Document Type : Research Paper
Authors
Department of Mechanical Engineering, Razi University, Kermanshah, Iran
Abstract
Keywords
Main Subjects
Research Article
Analytical Modeling of Functionally Graded Carbon NanotubeReinforced Composite Plates
under LowVelocity Impact
Saeid Feli ^{*}, Bromand Karimi, Seyed Sajad Jafari
Department of Mechanical Engineering, Razi University, Kermanshah, Iran
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20230701 Revised: 20231209 Accepted: 20240201 

In this paper, analytical and numerical modeling has investigated the behavior of Functionally Graded Carbon NanotubeReinforced Composite (FGCNTRC) plates under LowVelocity Impact (LVI). The mixed law is employed to obtain the mechanical properties of carbon nanotubes (CNT) and methyl methacrylate. Nonlinear equations of analytical modeling are derived based on the Thirdorder Shear Deformation Theory (TSDT), Hertz law (to define the contact force), and the energy principle. After calculating the strain and stress fields, the Ritz method is used to obtain motion equations. Fourth order RungeKutta method is applied for solving equations of motion. In the following, LVI on FGCNTRC plates is simulated using the FE code, ABAQUS software. The results of the analytical and numerical model are compared with the other results of LVI on FGCNTRC, and there is good agreement between them. In addition, the effects of CNTs graded profile, volume fraction, impactor velocity, impactor radius, and geometrical parameters are investigated. 



Keywords: Functionally graded; Carbon nanotube; Lowvelocity; Impact; Analytical model. 


© 2024 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an openaccess article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
Functionally graded carbon nanotubes reinforced composite (FGCNTRC) structures have widely been used in industrial, aerospace, and ... applications. FGCNTRCs have good mechanical, thermal, and vibration characteristics properties.
In recent decades, many types of research have been carried out about the behavior of the materials under different loadings, such as impact problems including LowVelocity Impact (LVI) [1], HighVelocity Impact (HVI)[2, 3], high strain rate loadings[4,5], etc. The projectile velocity is very low so that the strain rate value is less than one so that the behavior of the materials under LVI is nearly similar to the material's response under quasistatic loading. Abrate [4] has studied various models of LVI of the composite and sandwich panel structures which are used in the Hertz contact law. The effect of boundary conditions on LVI on Fiber Metal Laminates (FMLs) is investigated by Zarei et al. [6]. They used Hamilton’s principle, Hertzian contact law, and the fourthorder RungKutta method to study and showed the contact duration and the number of contacts between impactor and FMLs will vary when the boundary conditions change.
Taghipoor et al. [7] used the response surface methodology (RSM) to study the impact of graphene nanosheet, basalt fiber, and polypropylenegrafted maleic anhydride reinforcements on the mechanical behavior of polypropylene polymer nanocomposites, experimentally. They showed graphene nanosheets at 1.5 wt% improved the elastic modulus by about 70%. In addition, Niaraki et al. applied RMS to find the effects of graphene nanosheets, nano clay, and glass fibers on the hybrid polypropylene and ethylenepropylenemonomer (EPDM). According to the results of the article, enhancement of the nanosheets of graphene has improved the elastic modulus.
Taghipoor and Mirzaei [8] studied the effects of the natural kenaf fibers, basalt fiber, and nanographene on the tensile strength of the polypropylenebased biocomposites. They showed kenaf fiber has the most effect on improving the tensile strength, and basalt fiber has the most effect on the improvement of the elastic modulus. Sadeghian et al. [9] conducted an experimental study and used optimization to define the absorption and impact strength of fiberreinforced concrete (FRC).
Lei and Tong [10] analyzed the LVI response of the graphenereinforced composite functionally graded (GRCFG) cylindrical shells in thermal environments by using the Halpin–Tsai micromechanical model, Fourier series expansion and Laplace transforms. Fallah et al. [11] presented a semianalytical model to study the LVI impact response of temperaturedependent CNTRC plates by using the freemesh Ritz method. They showed the distribution profile of the type X pattern results in lower maximum deflection and higher maximum contact force.
Feli et al. [12] suggested a new analytical model based on the firstorder shear deformation theory (FSDT) and springmass model for the behavior of CNTRC under LVI. According to the analytical model, by increasing the CNTs volume fraction, enhancement of the axial stress and reduction of the axial strain has occurred.
Yang and Ma [13] analyzed the LVI response of an FGCNTRC plate with negative Poisson ratios. They combined Reddy's higherorder shear deformation theory, the rule of a mixture model, and the Hertz contact law to derive the equations of motion, material properties, and contact force, respectively. They showed that temperature, initial impact velocity, and contact force increase are effective parameters for FGCNTRC behavior.
In this paper, the LVI nonlinear response of an FGCNTRC plate is calculated by analytical and numerical methods. Four different kinds of nanotube distribution layers (i.e., UD, FGV, FGX, and FGO) are studied in the analysis. The mechanical properties of the plate are estimated by using the mixed law. Thirdorder Shear Deformation Theory (TSDT), Hertz law, and the energy principle are used in the equations of motion. Finally, the fourthorder RungeKutta method is used to solve equations of motion. In addition, the numerical method is employed to investigate the behavior of the FGCNTRC plate under LVI. The results of the analytical and numerical methods are compared with the other wellknown models and there is good agreement between them. The effects of CNRTC types, volume fraction, initial velocity, and radius impactor are investigated.
The schematic view of LVI on a rectangular FGCNTRC plate is shown in Figure 1. The length, width, and thickness of the plate are defined as
, and , respectively. In addition, the radius and initial velocity of the spherical rigid impactor are designated as and , respectively.
Fig. 1. Configuration of an FGCNTRC plate subjected to LVI.
The principal energy method was employed to derive equations of motion. By considering the plate and impactor as a system, the total potential energy ( ) and kinematic energy ( ) of the system can be expressed as:

