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THREE-DIMENSIONAL NUMERICAL FLOW SIMULATION OF RESIN TRANSFER
MOLDING PROCESS WITH DRAPING ANALYSIS
Sejin Han, *Mehran Ebrahimi, *Massimiliano Moruzzi, **Doug Kenik
Autodesk, 2353 North Triphammer Road, Ithaca, NY 14850, USA
* Autodesk, 210 King Street East, Toronto, ON M5A 1J7, Canada
**Autodesk, 203 S. 2nd St, Laramie, WY 82070, USA
Abstract
In this paper, the numerical flow simulation of
thermoset materials in resin transfer molding (RTM)
process with draping analysis is described. It gives an
introduction, theory and methods of analysis for the flow
and draping. Then, two example cases are shown. One is
a simple case where the accuracy of the solution can be
checked against analytical solutions. Another is the case
where the effect of using draping analysis in the RTM
flow simulation is shown. The simulation results in this
study are in good agreement with analytical solutions
where analytical solutions are available. They also verify
the significance of considering draping analysis results in
RTM flow simulations.
Introduction
The fiber composite materials have several
advantages over other materials (such as metals) in terms
of weight reduction, design flexibility, corrosion
resistance and reduced noise transmission [1]. The Resin
Transfer Molding (RTM) is one of the most popular
methods used in producing parts with fiber composite
materials. In the RTM process, the resin is forced to flow
through a cavity in which reinforcing preform (also called
fiber mat) is present [1]. The preforms (reinforcements)
are present in the mold as dry form. RTM process has the
advantages such as applicability to wide range of
components, and adjusting the fiber orientation to meet
the structural requirements and the use of lightweight
molds for the production [1]. This study is to analyze the
flow of thermoset materials in RTM process.
Nowadays, in the era of automation and advanced
manufacturing techniques, computer simulations imitating
real physical phenomena could help engineers to avoid
the time-consuming process of trial and error for creating
new designs. Consequently, developing more reliable
computer-aided-design (CAD) tools is a necessity in
today’s boom of new fabrication technologies. Thus,
exploiting numerical simulations, such as draping and
RTM simulations, in composite industries can extensively
reduce the manufacturing costs and accelerate the
processes.
In this study, three-dimensional numerical
simulations will be used to analyze the flow during RTM
processes. The three-dimensional analysis has the
following advantages compared to those using mid-plane
or dual-domain meshes. First, it can simulate anisotropy
of permeability in the thickness direction. Second, it can
handle complicated geometry cases better. The simulation
method developed in this study can also handle gravity
and venting effects.
The fiber mats undergo deformations as they are
shaped into complex geometries. The draping analysis is
utilized to analyze the deformation of the fiber mat [1].
For more accurate analysis of flow during RTM process,
the draping analysis needs to be used. In this study,
draping analysis is performed, and the result from the
draping analysis is used in the RTM flow simulation.
In the next sections, the simulation methods for the
flow and draping will be described, followed by some
example cases.
Governing Equations for Flow Analysis
To analyze the flow of resin during RTM process, we
need to solve the following set of equations [1, 2, 3].
First, we solve the continuity equation.
0)(
u
t
(1)
For the momentum equation, we solve the following
simplified equation (Darcy’s equation) [1].
pKu
1 (2)
In the above equation, u
is the superficial velocity, 𝜂
the viscosity, 𝐾 permeability and 𝑝 is the pressure [1].
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The superficial velocity is related to the macroscopic
velocity (�⃗�) through the following equation:
�⃗� = �⃗⃗�/𝜙 (3)
where 𝜙 is the porosity. The permeability 𝐾 is a tensor
which can be represented in 3D as components in 𝑥, 𝑦 and
𝑧 directions as below:
𝐾 = [
𝐾𝑥𝑥 𝐾𝑥𝑦 𝐾𝑥𝑧
𝐾𝑦𝑥 𝐾𝑦𝑦 𝐾𝑦𝑧
𝐾𝑧𝑥 𝐾𝑧𝑦 𝐾𝑧𝑧
] (4)
The temperature will be solved using a similar
equation as that for reactive molding simulation, but some
terms are modified to account for the presence of fiber
mat and resin in the system. In this equation, the term
without a subscript r is for the composite, and the term
with subscript 𝑟 is for the resin.
tHTkTuC
t
TC rprrp
)(
(5)
The curing will be analyzed by solving the following
equation due to Kamal [4]:
)1)(( 21 QQDt
D (6)
where the terms Q1 and Q2 are described below:
))((
11
1
T
E
eAQ
))((
22
2
T
E
eAQ
(7)
A three-dimensional finite element method is used to
solve the above set of equations as in [3]. Tetrahedral
elements are used in the current simulation.
Draping Analysis
In this paper, a hybrid finite element-geometric
algorithm for draping simulation of woven fabric
composites over a triangulated 3D surface is used. In this
algorithm, a fabric, before draping, is considered as a
group of square cells, and each cell is modeled as four
side-springs and two diagonal springs connected together
(Figure 1). These assumptions cause a trade-off between
accuracy and speed of draping simulation. However, the
generated results could be trustworthy for the majority of
cases, and if more accurate modeling is required, other
techniques exploiting pure finite element analysis (FEA)
[5]–[9] could be implemented with considerably higher
computational costs.
