# Modeling of Moisture Diffusion in Carbon Braided Composites

Authors: S. Laurenzi; T. Albrizio; M. Marchetti
Journal: International Journal of Aerospace Engineering
 (2008)

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## Abstract

In this study, we develop a methodology based on finite element analysis to predict the
weight gain of carbon braided composite materials exposed to moisture. The analysis was based
on the analogy between thermal conduction and diffusion processes, which allowed for a
commercial code for finite element analysis to be used. A detailed finite element model using a
repetitive unit cell (RUC) was developed both for bundle and carbon braided composites.
Conditioning tests were performed to estimate the diffusivity of both the resin and composite.
When comparing numerical and experimental results, it was observed that the procedure
introduces an average error of 20% and a maximum error of 31% if the RUC is assumed to be
isotropic. On the other hand, the average error does not exceed 10% and the maximum error is
less than 20% when the material is considered as orthotropic. The procedure is independent of
the particular fiber architecture and can be extended to other composites.


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## Body

## 1. Introduction

Moisture absorption is a critical factor in the
design of polymer composite materials applied to
aerospace structure, as
it affects the dimensional
stability of the pieces in long and short times.Even small quantities of moisture absorbed
from the environment can modify significantly the mechanical
and physical properties of composite materials
[1–4].
Moisture produces swelling, reduces the glass transition
temperature of the matrix, and affects matrix-dominated
properties such as interlaminar shear strength,
toughness to fracture, impact resistance, and possibly debonding
[1].
Other properties related to the life cycle of the structure,
such as resistance to fatigue, are affected as well:
the fatigue behavior of any composite is
influenced by the toughness of the
resin and its resistance to microcracking.The degradation of mechanical
properties of composites is strictly correlated
with the weight change due to moisture absorption.
Previous studies show that the weight change and the amount
of moisture at equilibrium depend on the environmental
conditions and the fiber architecture. Bao and Yee
investigated temperature effects on the moisture
absorption process for a BMI resin and its composites
[4]. Experimental results
verified that an increment of temperature corresponds to
an acceleration of the diffusion process and to a
decrease of the maximum amount of moisture at
equilibrium. The authors determined the diffusivity
by measuring the weight
variation as a function of time at different
temperatures, and found that the
activation energy for the reaction depends on the resin,
whereas the initial
diffusivity depends strongly on the presence of the fibers.
Some authors highlight a previously neglected problem,
which is the dependence of the diffusivity
on the humidity concentration in the material
[5]. Experimental results show
that the rate of absorption is faster during
the initial transition and then it
tends to slow down as it approaches equilibrium.
This behavior is justified by
the authors with the presence of voids and
fractures created during the curing
process. During the final stage, the increase
of the humidity content in the
composite induces swelling which causes the
closure of such spaces and
therefore a decrease in diffusivity.Zhou and Lucas [6]
conditioned a
unidirectional graphite/epoxy composite by immersing
it in distilled water at
various temperatures. They found that the
difference between theoretical and
experimental results increases with the
temperature of conditioning. The
authors individuated the limit temperatures,
and concluded that if
T/Tg≤0.25, the absorption process follows Fick’s law; if T/Tg≥0.5,
the anomalies are present in the mechanisms of
diffusion. Similar results were obtained from Wan et al.
for a three-axial braided composite
[7].
The complex fiber architecture and the presence of voids
inside the composite justify the divergence
between Fick's law and the
experimental data. The authors also investigated
the effects of external loads on
absorption and diffusion processes. The preloaded
material showed an acceleration
of the initial diffusion process with
respect to unloaded material and a
subsequent reduction of the mechanical
performances. On the other hand, an
external load applied to the material reduces
the absorption process and
improves the mechanical performance.
Initially, the external load produces an
increment of microcracks with an enhancement
of diffusivity; then the load
induces plasticity that blocks the microcracks
reducing the rate of absorption.Conditioning testing has been a
useful way to understand the mechanism of
moisture diffusion and the effects on
composite behavior and it is still the popular
technique for designing
composites. However, testing can require months of
experiments with obvious
costs. Prediction models for composite moisture
diffusion and long term
behavior would be powerful tools for the designer.
In the past, some efforts
have been made to develop numerical and analytical
methods to predict degradation
properties and moisture absorption.
These studies give an acceptable prediction
of the final moisture content and the swelling
coefficient for unidirectional
fiber composites [3].
However, it is difficult to build effective methods for
complex fiber architecture, due to the intrinsic
nature of dual-scale phenomenon. Kondo and Taki and
Lundgren and Gudmundson
focus their attention on the analogy and the importance of
the fiber distribution in the composite
[8, 9]. Tang et al. investigated the
effects of the tow microstructure on the moisture
diffusion for woven composites
and determined the cross-sectional diffusivity of a
single bundle adopting a 3D
unidirectional composite model with random
fiber distribution [10].Vaddadi
et al.
developed a methodology based on inverse analysis
techniques in order to estimate the parameters of
moisture diffusion processes
for unidirectional carbon reinforcements
[11, 12]. Both diffusion coefficient
and humidity content at equilibrium were determined
using few and simple
measurements of weight gain. This method used a
cyclic prediction process based
on the Kalman filter theory, which is particularly
effective in determining
variables under nonlinear conditions.
The authors adopted the repetitive unit
cell model for determining composite properties in
equilibrium conditions, but
this analysis cannot be extended to the transitory
period because the material cannot be analyzed uniformly.Here, we propose a numerical method
based on finite element analysis to predict the
weight gain as a function of
the exposure time to moisture for composites with
complex fiber architecture
such as a carbon braid. The analogy between
Fourier's and Fick's laws has been
adopted to use an available commercial code,
Femap 8-Maya TMG, which implements
the finite element analysis. To account for the moisture
absorption in a composite as a dual-scale phenomenon,
bundle- and carbon-braided RUCs were used
for the finite element analysis. Conditioning tests
were performed to validate
the numerical model. Future work will determine
stress and strain produced in the composite by the moisture
concentration field assessed using this model.


