Nonlinear dynamic analysis of functionally graded carbon nanotube-reinforced composite plates using MISQ20 element
Quoc-Hoa PHAM1, Trung Thanh TRAN2, Phu-Cuong NGUYEN1()
1. Advanced Structural Engineering Laboratory, Department of Structural Engineering, Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City 700000, Vietnam 2. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi 100000, Vietnam
The main objective of this study is to further extend the mixed integration smoothed quadrilateral element with 20 unknowns of displacement (MISQ20) to investigate the nonlinear dynamic responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates with four types of carbon nanotube distributions. The smooth finite element method is used to enhance the accuracy of the Q4 element and avoid shear locking without using any shear correction factors. This method yields accurate results even if the element exhibits a concave quadrilateral shape and reduces the error when the element meshing is rough. Additionally, the element stiffness matrix is established by integrating the boundary of the smoothing domains. The motion equation of the FG-CNTRC plates is solved by adapting the Newmark method combined with the Newton–Raphson algorithm. Subsequently, the calculation program is coded in the MATLAB software and verified by comparing it with other published solutions. Finally, the effects of the input parameters on the nonlinear vibration of the plates are investigated.
E T Thostenson, Z Ren, T W Chou. Advances in the science and technology of carbon nanotubes and their composites: A review. Composites Science and Technology, 2001, 61(13): 1899–1912 https://doi.org/10.1016/S0266-3538(01)00094-X
2
B Fiedler, F H Gojny, M H G Wichmann, M C M Nolte, K Schulte. Fundamental aspects of nano-reinforced composites. Composites Science and Technology, 2006, 66(3): 115–125
3
C A Cooper, S R Cohen, A H Barber, H D Wagner. Detachment of nanotubes from a polymer matrix. Applied Physics Letters, 2002, 81(20): 3873–3875 https://doi.org/10.1063/1.1521585
4
A H Barber, S R Cohen, H D Wagner. Measurement of carbon nanotube–polymer interfacial strength. Applied Physics Letters, 2003, 82(4): 140–142
5
J GouB MinaieB WangZ LiangC Zhang. Computational and experimental study of interfacial bonding of single-walled nanotube reinforced composites. Computational Materials Science, 2004, 31(3−4): 225−236
6
S J V Frankland, A Caglar, D W Brenner, M Griebel. Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube polymer interfaces. Journal of Physical Chemistry B, 2002, 106(12): 3046–3048 https://doi.org/10.1021/jp015591+
7
P C Ma, S Y Mo, B Z Tang, J K Kim. Dispersion, interfacial interaction and reagglomeration of functionalized carbon nanotubes in epoxy composites. Carbon, 2010, 48(6): 1824–1834 https://doi.org/10.1016/j.carbon.2010.01.028
8
J N Coleman, U Khan, W J Blau, Y K Gun’ko. Small but strong: A review of the mechanical properties of carbon nanotube–polymer composites. Carbon, 2006, 44(9): 1624–1652 https://doi.org/10.1016/j.carbon.2006.02.038
9
H D Wagner, O Lourie, Y Feldman, R Tenne. Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix. Applied Physics Letters, 1998, 72(2): 188–190 https://doi.org/10.1063/1.120680
10
D Qian, E C Dickey, R Andrews, T Rantell. Load transfer and deformation mechanisms in carbon nanotube–polystyrene composites. Applied Physics Letters, 2000, 76(20): 2868–2870 https://doi.org/10.1063/1.126500
11
G M Odegard, T S Gates, L M Nicholson, K E Wise. Equivalent-continuum modeling of nano-structured materials. Composites Science and Technology, 2002, 62(14): 1869–1880 https://doi.org/10.1016/S0266-3538(02)00113-6
12
G M Odegard, T S Gates, K E Wise, C Park, E J Siochi. Constitutive modeling of nanotube-reinforced polymer composites. Composites Science and Technology, 2003, 63(11): 1671–1687 https://doi.org/10.1016/S0266-3538(03)00063-0
