Science Direct e85fed05-6cf5-20141026012456

ELSEVIER

Composites Science and Technology 57 (1997) 1-22

0 1997 Elsevier Science Limited

Printed in Northern Ireland. All rights reserved PII: SO266-3538(96)00098-X 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, Sakyo-ku, Kyoto 606, Japan

Kyoto Institute of Technology, Matsugasaki,

(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 Produc- tion 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 Linear density of unimpregnated yarn (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 LY orientation 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 fabric- reinforced composites have been investigated exten- sively. 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 composites 3

F

Woven -I

Biaxial Weaving

Triaxial Weaving

Braid i

Flat Braiding

Circular Braiding

--I Warp Knitting

Weft Knitting

-I

Mechanical Process

Chemical Process

LCombination Knitting + Weaving

Knitting + Nonwoven

- Stitched --I Lock Stitching

Chain Stitching

- Woven -t

Biaxial Weaving

Triaxial Weaving

Multiaxial Weaving

3D Preforms - Braid

i

2 Step Braiding

4 Step Braiding

Solid Braiding

- Knit -I

Warp Knitting

Weft Knitting

- Combination -1

Knitting + Weaving

Knitting + Stitching

Fig. 1. Various techniques of manufacturing textile fiber preforms.

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 d I

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).

COURSE c

(4 (b)

Fig. 2. Schematic diagrams of (a) weft- and (b) warp-knit fabrics.

4 S. Ramakrishna

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.


Weft Knitted Fabrics

7 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

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

I

Milanese 1

Single Stroke Double Stroke

Tricot

Bearded Needle

Single Needle Bed

Raschel

Latch Needle

Crochet

Double - Raschel - Needle Bed

Compound Needle

Double Guide Bar Triple Guide Bar Multiple Guide Bar

Pentihose Cut Presser Chain Raschel

Jacquard -1

Single Stroke

Double Stroke

Special Type

Knock-off Lap Tricot Chain Automat Tricot Swan Warp Tricot Guide Bar Transfer Sinker Loop

L Double Needle Bed - Simplex Machine

I Circular Milanese

Multi-target Jacquard Circular Needleless

Lace Type i

Less Guide Bars

Multiple Guide Bars

i

Tulle Machine Net Type Fish Net Machine

Power Net Machine

i

Weft Insert Special Type Co-We-Nit

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

i

Single Needle Bed Raschel Self-Forced Beard Needle Crochet

1

.Multi-target Fine Net Carpet Type Jacquard Crochet

Fig. 5. Broad classification of warp knitting machines.

assumption that the yarn orientation in dry knitted fabric is exactly same as that in the composite form. Mundenz3 and Postlez4 demonstrated that the knitted-fabric geometry is influenced by the medium in which it is kept. The yarn orientation in a 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

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


Latch Needle - Traverse Knitting Machine

Flat Bed

i

Milanorib Knitting Machine

Beard Needle Jacquard Knitting Machine

Semi-Jacquard Knitting Machine

Rohben Knitting Machine

r Single Needle Bed

i Circular Bed 1

Latch Needle_ Sinker Wheel with Jacquard Knitting Machine

1 Sinker Top Knitting Machine

Double Needle Bed -

- Dial & Cylinder Latch Needle

Double Cylinder -

V-bed Knitting Machine

Beard or Cowper Knitting Machine Double Needle Jersey Knitting Machine

Hosiery Knitting Machine

L Beard Needle

1

Loop Wheel Knitting Machine

Tompkin Knitting Machine

Silver Knitting Machine

i

Fraise Knitting Machine

Double Side Knitting Machine

Seal Knitting Machine Double Jersey Knitting Machine

Fig. 6. Broad classification of weft knitting machines.

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 knitted- fabric composites the yarns are oriented three- dimensionally.

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 knitted- fabric composites. Efforts were also made to develop analytical procedures for predicting tensile strength of knitted-fabric composites.

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.

8 S. Ramakrishna

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 thick- nesses 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

1.0

Strain, %

2.0

Fig. 8. Tensile stress/strain curves for single-ply knitted- fabric 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 non- linearity 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.


. z 7

7 3 !-

COURSE

Fig. 9. Optical photographs and schematic diagrams of tensile tested specimens: (a) wale; (b) course.