## 2. Theory

For most composites, such as carbon-glass/epoxy-polyester,
it is assumed that the fibers are impermeable and
that moisture diffuses only in
the matrix. This hypothesis is not valid for
composites with aramid fibers, which can absorb significant amount
of moisture.Considering that the matrix is homogeneous and isotropic, the
moisture mass diffusion through the composite
material can be described using Fick's law
(equation (1)).
The adoption of Fick's law implies
that the composite does not have any defects,
as voids and microcracks.Fick's law (equation (1))
[2, 3, 13]
is the equivalent of Fourier's law
(equation (2))
[13] for thermal diffusion, so
it is possible to use this analogy to study the
diffusion of moisture through a material(1)∂C∂t=D¯⋅(∂2C∂x2+∂2C∂y2+∂2C∂z2),(2)∂T∂t=k¯ρ⋅c⋅(∂2T∂x2+∂2T∂y2+∂2T∂z2).The concentrationC in
(1) is
equivalent to the temperature T in
(2),
whereas the diffusivity D is equivalent
to the thermal diffusivity α=k/ρ⋅c.
However, the correspondence between the boundary conditions
is not so straightforward.
The major problem is the discontinuity at the
interface between fiber and matrix.
In fact, while the contact temperature between
two materials can be assumed to
be the same, this is not the case for moisture
diffusion. Kondo and Taki and Lundgren and Gudmundson
[8, 9] proposed to
introduce the relative concentration
r,
which represents the ratio between moisture
amount at the instant t and the
moisture amount at equilibrium. This parameter can be
considered approximately
continuous through the interface and can be
used to set the boundary condition.
The other boundary condition can be obtained
from the flux, which is constant
through the interface for both
thermal and diffusion processes.
The moisture diffusion inside the material is slower than the heat flow by many orders of magnitude.The analogy above is used to describe
two different levels of moisture flow: inside the tow
and through the tow, that is, in the microscale and in
the mesoscale. Indeed, the textile structure is made of
fiber tows that are braided together to define
the macrostructure of the composite. The fiber tows are
formed by bundles of
well-aligned fibers, yet significant empty volume is
present in a tow. The
arrangement within the bundles represents the
microstructure of the material. We assumed that the empty
space is completely replaced by the matrix, and that the
adhesion between fiber and matrix is perfect. Therefore,
moisture diffuses through the macrostructure and through
the space among the fiber tows.


## 3. Materials and Methods

### 3.1. Materials

T300 biaxial carbon-braided 12 K was produced in a tabular
shape with braided angle of ± 45°.
This material is identified
with the trade name 144/11 by Eurocarbon and it is
shown in Figure 1.
The material is characterized by an average
density of 1.76 g/cm3, a weight per
unit area of 6.6 g/cm2 and a fiber
diameter of about 7 μm. The resin was polyester
from Aropol with an average density of
1.2 g/cm3, Tg of
69°C,
and cure cycle of 24 hours at room temperature.
The resin was degassed for 12 hours to eliminate entrapped
air bubbles before use. The laminates have a
uniform thickness of 3 mm, with fiber volume
fraction vF
of 0.5. The fiber volume
fraction was set to have an even
number of layers; this choice simplifies the
finite element model
of the RUC-braided composite.Procedure for cutting a preform to a given
braided angle using a PVC mandrel.
(a)(b)(c)(d)(e)(f)(g)