13
Y J Liu, X L Chen. Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element. Mechanics of Materials, 2003, 35(1−2): 69–81 https://doi.org/10.1016/S0167-6636(02)00200-4
14
N Hu, H Fukunaga, C Lu, M Kameyama, B Yan. Prediction of elastic properties of carbon nanotube reinforced composites. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2005, 461(2058): 1685–1710 https://doi.org/10.1098/rspa.2004.1422
15
S J V Frankland, V M Harik, G M Odegard, D W Brenner, T S Gates. The stress–strain behaviour of polymer–nanotube composites from molecular dynamics simulation. Composites Science and Technology, 2003, 63(11): 1655–1661 https://doi.org/10.1016/S0266-3538(03)00059-9
16
M Griebel, J Hamaekers. Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites. Computer Methods in Applied Mechanics and Engineering, 2004, 193(17−20): 1773–1788 https://doi.org/10.1016/j.cma.2003.12.025
17
J M Wernik, S A Meguid. Multiscalemodeling of the nonlinear response of nanoreinforced polymers. Acta Mechanica, 2011, 217(1−2): 1–16 https://doi.org/10.1007/s00707-010-0377-7
18
J Wuite, S Adali. Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis. Composite Structures, 2005, 71(3−4): 388–396 https://doi.org/10.1016/j.compstruct.2005.09.011
19
T Vodenitcharova, L C Zhang. Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube. International Journal of Solids and Structures, 2006, 43(10): 3006–3024 https://doi.org/10.1016/j.ijsolstr.2005.05.014
20
M C Ray, R C Batra. A single-walled carbon nanotube reinforced 1–3 piezoelectric composite for active control of smart structures. Smart Materials and Structures, 2007, 16(5): 1936–1947 https://doi.org/10.1088/0964-1726/16/5/051
21
G Formica, W Lacarbonara, R Alessi. Vibrations of carbon nanotube-reinforced composites. Journal of Sound and Vibration, 2010, 329(10): 1875–1889 https://doi.org/10.1016/j.jsv.2009.11.020
22
A Arani, S Maghamikia, M Mohammadimehr, A Arefmanesh. Buckling analysis of laminated composite rectangular plates reinforced by SWCNTS using analytical and finite element methods. Journal of Mechanical Science and Technology, 2011, 25(3): 809–820 https://doi.org/10.1007/s12206-011-0127-3
23
S H Shen. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 2009, 91(1): 9–19 https://doi.org/10.1016/j.compstruct.2009.04.026
24
Z X Wang, H S Shen. Nonlinear vibration of nanotube-reinforced composite plates in thermal environments. Computational Materials Science, 2011, 50(8): 2319–2330 https://doi.org/10.1016/j.commatsci.2011.03.005
25
Z X Wang, H S Shen. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. Composites. Part B, Engineering, 2011, 43(2): 411–421 https://doi.org/10.1016/j.compositesb.2011.04.040
26
L Zhang, W Cui, K Liew. Vibration analysis of functionally graded carbon nanotube reinforced composite thick plates with elastically restrained edges. International Journal of Mechanical Sciences, 2015, 103: 9–21 https://doi.org/10.1016/j.ijmecsci.2015.08.021
27
S Natarajan, M Haboussi, G Manickam. Application of higher-order structural theory to bending and free vibration analysis of sandwich plates with CNT reinforced composite facesheets. Composite Structures, 2014, 113: 197–207 https://doi.org/10.1016/j.compstruct.2014.03.007
28
A Sankar, S Natarajan, M Haboussi, K Ramajeyathilagam, M Ganapathi. Panel flutter characteristics of sandwich plates with CNT reinforced facesheets using an accurate higher-order theory. Journal of Fluids and Structures, 2014, 50: 376–391 https://doi.org/10.1016/j.jfluidstructs.2014.06.028
29
A Sankar, S Natarajan, T B Zineb, M Ganapathi. Investigation of supersonic flutter of thick doubly curved sandwich panels with CNT reinforced facesheets using higher-order structural theory. Composite Structures, 2015, 127: 340–355 https://doi.org/10.1016/j.compstruct.2015.02.047
30
