This paper carries out forced vibration analysis of graphene nanoplateletreinforced composite laminated shells in thermal environments by employing the finite element method (FEM). Material properties including elastic modulus, specific gravity, and Poisson’s ratio are determined according to the Halpin–Tsai model. The firstorder shear deformation theory (FSDT), which is based on the 8node isoparametric element to establish the oscillation equation of shell structure, is employed in this work. We then code the computing program in the MATLAB application and examine the verification of convergence rate and reliability of the program by comparing the data of present work with those of other exact solutions. The effects of both geometric parameters and mechanical properties of materials on the forced vibration of the structure are investigated.
Nowadays, due to the development of science and technology, a variety of new materials have been applied widely to engineering applications such as composite materials, functionally graded materials (FGM), and piezoelectric materials (PZT). To enhance the strengths of structures, there are many common ways like adding stiffeners, using folded structures, or reinforcing the structures with smart materials. In these techniques, the carbon nanotube (CNT)reinforced composite shell is one of the most modern structures [
In [
Shen and his coworkers [
In the above studies, to our knowledge, most plate and shell structures are reinforced with graphenes, where a few of which dealt with graphenereinforced composite shells. In addition, the forced vibration problems of graphenereinforced composite shells with the different mass fractions of graphene of each component in thermal environments are still limitations. The main goals of this work are to analyze the forced vibration of the shell structures (Figure
Composite shell with 4 graphene reinforcement forms.
The organization of this paper is divided into 4 main sections. Section
Consider a curvilinear shell model reinforced graphene nanoplatelets with the following dimensions: length
CYL:
SPH:
The displacement field is defined through FSDT as follows:
The stressstrain relationship is expressed as follows:
Equation (
The stress and strain fields of layer
The parameters of the thermal development along the longitudinal and transverse directions of composite structures reinforced with graphene can be defined as follows:
In this work, we employ the 8node isoparametric element, where each node consists of five degrees of freedom (DOF); the degree of freedom of node
By substituting in the expression for verifying displacement of element, we have
To obtain the dynamic equation, we employ the weak form for each element, and then we obtain
By substituting equations (
For the case of taking into account the structural damping ratio, we obtain the forced oscillation connection of the structure element as follows:
From the differential equation of forced vibration of the shell element (
Flowchart of Newmarkbeta method for solving the dynamic response problem of the shell.
When
By assuming that
Now, we have the flowchart of the Newmarkbeta method with symbols “
Determine the initial conditions
From the initial conditions, we obtain
By approximating
The condition to stabilize the roots is expressed as follows:
Compute the efficiency stiffness matrix and the efficiency nodal force vector:
Determine the nodal displacement vector
Based on the finite element equations, the authors then code the calculation program in the MATLAB environment to examine the influence of some geometric parameters and mechanical properties on the forced vibration of the shell. In this study, we consider four different cases of graphene reinforcement: UDtype, Xtype, Otype, and Vtype with ten composite layers, which depend on the mass fraction of graphene. For the whole analysis, the volume component of graphene in the all considered cases of ten composite layers is expressed as follows [
Case 1: UDtype: [0.07/0.07/0.07/0.07/0.07]_{s}
Case 2: Xtype: [0.11/0.09/0.07/0.05/0.03]_{s}
Case 3: Otype: [0.03/0.05/0.07/0.09/0.11]_{s}
Case 4: Vtype: [(0.11)_{2}/(0.09)_{2}/(0.07)_{2}/(0.05)_{2}/(0.03)_{2}]
Herein, we choose polymethyl methacrylate (PMMA) as the matrix, which has temperaturedependent material properties as
Temperaturedependent material characteristics of monolayer graphene.
T (K) 






300  1.812  1.807  0.683  −0.90  −0.95 
400  1.769  1.763  0.691  −0.35  −0.40 
In Table
Efficiency parameters related to volume component of graphene.
T (K) 