### 3.2. Methods

To manufacture the laminate, the flat layers of the stack
sequence were obtained from the tubular braid.
Particular attention was used to cut the braided sleeves
while maintaining the braided angle. Changes in the braided
angle can generate stretching or compression along
the fiber, resulting in fiber deformation.
To avoid such defects, the tubular braid was put on a PVC
cylindrical mandrel, of a certain diameter, and fixed
with tape along the edges. The preparation
sequence is reported in Figure1. The layers were then
positioned inside the metallic mould for RTM process.Composite laminates were manufactured using
resin transfer molding (RTM) process to obtain
finished surfaces and to minimize the formation of internal
microvoids. The apparatus for the fabrication
process is shown in Figure2: a pressure
vessel pumps the resin inside the mold via the
injection gates; the resin forces the air out
from the vent, while saturating the preform. After the
mould is filled, gate and vent were clamped to
permit the cure of the laminate. When the polymerization
of the resin is complete, the mould was open to remove
the final piece.Figure 2
Experimental apparatus for RTM
process.After curing, the absence of dry spots, porosity,
resin pocket, and other defects, classified in the standard
ASTMD 2562, was verified by the visual inspection.
The void content was calculated according to the standard
ASTM D 2734. The highest
percentage of void content was 0.1%.
The fiber volume fraction was confirmed to be0.5±0.01 using the method
resin burn-off according to
the ASTM D 2584. Figure 3
shows a carbon-braided/polyester laminate used to obtain
the specimens for the conditioning test.
Samples were cut from the composite laminate in agreement
with the ASTM D 6856-03 and ASTM D 5229/D 5229M-92 standards.
According to these standards, the specimen dimensions must be at
least twice the thickness of the unit cell in the
direction of the moisture flow, and the test must be
performed with a relative humidity RH of 85% and
a conditioning temperature lower than Tg
-25°C.
The used polyester has a Tg of 69°C, and conditioning
tests were performed at the maximum allowable
temperature of 35°C to accelerate the slow dynamic
of the moisture diffusion process.
As visible in Figure 4,
the lateral surfaces of the samples were insulated
in order to avoid the direct exposure to the moisture.
This precaution was adopted in order
to simplify the boundary condition in the FEM analysis.Figure 3
12 K carbon-braided/polyester laminate.Figure 4
Running conditioning test with a particular
of internal chamber.A CHALLENGE-CH 250 climatic chamber was used, and the weight
changes of the samples were documented with a
precision balance (±0.1 mg). Each sample
was weighted every 24 hours.
Figure4 shows the
climatic chamber used to perform the conditioning tests; also
shown is the internal chamber during test,
with resin and composite specimens. Neat resin specimens were
obtained injecting the uncured resin into the RTM mold,
which allowed to have a controlled thickness.
The conditioning test of neat resin was used to
calculate the resin diffusivity DM and the relative weight
gain at equilibrium
weM.
DM and
weM
were calculated
from (3)
[3]
implementing an iterative procedure in MATLAB based on the
minimization of the error between
calculated weight gain and the relative weight gain
wS(t) after exposure to moisture(3)wC(t)=(weM−wi)⋅{1−exp⌊−7.3⋅(DM⋅t/h2)0.75⌋}+wi,(4)C=We−W0W0⋅ρ⋅100.
Once the relative weight gain at equilibrium is
determined, the moisture concentration CM
can be calculated using
(4), where
W0 is the initial weight.


## 3.1. Materials

T300 biaxial carbon-braided 12 K was produced in a tabular
shape with braided angle of ± 45°.
This material is identified
with the trade name 144/11 by Eurocarbon and it is
shown in Figure 1.
The material is characterized by an average
density of 1.76 g/cm3, a weight per
unit area of 6.6 g/cm2 and a fiber
diameter of about 7 μm. The resin was polyester
from Aropol with an average density of
1.2 g/cm3, Tg of
69°C,
and cure cycle of 24 hours at room temperature.
The resin was degassed for 12 hours to eliminate entrapped
air bubbles before use. The laminates have a
uniform thickness of 3 mm, with fiber volume
fraction vF
of 0.5. The fiber volume
fraction was set to have an even
number of layers; this choice simplifies the
finite element model
of the RUC-braided composite.Procedure for cutting a preform to a given
braided angle using a PVC mandrel.
(a)(b)(c)(d)(e)(f)(g)


