
the ideal packing conditions.
Af is given by:
D
A,=2
c
dPf
(22)
where D, is the linear density of the yarn, measured
by the Denier count method (Denier is defined as
g/9000 m of yarn), Cd = 9 X lo5 is a constant, and pf is
the density of fiber (g/cm”).
Combining eqns (21) and (22):
Fig. 12. Representation of vector of a short segment of
yarn.
A,=--J-
D
CdP&
(23)
Tensile properties of fabric-reinforced
The yarn diameter, d, is given by:
(24)
6.2 Estimation of fiber volume fractions
Owing to the large resin-rich
regions in the
knitted-fabric composite, it is reasonable to assume
that the volume fraction of fibers in the composite, V,,
is smaller than the volume fraction of fibers in the
impregnated yarn, Vyf. Assuming the yarn is uniform
along the length, V,, is given by:
V,,, = K
(25)
and V, is given by:
v,=
nkD,L,CW
(26)
C,pfAt
where L, is the length of yarn in one loop or stitch and
is given by eqn (18), t is the thickness of composite
specimen, Q is the number of layers of knitted fabric
in the composite, and A is the planar area of the
(a>
rY
X
Wale
t
Course
13
composites
composite over which W and C are measured. In the
present work, both C and W are measured over a unit
length of 2 cm and hence A = 4 cm’.
Since W X C = N, eqn (26) can be rewritten as:
Vf =
@JU’
Gp&
7 ANALYTICAL
MODEL
ELASTIC CONSTANTS
(27)
FOR ESTIMATING
The unit cell of a plain-knit fabric can be divided into
four identical sub-structures shown in Fig. 13(a). Each
sub-structure consists of two impregnated yarns that
cross over each other. This sub-structure is called a
‘cross-over model’. A three-dimensional
representation of the cross-over model is shown in Fig. 13(b).
The unit cell can be constructed by means of the
cross-over model. Repeating the unit cell in the fabric
plane obviously reproduces the complete plain-knit
fabric structure. Hence, analysis of the cross-over
model was carried out. Because of the curved yarn
architecture, the cross-over model consists of fiberand resin-rich regions. The effective elastic properties
of the composite can be obtained by combining the
elastic properties of the fiber- and resin-rich regions.
First, the analytical procedure for estimating the
effective elastic properties of the curved yarns is
described.
In the modeling process, each impregnated yarn is
further idealized as a curved unidirectional lamina.
Elastic properties of unidirectional lamina are given
by:*’
1
-r
& -
&
1.36 --( l-ti
(--&
Em 2_
l-%j
(b)
&,
1-v;
em&, *
v&f
1
1-v;
..
( l--y:
l-v”,
+ 1 - 1.05E,
-
(28)
1.36
1
-=
G2
Ef
Y
Vf)
yt2
2(1+ vrn)
L
=
(
Schematic diagrams of (a) unit cell and (b)
cross-over model.
Em
+
2(1+ v,)
(Vf - Vm,( 2)
Ef
13.
1 - 1*05*
Em
-
2(1+
l*osqf
Fig.
1
1+$
--
fv,
Em
1+v;
1
where Ell, E22, G12 and v12 represent the longitudinal
elastic modulus, transverse elastic modulus, in-plane
shear modulus and Poisson ratio of a unidirectional
14
S. Ramakrishna
Table 2. Material constants of glass fiber and
plain epoxy resin
Elastic
modulus,
GPa
Glass fiber
Epoxy resin
The stiffness matrix of the curved yarn is:29
Poisson
ratio
(31)
E,=74
3.6
V~= 0.23
Vm= 0.35
E, =
lamina, respectively. Ef, vf and E,, Y, are the elastic
moduli and Poisson ratios of fiber and resin matrix,
respectively. Material constants of the glass fiber and
epoxy resin are given in Table 2.
From the assumption that the yarn is an assemblage
of short segments, the orientation, CI, of each segment
with respect to the x axis is given by eqn (20). The
effective elastic properties of each segment in the x
direction can be derived as follows:29,30
&=
[F+
(+--~)sin2ucoa’u
+e]
&=
[F+
(k-~)sin’acos’u
+g]
The above equations have been developed for one
curved yarn. The cross-over model consists of two
curved yarns. From the geometry of knit loops, the
orientation of second yarn can be obtained by rotating
the first yarn by 180” in a clockwise direction. From
(Y= (US+ z) and eqns (29)-(31), the effective elastic
properties of the second yarn can be obtained.
