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A Finite Element Approach for Study of Wave Attenuation Characteristics of
Epoxy Polymer Composite
Conference Paper · November 2018
DOI: 10.1115/IMECE2018-87873
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Proceedings of the ASME 2018 International Mechanical Engineering Congress and Exposition
IMECE/CIE 2018
November 9-15, 2018, Pittsburgh, USA
IMECE2018/MESA-87873
A FINITE ELEMENT APPROACH FOR STUDY OF WAVE ATTENUATION
CHARACTERISTICS OF EPOXY POLYMER COMPOSITE
Shank S. Kulkarni
Alireza Tabarraei∗
Department of Mechanical Engineering and Engineering Science
University of North Carolina at Charlotte
Department of Mechanical Engineering and Engineering Science
Charlotte, NC 28223, USA
University of North Carolina at Charlotte
Email: skulka17@uncc.edu
Charlotte, NC 28223, USA
Email: atabarra@uncc.edu
Pratik P. Ghag
Department of Mechanical Engineering and Engineering Science
University of North Carolina at Charlotte
Charlotte, NC 28223, USA
Email: pghag@uncc.edu
ABSTRACT
The properties of the inclusions, viz. size, shape, and distribution significantly affect macroscopic properties of a polymer composite. Finite element (FE) modeling provides a viable
approach for investigating the effects of the inclusions on the
macroscopic properties of the polymer composite. In this paper, finite element method is used to investigate ultrasonic wave
propagation in polymer matrix composite with a dispersed phase
of inclusions. The finite element models are made up of three
phases; viz. the polymer matrix, inclusions (micro constituent),
and interphase zones between the inclusions and the polymer
matrix. The analysis is performed on a three dimensional finite
element model and the attenuation characteristics of ultrasonic
longitudinal waves in the matrix are evaluated. The attenuation
in polymer composite is investigated by changing the size, volume fraction of inclusions, and addition of interphase layer. The
effect of loading frequency of the wave on the attenuation characteristics is also studied by varying the frequency in the range
of 1 - 4 MHz.
Results of the test revealed that higher volume fraction of
inclusions gave higher attenuation in the polymer composite as
∗ Alireza
Tabarraei.
compared to the lower volume fraction model. Smaller size of inclusions are preferred over larger size as they give higher wave
attenuation. It was found that the attenuation characteristics of
the polymer composite are better at higher frequencies as compared to lower frequencies. It is also concluded that the interphase later plays a significant role in the attenuation characteristics of the composite.
NOMENCLATURE
PMC Polymer Matrix Composite
RSA Random Sequential Absorption
FEM Finite Element Method
XFEM Extended Finite Element Method
G shear modulus
K bulk modulus
ρ density
ν Poisson’s ratio
α attenuation coefficient
τi relaxation time
G0 instantaneous shear modulus
E Young’s modulus
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Copyright c 2018 by ASME
c volume fraction
n number of inclusions
f loading frequency
P pressure
t time
h·iJ ensemble average from J samples
Liu et al. [1] studied the wave propagation in polymer composite using the extended finite element method (XFEM). They
studied the effect of volume fraction and loading frequency on
the wave attenuation characteristics of the polymer composite
with spherical and cylindrical inclusions. In the case of spherical
inclusions, the results were in good agreement with the experimental and analytical results for lower volume fractions, while at
higher volume fractions only at lower frequencies. In the case of
cylindrical inclusions, the XFEM simulations predict that maximum attenuation occurs when the cylinders are oriented in the
direction of loading. The XFEM results also indicates that attenuation increases by increase in the loading frequency. One
shortcoming is, effect of interphase layer was not considered in
this paper.
INTRODUCTION
Polymer materials such as polyurea and epoxy display remarkable attenuation characteristics which makes them excellent candidates for coating structures to protect them from damages incur by compressive waves induced by blast. To increase
the mechanical strength and stiffness of such polymers and for
the purpose of improving their attenuation characteristics, fillers
are added to polymer matrix. The propagation of periodic and
transient waves is complicated in polymer matrix composites
(PMCs) due to the scattering of waves occurring at the material
interfaces and because of the dissipation in the matrix [1]. The
shape, size, and distribution of the inclusions in the polymer matrix [1] can significantly impact the wave propagation characteristics of polymer composites. The effective utilization of polymer composites for the purpose of wave attenuation requires a
thorough understanding of the factors that impact the wave propagation and dissipation in polymer composites.