## 3.2. Methods

To manufacture the laminate, the flat layers of the stack
sequence were obtained from the tubular braid.
Particular attention was used to cut the braided sleeves
while maintaining the braided angle. Changes in the braided
angle can generate stretching or compression along
the fiber, resulting in fiber deformation.
To avoid such defects, the tubular braid was put on a PVC
cylindrical mandrel, of a certain diameter, and fixed
with tape along the edges. The preparation
sequence is reported in Figure1. The layers were then
positioned inside the metallic mould for RTM process.Composite laminates were manufactured using
resin transfer molding (RTM) process to obtain
finished surfaces and to minimize the formation of internal
microvoids. The apparatus for the fabrication
process is shown in Figure2: a pressure
vessel pumps the resin inside the mold via the
injection gates; the resin forces the air out
from the vent, while saturating the preform. After the
mould is filled, gate and vent were clamped to
permit the cure of the laminate. When the polymerization
of the resin is complete, the mould was open to remove
the final piece.Figure 2
Experimental apparatus for RTM
process.After curing, the absence of dry spots, porosity,
resin pocket, and other defects, classified in the standard
ASTMD 2562, was verified by the visual inspection.
The void content was calculated according to the standard
ASTM D 2734. The highest
percentage of void content was 0.1%.
The fiber volume fraction was confirmed to be0.5±0.01 using the method
resin burn-off according to
the ASTM D 2584. Figure 3
shows a carbon-braided/polyester laminate used to obtain
the specimens for the conditioning test.
Samples were cut from the composite laminate in agreement
with the ASTM D 6856-03 and ASTM D 5229/D 5229M-92 standards.
According to these standards, the specimen dimensions must be at
least twice the thickness of the unit cell in the
direction of the moisture flow, and the test must be
performed with a relative humidity RH of 85% and
a conditioning temperature lower than Tg
-25°C.
The used polyester has a Tg of 69°C, and conditioning
tests were performed at the maximum allowable
temperature of 35°C to accelerate the slow dynamic
of the moisture diffusion process.
As visible in Figure 4,
the lateral surfaces of the samples were insulated
in order to avoid the direct exposure to the moisture.
This precaution was adopted in order
to simplify the boundary condition in the FEM analysis.Figure 3
12 K carbon-braided/polyester laminate.Figure 4
Running conditioning test with a particular
of internal chamber.A CHALLENGE-CH 250 climatic chamber was used, and the weight
changes of the samples were documented with a
precision balance (±0.1 mg). Each sample
was weighted every 24 hours.
Figure4 shows the
climatic chamber used to perform the conditioning tests; also
shown is the internal chamber during test,
with resin and composite specimens. Neat resin specimens were
obtained injecting the uncured resin into the RTM mold,
which allowed to have a controlled thickness.
The conditioning test of neat resin was used to
calculate the resin diffusivity DM and the relative weight
gain at equilibrium
weM.
DM and
weM
were calculated
from (3)
[3]
implementing an iterative procedure in MATLAB based on the
minimization of the error between
calculated weight gain and the relative weight gain
wS(t) after exposure to moisture(3)wC(t)=(weM−wi)⋅{1−exp⌊−7.3⋅(DM⋅t/h2)0.75⌋}+wi,(4)C=We−W0W0⋅ρ⋅100.
Once the relative weight gain at equilibrium is
determined, the moisture concentration CM
can be calculated using
(4), where
W0 is the initial weight.


## 4. Experimental Results

The experimental results of the conditioning test for matrix
and braided composite are shown in Figure5. The graphs represent
the trend of the percentage weight gain as a function
of the square root of time. The regression analysis
shows that the weight gain is well fitted
with cubic interpolation, as confirmed by the coefficient
of determination R2. The constants a1,
a2, and a3 used in the
regression analysis are, respectively, 0.0236,
0.0077, and −0.0009.Figure 5
Weight gain as a function of
the time of exposure to moisture for resin and composite.Moreover, the trend relative to the transient period is
closely linear (Figure5).
The slope of the linear curve is related to
the moisture diffusion, which increases with the slope.
Since the moisture absorbed approaches the
moisture equilibrium content, the local slope
decreases until approximately zero at the
saturation point, where no significant weight
variation is recorded. The saturation point is reached
after 600 hours and  the percentage weight
gain at this time is 0.195%. Moisture
diffuses more easily in the specimens of neat
resin than in composite ones; therefore, the amount
of moisture at equilibrium is reached earlier in the
resin with respect to the composite.
The different response is due to the presence of fiber
reinforcements. As described above, the matrix
parameters used in the numerical model were calculated
from experimental data: DM=5.182
10−9 cm2s−1, weM=0.782%,
CeM=9.386
10−3 g/cm3.


## 5. Finite Element Analysis

The finite element numerical analysis was performed using the
commercial FEMAP TMG software, adopting the
analogy between the heat conduction and the moisture
diffusion processes. The software solves the thermal conduction
problem using a FEM/control volume approach, determining the
temperature at the calculation points. The calculation
points are the gravity center, where boundary conditions
are applied, and the boundaries of an element.
The position of the boundary elements depends on
the type of element. For 3D elements,
the boundary elements are positioned in the middle of volume
surfaces, whereas for 2D elements, the boundaries are
in the middle of the surfaces edge. Once the temperature
matrix is solved at these points, temperatures are
interpolated to the mesh nodes. The temperature
field inside the elements is assumed to be linear.The numerical model shows some complexities due to the
different length scales of the diffusion process
inside the composite. To take into account this aspect,
two different models have been developed.
We assumed that fiber and matrix are perfectly
bonded and that the carbon fibers are impermeable
to the moisture, so their diffusivity is zero.
With this hypothesis, moisture flows only through
the resin. Assuming the bundle as a unidirectional
composite, the diffusivity parallel to the bundle direction
is equivalent to the matrix diffusivity, while
the diffusivity normal to the bundle direction is
coincident with the diffusivity into the bundle, that is,
the matrix diffusivity in microscale. This parameter
was determined performing the thermal analysis of
the RUC bundle model and then setting the material
properties of the carbon-braided RUC. Simulations
were run after defining the RUC models, the
material properties, and the numerical convergence.In order to reduce the computational cost due to the large
number of elements of carbon-braided RUC, we
considered the tow as an isotropic material with diffusivity
equal to the diffusivity normal to the tow directionsDFn. This assumption was supported
considering that the moisture flow is directed along
the thickness of the material, the z-axis. In fact, the
angle between the z-axis and the fibers direction varies
from 90° to 81°.The postprocessing was carried out using two programs: the
first one imports the mass of the elements in two matrices,
for tow and resin, respectively, and calculates the
total mass. The second program receives in input the
vector of the temperature of the elements and the
matrices determined from the first program.
The mass variation is calculated by multiplying the
temperature of every element forWe and the
corresponding mass. Then, the relative mass gain is
determined by dividing such variation for the
starting mass. These two programs have been
implemented in MATLAB.


