On Moving Harmonic Load and Dynamic Response of Carbon Nanotube-Reinforced Composite Beams using Higher-Order Shear Deformation Theories

1. Eghbali ^a , S.A. Hosseini ^b*

^a Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran

^b Department of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University, Qazvin, Iran

1.     Introduction

The mechanical, thermal, and electrical properties of carbon nanotubes (CNTs) make them suitable for reinforcing polymers and polymer nanocomposites. The laboratory research results indicate that only by adding 1% by weight of carbon nanotubes to the polyester resinYoung’s composite modulus is increased by 35% to 43%. The study of the dynamic behavior of carbon nanotubes plays an important role in developing their application in a wide range of non-mechanical equipment such as oscillators, clocks, and nanosensors. Reliability analysis of nanoscale equipment based on carbon nanotubes requires identifying the nanotube response to the applied mechanical forces. The application of carbon nanotubes in many existing fields requires a detailed understanding of their mechanical behavior. Since nanoscale testing is very challenging and complex, theoretical modeling is particularly important for predicting carbon nanotube mechanical behavior [1-3].

As we know, various theories have been presented to investigate the dynamic displacement of beams, among which we can refer to higher-order shear deformation theories. These theories are presented to reach a more accurate answer and reduce the error caused by the previous theories. In this article, various higher-order shear deformation theories have been investigated to present their differences. The difference between these theories is in the choice of shape function, according to which they have upper limit and lower limit values.

Zhao et. al. [4] analyzed the vibrations of a purposeful functionally graded plate with porosity. The equations were derived using high-order shear deformation theory and Jacobi-Ritz theory. The effect of parameters on the vibrations of the circular plate was investigated. Xiao et al. [5] analyzed the nonlinear vibrations of nanobeam in a magnetic-electric-thermal environment. Temperature-dependent material properties are defined. The equations were extracted using nonlocal and higher-order shear deformation theories, and the Galerkin method was solved. Finally, various parameters were investigated. Ramezani et al. [6] analyzed the nonlinear stability of a cylindrical shell reinforced with carbon nanotubes in a thermal environment. The von Kármán strain field is used to describe nonlinearly. The equations are derived using higher-order zigzag theory. Finally, the stress of cylindrical shells in a thermal environment is analyzed. Malabari et al. [7] presented a continuous mathematical model for analyzing the free vibrations of a CNT multilayer composite nanoplate. The relations were obtained using nonlocal strain gradient theory (NSGT). The relations were solved by the Galerkin method. Finally, various geometries on the nanoplate frequency were investigated.

Van Quyen et al. [8] analyzed the nonlinear vibrations of a carbon-nanotube-reinforced sandwich cylindrical with a honeycomb core. The sandwich panel is located in a thermal environment, and a negative Poisson’s ratio is used. The Reddy higher-order theory extracted the equations, and the equations were solved using the Runge-Kutta method. Finally, various free and forced vibration parameters are investigated. Hosseini et al. [9] investigated the response of an FG nanobeam with a moving force in a thermal environment. The relations were obtained by the Hamilton principle and nonlocal theory and then solved using Laplace transform. Finally, the effect of different parameters on the response of nanobeam was investigated.

Dat et al. [10] investigated the analytical solutions for nonlinear vibration of the sandwich plate with CNT nanocomposite core in the hygro-thermal environment. Dat et al. [11] investigated the vibration analysis of FG-CNTRC plate subjected to thermo-mechanical load based on higher-order theory. Dat et al. [12] investigated the vibration of CNT plates via refined higher-order theory. Dat et al. [13] studied the geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order NSGT.

Also, in another study, Civalek [14] investigated multi-layer composite plates using the discrete singular convolution (DSC) method. Akgöz et al. [15] investigated the vibrations of the thermo-elastic micro-beam located on the elastic foundation using different theories and numerical methods. Demir et al. [16] analyzed the static analysis of nanobeam under uniform load using the finite element method. The equations are derived using the non-local Eringen theory and the Galerkin method. Finally, various parameters have been investigated.

In another study, Daikh et al. [7] investigated free vibrations, buckling, and static displacement [17] and dynamic analysis [18] of carbon nanotubes. In another study, Eltaher et. al. [19, 20] investigated the post-buckling of carbon nanotubes on a nonlinear elastic substrate. Daikh et al. [21-23] investigated free and forced vibrations of carbon nanotubes with a moving load. Karami et al. [24-27] investigated curved structures’ vibrations and dynamic and static responses reinforced with carbon nanotubes and graphene.

This paper aimed to analyze the dynamic response of carbon nanotube-reinforced composite (CNTRC) beams under a moving harmonic load. The governing equations of the CNTRC beam are obtained based on the shear deformation beam theory and Laplace transform to solve the derived differential equations. Due to this effort, an exact solution for both transverse and axial responses is obtained. Through parametric study, valuable results have been concluded related to the effect of essential parameters such as aspect ratio, different higher-order theories, harmonic frequency, and load velocity on the dynamic response of the thermoelectric CNTRC beams, and the volume fraction of CNTs in axial and transverse modes.