Biwa et al. [2] carried out numerical analyses of longitudinal wave attenuation in a glass-epoxy composite and a rubberparticle toughened poly(methyl methacrylate) (PMMA) blend.
Kim [3] conducted a comparative study on eight existing theoretical models to create some benchmark results for wave propagation in two-dimensional composite materials. The models include Waterman and Truell [4], Llyod and Berry [5], Varadan et
al. [6], Kanaun and Levin [7], Sabina and Willis [8], Kim [9],
Beltzer and Brauner [10] and Yang and Mal [11]. Kim conducted numerical calculations for different composites by varying the material properties, volume concentration of the microinclusions and the loading frequency. Kim concluded that the
Llyod and Berry [5] model is more accurate than the Waterman
and Truell [4] model if the point scattering approximation is relevant. Kim further reported that the Kim [9] and Kanaun and
Levin [7] models predict values close to each other possibly because they are based on a common hypothesis and failure did not
occur in the cases considered.
Kinra et al. [12] studied ultrasonic wave propagation through
an epoxy-glass composite by conducting experiments in a frequency range of 0.3 - 5 MHz by varying the volume fraction of
glass from 8.6 % to 53 %. The composite studied in the experiment consists of spheres of glass dispersed in a random homogeneous manner in an epoxy. Kinra reported that higher volume
fractions of the inclusion yields better attenuation at lower frequencies.
Another important aspect that plays a vital role in the wave
attenuation characteristics of the polymer composite is the interphase region. Interphase region is formed due to cross-linking
or crystallization between the polymer matrix and the inclusion.
Interphase region can be formed due to mechanical imperfections, unreacted polymer components, fiber treatments, restricted
macromolecular mobility due to the fiber surface, and other inconsistencies [13–15]. The thickness of the interphase region
depends upon the material bonding properties [16]. Many efforts
have been made to study the characteristics of the interphase.
Techniques like nanoindentation, nano-scratch and atomic force
microscopy (AFM) are used to study the properties and thickness
of the interphase [15, 17–19]. An explicit study of the interphase
region is essential as the region is considered to be one of the
weakest regions in the polymer composite. It has been reported
that the structural integrity and modes of failure for the polymer
composites are dependent on the properties and the thickness of
the interphase [16].
In this paper, finite element method is used to investigate
the ultrasonic wave propagation in epoxy polymer matrix composite. The polymer composite has a dispersed phase of glass
inclusions intended to improve the mechanical properties. Since
the macroscopic properties of a composite are affected by the
inclusion size, shape and distribution, the effect these parameters on the wave attenuation characteristics of polymer composite
is evaluated. Multiple iterations were performed in order to remove the effect of randomness of inclusions position. All FEM
simulations are performed with the help of python scripting in
ABAQUS [20]. Furthermore, the role of interphase is studied
along with the effect of loading frequency on the attenuation of
waves in polymer composite. Study of the attenuation properties
of composite is not done yet with considering all these factors
together. The goal is to gain fundamental insights on designing
polymer composite with high attenuation capabilities. This will
help engineers and designers to simulate new material before its
production saving the cost associated with prototyping.
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TABLE 1: PRONY SERIES PARAMETERS FOR EPOXY MATRIX .
gi
τi
0.0738
463.4
0.1470
0.06407
0.3134
0.0001163
0.3786
7.321e-7
1 mm
[1]
Z
0.5 mm
MATERIAL MODEL
0.1 Epoxy matrix
The epoxy matrix is considered to be a viscoelastic material. Viscoelasticity is the property of materials which exhibit
both viscous and elastic characteristics when undergoing deformation. Viscoelastic materials show both stress relaxation and
creep deformation [21]. Their behavior is strain rate dependent
and the loading history affect the response of the material to current loading. Prony series is used to define the stress relaxation
modulus G(t) of epoxy as
!
−t τ
,
1 − ∑ gi 1 − e i
2 mm
FIGURE 1: Schematic of the finite element model.
FINITE ELEMENT MODEL
This study is conducted using commercial finite element
software ABAQUS [20]. The finite element analysis are conducted using three dimensional finite element models. The inclusions are dispersed randomly in the polymer matrix. A viscoelastic material model is used to model the polymer matrix
and linear elastic material model is used for inclusions.