The stiffness constants of both the yarns of the
cross-over model are known. The elastic properties of
the cross-over model can be derived by assuming that
each of the curved yarns is subjected to the same
strain in the x direction. The total effective stiffness
parameters of both the yarns of the cross-over model
are given by:29
A, = 2 L,Qii(,,
(i, j = 1, 2, S)
n=l
2
-
(
r
(sin4 (Y+ cos4 CZ)
11
1
1
1
sin2 a cos2 LY
+- - F
I
12>
II &I
1
p=
G,,(a)
(29)
where n represents the yarn number.
Thus, the stiffness matrix of the yarns
cross-over model is:
in the
(33)
sin2 (Y~0s’ (Y
The effective elastic properties
are given by:29
I
+$-(sin’a+cos”a)
12
(32)
A1422
I
where E,(a) and EJ(Y) are the elastic moduli of the
segment in the x and y directions, respectively. G,,(a)
and v~,,((Y)represent the shear modulus and Poisson
ratio of the segment, respectively.
Assuming that all the segments are subjected to the
same strain conditions:
of the cross-over yarns
-
82
lExl’
=
(L, + Lz)Az2
[Ey’b
=
AIIAZ - A:2
(L, + LJA,
[%y]b
=
2
,vyVyx,b =
1
(34)
yb
$
[Cylh
xh
A
LGxy]b= (L, pL2)
(30)
where L is given by eqn (17).
where L, and L2 are the lengths of yarns in the
cross-over model. [E,], and [Eylb are the combined
elastic moduli of the yarns in the x and y directions,
respectively. G,,(b), and v_,(b) and VJb) represent
planar shear modulus, and Poisson ratios of yarns,
respectively.
Equation (34) gives the combined elastic properties
of the yarns only. The elastic properties of the
composite can be determined by combining the elastic
properties of the yarns and resin-rich regions. For this
purpose we need to know the relative volume
Tensile properties of fabric-reinforced
fractions of yarns and the resin-rich regions in the
composite.
The volume fraction of yarns in the composite, V,,
can be determined as follows. V,, is the ratio of
volume of fibers and volume of impregnated yarns.
Similarly, V, is the ratio of volume of fibers and
volume of composite. Hence, the volume fraction of
impregnated yarns in the composite, V,, is given by:
Vb =
$
15
composites
failure strength of yarns bridging the fracture plane.
The number of yarns bridging the fracture plane
would depend on the testing direction with respect to
the knitted fabric. The number of yarns bridging the
wale, [n,],,, and course, [L&, fracture planes are given
by:
(35)
Yf
where V,,, and V, are given by eqns (25) and (27),
respectively.
The effective elastic properties of the composite are
given by:*’
where B is the width of tensile specimen in cm.
The area fraction of yarns bridging the wale, [Awlb,
and course, [A&,, fracture planes are given by:
J%,= [-Cldv,) + C%N - W
+ EJ(1 - v,)
E, = PylbW
Gvc=
1
r
1
1.36
Mb
_
I 1 - 1*05*1
Em
’
E,
1
Lv + b&lb) w + YnJ w + YnJ J
where t is the specimen thickness in cm and d is the
yarn diameter given by eqn (24).
The tensile strengths of a knitted-fabric composite
in the wale (a,) and course (a,) directions are given
by:
(T =
w
~,T~d*[o,l
At
(40)
(36)
where E, and E, are the elastic moduli of the
composite
in the wale and course directions,
respectively. G,, and vWCrepresent the shear modulus
and Poisson ratio of the knitted-fabric
composite,
respectively.
8 ANALYTICAL PROCEDURE FOR
ESTIMATING TENSILE STRENGTH
ab = Ul
By assuming that the ultimate fracture of the
composite occurs due to the simultaneous fracture of
matrix and reinforcement fibers, the tensile strength of
a knitted-fabric composite, g,,,, can be estimated from
the rule of mixtures:
u, =
(Vf)(Uf)
cos*
LT!+ ((Tm)(l -
where c is the mean strength of the set of yarns
bridging the fracture plane. c can be estimated by
using the following procedure.
If we assume that all of the bridging yarns possess
the same tensile strength and are aligned perfectly in
the loading direction, the z will be equal to the
longitudinal
tensile strength of a unidirectional
lamina, fll:
Vf)
(37)
where (TV and U, are the tensile strengths of
reinforcement
fibers and matrix resin, respectively,
and 6 is the average orientation of yarn with respect
to the loading direction.
