RESEARCH ON THE SHEAR BEHAVIOUR OF PLAIN WEAVE PREPREG AT
LATERAL COMPACTION STAGE
Lin Guo-chang, Wang You-shan, Du Xing-wen and Wang Chang-guo
Center for composite materials
Harbin Institute of Technology
Harbin, CH. 150001
ABSTRACT
An investigation of the shear behavior of plain weave prepreg at lateral compaction stage was undertaken by the picture-frame
test. A unit cell of weave prepreg is modeled and three moments were assumed on the unit cell, they are moment of shear
force that picture frame applies on the unit cell, the friction moment at crossover of yarns and the compaction moment between
the yarns. Through the moment balance, a relationship between the load and the shear rate was deduced. A plain sheet
shearing film resin model was established to obtain the friction moment at crossover. To get the compaction moment, a
constitutive equation of single prepreged yarn was deduced and a transverse viscosity coefficient equation was given. Also,
through an equivalent cross-section area model, the lateral contact thickness between the yarns was obtained. The
comparison between the experimental and theoretical results shows that the presented model can describe the shear behavior
of weave prepreg at lateral compaction stage well.
Introduction
The weave prepreg is composed by fabric and precured resin, which loads in the gaps between adjacent yarns and the gaps
between the fibers of the yarn. Because the resin is uncured, the yarns in the weave prepreg could rotate on their crossover
when it is under the shear state. The shear of the weave prepreg is divided into two stages. At the first stage, the gaps
between adjacent yarns decrease with the increase of the shear angle, and the precured resin in the gaps will be extruded
gradually. When the gaps disappeared completely, the second stage begins, which is called lateral compaction stage, and the
adjacent yarns compact laterally, which results in the increase of the load rapidly. McGuinness [1], Spencer [2] and Harrison [3]
has deduced the constitutive equations to describe the shear behavior of the weave prepreg at first shear stage, but the shear
behavior at the lateral compaction stage could not be described by their constitutive equations. The purpose of this paper is to
research the shear behavior of the weave prepreg at the lateral compaction stage and establish a model to predict the
relationship between the load and shear rate.
Experiment
Experimental research of the weave prepreg was performed by a picture-frame apparatus. The aim of this experiment is to
determine the relationship between shear angle and load for weave prepreg. Figure 1 is the picture-frame shear apparatus
which is a square four-bar linkage. The size of the internal square of the picture-frame apparatus is 135mm×135mm and outer
square is 165mm×165mm. The maximum shear angle of the apparatus is 60°.
Experiments were carried out at ambient temperatures, with force and displacement history recorded into a computer. The test
material is carbon plain weave prepreg which is comprised from T300 yarns and impregnated with epoxy resin. Some
parameters of the prepreg are listed in Table 1. The prepreg test size is 165mm×165mm, minus a 25mm radius quadrant cut
from each corner. This quadrant is required so that rotation of the hinges can occur freely.
The curve of load-displacement of weave prepreg is shown in Figure 2. Two shear stages are determined from the curve and
shear lock angle is the demarcation point. Prodromou [4] established a model to predict the shear lock angle. Using the
parameters in Table 1, the shear lock angle is determined by Prodromou’s model and the value is 32.3 ° , whose
corresponding displacement is 46mm.
Figure 1. Picture-frame shear apparatus
Table 1 Parameters of weave prepreg
2
Weight of prepreg (g/cm )
420±20
Side length of gap (mm)
0.31
Number of the fiber in yarns
3000
Width of yarn (mm)
1.7
Weight persent of resin(％)
54±4
Thickness of yarn (mm)
0.11
0.40
0.35
Load/kN
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
10
20
30
40
50
60
displacement/mm
Figure 2 Curve of the load-displacement for weave prepreg in the picture-frame test
Figure 3 shows the geometry of the picture frame experiments. In the test, the displacement δ and the load F are measured
at the crosshead. The relationship between the shear angle γ and the displacement δ is as follows:
δ
π
+ cos
2L
4
γ = π / 2 − 2θ
cos(θ ) =
(1)
(2)
From the law of instantaneous center of velocity, the angular velocity ω1 of CD-frame is
ω1 = ω2 =
δ
δ
=
AM 2 L sin(θ )
(3)
where
ω2 = the angular velocity of AD frame
δ = the velocity of the crosshead of the picture frame
Then the angular velocity of CB and AB is also ω1 because of the structure symmetry.
δ F
A
FS
FS
θ
δ
B
D
L
C
γ/2
Figure 3 Geometry of the picture frame experiment.