### 5.1. Model of the RUC Bundle

The particular geometry of the bundle permits
to consider only the cross section, reducing
the analysis to a two-dimensional problem. The first step to
build the model is to determine the area that represents the RUC.
In order to do that, the fibers distribution,
the number of fiber filament, and
the correspondent fiber volume fraction have
to be defined. Previous works on unidirectional
fibers have shown that a random distribution gives closer
agreement with the experimental data than a regular one.
This can be explained
considering that in a random distribution
there are zones with high fiber
density that creates a barrier for moisture
diffusion. On the other hand, a regular
fiber distribution is characterized by a
uniform distance among the fibers, which
generates paths for the moisture flux.Previous works determined that the minimum number of fiber
filaments necessary to describe the behavior
of the bundle is eight [9],
and that the fiber volume fraction for bundle
of 12 K carbon braided at ±45° is in
the range of 0.7 to 0.78
[10, 11].
Assuming that the values for vfF is 0.747 and that the fiber diameter is 7 μm,
the section areas of the fiber filaments
and the total area of the RUC ARUC
can be easily calculated. The moisture absorption
process relative to the bundle can be simulated using
a quadratic two-dimensional model having edge l =
20.3 μm, with eight randomly
distributed fibers. The fibers are modeled as
voids because the diffusivity and the moisture content
at equilibrium are approximately zero, so these regions
can be considered as invisible to the diffusion process.
The model was meshed with plate elements as shown in
Figure 6; the total
number of elements was 4839 and the total
number of nodes was 5381. The material properties
of the matrix used for the analogy are reported
in Table 1.Table 1
Material properties associated with the
bundle RUC used for the analogy.
Thermal conductivitykM=DM⋅CeR=4.864⋅10−15Specific
heatcM=CeMρM=7.822⋅10−3Matrix densityρM=1.2⋅10−12Figure 6
Two-dimensional model of the bundle with
eight fibers randomly distributed.


### 5.2. Model of the RUC-Braided Composite

The RUC-braided model is three-dimensional and presents some
geometric complexities due to the braided architecture.
In order to create a true representation of the material,
bundles are considered having lenticular sections.
Each section is built by the intersection of two
circumferences. Figure7
shows the geometrical construction and the relative
parameters used in the model; when the thickness and
the length of the tows are known, the tow geometry can be
completely defined.Figure 7
Construction and geometrical
parameters of the tow section.The thickness can be determined by observing that the real
composite is a 3 mm thick laminate with
4 layers. Each layer has an average thickness
of 0.75 mm and two tows overlap for every layer.
Therefore, the thickness of the lenticular
sectionhf is 0.375 mm.The determination of the bundle length is
more complex and is based on the repetitive unit cell
shown in Figure8.
Here, we show the top and bottom surfaces of a braided
RUC schematization, which for simplicity is rotated of
45° with respect to the material axes; the tows in
the y-direction are in blue, whereas the tows in the
x-direction are in yellow. The tows were numbered
to define their respective position on the top
and bottom surface. Indicating with l
the tow width, the RUC volume VRUC, and the tows total
volume VfTOT can be
expressed as a function of this; l is
then calculated by setting the fiber volume fraction of the composite vF=0.5, as illustrated in the
equation
(5)vF=VfTOTVCRU=VFTOT⋅vfFVCRU=0.5.
Equation(5)
was numerically solved using an iterative procedure
implemented in MATLAB, after expressing
VRUC and
VfTOT as function of the geometric factor f (Figure 7) and considering that the tow total volume
in the carbon-braided RUC can be obtained by multiplying the total length
of
the tows dfTOT
(equation (6))
with the cross-section area:
(6)dfTOT=∑i=l6di=8⋅(l+l2+0.3752).
The results are hf=10.759 mm
and l = 4 mm, which are in agreement
with the experimental observations.Figure 8
Top and bottom view of the RUC with tows in they-direction
(blue) and in the x-direction (yellow).
Numbers indicate the tows.The dimensions of a basic RUC are
5.657 mm × 11.318 mm ×
0.375 mm. The RUC is not unique, but to minimize
computational costs, we considered the RUC having the
smallest dimensions. In any case, a single RUC is
sufficient to investigate the moisture diffusion process
in stationary conditions, yet not during the
transitory regime. In fact, moisture already diffuses in
the adjacent RUC before reaching the equilibrium:
in this case, the absorption occurs through the thickness.
Considering that the composite is manufactured with four
layers, a model based on four overlapping RUCs should be adopted.
The symmetry of the laminate and the boundary conditions
of the medium plane reduce the number of RUCs to
two overlapping unitary cells. The model was meshed using 22480 elements and 6233 nodes, using the three-dimensional solid elements CHEXA and CTETRA. Figure9 shows the
carbon-braided RUC model used for the analysis: the matrix
zones are in blue, while the yellow indicates the fiber zones.Figure 9
View of the carbon-braided RUC with
matrix elements (blue) and fiber tow (yellow).After defining the model, it is necessary to determine the
thermal properties of the materials using the analogy
between absorption and heat conduction; these properties
are reported in Tables2
and  3.Table 2
Material properties
associated with the diffusion process for carbon-braided RUC.
Diffusivity [mm2 /h]Weight gain %Concentration to theDensity [g/mm3]to the equilibriumequilibrium [g/mm3]Matrix1.866·10−30.7829.386·10−61.200·10−3Fiber bundleDfn=1.307·10−40.1472.375·10−61.618·10−3Dfp=1.866·10−3Table 3
Material properties of carbon-braided
RUC from the heat-diffusion analogy.
Thermal conductivitySpecific heatDensity
[g/mm3]MatrixK=D·Ce=1.751·10−8c=Ce/ρ=7.822·10−31.200·10−3Fiber bundlekFn=3.104·10−101.468·10−31.618·10−3kFp=1.751·10−8