4 ANALYSIS APPROACH W can be determined experimentally. The procedure for estimation of d is outlined in Section 6.1. A

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

‘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

Table 1. Tensile properties of plain knitted glass-fiber-fabric/epoxy composites obtained from experiments

Number of plies Fiber content, of knitted fabric, vol. %

nk

Elastic modulus, GPa

Wale Course

Tensile strength, MPa

Wale Course

1 9.5 5.35 (0.33) 4.37 (0.07) 62-83 (7.1) 35.5 (2.21) 4 32.35 10.28 (O-35) 8-49 (0.21) 152.7 (9.5) 75.4 (4.5)

10 S. Ramakrishna

Vyf - Ef, Y f -j

E rn’ >,r

ANALYTICAL CROSS-OVER MODEL

\ UD Lamina Properties \ i E11,E2>G12 y12 ; L~_~~.__.__________.._________,

4 .--._--...______ ..__.__...________, Yam Properties

[Exlb, lEylb. [Gxylb, i

._.. !_%i!~.T “.??...........~

i ,---.._.__.___ ____._.._._____.

: Rule of Mixtures j

N=C*W

Ls “k I DY t Pf

Lack..____._____._____._______,

+ Composite E astic Properties

Ew, EC, Gwc, y WC

Fig. 10. Flow chart of the analysis method for predicting the elastic properties of knitted-fabric reinforced composite.

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 MODEL

5.1 Geometric 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

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


Fig. 11. Schematic representation of a unit cell of plain-knit fabric.

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 of the central axis in space are as follows:

1.

2.

3.

at 0 the height of the central axis above the plane of the fabric is zero, i.e.

z=O when 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 e=Oorcp

where s is the distance OP measured along the loop.

A simple function which satisfies all the above mentioned conditions is:

z = y (1 - cos 7&/p) (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

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 (4)

1 a=

4Wd sin cp (5)

and l/C is given by:

l/C=AC+CE=(2a-l)dcos$-2adcosrp

(6)

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

(2a - 1)d sin I++ = 2ad sin p (7)

From eqns (6) and (7):

l/C = (d(4a2 cos2 q - 4a + 1))d - 2ad cos cp (8)


From eqn (7):

Cc, = sin-’ ( 2a . - za - 1 lslIl q >

(9)

(10)

The parameter h can be determined as follows. The height of the central axis at H is:

z,,=F[l-cos+)]

Similarly, the height at J is:

z,=h$ l-cos;(+++) [ 1

(11)

(12)

But

ZJ - ZH = d

Rearranging eqns (II)-(13) gives:

(13)

h = [sin(T) sin(y)l-l (14)

To determine h we need to know the angles 4 and

12 S. Ramakrishna

+. $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

Hence, 4 is given by:

2a - 1 6 = coscl __ ( > 2a

(15)

(16)

5.2 Determination of loop length The length of yarn from 0 to Q, is given dy:

L-adcp (17)

Considering the symmetry of the loop, the length of yarn in one loop (MNOQR) is given by:

L, = 4adq (18)

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 three- dimensional space can be represented by the vector 6s in Fig. 12. (Y, /3 and y represent the orientations of the

Fig. 12. Representation of vector of yarn.

a short segment of

vector, as, with respect to the x, y and z axes, respectively. From the vector analysis principles, the Cartesian components of the vector are:

x n -x,-1 = sscosff

yn - y,_, = sscosp

2, - i&l = Gscosy

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:

p = cos -fyn -r) y=cos -fZn -Z-‘1

(20)

6 ESTIMATION OF YARN DIAMETER AND FIBER VOLUME FRACTION

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 fabric- reinforced composites K = 0.45, much smaller than the ideal packing conditions.

Af is given by:

D A,=2

c (22)

dPf

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):

D A,=--J-

CdP& (23)


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:

and V, is given by:

V,,, = K (25)

v,= nkD,L,CW

C,pfAt (26)

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>

Y r- X

Wale

t Course

(b)

- Y

Fig. 13. Schematic diagrams of (a) unit cell and (b) cross-over model.

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&

(27)

7 ANALYTICAL MODEL FOR ESTIMATING ELASTIC CONSTANTS

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 fiber- and 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 1.36 ---

( & &,

l-ti 1-v; . . -r

& - & ( -- Em 2 _ v&f em&, * l-%j 1-v; 1 ( l--y: l-v”, 1

+ 1 - 1.05-

E, (28)

1 1.36 -=

1 - 1*05*

G2 Ef Em + Em -

2(1+ Vf) 2(1+ v,) 2(1+ vrn)

l*osqf (Vf - Vm,( 2)

yt2 =

(

Ef Em

L fv,

--

1+$ 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

Poisson ratio

Glass fiber Epoxy resin

E,=74 V~ = 0.23 E, = 3.6 Vm = 0.35

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]

2 (sin4 (Y + cos4 CZ) 11

- ( 1 1 1 (29)

r + - - F II &I >

sin2 a cos2 LY 12 I

1 p= G,,(a) I

sin2 (Y ~0s’ (Y

+$-(sin’a+cos”a) 12 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:

where L is given by eqn (17).