Mechanics model
The gap between adjacent yarns disappeared when the prepreg is at lateral compaction stage. The original resin in the gap
distributed on the two up-down sides of the prepreg and has little influence on the prepreg deformation, so these resins can be
ignored. Figure 4 shows the unit cell of the weave prepreg with side length S at lateral compaction stage. The w is the
current width of the yarn and will decrease as the yarns compress each other deeply. From the Figure 4, the current width of
the yarn w can be calculated as:
w=
S
cos(γ)
2
(4)
S
γ 2
w
γ 2
Figure 4 The unit cell of a plain prepreg weave fabric at lateral compacton stage
Fiure 5 shows all the moments on the unit cell. M P is the moment of shear force that picture frame applies on the unit cell,
M F is the friction moment at crossover of yarns and M C is the compaction moment, then the moment balance equation is
M P − ( M C + 4M F ) = 0
(5)
MP
MF
MC
MP
Figure 5 Moments on the unit cell
The sequent analysis is focus on the solution of these three moments. From the Figure 3, the component FS can be calculated
as follow:
FS =
F
2cos(θ )
(6)
It assumed that the stress and strain are distributed uniformly within the weave prepreg, the shear force applied on the unit cell
is FS S L , then the moment of the shear force can be calculate as follow:
MP = 2
Fs S
FS
⋅ l = 2 s ⋅ 2S cos(θ )sin(θ )
L
L
(7)
where
l = the vertical distance from the center of unit cell to boundary
Substituting Eqs. (1), (2) and (6) into Eq. (7) results in:
MP = 2
FS 2
FS 2
sin(2θ ) = 2
cos(γ)
L
L
(8)
To calculate the friction moment, we took the warp and weft yarns as plain sheet, and there is a film thickness of resin between
the crossovers of warp and weft yarns. The film thickness was taken as equal to the initial gap between fibres in yarns [3], then
the friction is from the crossovers shear of the resin film.
Figure 6(a) shows one warp and one weft yarn and their crossover with A is area of crossover. The angular velocity of these
two yarns is ω1 , because they are parallel to the picture frame. To simply the analysis, we assume that one yarn is fixed and
the other’s angular velocity is 2ω1 . Figure 6(b) shows quarter area of the crossover in polar coordinate, the velocity of one
point in the quarter area is VR = 2ω1 R , then the shear rate of film resin can be calculate as follow:
γ=
VR 2ω1 R
δR
=
=
d
d
dL sin(θ )
(9)
where
d = the thickness of the film resin
ω1
ω1
B
θ
θ
VR
B
A
0
C
R
0
(a)
C
φ
(b)
Figure 6 Crossover of weft and warp
The shaded area in the Figure 6 is RdRdφ , on which the resistant force is τRdRdφ and τ is the shear force of film resin, then
the friction moment is:
M F = 4∫
π/ 2
0
∫
w0 / 2cos( θ-φ )
0
τR 2 dRdφ
(10)
It assumed that the film resin is Newtonian fluid, and then its shear force is
τ = ηm γ
(11)
where ηm is the constant viscosity coefficient. Usually, the shear force of high polymer resin is not linear to shear rate,
whereas, if the shear rate is very small, the resin can be take as Newtonian fluid. In the test, the angular velocity of picture
frame is small, so it is reasonable to take the resin as Newtonian fluid. Substituting Eqs. (9) and (11) into Eq. (10) results in:
4
ηm δw
0
I
16dL sin(θ )
MF =
(12)
where
I =
∫
π/ 2
0
1
dφ
cos (θ-φ)
4
When the ajacent yarn contact each other, A uniformly distributed force along the yarns is then assumed to represent this
lateral compaction force, as seen in Figure 7. Then the compaction moment can be obtained as
M C = 2∫
(sin( γ ) +1) S / 2
(sin( γ ) −1) S / 2
FC xdx = S 2 FC sin(γ)
(13)
where
FC = the lateral compaction force
x
FC
y
0
dx
S
FC
Figure 7 Deformed unit cell with lateral compaction force applied on the tows
To get the lateral compaction force, the single prepreged yarn is assumed transversely isotropic material and its model is
shown in Figure 8. A is axial direction and T is transverse direction. The axis stiffness and transverse stiffness of the prepreg
yarn is expressed by radial viscosity coefficient ηA and transverse viscosity coefficient ηT , separately, because of the
existence of the resin in the internal gap between the fibres in the yarn. Also, it is assumed that the stress is the function of the
strain ratio, and then the constitutive equation of the prepreged yarn can be expressed as:
σ AA   ηA
 σ  = η
 TT   TA
ηAT  εAA 
ηT  εTT 
(14)
The lateral strain can be expressed as
ε TT =
w0 − w
w0
(15)
where
w0 = the initial width of the yarn in the prepreg
Substituting Eq.(4) into Eq.(15) and derivative with time, then the lateral strain ratio can be obtained as
εTT =
Sγ
sin(γ)
2w0
(16)
In the picture-frame test, the weave prepreg is under pure shear state, so εAA = 0 , then the lateral compaction force can be
obtained as
FC = σTT St = ηT
where
S 2 γ
t sin(γ)
2w0
(17)
t = the lateral contact thickness of yarn each other
A
T
w
Figure 8 A transversely isotropic single prepreged yarn
Many researchers has research the transverse viscosity coefficient [5-7] and here a equation is given to predict the transverse
viscosity coefficient:
ln
ηT
kVf
=
ηm 1 − Vf Vfmax
(18)
where
Vf
= the current volume content of fibres in the yarn
Vfmax = the maximum volume content of fibres in the yarn
=undetermined coefficient
k
The maximum volume content of fibre can be obtained as Vf max = π 2 3 by assumption of hexagonal close packing. The
current volume content of fibre is
Vf =
πd 2 N
4 Af
(19)
where
N = the current volume content of fibre in the yarn
Af = the cross-sectional area of the current yarn
Usually, the cross-section of the yarn is elliptical shape as shown in Figure 9(a). At the lateral compaction stage, the long
axis(the width of the yarn) and minor axis(the thickness of the yarn) of the ellipse will change, but it is still assumed that the
ross-section of the yarn is elliptical shape. The width of the yarn changed according to the Eq.(4) and thickness of yarn
changed minimally, so we surppose it as a constant t y .
