1. 2nd
International Conference on Engineering Optimization
September 6-9, 2010, Lisbon, Portugal
Mechanical Properties of Nanocomposite Laminated Structures by Modal
Method
Horacio V. Duarte, L´azaro V. Donadon, Antˆonio F. ´Avila
Universidade Federal de Minas Gerais, Av. Antˆonio Carlos 6627, Belo Horizonte, Brasil
Abstract
The Finite Element model of the laminate plate is used to approximate the experimental modal results
by a optimization procedure. From the modal properties the tensile and shear modulus and the Poisson
coeﬃcient of a woven orthotropic composite plate were determined by this method. The plates used
in the experiment are made of S2-glass/epoxy with 16 layers are manufactured by vacuum assisted wet
lay-up. The matrix of the nanocomposite has been obtained by adding 0%, 1%, 2% nanoclays in weight
into epoxy matrix. Remarks are made about the results, analysis methodology and its limitations. .
Keywords:Nanocomposites, Laminated Composites, optimization, Modal analysis .
1. Introduction
There is a numbers of researchers who studied the eﬀect of nanoparticles (organically modiﬁed montmo-
rillonite - Cloisite 30B) into epoxy systems. One aspect of the nanocomposite structure is the damage
reduction due to impact loadings. Among researchers who studied the eﬀect of nanoparticles, Yasmin et
al. [1] and Isik et al. [2] found also an increase in the elastic moduli and toughness [2].
A more comprehensive study on clay-epoxy nanocomposites was performed by Haque et al. [3], since they
evaluated both mechanical and thermal properties. Their main conclusions were that thermo-mechanical
properties mostly increase at low clay loadings ( 1-2% in weight) but decrease at higher clay loadings
( 5% in weight). In addition, the uses of nanoclays also decrease the coeﬃcient of thermal expansion
(CTE). They also observed a degradation of properties at higher clay loadings. This phenomenon can be
due to the phase-separated structures and defects in cross-linked structures.
The objective of this paper is to study the nanoparticle inﬂuence into the plate vibration behavior and
to determine the elastic constants using the optimization procedure. To analyze only the nanoparticle
inﬂuence all manufacturing parameters are kept ﬁxed but the nanoparticle concentration. The modal
analysis was made with the objective to determine the structural mode shape to compare the frequencies
and modal properties for diﬀerent nanoclay composite plates.
A commercial code, Ansys, which oﬀers the function of design optimization, is employed to model the
woven laminate a orthotropic model. The optimization allows to approximate the theoretical laminate
model modal results to experimental ones and by this way determining the elastic properties.
2. Testing Procedures and Results
The nanocomposite prepared for this work is a S2-glass/epoxy-clay. The epoxy formulation is based on
two parts, part A (diglycidyl ether of bisphenol A) and part B - hardener aliphatic amine- (triethylenete-
tramine). The nanoclay particles used in this study are organically modiﬁed montmorillonite in a platelet
form, while the S-2glass ﬁber has a plain-weave woven fabric conﬁguration with density of 180 g/m2
from
Texiglass. The S2-glass/epoxy-nanoclay composite is a laminate plate with 16 layers and 65% ﬁber vol-
ume fraction. The nanocomposite synthesis followed the methodology proposed by ´Avila et al. [?] [5]. In
order to be able to investigate the nanoparticles inﬂuence into vibration analysis, samples with 0%, 1%,
2% of nanoclay with respect to the matrix mass were employed. All plates were rectangular and had the
same dimensions 136mm x 116mm x 2.4mm.
The vibration analyses were performed to determine the shape modes and its natural frequency, shape
mode amplitude and the damping coeﬃcient. The vibration testing were performed using a Laser Doppler
Vibrometer, laser model OFV 303.8 and controller model OFV 3001 S from Politec, a Hewlett Packard
data acquisition system model 35670A, a nini-shaker, a shaker power ampliﬁer and force transducer from
PCB. The test plates were hanged by a ﬁne nylon wire and excited by a random signal (white noise). A
piezoelectric force transducer was used as the reference for the force bonded to the plate and linked to
the stinger/shaker exciter, which transform the ampliﬁed electrical signal in force. There was only one
force excitation point at the same position for all plates. The velocity of the plate surface was measured
1
2. using a grid of 35 points by the laser Doppler vibrometer. The data acquisition system processed the
signal response of the measurement point generating the Mobility (velocity/force) Function Response
Frequency (FRF), for each point of the plate.