However, the tensile failure mechanisms observed
from experiments indicate that the tensile strength of
the knitted-fabric composite mainly depends on the
=
W(Vyf) + (flnl)(l - Vyf)
(41)
However, owing to their looped architecture, it is
reasonable to assume that the yarns in the fracture
plane orient at an angle (Ywith respect to the loading
direction. An approximate estimate of yarn orientation in the fracture plane can be obtained from eqn
(20). The yarn can be treated as an off-axis loaded
unidirectional lamina. Hence, the tensile strength of a
yarn is given by:30
cos4 sin:
gb = L+y+
L g:
(72
sin: ~0s:
62
sin2,cost ~ “’
I
4
(42)
where ul, u2 and z12 are the longitudinal, transverse
and shear strengths of unidirectional lamina, respectively (Table 3).
S. Ramakrishna
16
Table 3. Tensile
properties
fiber/epoxy
of unidirectional
lamina
glass-
Longitudinal
strength,
u1 (Ml%
Transverse
strength,
c2 (MPa)
Shear strength,
rLZ(MPa)
885
45
35
the range 0 < cy < (Yk.The maximum orientation, ak,
can be determined from the fracture surfaces. Let gi,k
be the bundle strength corresponding to the maximum
orientation, (Yk.From eqns (43) and (44):
(Tbk= aleeQak
which, when rearranged,
The typical variation of cb with (Yis shown in Fig.
14. (TVdecreased with increasing (Y, and the decrease
of gb was significant in the range 0” < LY< 15”. Hence,
the variation of (+b with (Y in this range on the
composite strength was analyzed. All of the yarns in
the fracture plane may not have same (Yvalue, since
the fracture path is irregular and occurs at different
positions of the knit loops. During tensile testing the
yarns are peeled (debonded) from the fracture surface
and stretched before their failure. As a consequence
of the peeling and stretching effect, the yarns try to
align in the testing direction. Determination
of the
actual LYjust before the failure of a yarn is a difficult
task. It may be the case that different yarns orient at
different CYwith respect to the loading direction.
Because of these different (Y values, it may be
expected that yarns bridging the fracture plane possess
different strengths. The yarns may possess different
strengths owing to the statistical nature of fiber
strength. Many researchers
have investigated the
statistical nature of fiber bundle strengths, but the
present study is mainly concerned with the variation
of Crb with CX. From Fig. 14, an exponential
relationship between (TVand (Yis given by:
ub = Pe-“”
(43)
where P and Q are parameters
function and can be determined
and (45). When CY= 0:
of the exponential
from the eqns (44)
Equation (43) indicates the changes in (Tb with (Y.
Equation (38) gives the number of yarns bridging the
fracture plane. It is necessary to know how many of
these yarns orient at each value of CY.The following
exponential function, f(a), was assumed for expressing the orientation distribution of yarns in the fracture
plane:
f(a) = RemS”
(46)
where R and S are the parameters of the exponential
function.
This function suggests that more yarns orient close
to the testing direction. This assumption is reasonable
as the yarn bundles try to align in the loading
direction as a result of the debonding and stretching
mechanisms. Typical curves for the function f(a) are
shown in Fig. 15. Assuming the area under a curve is
(4
2.5
I
0.5 %A
11
0
I,,,
0"
5"
iL--&-a
(44)
We assume that all the yarn bundles are oriented
--_-S
= 0.6
+s=i.o
eSs1.6
"k=lo"
-0.5.u
P = c71
gives:
IO"
15c
in
(b)
q
1000
-a”k=F
01
800
*ak=
I
-a
-0.2 C.I__LL
0"
a
Fig. 14. Typical variation
of CT,,with a!
Fig. 15. Typical
1,,
/
5"
10
=15
Li_l_LL!
a
10"
curves of function
15"
f(u).
Tensile properties of fabric-reinforced
unity, we see that:
da = - ; [e-set - I]= 1
where R is dependent on the values of S and (Ye.
Typical f(a) curves for different S and (Yeare shown
in Fig. 15. These curves indicate that f(a) is more
sensitive to the parameter S than to (Ye. When S is
small, the yarn orientation distribution is spread out.
For large values of S, the distribution is skewed and
more yarns are aligned close to the loading direction.