A schematic diagram of the model used in the analysis is
shown in Fig. 1. The element data for all the time steps are
recorded for all the simulations at z = 2.0 mm and z = 0.5 mm
as shown in the Fig. 1. The time history plot for stress σzz (in
the loading direction) is generated using the history data at these
surfaces. The amplitude of the wave reduces as it travels through
the matrix due to scattering and reflections from the inclusions
and material damping of the polymer matrix. The attenuation
coefficient (α) is calculated using [1]
n
G(t) = G0
P = Posin(2πft)
Y
(1)
i=1
where, G0 is the instantaneous shear modulus and gi and τi are
material constants defined in Table 1 taken from [1]. The instantaneous shear modulus G0 of epoxy considered in this paper is
1481.8 MPa, its instantaneous Young’s modulus E0 is 4060.11
MPa and has a density ρ of 1.18 g/cm3 .
0.2
Inclusions (Glass)
The inclusions considered in this study are made from glass.
Since the glass inclusions are much stiffer than the polymer matrix, a linear elastic model using Hooke’s law is used to model the
inclusion’s behavior. The glass inclusions are assumed to have a
Young’s modulus E of 64890 MPa, Poisson’s ratio ν of 0.249
and a density ρ of 2.47 g/cm3 .
α=
ln( SS21 )
(z1 − z2 )
,
(2)
where,
z1 = 2 mm,
z2 = 0.5 mm,
S1 - Stress at z1 (MPa) and
S2 - Stress at z2 (MPa).
There are number of techniques which are used for the generation of computational finite element models. Random sequential absorption (RSA) algorithm [22–27] is commonly used
in generating composites with random microstructures. This
technique has limitation in achieving high volume fractions ( >
50%) due to jamming issue. Another technique used widely is
the Monte Carlo (MC) [28–30] technique which is a two-step
scheme. In the first step all the filler particles or the inclusions
0.3
Interphase
The interphase is modeled as a linear elastic material. The
objective is to study the attenuation due to scattering from the
surfaces of the inclusions and not due to viscoelastic nature of the
interphase. Young’s modulus is taken to be 33 GPa, the Poisson’s
ratio is assumed to be 0.249 and the density ρ of the interphase
is assumed to be 1.18 g/cm3 , which is the mean of the densities
for the inclusion and the polymer matrix.
3
Copyright c 2018 by ASME
are deposited in the simulation box then in the second step the
location and orientation of the inclusions are changed randomly
until all overlap is removed. The removal of overlaps is slow
in the MC technique as the movements are random. Molecular
dynamic based processes [31, 32] can be used to accelerate the
removal of overlaps.
For the purpose of this research the RSA algorithm is used
for the generation of inclusions by employing Python scripting
in ABAQUS [20]. The volume fraction c is varied between 5%
to 20% to study its effect on attenuation characteristics of polymer matrix. The number of inclusions n required to achieve the
desired volume fraction is determined using
n≈
c ×V
,
v
FIGURE 2: Representation of meshed finite element model for
12% volume fraction.
(3)
where,
is created on a plane at z = 0.5 mm where the readings for output
are recorded. The sinusoidal pressure P is given as follows:
n - Number of inclusions,
c - Volume fraction,
V - Total volume and
v - Volume of the inclusion.
Location of the inclusion is generated by random number
generation in Python. The point generated in Python serves as
the center point of the inclusion. The distance of this point is
checked with any point previously accepted by the algorithm to
ensure there is no overlap between the inclusions.
A minimum gap of 12.5 % of radius is maintained between
the inclusions to ensure proper meshing of the finite element
model. The process continues until the required number of inclusions to satisfy the volume fraction are achieved.
Simulations on three-dimensional polymer composites are
conducted with spherical inclusions. The geometry of the specimen used in the simulation is a 1 × 1 × 2 mm3 cuboid. The
inclusions are modeled as solid spheres. The radius of the inclusions varies from 120µm to 180µm. The thickness of interphase
is 12.5 % of radius of inclusion.
The interphase and the inclusions are meshed with tetrahedral elements with a mesh size of 0.025 mm and the polymer
matrix is meshed with tetrahedral elements with a mesh size of
0.05 mm. A node to node connectivity is maintained in all phases
of the model. Figure 2 shows the meshed model for PMC with
inclusions as well as the interphase. From the ABAQUS element
library tetrahedral C3D10 elements and hexahedral C3D20 elements are used for the finite element simulations.