Figure 1: Fabric cells represented by six springs used in
draping analysis.
This draping technique is built upon the fact that the
fabric draping is optimum when the amount of distortions
(wrinkles) in it is minimum and the magnitude of forces
applied to each fabric node after draping, due to the
change in spring lengths, is zero.
After draping, depending on the surface geometry,
the fabric can wrinkle and cells may distort and no longer
be a perfect square. Shear angle, defined as 𝛾 by Equation
(8), at each fabric node is used as a representation of
wrinkles in the fabric at that location (Figure 2). In other
words, the higher the shear angle is, the more severe the
wrinkles are at that location. In the following equation, 𝛼′
is the angle between 2 sides after the draping.
𝛾 =𝜋
2− 𝛼′ (8)
(a) (b)
Figure 2: (a) undistorted cell before draping and (b)
distorted cell after draping.
Unlike FEA-based methods that require an initial flat
(2D) fabric as an input, the algorithm used in this paper
does not initiate the simulation from a 2D pattern. This is
another advantage of this method over these techniques.
The process begins from a given starting point (seed
point) and propagation direction (local fiber orientation at
the seed point). Other required inputs are:
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The triangulated surface to be covered by the
fabric
Fabric cell size along the sides
The ratio of spring constants along the sides and
diagonal directions. The higher this value, the
more resistant the fabric is against stretching
along the sides versus that along diagonals,
which results in shear.
Example Cases
In this section, two example cases of RTM will be
presented. One example is a simple case where draping
analysis is not needed. Another example is where draping
analysis is used.
1. Flat Plate Case
In the first example, a flat-plate case is used. The
geometry of the part used in this case is shown in
Figure 3. The part has dimension of 300 x 100 x 2
mm. The injection is done along one side of the part.
The fluid is Newtonian, and the flow is isothermal.
The viscosity of the fluid is constant at 0.4 Pa-sec. The
permeability is isotropic with a value of 1.0x10-9
[m2/sec]. The porosity is set at 0.5. This simple case is
chosen as an example because the simulation results
can be compared with analytical solutions.
Figure 3: Mesh used in the analysis of 3D RTM case.
The analysis was done for a constant flow rate
with and without venting analysis. Because of the
simple geometry, no draping analysis was done for
this case. For the case of constant flow rate without
venting analysis, the injection pressure (pressure at the
end of filling) can be calculated from the following
equation [1]:
𝑝 = 𝑄𝜂𝐿/𝐴𝐾 (9)
where 𝑄 is the flow rate, 𝜂 is the viscosity, , L is the flow
length, 𝐴 is the cross-sectional area and 𝐾 is the
permeability. A constant flow rate of 0.5 cm3/sec was
applied. By substituting the values for this case, we get
0.3 MPa as the injection pressure from the analytical
equation.
Figure 4 shows the pressure near the end of filling for
this case obtained from the simulation. This result shows
the pressure value of 0.299 MPa. The simulation result is
very close to the value from the analytical equation.
Figure 4: Pressure (in MPa) plot near the end of filling
obtained from a simulation for the case of a constant flow
rate of 0.5 cm3/sec.
Next, a venting analysis was done for the constant
flow rate of 0.5 cm3/sec. For this case, air pressure was
applied at the flow front during filling stage. The venting
pressure at the exit was set at -0.03 MPa. The negative
pressure means that a partial vacuum is applied. Figure 5
shows the venting pressure (which is the pressure in the
region not occupied by resin) obtained from the
simulation. Figure 6 is the pressure near the end of filling.
The value is 0.269 MPa. The analytical solution is 0.27
MPa. The calculation and the analytical solution show
good agreement.
Figure 5: Vent region pressure (in MPa) plot.
Figure 6: Pressure (in MPa) plot near the end of filling
obtained from a simulation for the case of a constant flow
rate of 0.5 cm3/sec with venting pressure of -0.03 MPa.
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2. Hemisphere Case
The second case used in this study is for a part
with hemispherical geometry. The geometry of the
part used in this case is shown in Figure 7(a). The part
has a diameter of 100 mm and thickness of 3 mm. The
injection is done at the center of the hemisphere. The
resin is Arotran Q6055 from Ashland Chemical, and
the fluid is non-Newtonian. The flow is non-
isothermal with the initial melt temperature of 30oC
and the mold temperature of 90oC. The injection time
is 10 sec. The permeability is anisotropic with the
value of 3.1x10-11
[m2/sec] in the first principal
direction, 1.9x10-11
[m2/sec] in the second principal
direction and 5.7x10-11
[m2/sec] in the third principal
direction. The porosity is set at 0.43. The first and the
second principal directions are along planar directions,
and the third principal direction is in the thickness
direction.
For this case, draping analysis is performed. The
results from the draping analysis are shown in Figure
7. Figure 7(a) shows the triangulated surface for the
draping analysis. Figure 7(b) shows the draping results
and shear angle distribution after draping. As can be seen
from this plot, the shear angle is high along some of
the edges. Figure 7(c) shows the plot of triangulated
surface and draped composite together.