A Sankar, S Natarajan, T Merzouki, M Ganapathi. Nonlinear dynamic thermal buckling of sandwich spherical and conical shells with CNT reinforced facesheets. International Journal of Structural Stability and Dynamics, 2017, 17(9): 1750100 https://doi.org/10.1142/S0219455417501000
31
J Rodrigues, S Natarajan, A Ferreira, E Carrera, M Cinefra, S Bordas. Analysis of composite plates through cell-based smoothed finite element and 4-noded mixed interpolation of tensorial components techniques. Computers & Structures, 2014, 135: 83–87 https://doi.org/10.1016/j.compstruc.2014.01.011
32
T J R HughesJ A CottrellY Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39−41): 4135−4195
33
M J BordenM A ScottJ A EvamsT J R Hughes. Isogeometric finite element data structures based on Bezier extraction of NURBS. International Journal for Numerical Methods in Engineering, 2011, 87 (1−5): 15−47
34
P Phung-Van, C H Thai, H Nguyen-Xuan, M Abdel-Wahab. An isogeometric approach of static and free vibration analyses for porous FG nanoplates. European Journal of Mechanics. A, Solids, 2019, 78: 103851 https://doi.org/10.1016/j.euromechsol.2019.103851
35
C L Thanh, T N Nguyen, T H Vu, S Khatir, M Abdel Wahab. A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate. Engineering with Computers, 2022, 38(Suppl 1): 449–460 https://doi.org/10.1007/s00366-020-01154-0
36
T Cuong-Le, K D Nguyen, M Hoang-Le, T Sang-To, P Phan-Vu, M A Wahab. Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate. Physica B, Condensed Matter, 2022, 631: 413726 https://doi.org/10.1016/j.physb.2022.413726
37
Q H PhamP C NguyenV K TranQ X LieuT T Tran. Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates. Engineering with Computers, 2022: 1−26
38
Q H Pham, P C Nguyen, T T Tran. Dynamic response of porous functionally graded sandwich nanoplates using nonlocal higher-order isogeometric analysis. Composite Structures, 2022, 290: 115565 https://doi.org/10.1016/j.compstruct.2022.115565
39
H Guo, H Zheng. The linear analysis of thin shell problems using the numerical manifold method. Thin-walled Structures, 2018, 124: 366–383 https://doi.org/10.1016/j.tws.2017.12.027
40
H Guo, H Zheng, X Zhuang. Numerical manifold method for vibration analysis of Kirchhoff’s plates of arbitrary geometry. Applied Mathematical Modelling, 2019, 66: 695–727 https://doi.org/10.1016/j.apm.2018.10.006
41
X Zhuang, H Guo, N Alajlan, H Zhu, T Rabczuk. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. European Journal of Mechanics. A, Solids, 2021, 87: 104225 https://doi.org/10.1016/j.euromechsol.2021.104225
42
H Guo, X Zhuang, T Rabczuk. A deep collocation method for the bending analysis of Kirchhoff plate. Computers Materials & Continua, 2021, 59(2): 433–456
43
E Samaniego, C Anitescu, S Goswami, V M Nguyen-Thanh, H Guo, K Hamdia, X Zhuang, T Rabczuk. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790 https://doi.org/10.1016/j.cma.2019.112790
44
M Ganapathi, T K Varadan, B S Sarma. Nonlinear flexural vibrations of laminated orthotropic plates. Computers & Structures, 1991, 39(6): 685–688 https://doi.org/10.1016/0045-7949(91)90211-4
45
T Kant, J R Kommineni. Large amplitude free vibration analysis of cross-ply composite and sandwich laminates with a refined theory and C0 finite elements. Computers & Structures, 1994, 50(1): 123–134
46
K D Anil. Large amplitude free vibration analysis of composite plates by finite element method. Thesis for the Master’s Degree. Rourkela: National Institute of Technology, 2010
47
A Javed. Dynamic stability of delaminated cross ply composite plates and shells. International Journal of Mechanical Sciences, 1998, 40(8): 805–823
48
P K Parhi, S K Bhattacharyya, P K Sinha. Hygrothermal effects on the dynamic behaviour of multiple delamated composite plates and shells. Journal of Sound and Vibration, 2001, 248(2): 195–214 https://doi.org/10.1006/jsvi.2000.3506
49