(30)

The stiffness matrix of the curved yarn is:29

(31)

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) (32) n=l

where n represents the yarn number. Thus, the stiffness matrix of the yarns in the

cross-over model is:

(33)

The effective elastic properties of the cross-over yarns are given by:29

A1422 - 82

lExl’ = (L, + Lz)Az2

AIIAZ - A:2

[Ey’b = (L, + LJA, 1

[%y]b = 2

,vyVyx,b = $ yb [Cylh xh

A LGxy]b = (L, pL2)

(34)

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


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 = $ Yf

(35)

where V,,, and V, are given by eqns (25) and (27), respectively.

The effective elastic properties of the composite are given by:*’

J%, = [-Cldv,) + C%N - W

E, = PylbW + EJ(1 - v,)

Gvc = 1

r 1.36 I 1 - 1*05*1

1 Mb _ Em ’ E, 1

Lv + b&lb) w + YnJ w + YnJ J

(36)

where E, and E, are the elastic moduli of the composite in the wale and course directions, respectively. G,, and vWC represent the shear modulus and Poisson ratio of the knitted-fabric composite, respectively.

8 ANALYTICAL PROCEDURE FOR ESTIMATING TENSILE STRENGTH

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 - 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.

ab = Ul = 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 (Y with 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: sin: ~0s: sin2,cost gb =

L L+y+

~ “’

g: (72 62 4 I (42)

However, the tensile failure mechanisms observed where ul, u2 and z12 are the longitudinal, transverse from experiments indicate that the tensile strength of and shear strengths of unidirectional lamina, respec- the knitted-fabric composite mainly depends on the tively (Table 3).

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:

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:

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 = ~,T~d*[o,l w At

(40)

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:

16 S. Ramakrishna

Table 3. Tensile properties of unidirectional glass- fiber/epoxy lamina

Longitudinal Transverse strength, strength, u1 (Ml% c2 (MPa)

Shear strength, rLZ (MPa)

885 45 35

The typical variation of cb with (Y is shown in Fig. 14. (TV decreased 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 (Y value, 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 LY just before the failure of a yarn is a difficult task. It may be the case that different yarns orient at different CY with 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 (TV and (Y is given by:

ub = Pe-“” (43)

where P and Q are parameters of the exponential function and can be determined from the eqns (44) and (45). When CY = 0:

P = c71 (44)

We assume that all the yarn bundles are oriented in

1000

01

800 I

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, gives:

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

--_-S = 0.6 +s=i.o "k=lo" eSs1.6

0.5 %A I

0 1 1

-0.5.u I,,, iL--&--

0" 5" IO" 15c a

(b)

-a”k=F

q *ak= 10

-a =15

-0.2 C.I__LL / 1,, Li_l_LL!

0" 5" 10" 15" a

Fig. 15. Typical curves of function f(u).

a

Fig. 14. Typical variation of CT,, with a!


unity, we see that:

[‘f(cz)d” =/a* 0

Re+ da = - ; [e-set - I]= 1

S (47)

R = [l - e-$1

where R is dependent on the values of S and (Ye. Typical f(a) curves for different S and (Ye are 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 transforma- tion technique, g(r,,)da,, = f (cY)da

which, on rearranging, gives:

g(g.b) =f(a) 121 (48)


g(ab) = & e(Q-s)a (49)

From eqn (43):

-1 a=-ln 3 ( 1 Q P

(50)

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:

[I- G(o-Jl= /-“I g(gd dab fn

(1 - G(a,)] = & [dTIQ - +?“I

(52)

0.015

0.01

9 @b)

0.005

+ S = 0.6

+ss=l.o

*S=l.6

-0.005 200 300 400 500 600 700 600 900

q,, ma

Fig. 16. Typical curves of function g(gJ.

Let (+bm be the value of yarn stress, gb, which gives gb[l - G((Tb)] its maximum Value, namely:

& b-d1 - G(6)])mb=~,,, = 0 b

(53)

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 gmb (54)

The maximum mean strength, Z& of surviving yarns can be obtained by substituting the value of cbrn in

(+b[l - G(cb)]:

RP 1 ((Q/9+1)

K=- - Q

[ I I+;

(55)

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 (Yk is 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))

600

700

I b” 600

500

400

300 1

0 2 4 6 6 10

Parameter, S

Fig. 17. Typical variation of a, with parameters S and (Ye.