P = Po sin(2π f t),
(4)
where,
Po = 10 MPa,
f - Frequency (MHz) and
t - Time (ms).
For the purpose of this study the frequency is varied between
1 MHz to 4 MHz while the force amplitude Po remains the same
for all finite element solutions.
RESULTS AND DISCUSSION
The wave attenuation coefficient is calculated using Eq. (2)
by considering the peak value of the stresses on both planes as
specified above. For the case of this study the following parameters are evaluated and the results of each of the these parameters
are discussed in the subsequent sections:
1. Effect of volume fraction
2. Effect of loading frequency
3. Effect of size of inclusions
4. Effect of interphase layer
0.5
Validation
The results from the FEM are plotted in Fig. 3 with theoretical predictions [2] and results obtained using extended finite element method (XFEM) [1]. Analytical predictions are calculated
by considering scattering characteristic using Rayleigh scattering
behavior and viscoelastic properties of matrix, details of which
can be found in [2]. XFEM results are calculated by representing discontinuous material properties by XFEM. As can be seen
0.4
Boundary conditions
A uniform sinusoidal compressive pressure with Po = 10
MPa is applied at the surface z = 2 mm as shown in Fig. 1.
All other surfaces are constrained in two translational degrees
of freedom in the X and Y direction. Translational motion is allowed in the direction of loading i.e. the Z direction. A node set
4
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FIGURE 3: Attenuation in the particulate composite with volume
FIGURE 4: The ensemble average of the attenuation co-efficient
fraction 0.086 vs frequency.
(α) for 5% PMC verses the realization number.
from the figure the FEM results are closer to the analytical results than the XFEM results. Simulations were also performed
in which the matrix material was replaced by the inclusion material (glass) without any inclusions to see the effect of elastic
material on the wave attenuation of the polymer composite. No
attenuation was observed in these simulations which proves that
the attenuation occurs only by scattering from the surface of the
inclusions.
0.6
Ensemble averaging
As the position of the inclusion plays an important role in
determining the mechanical properties of polymer matrix composite, ensemble averaging was performed to remove the effect
of randomness of the positioning of the inclusions. The following convergence criterion is used to verify that the number of
samples in the ensemble is adequate
hαi2J − hαiJ
hαi2J
< TOL,
FIGURE 5: Attenuation Coefficient for PMC with inclusion ra-
dius of 120 µm.
0.7
Effect of volume fraction
Volume fraction is one of the vital factors which affects the
properties and behavior of the polymer matrix composite. To
evaluate the effect of volume fraction on the wave attenuation,
finite element simulations are performed by changing the number
of inclusions while keeping the radius of the inclusion constant.
The impact of inclusion’s volume fraction on attenuation coefficient for loading frequencies ranging from 1 MHz to 4 MHz
is shown in Figs. 5 –7. It can be seen from these figures that, the
attenuation coefficient increases as the volume fraction increases.
The higher number of inclusions leads to a higher scattering of
waves. It is also observed from the figures that the attenuation
increases substantially for higher frequencies as the volume fraction increases.
(5)
where, h iJ implies an ensemble average using J realisations, and
h i2J represent the same quantity obtained using twice this number of realisations. The tolerance TOL used in this equation is
1 %. The ensemble average of the attenuation coefficient obtained for a polymer matrix with 5% inclusion volume fraction
versus the number of samples in the ensembles are shown in
Fig. 4. It can be deduced from the figure that convergence is
achieved when 50 samples are available in the ensemble with a
convergence error of less that 0.7 %. As a result, 50 iterations
are performed for each design parameter to remove the effect of
randomness of the positioning of the inclusions.
0.8
Effect of loading frequency
The attenuation characteristics are affected by the loading
frequency. To study the impact of loading frequency on attenu5
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FIGURE 6: Attenuation Coefficient for PMC with inclusion radius of 150 µm.
FIGURE 7: Attenuation Coefficient for PMC with inclusion radius of 180 µm.
