3.2. Free Vibration of Epoxy/GPLs Beams in Magnetic Fields
In what follows, detailed parametric studies were carried out to examine the free vibration of the FG-GPLRC beams resting on an elastic foundation in magnetic fields. Herein, the epoxy as the matrix phase and GPLs as the reinforced phase are selected as numerical examples, and the corresponding material constants are listed in
Table 3.
Unless otherwise stating, the geometry and size of the GPL nanofillers are respectively
,
, and
, and the weight fraction of GPLs is fixed at
. The ideal graphene is intrinsically nonmagnetic and lacks localized magnetic moments due to a delocalized π-bonding network [
50]. Therefore, the magnetic field permeability of GPLs is assumed to be zero in the present discussion. Although the intrinsic graphene cannot be affected by the magnetic field, the additions of GPLs into a magnetic polymer matrix can significantly improve the mechanical properties of the nanocomposites. Therefore, the addition of GPLs into a magnetic matrix can meaningfully affect the dynamical behaviors of the nanocomposite beams in magnetic fields. Moreover, it should be pointed out that the unusual positive magnetic signals (paramagnetic and/or ferromagnetic) have been experimentally reported for synthesizing magnetic graphene in recent years [
50]. In addition, the reference physical quantities
,
, and
are chosen as the values of the epoxy matrix.
Figure 3 plots the effects of the total layer number
N of the FG-GPLRC beam (
) on the dimensionless fundamental frequency
for different GPL distribution patterns. For the UD pattern, the vibration frequencies are independent of the total number
N due to the uniform distribution of GPLs. However, the total layer number
N plays a critical role in the vibration frequencies of the patterns with GPLs dispersed nonuniformly. For a fixed total volume fraction
WGPL, with the increasing total layer number
N, the fundamental frequencies increase distinctly first and then vary slightly for the FG-X pattern. However, the fundamental frequencies decrease significantly first and then remain nearly unchanged for the FG-O and FG-V/A patterns. In the FG-X pattern, more GPLs are dispersed near the top and bottom layers with the increasing total layer number
N, which is more powerful for promoting stiffness and hence increases the vibration frequencies of the beam. Moreover, it can be concluded that the multilayer beam with the sufficiently large number of individual layers is an excellent alternative for the functionally graded GPL-reinforced nanocomposites. In the following analysis, the total layer number
N = 20 is adopted.
Table 4 gives the first-five order vibration frequencies
of FG-GPLRC beams without considering the effects of magnetic fields and elastic foundation. The corresponding vibration frequencies of pure epoxy beams are involved for comparisons. It can be seen that, regardless of GPL distribution patterns, the vibration frequencies of the beams increase significantly even by adding a low content of GPLs into the epoxy matrix. As expected, the vibration frequencies of FG-V and FG-A are identical when neglecting the effects of an elastic foundation. The FG-X GPLRC beam has the largest, while FG-O GPLRC beam has the lowest fundamental frequencies among the five beams. In the FG-X pattern, more GPLs are dispersed near the top and bottom surfaces of the beam where the normal stresses are higher. Therefore, the GPL nanofillers can maximize the reinforcing effects to increase the stiffness of the beam.
Table 5 demonstrates the effects of the elastic foundation on the fundamental frequencies
of the FG-GPLRC beams in absence of magnetic fields. It is obvious that both the Winkler and shearing layer elastic coefficients have significant effects on the fundamental frequency parameters. All the fundamental frequencies increase dramatically when promoting the elastic coefficients of the foundation. However, for the large values of the Winkler elastic coefficient, the shearing layer elastic coefficient has less effect on the fundamental frequencies. The effects of the GPL distribution pattern are also listed in the table. It was found again that the distribution pattern of GPLs can critically influence the fundamental frequencies. The vibration frequencies of FG-A and FG-V are quite different when considering the elastic foundation. In the FG-A pattern, the bottom surface of the beam is GPL-rich, and the fundamental frequency is higher than that of FG-A. Moreover, among all the GPL patterns, the FG-X pattern holds the highest fundamental frequencies. It can be concluded that the FG-X pattern can sufficiently utilize the reinforcing effects of GPLs and increase the bending stiffness of the FG-GPLRC beams more powerfully.
