Kauchuk i Rezina, No. 5, 2013, pp. 24–26
Investigation of impregnation fibrous materials in
pultrusion process of polymer composite materials
Grigoriev S.N., Krasnovskii A.N., and Kvachev K.V.
Moscow State Technological University «STANKIN», 127055, Moscow, Russia
Selected from the journal Vestnik MSTU STANKIN
Translated by P. Curtis
Summary
In this article the results of theoretical investigation
of fibers impregnation in manufacturing high filled
composite by pultrusion are represented. A criterion
is maximum intention of technological process. An
analytical investigation of intention depending on main
physical factors is proposed.
The main impregnation problem of fibrous materials is
replacement by liquid binding agent an air and moisture
in the fibers micropores. Insufficient fibers impregnation
leads to decreasing strength and untimely destruction
manufactures from polymer composite materials (PCM).
Formation air inclusions during impregnation process
essentially decreasing operational characteristics of
manufactures from PCM. Solution an impregnation
problem is fundamental to guarantee a stable quality
of composite materials [1–9].
There are different technological ways to facilitate
a penetration of binding agent in interfibrillar space
of stuff. To improve an impregnation of fibers we use
binding agents with low viscosity and surface tension
coefficient, we heat up for the same purpose. A decrease
velocity of a stuff through impregnating bath, which
is an increase of time impregnation, also increases a
content of binding agent in a stuff. At the same time
essentially decreases an output of a process because a
pultrusion velocity too much greater then capillary fiber
impregnation velocity. Increase a size of an impregnation
bath in motion direction has the same purpose, but leads
to overall dimensions increase and material capacity.
To improve strength of composite materials it is
necessary to improve essentially an effectiveness
© 2014 Smithers Information Ltd.
of impregnation process. For that we should set up
complementary conditions to simplify a binding agent
penetration in interfibrillar stuff area. An infill micropore
by liquid bending agent occur because of capillary effect.
One of the wide spread ways to improve an
impregnation of capillary-porous materials by liquid
medium is impregnation in vacuum. But combine pumping
of impregnation process with continuous fibers feeding is
technologically sufficient difficult. To travel fibers through
vacuum impregnating bath it is necessary to have gaps.
At the same time to make vacuum in impregnating bath
gaps and mobile connections must be excluded. Gaps
compacting and decrease in that case will lead to
abrasion and damage of fibers at entry and at exit of
impregnating bath. A stuff impregnation during extraction
by traveling through leading shoes and rotating rollers
(calendering) is one of the simple and effective solution.
An impregnation degree of fibres by binding agent is
improve, occluded gas are squeeze. But fibers as a
result of such elaboration are subject to extra effect of
leading cylinders (Figure 1).
Figure 1. Impregnating bath
T/59
The most effective way to improve binding agent
penetration in interfibrillar space of a stuff is a physical
modification of a binding agent by influence of ultrasonic
oscillations on fibers and binding agent [10, 11].
An ultrasonic impregnation mechanism is basis on
acoustocapillary effect. At the same time liquid bending
agent fill up micropores in fibrous material with high rate
and impregnation time decrease in dozens times. The
most important advantage of ultrasonic impregnation
is a capability of bending agent viscosity decrease in
several times. This ensured that bending agent penetration
make easier in interfibrillar stuff area. Under cavitational
stream in bending agent intense stuff degassing in
impregnation area occur. Ultrasonic degassing leads
to decrease occluded gas in manufacture and increase
their strength.
To define an intention of technological process we
use the following correlation [12]:
A = πR ρ (Pk + P2 − P1lk / (lk −1)) / 8ηlk Sm
3
1
3 '2 1
2σ
R
R + P∞ − P + Pa sinωt +
+ 4µ  = 0
R
2
ρ
R
'
3γ 

3 '2 1
2σ
R' 
2σ   R 0  
R + P∞ + Pa sinωt +
+ 4µ − P∞ +
  = 0
2
ρ
R
R 0   R  
R 


(6)
After substitution (4) in (6) and linearization we have:
x'' +

1 
2σ
2σ  
'