Let g(ab) be the function of yarn distribution with
respect to the bundle strength. Typical g(a,) curves
are shown in Fig. 16. Using the variable transformation technique, g(r,,)da,, = f (cY)da
which, on rearranging,
121
(48)
e(Q-s)a
(49)
From eqns (43) and (48):
= &
From eqn (43):
-1
a=-ln
3
(50)
(P1
Q
Combining eqns (49) and (50):
R
sw=&PSIQcJ
((.W-1,
(51)
Let G((T,J indicate the yarns fractured by the
applied stress, (+,,.The surviving yarns [l - G(a,)] are
given by:
G(o-Jl= /-“Ig(gd dab
fn
(1 - G(a,)]
=&
Equation (53) implies that the maximum yarn stress,
gb,,,, is found from the condition that at failure the
load borne by the yarns is the maximum. Hence:
r 1 lQ’s
The maximum mean strength, Z& of surviving yarns
can be obtained by substituting the value of cbrn in
(+b[l
-
G(cb)]:
K=-
RP
-
Q
((Q/9+1)
1
(55)
I+;
I
For a given composite system, the parameter P is
constant (eqn (44)). Q is mainly dependent on the (Ye
and (+bk(eqn (45)). The parameter R is dependent on
S and ak (eqn (47)). In other words, & mainly
depends on S and (Yk.Typical variation of ab with S
and (Ykis shown in Fig. 17. Figure 17 clearly indicates
that c is mainly influenced by the parameter S. c
initially increased rapidly with increasing S from O-2 to
2.5 above which it increased only marginally. This
behavior is expected since large S means a greater
number of yarns aligned close to the loading direction
and hence higher mean yarn strength. Small values of
S indicate that the yarn orientation distribution is
spread out and, hence, there is a lower mean yarn
strength.
Substituting eqn (55) in eqn (40), the knitted-fabric
composite tensile strength is given by:
_=[~]{
$[$r)+l))
(56)
(52)
S = 0.6
+ss=l.o
600
*S=l.6
0.01
(54)
gmb
[dTIQ - +?“I
+
0.015
(53)
b
[
gives:
g(g.b) =f(a)
g(ab)
& b-d1- G(6)])mb=~,,,= 0
(47)
S
R = [l - e-$1
[I-
17
Let (+bmbe the value of yarn stress, gb, which gives
gb[l - G((Tb)] its maximum Value, namely:
=/a* Re+
0
[‘f(cz)d”
composites
700
9 @b)
0.005
I b”
600
500
400
300
-0.005
1
200
300
400
600
500
q,,
700
600
ma
Fig. 16. Typical curves of function g(gJ.
900
0
2
4
6
Parameter,
S
6
Fig. 17. Typical variation of a, with parameters
10
S and (Ye.
S. Ramakrishna
18
where
u,
knitted-fabric
directions,
and CT, are the tensile strength of the
composite in the wale and course
respectively.
9 ANALYTICAL
9.1 Estimation
RESULTS
of volume
AND
fraction
DISCUSSION
of fibers, V,
Table 4 summarizes the volume fraction of fibers, V,,
of knitted-fabric
composites obtained from experiments and theoretical predictions. Vf predictions were
made from eqn (27). The good correlation between
the predicted and experimental results suggest that the
volume fraction of fibers of knitted-fabric composites
can be predicted by eqn (27).
V, can be increased in three ways: (1) by increasing
the linear density of yarn, D,; (2) by increasing the
stitch density of the knitted fabric, N; and (3) by
increasing the number of plies of the knitted fabric,
nk. Typical variations of V, with D,, nk and N are
shown in Fig. 18. Figure 18(a,b) suggests that for
constant N and t, V, increases linearly with increasing
D, and fik. This behavior can be expected from eqn
(27). However, V, increases non-linearly with increasing N (Fig. 18(c)). This can be understood
by
examining eqn (27). &, D, and other parameters in
the denominator
of eqn (27) are assumed to be
constant. Hence, V, is proportional to the product of
15, and N. An increase of N means smaller knit loops.
Smaller knit loops mean shorter L, (eqn (18)). In
other words, L, decreases with increasing N. The
inverse relationship between N and L,, results in a
non-linear variation of V, with increasing N. Figure 18
gives an approximate idea of the variation of V, with
D,, N and &. The maximum V, that can be achieved
in knitted-fabric composites is yet to be estimated, as
it is dependent on many other parameters such as yarn
jamming
conditions,
knitting
needle
size, knitting
of knitted
fabrics,
machine
gauge, compressibility
composite
fabrication
conditions,
etc. Efforts
are
being made to predict theoretical
maximum
V, that
can be achieved in knitted-fabric
composites.