Flow analysis has been performed using the principal
directions calculated from the draping analysis. As for the
effect of shear angle, analyses were performed with and
without the effect of shear angle. This is to see the effects
of shear angle on the flow results. The effect of shear
angle on flow has been considered in the analysis by
changing the fiber volume fraction and the permeability
as in [10].
Flow simulation results are shown in Figures 8 - 11.
Figure 8 shows the fiber-mat orientation calculated from
draping analysis. Figure 8(a) is for the first principal
direction, and Figure 8(b) is for the second principal
direction. The value is the component value in y-
direction. Figure 9 shows the shear angle calculated from
draping analysis. It shows that the maximum shear angle
is about 60 degrees.
Figure 10 shows the fill time results obtained from
the flow analysis. Figure 10(a) is with the effect of shear
angle, and 10(b) is without. As can be seen, the shear
angle changes the filling pattern where the shear angle is
high.
Figure 11 shows the pressure results near the end of
filling obtained from the flow analysis. Figure 11(a) is
with the effect of shear angle, and 11(b) is without. As
can be seen, including the effect of shear angle increases
the pressure for the current case.
(a)
(b)
(c)
Figure 7: (a) A triangulated hemisphere, (b) shear angle
distribution of the draped composite and (c) the
hemisphere and draped composite together.
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(a)
(b)
Figure 8: (a) Mat orientation in the first principal
direction and (b) the second principal direction.
Figure 9: Shear angle calculated from draping analysis.
(a)
(b)
Figure 10: (a) Fill time calculated with the effect of shear
angle and (b) without the effect of shear angle.
(a)
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(b)
Figure 11: (a) Pressure near the end of fill calculated with
the effect of shear angle and (b) without the effect of
shear angle.
Conclusion
In this paper, the method of solving the flow during
RTM process with draping analysis has been described.
Two test cases were used for the verification of RTM
simulation. The results from a case show good agreement
between the simulation and the analytical results. This
paper also showed the effect of including the draping
analysis in the RTM flow simulation.
Nomenclature
A Cross sectional area for the flow
, A1, A2, E1, E2 Curing kinetics parameters
PC Specific heat
K Permeability
k Thermal conductivity
L Flow length
p Pressure
Q Flow rate
T Temperature
t Time
u
Superficial velocity vector
�⃗� Macroscopic velocity vector
Degree of cure
’ Angle between 2 sides after the draping
e Expansivity
Porosity
Shear angle
Shear rate
Viscosity of the resin
Density
References
1. Rudd, Long, Kendall and Mangin, “Liquid molding
technologies”, Woodhead Publishing Limited,
1997.
2. L. Skartsis, J. L. Kardos, and B. Khomami, “Resin
Flow Through Fiber Beds During Composite
Manufacturing Processes. Part 1: Review of
Newtonian Flow Through Fiber Beds”, Pol. Eng.
Sci, 1992, vol. 32.
3. S. Han, F. Costa, P. Cook, S. Ray, “Three-
dimensional Simulation of Thermoset Molding
Applied to Semiconductor-chip Encapsulation”,
SPE ANTEC Paper, 2002.
4. Kamal, M. R. and Ryan, M. E., Chapter 4 of
Injection and Compression Molding Fundamentals,
A. I. Isayev (editor), Marcel Dekker, New York
(1987).
5. P. Boisse, Y. Aimène, A. Dogui, S. Dridi, S.
Gatouillat, N. Hamila, M. A. Khan, T. Mabrouki, F.
Morestin, and E. Vidal-Sallé, “Hypoelastic,
hyperelastic, discrete and semi-discrete approaches
for textile composite reinforcement forming”,
International journal of material forming, vol. 3 (2),
pp. 1229–1240, 2010.
6. P. Boisse, N. Hamila, F. Helenon, B. Hagege, and
J. Cao, “Different approaches for woven composite
reinforcement forming simulation”, International
journal of material forming, vol. 1(1), pp. 21–29,
2008.
7. N. Hamila and P. Boisse, “Simulations of textile
composite reinforcement draping using a new semi-
discrete three node finite element”, Composites
Part B: Engineering, vol. 39(6), pp. 999–1010,
2008.
8. X. Q. Peng and J. Cao, “A continuum mechanics-
based non-orthogonal constitutive model for woven
composite fabrics”, Composites part A: Applied
Science and manufacturing, vol. 36(6), pp. 859–
874, 2005..
9. Y. Aimene, E. Vidal-Salle, B. Hagege, F. Sidoroff,
and P. Boisse, “A Hyperelastic Approach for
Composite Reinforcement Large Deformation
Analysis”, Journal of Composite materials, vol.
44(1), pp. 5–26, 2010.
10. M. Grujicic, K.M. Chittajallu, Shawn Walsh,
“Effect of shear, compaction and nesting on
permeability of the orthogonal plain-weave fabric
preforms”, Materials Chemistry and Physics, 86
358–369, 2004.
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