P
sinωt
+
x
+
4µx
+
3γ
P
+

a
 ∞ R  x  = 0
R0
ρR 02 

0  
(7)
We simplify getting equation, suppose that we can
neglect s, m:
3γP∞
ρR 0
2
x=−
Pa sinωt
ρR 02
(8)
General solution has the following form:
x = C1 sinω1t + C2 cos ω1t + C3 sinωt
(9)
where w1:
ω1 =
3γP∞
ρR 0
2
ω3 = −
Pa
2
ω ρR 02 + 3γP∞
(10)
Initial conditions are:
(2)
where stroke is a time derivative; P∞ – static pressure in
luquid; Pï – a pressure of saturated fallow; s – surface
tension of liquid; R – radius of bubble; R0 – initial bubble
radius; m – binding agent viscosity; Pa – ultrasonic
pressure amplitude; w – ultrasonic frequency; r – density
of binding agent.
We can’t integrate equation (2) in general case. In this
article an approximate analytic calculation is applied.
We suppose, that there is a series expansion parameter:
(3)
From the physical reason we suppose that bubble radius
will change slightly. We write the following correlation:
R(t) = R 0 (1+ x(t)) , where x(t) <<1
RR'' +
(1)
We propose, that during an ultrasonic impact in
technological volume appear a cavitation bubble, which
behaviour describe by the Rayleigh-Plesset equation:
Pa
= ε , where e <<1
P∞
Rewrite (2) in the form:
x'' +
where R1 – radius of interfibrillar capillary, ρ – density of
liquid, lk – length of capillary, η – viscosity, S – specific
surface of capillaries, m – stuff mass, P1 and Pk – at
entry and at exit pressure of capillary correspondingly,
P2 – extra pressure under ultrasound.
RR'' +
where g = cp/cn – correlation of specific heat for gas
and steam in bubble.
(4)
x(0) = 0, x' (0) = 0
(11)
We get:
C1 = −
C3ω
, C2 = 0
ω1
(12)
To define pressure in liquid binding agent we use the
following correlation [13]:
 R2R'' + 2RR' 2 R 4R' 2 
− 4  + P∞ + Pa sinωt
P(r,t) = ρ 
r
2r 

(13)
where Pr– liquid pressure at distance r from center of
cavity. Taking in account (6) and conditions s = 0, m =
0 (13) we rewrite in the form:
3γ
Let suppose, that proceeding process inside a bubble
is polytropic:
 RR' 2 R 4R' 2  RP  R 
 R
P(r,t) = (P∞ + Pa sinωt ) 1−  + ρ 
− 4 + ∞  0 
2r  r  R 
 r
 2r
Pn (t)V γ = const (5)
(14)
T/60
International Polymer Science and Technology, Vol. 41, No. 7, 2014
where:
 R (1+ x) 
(1+ x)R 03 x'2
P(r,t) = (P∞ + Pa sinωt ) 1− 0
−
+ρ
2r
r


−ρ
R 0 6 (1+ x)4 x' 2
2r
4
+
R 0P∞
r(1+ x)3γ−1
(15)
For analysis of maximum technological process we
suppose that we can change binding agent parameters
and parameters of outside impact. Consider the following
function:
ρ(B + P(r,t))
(16)
where:
B = Pk − P1lk / (lk −1)
Figure 2. Dependence impregnation intensity during 30sek
from frequency of ultrasonic oscillations when R: 1 – 3*10-8
m; 2 – 6*10-8 m; 3 – 9*10-8 m
(17)
Take into account (15) and the fact, that x << 1, we
integrate (16) by domain time [0,T], during which fiber
will be in impregnation bath. Suppose that capillary is
a cylinder with altitude a and radius R1. A bubble center
is on cylinder axis and on its base. We integrate by the
cylinder volume excluding the bubble volume and get
the following term of intention of a technological process
as a function of parameters w, Pa, T, r:



P
R 
π2R 03ρ BT(R1 − R 0 ) + P∞ T + a (1− cos ωT )  R1 − R 0 − R 0 ln 1  +
ω
R0 



R 3 R R 6  1
R
1 
+R 0P∞ T ln 1 + C32ω2ρ  0 ln 1 + 0  3 − 3  ×
R0
6  R1 R 0 
 2 R0

sin2ωT sin2ω1T sin(ω1 + ω)T sin(ω − ω1)T 
×  T +
+
−
−

4ω
4ω1
ω1 + ω
ω − ω1 



R π
R 
+ρ 2BT  a × arcsin 1 − R1 − R1 × ln 1  +
a 2
2a 


∞


R π
R
(2k)!
+2  a × arcsin 1 − R1 − R1 × ln 1 − R 0 ∑ k 2
×
2
a 2
2a

k=0 4 (k!) (2k +1) 
∞


P
(2k)!
× P∞ T + a (1− cos ωT )  + 2R 0P∞ T∑ k 2
+
ω
4
(k!)
(2k +1)2


k=0

sin2ωT sin2ω1T sin(ω1 + ω)T sin(ω − ω1)T 
+C32ω2ρ  T +
+
−
−
×
4ω1
ω1 + ω
ω − ω1 
4ω