2.     CNTRC Beams

A CNTRC beam made from a mixture of a single-walled carbon nanotube (SWCNT) and an isotropic polymer matrix is considered. The beam has a length (L), width (B), and height (H), as shown in Fig. 1.

The expressions of the effective Young and shear modulus of CNTRC beams are as follows [28, 29].

Fig. 1. Geometry of a CNTRC beam and cross-sections of four patterns of reinforcement CNTRC beam

 and  are defined as Young and shear modulus.  And  as the polymer matrix’s corresponding material properties. Also,  and   are the volume fractions for CNT and the polymer matrix, with the relation of . By using the same rule, Poisson’s ratio ( ) and mass density ( ) of the beam are written as [30]:

where , ,  and  are the Poisson’s ratios, densities, and thermal expansion coefficients of the carbon nanotube and polymer matrix. Different patterns of CNT reinforcement distribution are given below [31]:

where volume fraction ( ) is the given volume fraction of carbon nanotubes [31]:

where  is the mass fraction of CNTs  [29].

3.     Equations of Motion

According to shear deformation beam theory, the displacement field at any point is as follows [32]:

where u₀ and w₀ are the axial and transverse displacement at the reference plane of the beam,  is the shape function. f₀ and , defined as [33]:

For other shear strain shape functions, see Table 1 [33].

Table 1. Shear strain shape function  values.

where  is the bending rotation of the reference plane,  and t is time. The expression of shear and normal strain components in Eq. .

The normal and shear stress, respectively  and  as:

Using the Hamilton principle as follows [34]:

where  is the virtual kinetic energy, and is the virtual of the total strain energy,  is the virtual work done by external loads. The initial and final time is defined as t₁ and t₂.

The virtual work done ( ) can be expressed as:

where the transverse load (q),   is concentrated moving the harmonic load,  is excitation frequency, and  is Dirac delta function.

For the dynamic model, ( ) is required for the equations of motion:

The stress results are extracted as:

N_x, M_x, P_x, and Q_x are the stress resultants in the normal bending moment, higher-order generalized, and shear force. I_i (i = 0, 1, 2...5) are the mass moments of inertia:

Using Eq. and, Eqs. - with solving the relationship and factorization, the equilibrium equation results in the following effect:

From the above equations:

where:

The stress resultants of Eqs. - are substituted into Eq. to obtain the relations of motion:

Assuming [35]:

,  and  are the unknown Fourier coefficients to be determined for each n value. By using the general property of the Dirac Delta function [35]:

Assuming:   

According to the exact solution provided to investigate the forced vibrations and the difficulty of solving a couple of equations for the moving load state, it should be noted that the provided method can only be solved for simply supported boundary conditions and other boundary conditions, solving the inverse Laplace equations, It is not possible. For initial conditions, Simply-support:

To solve the system of the differential Eqs. - in the time domain. By recalling this transform:

where applying Laplace Transform in Eqs. -, the system of equation is obtained as follow:

where:

By solving the Eq. , ,  and  are obtained as:

By applying an inverse Laplace transform to Eq. , the responses of the thermoelectric CNTRC beam are obtained:

4.     Results and Discussions

The results presented in this chapter are related to analyzing the axial and transverse dynamic response of CNTRC beams under moving harmonic force. Also, the influence of parameters such as aspect ratio, different higher-order theories, excitation frequency, the volume fraction of carbon nanotubes, and load velocity on the response of the thermoelectric CNTRC beams in axial and transverse modes were investigated.

The relations described in Eq. are performed to calculate dimensionless natural frequencies.

Frequency values were compared, and good validity in terms of frequency values was observed. The results are compared with Wattanasakulpong et. al. [32], given in Table 2. Also, the dimensions of the beam are: .

The effective material properties are given as follows. ;  and . For reinforcement material: ; ; ;  and  [29].

As shown in Table 2, among different types of CNTRC beams, frequency O-Beam is the lowest, and X-Beam is the highest.

Figures 2-4 show the axial displacement in terms of time. Figure 2 shows the axial displacements for different harmonic frequencies for V-Beam. As can be seen, the moving harmonic force decreases, and the axial displacement also decreases with increasing harmonic frequency.

Table 2. Comparisons of fundamental frequencies

Fig. 2. Variation of dimensionless deflection V-Beam versus time for three different excitation frequencies

Figure 3 shows the axial displacement for different aspect ratios for V-Beam. The axial displacement increases with the increasing aspect ratio.

Fig. 3. Variation of dimensionless deflection V-Beam versus time for three aspect ratios

Figure 4 shows the axial displacements for different velocities for V-Beam. As can be seen, the graph increases with the speed of the extreme points.