(1) 

(2) 
where and are components of axial plane stress in the x and y direction, respectively. In addition, and are transverse components of the stress field. and are components of axial strain and , and are components of the shear strain field. It is worth mentioning, in the above equation is the position in which impact takes place, the displacement of the impactor, is transverse displacement of the midplane, and the contact stiffness which is evaluated by [14]:

(3) 
in the above equation, , , and are impactor radius, the transverse elasticity modulus in the surface of FGCNTRC plate, Poisson ratio, and Young modulus of the impactor, respectively. It should be noted that the first and second parts in Eq.(1) demonstrate the potential energy of the plate and potential energy due to contact between the impactor and the plate.
In Eq.(2), , , are components of plate velocity in x, y, and z directions, respectively. In addition, is plate density and impactor velocity. The first part of Eq.(2) represented the kinetic energy of the plate and the second part showed the kinematic energy of the impactor.
To calculate the potential and kinetic energy, the stress and strain field and velocity component (due to displacement) must be defined. Highorder Shear Deformation Theory (HSDT) is employed to determine the total displacements
( , , ) along the three coordinate axes
( , , ) [15]:

(4) 
in which ( , , ) are the displacement components along the ( , , ) direction, of a point on the midplane, respectively. and represent rotations of a point on the midplane of the plate, while represents the transverse shear strain distribution along the thickness.
The straindisplacement relation of any point of the plate and the stressstrain relation is present may be written as [16]:

(5) 

(6) 

(7) 

(8) 
in which and represent the stress and strain components. Besides, the reduced stiffness matrix for the plate and defined below:

(9) 
where , and are Young’s modulus, Poisson’s ratio, and shear modulus of the FGCNTRC plate by combining the properties of the matrix (shown by m superscript) and carbon nanotubes (shown by CN superscript). These parameters are determined by the following relations[15]:

(10) 
The coefficients are introduced to account for the scaledependent material properties. Furthermore, and are the volume fractions of CNTs and matrix phase, respectively, and must satisfy the bellow condition:

(11) 
By considering Eqs. (19), the potential and kinetic energy can be expressed as below:

(12) 

(13) 
In the above equations, five parameters are undefined which include: , , , and . There are several methods to solve partial or ordinary equations [1618]. In the paper, the Ritz method is employed. separation of variables method is the first step for solving the equations:
(14) 








where, and are timedependent coefficients. In addition, , , , and are the shape functions have to be chosen according only to the essential boundary conditions. In this paper, the linear types of shape functions are used (Appendix A). Furthermore, the number of terms should be chosen to ensure the convergence of the total displacements. By considering Eq (1213) and appendix:

(15) 

(16) 
Motion equations can be determined based on the generalized Lagrange equations:
(17) 










By substituting the eq. (15) and eq. (16) to eq. (17):
(18) 








In addition, by applying Newton’s second law for contact force and considering Hertz’s contact law, the Equilibrium equation can be expressed as

(19) 
Finally, nonlinear coupled timedependent equations can be written as

(20) 
The relation between the mass and stiffness components is mentioned in Appendix (A). Eq. (20) can be solved by as fourthorder Runge–Kutta numerical method. The initial conditions are