 ∞
 R 6  π 2  1
(2k)!
 + 0 3  − + 
× R 03 ∑ k 2
2
4
(k!)
(2k
+1
)

 3R1  2 3  8ηaSm
k=0

(18)
This term show that intension of technological process is
an increasing function of Pa and T. We analyse a change
of impregnation intension from ultrasonic frequency,
density of binding agent and get the following results
(Figures 2 and 3):
© 2014 Smithers Information Ltd.
Figure 3. Dependence impregnation intensity from bending
agent density at 20kHz frequency of ultrasonic oscillations
when T: 1 – 10 s; 2 – 20 s; 3 – 30 s
It means that at the same bending agent density
impregnation intensity will increase with time. From
the correlation (18) follows that technological process
intensity depends on bending agent density as a quadratic
function and it properties define by parameters of system.
It is necessary to notice that change of parameters Pa,
R0, w almost doesn’t influence on behavior of graphics.
References
1. Krasnovskii A.N., Kazakov I.A., Kvachev K.V.,
Science foundation of continuous shaping
manufactures from polymer composite materials.
MSTU Stankin. 2012. – 65 p.
2. Grigoriev S.N., Krasnovskii A. N., Kazakov I.A.,
Kvachev K.V. An analytic definition of the border
polymerization line for axisymmetric composite
T/61
rods. Applied Composite Materials. DOI:
10.1007/s10443-013-9317-8.
8. Krasnovskii A.N., Kazakov I.A., Optimization of
die construction parameters to produce composite
rods in pultrusion way. Constructions from
composite materials. 4, 2012. 16-23.
3. S.N. Grigoriev, A.N. Krasnovskii, A study of the
process of continuous forming of nanocrystalline
composite powders. Metal Science and Heat
Treatment. 2012, 54(1-2). 13-16.
9. Krasnovskii A.N., Kazakov I.A. Investigation of
material deflected mode in pultrusion. Plastic
masses. 2012, 10 22-26.
4. S.N. Grigoriev, A.N. Krasnovskii, Distribution of
the density of material in the pressing channel in
continuous forming of nanocrystalline composite
powders. Metal Science and Heat Treatment.
2012, 54(3-4). 135-138.
5. S.N. Grigoriev, A.N. Krasnovskii, Study of the
Triboengineering Characteristics of Ultradispersed
Composite Powder Materials. Journal of Friction
and Wear, 2011, 32(3). 164-166.
6. A.N. Krasnovskii, Study of Circular Slip of
Superdispersed Powder Materials in Continuous
Formation Processes. Journal of Friction and
Wear, 2013, 34(3). 221-224.
7. Krasnovskii A.N., Kvachev K.V. Mathematical
modeling of pultrusion mechanical process in
manufactures from polymer composite materials.
Design. Materials. Technology. 5 (25), 2012.
78-81.
T/62
10.Hozin V.G., Karimov A.A., Cherevatskii A.M.,
Polanskii A.A., Murafa A.V., Murashov B.A.,
Pimenov N.V., Modification of epoxy composition
by ultrasonic //Mechanics of composite
materials. 4, 1984. 702-706.
11.Prohorenko P.P., Dezkunov N.V., Konovalov
G.E., Ultrasonic capillary effect. Minsk, Nayka i
tehnica, 1981. 135 p.
12.Materials of conference “Impregnation of
capillary-porous bodies”. 26-28 October, 2005.
Dnepropetrovsk, 65 p.
13.Bronin F.A., Investigation of cavitation destruction
and dispersion of solid bodies in ultrasonic field
high intension. Dissertation of the candidate of
technical sciences. 1966.
International Polymer Science and Technology, Vol. 41, No. 7, 2014