(56)

18 S. Ramakrishna

where u, and CT, are the tensile strength of the knitted-fabric composite in the wale and course directions, respectively.

9 ANALYTICAL RESULTS AND DISCUSSION

9.1 Estimation of volume fraction 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

machine gauge, compressibility of knitted fabrics,

composite fabrication conditions, etc. Efforts are being made to predict theoretical maximum V, that can be achieved in knitted-fabric composites. Experi-

mental research work reported in the literature suggests that a fiber volume fraction of 0.40 is realistically possible in knitted-fabric composites.

1000 2000 3000 4OKl 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 Number of Knitted fabric details Fibre content, vol.% thickness, plies of

t (4 knitted Yarn linear Fabric stitch Experimental Prediction fabric, density, density, N

nk Dy (Denier) loops/4 cm’

0.06 1 1600 20 9.5 9.25 0.07 4 1600 20 32.33 31.71


Hence, the elastic properties were predicted for the variations in the knit structure that greatly affect different fiber volume fractions less than 0.4. the composite properties.

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

(a) 20 VP

The wale and course tensile strengths of knitted- fabric 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)

0 L---L ” ” ” ” ” ” ’ “‘1 0 0. I 0.2 0.3 0.4 0.5

Volume Fraction of Fibers, VI

(cl

4 , I

0. I 0.2 0.3 0.4


0 0.1 0.2 0.3 0.4

J

0.5 Volume Fraction of Fibers, VI

(4

.s 0.6 - $ _m 8 0.4 - z

.o’ a 0.2

.l

J 0.1 0.2 0.3 0.4 0.5


Fig. 19. Typical variation of (a) wale elastic modulus, E,, (b) course elastic modulus, E,, (c) shear modulus, G,,, and (d) Poisson ratio, Y,,, of knitted-fabric composite with volume fraction of fibers, r/;.

S. Ramakrishna

(4 (b)

60 ~___-_-_-_-

40

20

0 0 2 4 6 6 10

Parameter, S

(c) (d)

300

250

--_-_-_-_-_- Experimental Tensile Strength

,‘,~“,,“““‘,““‘I 0 2 4 6 6 10

Parameter, S

300 m

2

g 250

P g 200 (I) Q) = 150 E I-” 100 Experimental Tensile Strength

P

.- ; 50

a’ 0

4 6 a 10 0 2 4 6 a 10

Parameter, S Parameter, S

Fig. 20. Variation of predicted tensile strength of knitted-fabric composite with parameters 5’ and (Ye: (a) wale specimen with V, = 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. composite, both the wale and course predicted tensile Comparing Tables 1 and 5, it can be said that strengths match approximately with respective ex- when S = 0.2 and 10.0 the predicted values indicate perimental results when S = 1.0 (Fig. 20(a,b)). In the lower and upper bounds of tensile strength of case of a 4-ply composite, when S = 0.5 the wale and knitted-fabric composites. The limit of the lower course predicted strengths match approximately with bound would depend on the parameter S. It is respective experimental strengths (Fig. 20(c,d)). In necessary to determine the parameter S precisely for other words, the critical value of parameter S is accurate estimation of composite tensile strength. For independent of the testing direction with respect to this purpose, the experimental tensile strengths are the knitted fabric. However, it appears to be shown as dotted lines in Fig. 20. From Fig. 20, the dependent on the number of plies of knitted fabric critical value of parameter S corresponding to which used for reinforcing the composite material. This may the predicted strength matches with the experimental be due to the mismatch between the adjacent plies of result can be identified. In the case of single ply knitted fabrics in composites reinforced with more

Table 5. Predicted tensile properties of plain knitted glass-fiber-fabric/epoxy composites

Number of plies Fiber content, of knitted fabric, vol.%

nk

Elastic modulus, GPa Tensile strength, MPa

Wale Course Wale Course

1 9.25 6.38 5,62 31.83 (S = 0.2) 19.85 (S = 0.2) 60.00 (S = 1.0) 36.00 (S = 1.0) 84.75 (S = 10.0) 52.96 (S = 10.0)

4 31.71 13.12 10.51 109.1 (S = 0.2) 68.2 (S = 0.2) 150.0 (S = 0.5) 85.0 (S = 0.5) 290.6 (S = 10.0) 181.6 (S = 10.0)


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

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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