Fig. 4. Variation of dimensionless deflection V-Beam versus time for five velocities

It should be noted that the axial displacement diagram has values for the states where the arrow is close to state FG and zero for other states, for example, UD-Beam.

Fig. 5. Variation of dimensionless deflection X-Beam versus time for five different velocities of moving load

Similarly, the variation of transverse dynamic deflection versus time for X-Beam for different velocities and aspect ratio of moving harmonic load is illustrated in Figs. 5-6.

The results of these figures show that as the speed increases, the peak point of the plot decreases to shorter times. Also, the transverse displacement has increased with an increasing aspect ratio.

Fig. 6. Variation of dimensionless deflection X-Beam versus time for five different aspect ratios of moving load

Fig. 7. Maximum dimensionless displacements
 of V-Beam versus excitation frequency for
four different velocity

Fig. 8. Maximum dimensionless displacements of V-Beam under moving load versus excitation frequency
for four different aspect ratio

Figure 7 shows the maximum axial displacements of V-Beam under moving force versus excitation frequency for velocities. By increasing the speed, maximum dimensionless axial displacements decrease.

Figure 8 presents the Maximum axial displacements of V-Beam under moving force versus excitation frequency for aspect ratios. By increasing the aspect ratio, maximum dimensionless axial displacements increases.

Fig. 9. Maximum dimensionless transverse displacements
of V-Beam versus excitation frequency
for three different velocity

Fig. 10. Maximum dimensionless transverse displacements of V-Beam versus excitation frequency for
three different aspect ratio

Fig. 11. Maximum dimensionless displacements of V-Beam versus aspect ratio for three different
excitation frequencies

Figure 9 demonstrates maximum transverse displacements of V-Beam under moving harmonic load versus excitation frequency for velocities. By increasing the speed, maximum dimensionless transverse displacements decrease.

Fig. 10 represents the maximum transverse displacements of V-Beam under moving harmonic load versus excitation frequency for aspect ratios. By increasing the aspect ratio, maximum dimensionless transverse displacements increase.

Figure 11 illustrates the maximum transverse displacements of V-Beam under moving harmonic load versus aspect ratio for excitation frequencies. By increasing the excitation frequency, maximum dimensionless transverse displacements decrease.

Fig. 12. Maximum dimensionless displacements of V-Beam versus velocity for three different excitation
 frequencies

Fig. 13. Maximum dimensionless displacements of V-Beam versus excitation frequency for five different
 theories

Figure 12 shows the maximum transverse displacements of V-Beam under moving harmonic load versus velocity for excitation frequencies. By increasing the harmonic frequency, maximum dimensionless transverse displacements decrease.

Figure 13 presents the maximum transverse displacements of V-Beam under moving harmonic load versus excitation frequency for five theories . As can be seen, the theory of HSDT is a higher limit theory than the theories by Mantari, Karama, Bonilla, and Reddy, respectively.

Fig. 14. Maximum dimensionless displacements
 of CNTRC beams versus excitation
frequency

Figure 14 demonstrates the maximum transverse displacements of CNTRC beams under moving harmonic load versus excitation frequency. As seen, beams O, V, UD, and X have the highest max transverse dynamic deflection.

Table 3. Maximum transverse displacements of V-Beam
 for five different theories.

Table 3 shows the maximum transverse displacement for three different excitation frequencies for different higher-order shear deformation theories. The difference between the theories is based on the choice of their shape function. HSDT theory has the highest displacement values. Reddy's theory has the lowest displacement values. As a result of the HSDT theory, the upper limit is also the Reddy theory, the lower limit. The type of theory chosen has an impact on the results.

5.     Conclusions

This paper analyzes the dynamic response of CNTRC beams under moving harmonic force. The governing equations of the CNTRC beam are obtained based on higher-order theory, the Hamilton principle, and Laplace transforms to solve the derived differential equations.

It was found that the parameters of beam thickness, excitation frequency, and moving load speed have a significant effect on forced vibrations and transverse and axial displacement of CNTRC beams. It should be noted that the effects created on the results are due to the changes in the stress results.

• Among CNTRC beams, the highest transverse and axial displacements are for O, V, UD, and X beams, respectively, so O-Beam has the weakest resistance to flexural loads. They increased the aspect ratio of the displacement of the CNTRC beams. Also, increasing the excitation frequency, velocity and , displacement of the CNTRC beams reduced.
• The axial displacement diagram has values for the states where the arrow is close to state FG and zero for other states, for example, UD-Beam.
• Theory HSDT is a higher limit theory than Theories by Mantari, Karama, Bonilla, and Reddy . Among different CNTRC beams, frequency O-Beam is the lowest, and X-Beam is the highest.
• The presented method for the boundary condition simply support.

Nomenclature

Appendixes

References

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