Experimental
research
work
reported
in the literature
suggests
that a fiber volume
fraction
of 0.40 is
realistically
possible
in knitted-fabric
composites.
1000
2000
Number of
plies of
knitted
fabric,
nk
0.06
0.07
1
4
5m
Linear Density of Yarn, DY(Denier)
(b)
Number of Plies of Knitted Fabrics, nk
50
100
150
200
Stitch Density, N (No. of Loop&km’)
Fig. 18. Typical variation of volume fraction of fibers, V,,
with (a) linear density of the yarn, D,, (b) number of plies of
knitted fabric, IZ~,and (c) stitch density of knitted fabric, N.
Table 4. Volume fraction of fibers, V, in knitted-fabric composites
Specimen
thickness,
t (4
4OKl
3000
Knitted fabric details
Yarn linear
density,
Dy (Denier)
Fabric stitch
density, N
loops/4 cm’
1600
1600
20
20
Fibre content, vol.%
Experimental
9.5
32.33
Prediction
9.25
31.71
Tensile properties of fabric-reinforced
Hence, the elastic properties
were predicted
different fiber volume fractions less than 0.4.
for
9.2 Elastic and tensile strength properties
With the analytical procedure outlined in the flow
chart of Fig. 10, the elastic properties of knitted fabric
composites were computed for different V,. Figure 19
shows the variation of elastic properties with V,. In
general, all the elastic properties increased linearly
with increasing V,. Similar to the experimental results,
the predicted values indicate that the wale direction
elastic modulus is higher than the course direction
elastic modulus (Tables 1 and 5). The predicted elastic
moduli were approximately
20% higher than the
experimental results.
In the cross-over model used for estimating the
elastic properties of a knitted-fabric composite, the
projection of the central axis of the yarn on the plane
of the fabric was assumed to be composed of circular
arcs. This may be an idealized situation. It is often the
case that the loop geometry may not be circular and
suitable assumptions have to be made according to the
knitted structure. The preliminary procedure outlined
in this paper has to be further developed to consider
composites
the variations in the knit structure that greatly affect
the composite properties.
The wale and course tensile strengths of knittedfabric composites were computed from eqn (56). The
main assumptions are: (1) in the fiber content range
investigated,
the tensile failure mechanisms
of
knitted-fabric
composites are similar and (2) the
composite strength is determined
mainly by the
fracture strength of yarns bridging the fracture plane.
In the present study, experiments were carried out
using knitted-fabric composites with V, = 0.095 and
O-323. Hence, tensile strengths were predicted for
composites with similar V,. Figure 20 shows the
variation predicted tensile strength with the parameters S and &!k. The predicted strength is more
sensitive to the parameter S than (_yk.This behavior is
similar to the variation of mean yarn strength, K,
with S and (Yk(Fig. 17). The predicted tensile strength
increased rapidly with increasing S from 0.2 to 2
above which it increased only marginally. Larger S
means that a greater number of yarns are aligned
close to the loading direction and hence higher tensile
strength. Smaller S indicates that yarn orientation
distribution is spread out and hence lower tensile
strength. Table 5 summarizes the predicted tensile
(b)
(a)
20
VP
L---L
0
0
”
0. I
”
0.2
”
”
0.3
”
”
’
0.4
“‘1
J
0
0.5
0.1
0.2
0.3
0.4
0.5
Volume Fraction of Fibers, VI
Volume Fraction of Fibers, VI
(4
(cl
4
19
,
I
.l
.s
$
_m
8
z
.o’
a
0.6
-
0.4
-
0.2
J
0. I
0.2
0.3
0.4
Volume Fraction of Fibers, VI
Fig. 19. Typical
0.1
0.2
0.3
0.4
0.5
Volume Fraction of Fibers, VI
variation of (a) wale elastic modulus, E,, (b) course elastic modulus, E,, (c) shear modulus,
composite with volume fraction of fibers, r/;.
Poisson ratio, Y,,, of knitted-fabric
G,,, and (d)
S. Ramakrishna
(b)
(4
~___-_-_-_-
60
40
--_-_-_-_-_Experimental Tensile Strength
20
0
0
2
4
6
6
,‘,~“,,“““‘,““‘I
10
0
2
4
Parameter, S
6
6
10
Parameter, S
(c)
(d)
300
m
2
250
g
P
g
(I)
=Q)
E
I-”
300
250
200
150
Experimental Tensile Strength
100
P
.;
50
a’
0
4
6
a
10
0
2
Parameter, S
Fig. 20.