The effects of the magnetic fields along different directions on the fundamental frequencies are tabulated in
Table 6. It can be observed that the magnetic field significantly influences the fundamental frequencies of the FG-GPLRC beam. The fundamental frequencies of the FG-GPLRC beam increase as we increase the values of the magnetic fields both along the
x and
y directions. However, the fundamental frequencies decrease as the
z-direction magnetic field is increased. As shown in Equation (10), the
x-direction magnetic field induces the transverse Lorentz force through the thickness direction of the beam, and the
y- and
z-direction magnetic fields induce the longitudinal Lorentz forces, respectively. In addition, the Lorentz force induced by the
z-direction magnetic fields is only decided by the longitudinal displacement
u. However, the Lorentz forces induced by the other two-direction magnetic fields are still concerned with the transverse displacement
w. Therefore, the trend of the
z-direction magnetic field on the fundamental frequencies differs from the others. It should be noted that the fundamental frequencies of the beam change slightly in a strong
x-direction magnetic field (larger value of
Mx). Regardless of the magnetic fields, the addition of GPLs into the matrix can improve the stiffness of the beams and hence promote the corresponding vibration frequencies. Moreover, the FG-X pattern gives the highest fundamental frequencies in all magnetic fields, and this will be focused on in the following discussion.
The variations of the fundamental frequencies of the FG-GPLRC beams with various magnetic parameters as well as the GPL weight fraction are plotted in
Figure 4,
Figure 5 and
Figure 6. Here, only the weak magnetic field in the
x direction is involved in
Figure 4. The fundamental frequencies of the beams increase with the promotion of magnetic fields, except for the case of the beams in the
z-direction magnetic field. As the GPLs are more dispersed, the fundamental frequencies of the beams increase for the
y-direction and
z-direction magnetic fields, while the fundamental frequencies of the beams in the
x-direction magnetic field increase first and then decrease. As stated before, the
x-direction magnetic field induces only the
z-direction Lorentz force, and the corresponding Lorentz force is determined by the magnetic field and the bending deformations. The transverse displacement of the beam decreases when adding more GPL nanofillers into the matrix, while the competition between the bending deformation and the magnetic field leads to the complicated variation of the fundamental frequencies.
The effects of size and geometry of GPL nanofillers on the fundamental frequencies of the FG-GPLRC beams with various magnetic parameters are depicted in
Figure 7,
Figure 8 and
Figure 9. The fundamental frequencies of the beams increase sharply and then change slightly with decreasing the thickness of the GPL nanofillers when the beams are absent of the magnetic fields or in
y-direction and
z-direction magnetic fields. The GPLs as nanofillers dispersed into the epoxy matrix can dramatically improve the bending stiffness of the beams, and thus the vibration frequencies will be promoted as expectedly. Moreover, thinner GPL nanofillers can increase the stiffness of the beams more powerfully. The magnetic fields applied along the
y direction and
z direction induce the Lorentz forces in the
x direction, which can be evaluated by the longitudinal and transverse displacements as shown in Equation (10) and can help to increase the stiffness of the beams. However, the fundamental frequencies of the FG-GPLRC beams in the
x-direction magnetic field reduce when the GPL nanofillers are thinner. It is also the result of the competition between the bending deformation and the magnetic field. The thinner GPL nanofillers can powerfully increase the bending stiffness of the beams, and hence the transverse displacements become smaller and smaller. Accordingly, the correlative displacement terms in Equation (18) and the corresponding fundamental frequencies decrease. It is also can be observed that GPL nanofillers with larger surfaces can increase the stiffness more efficiently. This is due to the fact that larger surfaces between the GPL nanofillers and the matrix can transfer loads better. It can be concluded that thinner and larger GPL nanofillers are preferred as nano-reinforcements for increasing the fundamental frequencies of FGGPLRC beams; however, it reverses for the case of the beams in the
x-direction magnetic field.
Figure 10 and
Figure 11 show the effects of the Winkler and Pasternak coefficients of the elastic foundation on the fundamental frequencies of the FG-GPLRC beams, respectively. According to the figures, by increasing the Winkler or Pasternak coefficients, all vibration frequencies increase first and then almost keep unchanged. Moreover, it was observed that the magnetic field has a critical effect on the fundamental frequencies of the beams. In lower values of the Winkler/Pasternak coefficient, the
x-direction magnetic field promotes the vibration frequencies more remarkably than that in
y-direction magnetic field. On the contrary, the z-direction magnetic field decreases the fundamental frequencies. However, in higher values of the Winkler/Pasternak coefficient, the fundamental frequencies of the beams nearly keep constant when increasing the elastic coefficients of the foundation. The effects of the
x-direction magnetic field are neglectable, while the magnetic field along the
z-direction increases the vibration frequencies more dramatically.