300  0.03  2.929  2.855  11.842 
0.05  3.068  2.962  15.944  
0.07  3.013  2.966  23.575  
0.09  2.647  2.609  32.816  
0.11  2.311  2.260  33.125  


400  0.03  2.977  2.896  13.928 
0.05  3.128  3.023  15.229  
0.07  3.060  3.027  22.588  
0.09  2.701  2.603  28.868  
0.11  2.405  2.337  29.527 
Let us consider a (0°/90°/0°/90°/0°)s composite shell with geometrical parameters:
In ref. [
The first nondimensional fundamental frequencies with different meshes.
Cases of GPL  UDtype  Xtype  Otype  Vtype  

BC  SSSS  CCCC  SSSS  CCCC  SSSS  CCCC  SSSS  CCCC  


Mesh  4 × 4  28.9466  45.7319  29.7675  44.5987  23.9914  39.9593  25.9421  41.9856 
6 × 6  28.9305  45.6370  29.7532  44.5122  23.9789  39.8547  25.9291  41.8878  
8 × 8  28.9279  45.6238  29.7508  44.4995  23.9769  39.8417  25.9270  41.8752  
10 × 10  28.9273  45.6202  29.7502  44.4959  23.9763  39.8384  25.9291  41.8719  
12 × 12  28.9270  45.6189  29.7500  44.4946  23.9761  39.8372  25.9264  41.8706  
14 × 14  28.9269  45.6183  29.7499  44.4940  23.9760  39.8367  25.9262  41.8701  
[ 
29.5920  44.0300  30.8370  43.2450  25.3060  39.4860  27.4230  40.5730  




Mesh  4 × 4  27.0804  43.7319  28.4903  41.3521  22.7309  38.5900  24.5928  40.6802 
6 × 6  27.0659  43.6467  28.4761  41.2750  22.7179  38.4899  24.5792  40.5859  
8 × 8  27.0635  43.6348  28.4737  41.2635  22.7158  38.4780  24.5770  40.5741  
10 × 10  27.0629  43.6316  28.4731  41.2603  22.7152  38.4750  24.5764  40.5711  
12 × 12  27.0627  43.6304  28.4729  41.2590  22.7150  38.4739  24.5762  40.5700  
14 × 14  27.0626  43.6299  28.4728  41.2585  22.7149  38.4734  24.5761  40.5695  
[ 
28.4430  42.4540  29.7630  41.7740  23.9830  37.9270  25.9830  39.0500 
Consider a fully clamped square plate with parameters that can be found in [
We can see from Figure
The deflection of the center point of the structure over time.
Next, we study the effects of some parameters on forced vibration of the shell structure; we consider a fully clamped composite shell consisting of ten composite layers with the following geometrical parameters: the length
Nondimensional displacement and velocity of the center point over time are given as follows:
The nondimensional frequencies are calculated as
The first five nondimensional fundamental frequencies of fully clamped shell.
UDtype  Xtype  Otype  Vtype  

 

55.0585  57.9374  46.0600  49.0584 

105.4688  111.5374  87.5778  93.7707 

105.7189  111.9037  87.6186  93.9600 

151.0943  160.8222  125.7792  135.2560 

184.6815  194.3923  154.0573  164.2965 


 

51.9456  54.5550  43.5648  46.7749 

99.3956  104.9268  82.7213  89.1410 

99.6000  105.2008  82.7251  89.4610 

141.8134  150.0833  118.5870  128.0876 

174.0027  182.9746  145.3327  156.2593 
First four vibration mode shapes of the shell structure are presented in Figure
First four vibration mode shapes of fully clamped SPH: (a) mode 1; (b) mode 2; (c) mode 3; (d) mode 4.
In this section, we investigate the effect of four different types of graphene reinforcement. Let us consider a fully clamped (CCCC) composite shell with geometrical parameters
Nondimensional deflection, stress, and velocity of the center point at
Nondimensional deflection, stress, and velocity of the center point at
Effect of four different types of graphene reinforcement on nondimensional velocity and deflection of the center point.
Temperature 