0
It is difficult to determin the contaction thickness of the yarns by experiment, so the elliptical shape is equivalent to a
rectanglure shape with long side is w and short side is t , which is assumed that the thickness of the yarns is a constant. So
the contaction thickness can be calculated as
t = πt y0 4
(20)
ty0
t
w
w
(a)
(b)
Figure 9 Cross-section of the prepreged yarn (a) the real cross-section; (b) equivalent cross-section
Substituting Eq. (20) into Eq. (17) results in:
FC = ηT
πt y0 S 2 γ
8w0
sin(γ)
(21)
The cross-sectional area of the current yarn can be calculated as:
Af = wt =
πSt y0
8
cos(γ)
(22)
Substituting Eqs.(8), (12), (13), (21), (22) into Eq. (5) results in:
F = ηT
πt y0 SLγ
16w0 cos(γ)
sin 2 (γ) +
4
ηmδw
0
I
8dS 2 sin(θ ) cos(γ)
(23)
where ηT can be calculated from Eqs.(18) and (19) as
ηT = ηm exp(
kπ d 2 N
)
π St y cos(γ) − 2 3d 2 N
(24)
Results
There are three undetermined coefficients in the Eq.(23) and they are ηT , the film thickness d and ηm . According to the
Eq.(18), ηT is the function of the d , ηm and k , therefore, we get these three undetermined coefficients d , ηm and k to be
determined. It is difficult to get these coefficients by experiments, but they are can be fit from the results of the picture-frame
test. From the Eqs.(1) and (2), the shear rate can be calculated as
γ = δ Lsin( π 4- γ 2)
(25)
Set the shear rate γ as x-axis and the Load as y-axis, and then the experimental values at the lateral compaction stage are
obtained (See Figure 10). From the experimental values, the ηm =0.00552GPa·s, d =0.00757mm and k =156.29. Substituting
these three coefficients and these parameters δ =0.5mm/min, w0 =1.5mm, L =135mm, t y =0.11mm, S =3.63mm into Eq.(23)
0
results the theoretical values of load F vs. shear rate γ (See Figure 10). From the Figure 10, the presented model (Eq. (23))
could describe the shear behavior of weave prepreg at lateral compaction stage well.
0.44
experimental values
theoretical values
0.40
Load/kN
0.36
0.32
0.28
0.24
0.20
0.0066
0.0068
0.0070
0.0072
0.0074
-1
shear rate/s
Figure 10 Experimental curve and predicted curve of the rate of load vs. shear rate
Conclusions
A study of the shear behavior of plain weave prepreg was made by picture-frame test and two shear stages were determined.
At the first shear stage, the gaps between adjacent yarns decrease gradually. When the gaps disappeared, then the two
adjacent yarns compact each other, the second sheer stage called lateral compaction stage begins. This paper is focus on the
research of the shear behavior of prepreg at lateral compaction stage. A unit cell model at lateral compaction stage was
established and a relationship between the load and shear rate was deduced from the moment balance on the unit cell. A plain
sheet sheering film resin model was introduced to get the friction force on the crossover of the warp and weft yarn. An
equation was given to obtain the transverse viscosity coefficient. By the cross-section of yarn equivalent assumption, the
lateral contact thickness between the yarns was given. The comparison between the experimental and theoretical results
shows that the presented model can describe the shear behavior of weave prepreg at lateral compaction stage well.
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