E Providas, M A Kattis. An assessment of two fundamental flat triangular shell elements with drilling rotations. Computers & Structures, 2000, 77(2): 129–139 https://doi.org/10.1016/S0045-7949(99)00215-1
50
A A Groenwold, N Slander. An efficient 4-node 24 DOF thick shell finite element with 5-point quadrature. Engineering Computations, 1995, 12(8): 723–747 https://doi.org/10.1108/02644409510104686
51
C K Choi, T Y Lee. Efficient remedy for membrane locking of 4-node flat shell elements by non-conforming modes. Computer Methods in Applied Mechanics and Engineering, 2003, 192(16–18): 1961–1971 https://doi.org/10.1016/S0045-7825(03)00203-2
52
G Pimpinelli. An assumed strain quadrilateral element with drilling degrees of freedom. Finite Elements in Analysis and Design, 2004, 41(3): 267–283 https://doi.org/10.1016/j.finel.2004.05.004
53
C Thai-Hoang, N Nguyen-Thanh, H Nguyen-Xuan, T Rabczuk. An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates. Applied Mathematics and Computation, 2011, 217(17): 7324–7348 https://doi.org/10.1016/j.amc.2011.02.024
54
H Phan-Dao, H Nguyen-Xuan, C Thai-Hoang, T Nguyen-Thoi, T Rabczuk. An edge-based smoothed finite element method for analysis of laminated composite plates. International Journal of Computational Methods, 2013, 10(1): 1340005 https://doi.org/10.1142/S0219876213400057
55
N Nguyen-Thanh, T Rabczuk, H Nguyen-Xuan, S Bordas. A smoothed finite element method for shell analysis. Computer Methods in Applied Mechanics and Engineering, 2008, 198(2): 165–177 https://doi.org/10.1016/j.cma.2008.05.029
56
H Nguyen-XuanT RabczukS BordasJ F Debongnie. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering, 2008, 197(13−16): 1184−1203
57
H S Shen. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially loaded shells. Composite Structures, 2011, 93(8): 2096–2108 https://doi.org/10.1016/j.compstruct.2011.02.011
58
H Nguyen-Van, N Nguyen-Hoai, T Chau-Dinh, T Nguyen-Thoi. Geometrically nonlinear analysis of composite plates and shells via a quadrilateral element with good coarse-mesh accuracy. Composite Structures, 2014, 112: 327–338 https://doi.org/10.1016/j.compstruct.2014.02.024
59
J N Reddy. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. Boca Raton: CRC Press, 2004
60
A K Chopra. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Upper Saddle River: Pearson Prentice Hall, 2007
61
P C Nguyen, S E Kim. Investigating effects of various base restraints on the nonlinear inelastic static and seismic responses of steel frames. International Journal of Non-linear Mechanics, 2017, 89: 151–167 https://doi.org/10.1016/j.ijnonlinmec.2016.12.011
62
TT TranQH PhamT Nguyen-Thoi. Dynamic analysis of sandwich auxetic honeycomb plates subjected to moving oscillator load on elastic foundation. Advances in Materials Science and Engineering, 2020, 6309130
63
H N Nguyen, T N Canh, T T Thanh, T V Ke, V D Phan, D V Thom. Finite element modelling of a composite shell with shear connectors. Symmetry, 2019, 11(4): 527 https://doi.org/10.3390/sym11040527
64
T T Tran, V K Tran, P B Le, V M Phung, V T Do, H N Nguyen. Forced vibration analysis of laminated composite shells reinforced with graphene nanoplatelets using finite element method. Advances in Civil Engineering, 2020, 2020: 1471037 https://doi.org/10.1155/2020/1471037
N Sundararajan, T Prakash, M Ganapathi. Nonlinear free exural vibrations of functionally graded rectangular and skew plates under thermal environments. Finite Elements in Analysis and Design, 2005, 42(2): 152–168 https://doi.org/10.1016/j.finel.2005.06.001
67
V Balamurugan, M Ganapathi, T Varadan. Nonlinear dynamic instability of laminated composite plates using finite element method. Computers & Structures, 1996, 60(1): 125–130 https://doi.org/10.1016/0045-7949(95)00368-1
68
P Phung-Van, M Abdel-Wahab, K Liew, S Bordas, H Nguyen-Xuan. Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory. Composite Structures, 2015, 123: 137–149 https://doi.org/10.1016/j.compstruct.2014.12.021