 

Maximum value 




UDtype  0.0705  0.1464  0.0873  0.1655 
Xtype  0.0616  0.1383  0.0768  0.1493 
Otype  0.1068  0.2062  0.1268  0.2251 
Vtype  0.0931  0.1846  0.1097  0.2009 
From Figures
Next, to investigate the influence of
Influence of
Influence of
Influence of

10  25  50  75  

Maximum value 








Temperature 

UDtype  0.0052  0.0182  0.0705  0.1464  0.4652  0.6832  1.1324  1.4665 
Xtype  0.0046  0.0171  0.0616  0.1383  0.4311  0.6469  1.0903  1.4413 
Otype  0.0074  0.0212  0.1068  0.2062  0.6239  0.8364  1.4910  1.6342 
Vtype  0.0065  0.0180  0.0931  0.1846  0.5753  0.7604  1.4027  1.6145 


Temperature 

UDtype  0.0072  0.0176  0.0873  0.1655  0.5418  0.6793  1.3206  1.5670 
Xtype  0.0064  0.0164  0.0768  0.1493  0.5058  0.6837  1.2767  1.5529 
Otype  0.0100  0.0237  0.1268  0.2251  0.7014  0.9191  1.6752  1.7314 
Vtype  0.0086  0.0210  0.1097  0.2009  0.6417  0.8327  0.9758  1.6705 
We can see in Figures
Next, we examine the influence of the lengthtowidth ratio on the nondimensional deflection and velocity of the center point of the shell with
Influence of lengthtowidth ratio on nondimensional deflection, velocity, and stress of the centroid at
Influence of lengthtowidth ratio on nondimensional deflection, velocity, and stress of the center point at
Influence of lengthtowidth ratio on the nondimensional deflection and velocity of the center point.

0.5  1.0  2.0  4.0  

Maximum value 








Temperature 

UDtype  0.1258  0.2321  0.0705  0.1464  0.0086  0.0340  0.0006  0.0036 
Xtype  0.1186  0.2465  0.0616  0.1383  0.0076  0.0311  0.0005  0.0035 
Otype  0.1892  0.2915  0.1068  0.2062  0.0129  0.0475  0.0008  0.0049 
Vtype  0.1671  0.2994  0.0931  0.1846  0.0112  0.0452  0.0007  0.0045 


Temperature 

UDtype  0.1531  0.2732  0.0873  0.1655  0.0107  0.0402  0.0007  0.0042 
Xtype  0.1377  0.2283  0.0768  0.1493  0.0095  0.0346  0.0007  0.0039 
Otype  0.2250  0.3244  0.1268  0.2251  0.0158  0.0415  0.0010  0.0055 
Vtype  0.1963  0.2949  0.1097  0.2009  0.0135  0.0473  0.0009  0.0051 
Now, we can see in Figures
In this paper, the numerical results of mechanical responses of laminated composite shells reinforced with graphene nanoplatelets are explored. The authors use FEM based on FSDT that has the following advantages:
Simple formulations for a theoretical representation
The computed results of free vibration and forced vibration obtained by this approach compared to other solutions show a good agreement
From new computed results, several conclusions may be drawn as follows:
With the same geometric parameters, the Xtype has the largest hardness and the smallest one is the Otype structure. In the temperature environment, the shells with all reinforcement types are greater in the displacement response, velocity, and stress than those of the shells in the room temperature. This demonstrates that heat reduces the stiffness of the shell, and thermal stresses appear in the structure. Thus, due to the effect of the external load, the displacement response tends to move away from position 0.
When increasing
We also understand that in all cases, the maximum displacement of the centroid of the structures performing in the temperature environment
Finally, based on achieved numerical results, the proposed program is able to analyze the static bending, dynamic response, nonlinear problems, etc., with complicated structures, which is not convenient to solve by analytical and other methods.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
DVT gratefully acknowledges the support of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 107.022018.30.