A modal analysis program has done the mode shape identiﬁcation from the 35 FRF for each plate.
This modal analysis program is based on polynomial interpolation and employs Chebycheﬀ Orthogonal
Polynomials method, Arruda et al. [6].
Mode 1 180.6321 [Hz] Damping=0.0297 Mode 2 344.4358 [Hz] Damping=0.0166
Mode 3 423.6049 [Hz] Damping=0.0098 Mode 4 568.7344 [Hz] Damping=0.0112
Mode 5 692.8686 [Hz] Damping=0.0137 Mode 6 735.2092 [Hz] Damping=0.0131
Figure 1: Shape modes for epoxy resin composite plate.
There is no diﬀerence between the modal shapes of the 1% nanoclay composite and those of the resin
reference plate, so the 1% resin shape are not showed. Figure (2) plots the shape modes for the composite
plate with 2% nanoclay weight. The latest shape mode, named 7th
mode, is out of the increasing frequency
order, and this mode is similar to 4th
one.
Table (1) brings a summary of the modal properties natural frequencies, the associated damping coeﬃ-
cients and Mobility amplitude for each experimental mode. The 7th
mode, for the 2% nanoclay plate, is
detached and presented in the appearing sequence.
2
4. Table 1: Matrix composite content and Modal Properties.
Composite Mass Modal Shape Modes
Matrix [gr] Properties 1th 2th 3th 4th 7th∗ 5th 6th
Pure Nat. Frequency [Hz] 180.6 344.4 423.6 568.7 692.9 735.2
Epoxy 66.37 Damping Coef. 0.0297 0.0166 0.0098 0.0112 0.0137 0.0131
Resin Amplitude [m/s]/kgf 57.0 17.0 6.5 4.8 12.1 8.5
Epoxy Nat. Frequency[Hz] 184.9 347.4 424.6 550.1 683.4 728.2
Nanoclay 65.27 Damping Coef. 0.0209 0.0247 0.0088 0.0136 0.0137 0.0138
1% weight Amplitude [m/s]/kgf 79.9 14.0 24.1 2.6 17.9 9.6
Epoxy Nat. Frequency [Hz] 173.6 361.3 424.0 578.0 655.3 ∗ 716.1 783.0
Nanoclay 66.92 Damping Coef. 0.023 0.0248 0.0116 0.0173 0.0052∗ 0.0134 0.0166
2% weight Amplitude [m/s]/kgf 32.9 5.8 4.7 21.5 44.3 ∗ 15.0 8.1
3. Data Analysis
To determine the elastic properties was used the inverse method. This method is based on optimization of
a Finite Element Model and is very popular. The optimization is a zero-order approach method, oﬀered
as a tool of the FE commercial code, and followed the Hu and Wang [8] procedure. The state variables
ξn are related to diﬀerence between FE, fF E
n , and experimental modal nth frequency fn:
ξn =
fF E
n − fn
fn
100 (1)
The cost function F is deﬁned as:
F(Exy, Gxy, νxy) =
k
n=1
ξ2
(2)
Where the mechanical properties are Exy the Tensile Modulus, Gxy the Shear Modulus and νxy the
Poisson’s ratio.
The element used in the Finite Element Model was the shell181 element. This shell element has four
nodes and six degrees of freedom in each node. This element has been formulated to thin to moderately
thick plates. This element also accepts multi-layers shell allowing the construction of complex models of
laminates. The FE model also employed the mass21 element, to consider the eﬀect of mass and rotary
inertia of the force transducer bonded to the plate. The structure was modeled by 320 shell elements and
by concentrated mass element.
For the reference resin plate and 1% nanocomposite plate the results are obtained by direct application
of the method. The 2% nanocomposite plate showed a singular situation. The 4t
h and 7t
h has identical
shape modes at diﬀerent frequencies. This behavior was not observed in the previous cases. The ﬁnite
element simulation showed a coupling mode between the plate and the force transdutor, this result has
appeared for all plate simulations but at a low frequency, below the experimental frequency range.