## 5.1. Model of the RUC Bundle

The particular geometry of the bundle permits
to consider only the cross section, reducing
the analysis to a two-dimensional problem. The first step to
build the model is to determine the area that represents the RUC.
In order to do that, the fibers distribution,
the number of fiber filament, and
the correspondent fiber volume fraction have
to be defined. Previous works on unidirectional
fibers have shown that a random distribution gives closer
agreement with the experimental data than a regular one.
This can be explained
considering that in a random distribution
there are zones with high fiber
density that creates a barrier for moisture
diffusion. On the other hand, a regular
fiber distribution is characterized by a
uniform distance among the fibers, which
generates paths for the moisture flux.Previous works determined that the minimum number of fiber
filaments necessary to describe the behavior
of the bundle is eight [9],
and that the fiber volume fraction for bundle
of 12 K carbon braided at ±45° is in
the range of 0.7 to 0.78
[10, 11].
Assuming that the values for vfF is 0.747 and that the fiber diameter is 7 μm,
the section areas of the fiber filaments
and the total area of the RUC ARUC
can be easily calculated. The moisture absorption
process relative to the bundle can be simulated using
a quadratic two-dimensional model having edge l =
20.3 μm, with eight randomly
distributed fibers. The fibers are modeled as
voids because the diffusivity and the moisture content
at equilibrium are approximately zero, so these regions
can be considered as invisible to the diffusion process.
The model was meshed with plate elements as shown in
Figure 6; the total
number of elements was 4839 and the total
number of nodes was 5381. The material properties
of the matrix used for the analogy are reported
in Table 1.Table 1
Material properties associated with the
bundle RUC used for the analogy.
Thermal conductivitykM=DM⋅CeR=4.864⋅10−15Specific
heatcM=CeMρM=7.822⋅10−3Matrix densityρM=1.2⋅10−12Figure 6
Two-dimensional model of the bundle with
eight fibers randomly distributed.


## 5.2. Model of the RUC-Braided Composite

The RUC-braided model is three-dimensional and presents some
geometric complexities due to the braided architecture.
In order to create a true representation of the material,
bundles are considered having lenticular sections.
Each section is built by the intersection of two
circumferences. Figure7
shows the geometrical construction and the relative
parameters used in the model; when the thickness and
the length of the tows are known, the tow geometry can be
completely defined.Figure 7
Construction and geometrical
parameters of the tow section.The thickness can be determined by observing that the real
composite is a 3 mm thick laminate with
4 layers. Each layer has an average thickness
of 0.75 mm and two tows overlap for every layer.
Therefore, the thickness of the lenticular
sectionhf is 0.375 mm.The determination of the bundle length is
more complex and is based on the repetitive unit cell
shown in Figure8.
Here, we show the top and bottom surfaces of a braided
RUC schematization, which for simplicity is rotated of
45° with respect to the material axes; the tows in
the y-direction are in blue, whereas the tows in the
x-direction are in yellow. The tows were numbered
to define their respective position on the top
and bottom surface. Indicating with l
the tow width, the RUC volume VRUC, and the tows total
volume VfTOT can be
expressed as a function of this; l is
then calculated by setting the fiber volume fraction of the composite vF=0.5, as illustrated in the
equation
(5)vF=VfTOTVCRU=VFTOT⋅vfFVCRU=0.5.
Equation(5)
was numerically solved using an iterative procedure
implemented in MATLAB, after expressing
VRUC and
VfTOT as function of the geometric factor f (Figure 7) and considering that the tow total volume
in the carbon-braided RUC can be obtained by multiplying the total length
of
the tows dfTOT
(equation (6))
with the cross-section area:
(6)dfTOT=∑i=l6di=8⋅(l+l2+0.3752).
The results are hf=10.759 mm
and l = 4 mm, which are in agreement
with the experimental observations.Figure 8
Top and bottom view of the RUC with tows in they-direction
(blue) and in the x-direction (yellow).
Numbers indicate the tows.The dimensions of a basic RUC are
5.657 mm × 11.318 mm ×
0.375 mm. The RUC is not unique, but to minimize
computational costs, we considered the RUC having the
smallest dimensions. In any case, a single RUC is
sufficient to investigate the moisture diffusion process
in stationary conditions, yet not during the
transitory regime. In fact, moisture already diffuses in
the adjacent RUC before reaching the equilibrium:
in this case, the absorption occurs through the thickness.
Considering that the composite is manufactured with four
layers, a model based on four overlapping RUCs should be adopted.
The symmetry of the laminate and the boundary conditions
of the medium plane reduce the number of RUCs to
two overlapping unitary cells. The model was meshed using 22480 elements and 6233 nodes, using the three-dimensional solid elements CHEXA and CTETRA. Figure9 shows the
carbon-braided RUC model used for the analysis: the matrix
zones are in blue, while the yellow indicates the fiber zones.Figure 9
View of the carbon-braided RUC with
matrix elements (blue) and fiber tow (yellow).After defining the model, it is necessary to determine the
thermal properties of the materials using the analogy
between absorption and heat conduction; these properties
are reported in Tables2
and  3.Table 2
Material properties
associated with the diffusion process for carbon-braided RUC.
Diffusivity [mm2 /h]Weight gain %Concentration to theDensity [g/mm3]to the equilibriumequilibrium [g/mm3]Matrix1.866·10−30.7829.386·10−61.200·10−3Fiber bundleDfn=1.307·10−40.1472.375·10−61.618·10−3Dfp=1.866·10−3Table 3
Material properties of carbon-braided
RUC from the heat-diffusion analogy.
Thermal conductivitySpecific heatDensity
[g/mm3]MatrixK=D·Ce=1.751·10−8c=Ce/ρ=7.822·10−31.200·10−3Fiber bundlekFn=3.104·10−101.468·10−31.618·10−3kFp=1.751·10−8