(21) 
To investigate the model, the FEM simulation is applied in the ABAQUS. The 937500 C3D8R element is used in the FEM model (Figure 2). Surface to Surfacetosurface contact is considered between the impactor and plate.
Fig. 2. FEM model of the LVI on the FGCNTRC.
For validating the accuracy of the analytical model and FEM simulation, contact force and central deflection history computed by the analytical model and simulation are compared with Wang et al. model [19] as shown in Figure 3.
Table 1 shows the material and geometrical properties of the composite plate and the impactor used in Refs [15]. Figures 3(a) and 3(b) show a good agreement in the results.
Considering that the FEM simulation is based on solving the governing constitutive equations with the numerical method, but the analytical model presented in this paper is based on the thirdorder shear deformation Theory, Hertz contact law, and the energy principle, the difference between the results of the analytical model and the numerical method is greater than the difference between the results of the analytical model and Wang et al. model [19].
(a) 

(b) 

Fig. 3. Comparisons between the present model and FEM with Wang et al.model [20] for impact response of FGX 
Table 1. Geometrical and material properties of
FGCNRTC Plate and impactor [15].
Geometrical properties of the plate Simply supported boundary condition Size: 200.0×200.0×20 mm 
Mechanical properties of Poly (methyl methacrylate) ; ; Mechanical properties of singlewalled CNT (SWCNT) ; ; ; ;

properties of steel sphere impactor ; ; Diameter= 12.7 mm Velocity= 3 m/s 
In this section, the results of FGCNTRC types (including UD, FGV, FGX, and FGO) effects on the singlelayer FGCNTRC plate’s contact force and central deflection are discussed. As shown in figure 4 (a), The volume fraction of CNT is equal to 0.28 ( ). It is obvious from the figure, that the history of contact force of type X and V is similar together while the value of the maximum values of the contact force are equal to 925 and 926 N for the FGCNTRC plate of type V and X, respectively. The behavior is due to enhancement of the volume fraction and (Eq. (3)). Besides, the lowest contact time belongs to X and V types. FGO type has the maximum contact time and lowest contact force among other types of CNTs. From Figure 4 (b), It can be seen the FGCNTRC plate of type X has the lowest deflection and shortest contact time duration.
(a) 

(b) 

Fig. 4. Comparisons of impact response of four types 
The volume fraction of CNT is one of the effective parameters on the FGCNTRC plate behavior under LVI.
In the section, the X type of FGCNTRC plate is considered with different volume fractions
( ). The effect on the contact force and central deflection are shown in Figure 5(a) and 5(b), respectively. According to the figure, by increasing the volume fraction the maximum value of the contact force is increased while contact time duration and central deflection are decreased. This response is due to the relation between E_{22} and (Eq. (10) so that enhancement of the cause increasing of the E_{22} and .
(a) 

(b) 

Fig. 5. Comparisons of impact response of volume fraction of X type of FGCNTRC square plate: (a) contact force 
As illustrated in Eq.(2), the initial velocity of the impactor is an effective parameter of the kinetic energy of the system. The efficacy of initial velocity in the contact force and central deflection is shown in Figure 6. As seen from this figure (for three values of initial velocity 1,2 and 3 m/s), an increase in the initial velocity of the impactor, maximum value of contact force, and lower contact time duration are concluded. Besides, the central deflection of the plate increases as the initial velocity of the impactor increases.
(a) 

(b) 

Fig. 6. Comparisons of impact response of volume fraction of X type of FGCNTRC square plate with different initial velocities: (a) contact force and (b) central deflection. 
For the X type of FGCNRTC, and different values of radius impactor ( ), the response of the plate under LVI is shown in Figure 7. It can be seen from the figure by increasing the radius impactor, the maximum value of the contact force and contact time duration increased. The central deflection of the plate is also increased by increasing the radius impactor. Enhancement of the impactor mass, which is due to increasing the radius, is the main reason for the plate response.
(a) 

(b) 

Fig. 7. Comparisons of impact response of volume fraction of X type of FGCNTRC square plate with different radius impactors: (a) contact force and (b) central deflection. 
In the paper, the behavior of an FGCNTRC plate under Lowvelocity impact (LVI) is investigated analytically and numerically. Properties of the plate are obtained by applying a rule of mixture. Hertz law and the energy principle are used to derive the governing equations of Thirdorder Shear Deformation Theory (TSDT). The Ritz method based on the linear functions is used to discrete the system's space domain equations of motion, which is solved by the fourthorder RungeKutta method. It is concluded that:
Acknowledgments
We would like to thank Razi University for supporting this research.
Funding Statement
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this article.
Appendix
The linear shape functions which are presented in Eq(14) are:
where and are dependent boundary conditions. For simple support:
A(2) 
In the study, and are defined as below:
A(3) 

The components of the mass and stiffness matrix can be calculated by the following equations:
































































A(4) 




A(5) 

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