V, =
4
6
a
10
Parameter, S
Variation of predicted tensile strength of knitted-fabric composite with parameters 5’ and (Ye:(a) wale specimen with
0.093; (b) course specimen with V,= 0.093; (c) wale specimen with V, = 0.317; (d) course specimen with V, = 0.317.
strengths for different S in the range 0.2-10.0.
Comparing Tables 1 and 5, it can be said that
when S = 0.2 and 10.0 the predicted values indicate
lower and upper bounds of tensile strength of
knitted-fabric
composites. The limit of the lower
bound would depend on the parameter
S. It is
necessary to determine the parameter S precisely for
accurate estimation of composite tensile strength. For
this purpose, the experimental tensile strengths are
shown as dotted lines in Fig. 20. From Fig. 20, the
critical value of parameter S corresponding to which
the predicted strength matches with the experimental
result can be identified. In the case of single ply
Table 5. Predicted tensile properties
Number of plies
of knitted fabric,
nk
Fiber content,
vol.%
composite, both the wale and course predicted tensile
strengths match approximately
with respective experimental results when S = 1.0 (Fig. 20(a,b)). In the
case of a 4-ply composite, when S = 0.5 the wale and
course predicted strengths match approximately with
respective experimental strengths (Fig. 20(c,d)). In
other words, the critical value of parameter S is
independent of the testing direction with respect to
the knitted fabric. However,
it appears to be
dependent on the number of plies of knitted fabric
used for reinforcing the composite material. This may
be due to the mismatch between the adjacent plies of
knitted fabrics in composites reinforced with more
of plain knitted glass-fiber-fabric/epoxy
Elastic modulus, GPa
composites
Tensile strength, MPa
Wale
Course
Wale
Course
31.83 (S = 0.2)
60.00 (S = 1.0)
84.75 (S = 10.0)
109.1 (S = 0.2)
150.0 (S = 0.5)
290.6 (S = 10.0)
19.85 (S = 0.2)
36.00 (S = 1.0)
52.96 (S = 10.0)
68.2 (S = 0.2)
85.0 (S = 0.5)
181.6 (S = 10.0)
1
9.25
6.38
5,62
4
31.71
13.12
10.51
Tensile properties of fabric-reinforced
than one ply. Further detailed experiments
are
necessary to establish clearly the dependence
of
critical value of the parameter S on the variables such
as number of plies of knitted fabrics, knitted-fabric
stitch density, linear density of yarn, etc. This will
enable accurate prediction of tensile strengths of
knitted-fabric composites with different fiber volume
fractions.
In the present study, only the variation
of
orientation of bridging yarns was considered. The
fracture process of a set of bridging yarns would
depend on the yarn orientation distribution as well as
on the yarn strength distribution. The preliminary
procedure outlined here may be further modified by
considering the statistical nature of yarn strengths for
accurate determination of composite strength.
As in the case of the experimental results, the
predicted values (Tables 1 and 5) also suggest that
plain weft-knit fabric-reinforced
composites possess
better tensile strengths in the wale direction than in
the course direction. This is mainly due to the greater
number of yarns oriented in the wale direction than in
the course direction (eqn (38)). Both the wale and
course tensile strengths increased with increasing
volume fractions of fibers of knitted-fabric composites.
10 CONCLUSIONS
Preliminary methodologies for predicting the tensile
properties of plain weft-knit fabric-reinforced
composites have been established. The elastic properties
were predicted by using laminate theory and a
cross-over model which considers the orientation of
yarns and resin-rich regions in the composite. Tensile
strength properties were predicted by estimating the
fracture strength of yarns bridging the fracture plane.
The predicted tensile properties compare favorably
with the experimental
results. A more detailed
analysis is necessary to assess fully the applicability
and limitations of these methods.
The tensile properties of knitted-fabric composites
increased with increasing fiber content. It has been
shown that the fiber content of the composite can be
increased by increasing (a) linear density of yarn, (b)
stitch density of knitted fabric and (c) number of plies
of knitted fabrics. In general, weft knitted-fabric
composites display superior tensile properties in the
wale direction than in the course direction.
ACKNOWLEDGEMENTS
The authors are grateful to Prof. Z. Maekawa of
Kyoto Institute of Technology, Japan and Dr K. B.
Cheng of National Taipei Institute of Technology,
Taiwan for their useful suggestions and technical
composites
21
discussions. The authors would like to acknowledge
Mr N. K. Cuong of Kyoto Institute of technology for
his help in carrying out the experiments.
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