## 6. Numerical Results

### 6.1. FEM Analysis of Moisture Diffusion in Bundle RUC

The simulation of the diffusion process of the
bundle model allows calculating the diffusivity normal
to the bundle direction. Figure10 shows the convergence
analysis performed to estimate the accuracy of the
numerical results. The numerical convergence was
considered as reached when the variation of the average
temperature as a function of the time step was less or equal
to 10−3. This variation corresponds
to 75000 time steps.
Figure 11 shows the
simulation at four different time steps from the
initial time to the start of stationary process.
The transitory stage of the diffusion process
is considered as finished at 3000 seconds,
when the temperature, which is equivalent to the
relative concentration for the heat-mass analogy,
reached a constant value. Subsequently, the material
can be assumed as saturated and the equilibrium
parameters can be determined. The different colors
represent the different values of the thermal field,
which represents the distribution of the moisture
concentration normalized with
respect to the moisture concentration at equilibrium.Figure 10
Temporal convergence
analysis for the bundle RUC.Simulation of the diffusion process of a bundle
RUC. The image shows the simulation at four
different time steps: 0 second, 100 seconds,
1000 seconds, and 3000 seconds. It is assumed
that the transitory regime ends at 3000 seconds.
The different colors represent the different moisture
concentration normalized with respect to the
moisture concentration at equilibrium.
(a)(b)(c)(d)Results of postprocess performed to
estimate the weight gain as a function of the square
root of the time are shown in Figure12.Figure 12
Weight gain as a function of the time for bundle
RUC by postprocessing of numerical results
from the FEM analysis.The tow diffusivity normal to the tow direction,
which is equal to the bundle diffusivity normal to the
bundle direction, was calculated by MATLAB(7)DFn=3.63⋅10−2μm2s.


### 6.2. FEM Analysis of Moisture Diffusion in Carbon-Braided RUC

As in the previous analysis, a convergence study
was also made. Convergence was reached when the variation
of the average temperature was smaller or
equal to 10−3, which corresponds
to time step 2000 (Figure 13).
The numerical results show that assuming the tow as
isotropic material gives a large gradient
of concentration in x-z plane
(Figure 14). A large gradient with
a large value of
DFx-y indicates
that the approximation of isotropic
bundles is not valid. The diffusion coefficient
DFx-y is a linear combination between
DFn and DFp.Figure 13
Temporal convergence analysis for the
carbon-braided RUC as orthotropic material.Figure 14
Section of carbon-braided RUC. The circle
highlights the gradient of concentration
through the thickness of the braided RUC considering
the tows as isotropic material.A finite element analysis was performed considering
orthotropic bundles with parameters of absorption as reported
in Tables2 and
3.
This analysis shows that the moisture reaches
equilibrium in 90 days
(Figure 15).Figure 15
Simulation of the diffusion process
in the RUC of a carbon-braided composite. The image shows the simulation at two different time steps: at 1 day (left) and at 90 days (right). It is assumed that the transitory regime
ends after 90 days.