There was not used a complex FE model to deal wiht the dynamical coupling between the elastic plate
the force trandutor, the elastic stinger and shaker. So the dynamical coupling appeared something like
a rigid body mode. But this model oﬀered a answer to the similar modes, one of them were a coupling
mode between the experimental apparatus and the elastic plate. The nanocomposite mass in the matrix
changed the elastic properties of the plates and the coupling frequency mode move inside the range of
the experimental plate natural frequencies.
The problem is to determine what mode is a plate mode and what is not. The solution presented here
is to ﬁnd the real mode by comparing the error between the experimental mode frequencies and the
frequencies of the optimized or minimized model.
4
5. Table 2: Frequency mode error in each iteration. Frequency for the 4th
mode 578.0 Hz
iteration 1th
mode 2th
mode 3th
mode 4th
mode 5th
mode 6th
mode
1th
0.872E-02 0 0 0 0 0 0.01
2th
0.423E-01 0.389E-01 0 0 0 0 0.08
3th
0.985 2.08 2.22 0 0 0 5.28
4th
0.984 2.10 2.20 0.789 0 0 6.08
5th
1.03 2.02 2.28 0.670 2.30 0 8.29
6th
1.02 2.03 2.27 0.692 2.28 3.92 12.21
The Tab. (2) and Tab. (3) show the frequency error for each mode in the minimization procedure. The
minimization was not done for all frequencies in a unique process. The procedure was to include one
mode frequency at time and ﬁnd the elastic properties. The latest elastic propeties are used as initial
value in a new minimization process, that includes the next frequency. This procedure will be called
an iteration. The latest iteration is performed using a small tolerance for the cost function F, Eq. (2),
for all cases F < |10−2
|. The Tab. (2) presents error for each mode frequency in each iteration, in this
case, the fourth mode is supposed to be the plate mode. The Tab. (3) presents the same error evolution
considering the seventh mode (655.3 Hz) as the plate mode.
Table 3: Frequency mode error in each iteration. Frequency for the 4th
mode 655.3 Hz
iteration 1th
mode 2th
mode 3th
mode 4th
mode 5th
mode 6th
mode
1th
0.872E-02 0 0 0 0 0 0.01
2th
0.423E-01 0.389E-01 0 0 0 0 0.08
3th
0.985 2.08 2.22 0 0 0 5.28
4th
1.0087 7.5204 2.0591 2.7306 0 0 13.3
5th
1.0262 2.4241 1.5333 8.8377 1.6721 0 15.5
6th
0.98710 2.4631 2.0953 9.5305 2.1113 4.1373 21.3
The Tab. (2 shows the lowest errors for the fourth and the sum of error of each frequency mode. So it is
used to determine the elastic properties showed on Tab(4).
4. Closing Remarks
The inverse method used here minimizing a ﬁnite element model showed to be a powerful, fast and
very ﬂexible method of analysis. Without the method it was necessary to repeat all the experimental
procedure to ﬁnd the problem.
In this work the sensor mass and mainly the rotatory inertia has made a strong inﬂuence on the results,
this parameter is diﬃcult to determine with precision. The excitation of the system is still a problem
and can introduce error in the results.
The Tensile Modulus Exy increases with nanoclay mass increase in the matrix epoxy resin. The Tensile
modulus for the 2% composite plate is 17% greater than the equivalent resin modulus. For the Shear
Modulus, Gxy, the maximum value is at 1 % nanoclay in matrix resin. The Poisson’s coeﬃcient νxy
showed a very low value. Part of the problem is that the Shear and Tensile modulus have highest values
compared to Poisson’s coeﬃcient value. Probably the shape modes used do not capture its inﬂuence [7].
Table 4: Elastic Properties
Composition Exy[GPa] Gxy[GPa] νxy
Resin 100% 27.3 4.2 0.10425E-07
Nanoclay 1% weight 28.3 5.1 0.10212E-03
Nanoclay 2% weight 31.9 3.8 0.75339E-03
5
6. All nanocomposite plates had a highest damping coeﬃcient and this properties must be included as it
can change the frequencies mainly the highest ones. The plates used in this work can not be considered
thin and some shape modes are diﬃcult to obtain, the synclastic mode for instance. The behavior of the
new materials can not be obtained by standard procedures.
Acknowledgements
This work was supported by the Funda¸c˜ao de Amparo a Pesquisa de Minas Gerais FAPEMIG, which
support the authors are grateful.
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