FIGURE 8: Time history of surface averaged stress σzz at z = 2
mm and z = 0.5 mm at different frequencies.
ation, finite element simulations are performed for loading frequencies from 1 MHz to 4 MHz while all other parameters are
kept constant. The simulations are conducted on polymer composites having inclusions with radius ranging from 120 µm to
180 µm and inclusion volume fraction from 5% to 20%. The
time histories of surface averaged stress σzz at surfaces z = 2 mm
and z = 0.5 mm are shown in Fig. 8 for 5% volume fraction. It
can be observed that peaks of the stress σzz at z = 0.5 mm is lower
compared with that at z = 2 mm. This is a result of scattering by
the inclusions and dissipation by the matrix.
The impact of the loading frequency on the wave attenuation
for inclusions with radius of 120 µm, 150 µm and 180 µm are
shown in Figs. 9–11. For the purpose of comparision, the attenuation coefficient of the epoxy matrix in the absence of inclusions
is also shown in these graphs. These figures show that the sensitivity of attenuation coefficient to the volume fraction increases
by increase in the loading frequency.
FIGURE 9: Effect of loading frequency on attenuation for a ra-
dius of 120 µm for different volume fractions.
6
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FIGURE 10: Effect of loading frequency on attenuation for a
FIGURE 12: Effect of size of inclusions on attenuation charac-
radius of 150 µm for different volume fractions.
teristics of PMC.
FIGURE 11: Effect of loading frequency on attenuation for a
FIGURE 13: Comparison of results for a polymer composite with
radius of 180 µm for different volume fractions.
and without interphase for the inclusions.
0.9
fraction for both the cases respectively. As can be seen from
the Fig. 13 that, the presence of interphase affects the attenuation
characteristics of the wave. The difference between the two cases
are minimal at lower frequencies while a considerable difference
is seen at higher frequencies. Thus, interphase properties play
an important role in the attenuation characteristics of the polymer composite and should be considered in studying attenuation
characteristics of the polymer composites.
The attenuation coefficients against the interphase Young’s
modulus are shown in Fig. 14. These plots show that increase in
the Young’s modulus of interphase leads to improvement in the
attenuation characteristics of polymer composites.
Effect of size of inclusions
For this study, finite element models were generated by varying the radius of the inclusions for the same volume fraction.
The simulations are carried for inclusions with radius 120 µm,
150 µm, and 180 µm. The results are shown in Fig. 12. It can be
seen that as the radius of inclusion increases the attenuation characteristics of the polymer composite decreases and this effect is
more pronounced at higher volume fractions. The higher attenuation capability of polymer composites with smaller inclusions
is due to the higher surface area of the inclusions which leads to
higher wave scattering.
0.10 Impact of interphase
Figure 13 shows the wave attenuation characteristics of a
polymer composite with radius of inclusion 150µm for two
cases, (a) no interphase between the inclusion and the polymer
matrix and (b) interphase between the inclusion and the polymer matrix. The results are shown for 5% and 8.6% volume
CONCLUSIONS
Using finite element simulations it is shown that the addition
of small sized glass inclusions in the epoxy matrix can significantly alter its attenuation characteristics. The volume fraction
and the size of the inclusions play an important role in the atten7
Copyright c 2018 by ASME
[3]
[4]
[5]
[6]
FIGURE 14: Comparison of results for 5% and 8.6% volume
fraction for loading frequency of 1 MHz.
[7]
uation characteristics of the polymer composite. Other factors
which play important roles in the attenuation characteristics of
the polymer composites are the loading frequency and the properties of the interphase zone between inclusions and polymer matrix.
There is a direct relationship between the volume fraction of
the inclusions and the attenuation characteristics of the polymer
composite. It is observed that as the volume fraction increases
the attenuation in the polymer composite also increases. There is
negligible attenuation observed in a polymer matrix with no inclusions which infers that the attenuation characteristics depend
on the presence of inclusions in the matrix. It is also observed
that the size of inclusions is a major factor in the attenuation
characteristics of a polymer composite. At the same inclusion
volume fractions, polymer composites with smaller inclusions
have higher attenuation coefficient. For all the cases considered
in this paper, polymer composites show better attenuation characteristics when the loading frequency is higher.
[8]
[9]
[10]
[11]
[12]
ACKNOWLEDGMENT
The finite element simulations were performed on the University Research Computing’s (URC) High Performance Computing (HPC) clusters at UNC Charlotte.
[13]
[14]
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