## 6.1. FEM Analysis of Moisture Diffusion in Bundle RUC

The simulation of the diffusion process of the
bundle model allows calculating the diffusivity normal
to the bundle direction. Figure10 shows the convergence
analysis performed to estimate the accuracy of the
numerical results. The numerical convergence was
considered as reached when the variation of the average
temperature as a function of the time step was less or equal
to 10−3. This variation corresponds
to 75000 time steps.
Figure 11 shows the
simulation at four different time steps from the
initial time to the start of stationary process.
The transitory stage of the diffusion process
is considered as finished at 3000 seconds,
when the temperature, which is equivalent to the
relative concentration for the heat-mass analogy,
reached a constant value. Subsequently, the material
can be assumed as saturated and the equilibrium
parameters can be determined. The different colors
represent the different values of the thermal field,
which represents the distribution of the moisture
concentration normalized with
respect to the moisture concentration at equilibrium.Figure 10
Temporal convergence
analysis for the bundle RUC.Simulation of the diffusion process of a bundle
RUC. The image shows the simulation at four
different time steps: 0 second, 100 seconds,
1000 seconds, and 3000 seconds. It is assumed
that the transitory regime ends at 3000 seconds.
The different colors represent the different moisture
concentration normalized with respect to the
moisture concentration at equilibrium.
(a)(b)(c)(d)Results of postprocess performed to
estimate the weight gain as a function of the square
root of the time are shown in Figure12.Figure 12
Weight gain as a function of the time for bundle
RUC by postprocessing of numerical results
from the FEM analysis.The tow diffusivity normal to the tow direction,
which is equal to the bundle diffusivity normal to the
bundle direction, was calculated by MATLAB(7)DFn=3.63⋅10−2μm2s.


## 6.2. FEM Analysis of Moisture Diffusion in Carbon-Braided RUC

As in the previous analysis, a convergence study
was also made. Convergence was reached when the variation
of the average temperature was smaller or
equal to 10−3, which corresponds
to time step 2000 (Figure 13).
The numerical results show that assuming the tow as
isotropic material gives a large gradient
of concentration in x-z plane
(Figure 14). A large gradient with
a large value of
DFx-y indicates
that the approximation of isotropic
bundles is not valid. The diffusion coefficient
DFx-y is a linear combination between
DFn and DFp.Figure 13
Temporal convergence analysis for the
carbon-braided RUC as orthotropic material.Figure 14
Section of carbon-braided RUC. The circle
highlights the gradient of concentration
through the thickness of the braided RUC considering
the tows as isotropic material.A finite element analysis was performed considering
orthotropic bundles with parameters of absorption as reported
in Tables2 and
3.
This analysis shows that the moisture reaches
equilibrium in 90 days
(Figure 15).Figure 15
Simulation of the diffusion process
in the RUC of a carbon-braided composite. The image shows the simulation at two different time steps: at 1 day (left) and at 90 days (right). It is assumed that the transitory regime
ends after 90 days.


## 7. Discussion

The finite element numerical analysis was performed to
simulate the moisture diffusion process in a
carbon-braided structure.
Figure16
shows experimental and numerical results for the
relative weight gain in percentage as a
function of the exposure time. There is good
agreement between the experimental trend and the
numerical evaluation. Moreover, we observed
that the initial content of moisture affects the result
considerably. Considering wi
equal to zero (equation (3)), the value of
DM is 54%
larger than the experimental result.Figure 16
Numerical and experimental results of relative
weight gain as a function of time.The first step of the
analysis was to determine the diffusion
coefficients and the moisture content of the fiber bundle.
The presence and the
arrangement of the fibers were taken
into account in the moisture absorption study. After the
determination of the diffusion parameters,
these were introduced in the repetitive unit
cell of the carbon-braided composite. The analysis of
the moisture diffusion in the RUC of carbon-braided composite
was approached in two different ways.
In the first one, the composite material was considered
isotropic with diffusivityDFn. In this case, the moisture flow is
orthogonal to the direction of the fibers. However, the analysis
highlighted that the concentration gradient in the
x-y plane,
due to the nonhomogeneity of the material,
could not be neglected. Therefore,
the composite was modeled as an orthotropic material.
The second analysis confirmed such hypothesis.
Figure 16 shows that
there is a large difference between the trends obtained
assuming the tow as isotropic or
orthotropic and the experimental results.
The isotropic hypothesis introduces an
average error of 20% and a maximum
error of 31%. On the other hand,
the average error does not exceed 10% and
the maximum error is less than 20% when
considering the material as orthotropic.The percentage weight gain is higher in the orthotropic model than in the isotropic model, due to the
concentration gradients in thex-y plane.
For concentration gradient, the diffusivity is a linear
combination of the diffusivities in the
x-y plane. Therefore,
in isotropic materials, the diffusivity is constant
in all directions, whereas in orthotropic materials,
the diffusivity is one order magnitude higher than the other
direction. The approximation introduced with the assumption of
isotropic material reduces the accuracy considerably.
However, the orthotropic model improves the accuracy
of the results, but it increases the computational
time cost of approximately four times with respect
to the isotropic case.


## 8. Conclusions

The methodology developed in this study allows simulating
the process of moisture diffusion in a composite
braided material with sufficient approximation
starting from a limited set of
experimental data. We developed an analysis that reduces
drastically the number of experimental tests necessary
to understand the behavior of a composite
material in moisture environment. At the same time,
the analysis decreases considerably
the design costs and times. A commercial
code for FEM analysis was used to estimate
in few hours and with good approximation the
weight gain as a function of time,
which normally requires several months of experiments.



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*Source: 294681-2007-11-19.xml*