Smart Structures and Systems

  • 제27권1호
  • /
  • Pages.1-18
  • /
  • 2021
  • /
  • 1738-1584(pISSN)
  • /
  • 1738-1991(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

2D magneto-mechanical vibration analysis of a micro composite Timoshenko beam resting on orthotropic medium

  • Mehrabi, Mojtaba (Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan) ;
  • Mohammadimehr, Mehdi (Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan) ;
  • Mousavinejad, Fatemeh S. (Department of Civil Engineering, Faculty of Engineering, University of Kashan)
  • 투고 : 2019.08.07
  • 심사 : 2020.11.15
  • 발행 : 2021.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

In the present study, the free vibration analysis of a size-dependent micro composite Timoshenko beam model reinforced by various distributions of carbon nanotubes under temperature changes and two-dimensional magnetic field is investigated based on modified strain gradient theory. Also, the effects of environment are simulated by orthotropic elastic foundation and it is assumed that the material properties are temperature-dependent. Mathematical formulations are obtained using Hamilton's principle and the governing equations of motion are derived based on energy approach and variation method. These equations are solved using semi-analytical and numerical methods such as Navier's type solution, finite element method and generalized differential quadrature method for various boundary conditions. The obtained results of this study are compared with the other previous researches and there is a good agreement between them. The main purpose of this work is the comparison of various solution methods on the problem outputs. Thus, the results are compared together and the effects of solution approach on the dimensionless natural frequencies is developed. Moreover, the effects of length-to-thickness ratio, magnetic field, temperature changes, elastic foundation and carbon nanotubes volume fractions on the dimensionless natural frequencies are studied. The results of this article demonstrate that the micro composite Timoshenko beam reinforced by FG-O and FG-X CNTs have lowest and highest dimensionless natural frequency, respectively. It is investigated that the dimensionless natural frequency enhances by increasing the magnetic field in x and z-directions.

키워드

과제정보

The authors would like to thank the reviewers for their valuable comments and suggestions to improve the clarity of this work. Also, they would like to thank the Iranian Nanotechnology Development Committee for their financial support and the University of Kashan for supporting this work by Grant No. 682561/11.

참고문헌

  1. AkhavanAlavi, S.M., Mohammadimehr, M. and Edjtahed, S.H. (2019), "Active control of micro Reddy beam integrated with functionally graded nanocomposite sensor and actuator based on linear quadratic regulator method", Eur. J. Mech.-A/Solids 74, 449-461. https://doi.org/10.1016/j.euromechsol.2018.12.008
  2. Anitescu, C., Atroshchenko, E., Alajlan, N. and Rabczuk, T. (2019), "Artificial neural network methods for the solution of second order boundary value problems". Comput. Mater. Contin., 59(1), 345-359. https://doi.org/10.32604/cmc.2019.06641
  3. Ansari, R., Gholami, R. and Sahmani, S. (2011), "Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory", Compos. Struct., 94(1), 221-228. https://doi.org/10.1016/j.compstruct.2011.06.024
  4. Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V. and Sahmani, S. (2013), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko micro beams based on the most general strain gradient theory", Compos. Struct., 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048
  5. Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R. and Sadeghi, F. (2014), "Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams", Compos. Struct., 113, 316-327. https://doi.org/10.1016/j.compstruct.2014.03.015
  6. Arani, A.G., Mobarakeh, M.R., Shams, S. and Mohammadimehr, M. (2012), "The effect of CNT volume fraction on the magnetothermo-electro-mechanical behavior of smart nanocomposite cylinder", J. Mech. Sci. Technol., 26(8), 2565-2572. https://doi.org/10.1007/s12206-012-0639-5
  7. Arani, A.G. and Amir, S. (2013), "Electro-thermal vibration of visco-elastically coupled BNNT systems conveying fluid embedded on elastic foundation via strain gradient theory", Physica B., 419, 1-6. https://doi.org/10.1016/j.physb.2013.03.010
  8. Ardestani, M.M., Zhang, L.W. and Liew, K.M. (2017), "Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates", Comput. Method. Appl. Mech. Engin., 317, 341-379. https://doi.org/10.1016/j.cma.2016.12.009
  9. Areias, P., Rabczuk, T. and Msekh, M. (2016), "Phase-field analysis of finite-strain plates and shells including element subdivision", Comput. Method. Appl. Mech. Engin., 312, 322-350. https://doi.org/10.1016/j.cma.2016.01.020
  10. Banerjee, J.R. (2001), "Frequency equation and mode shape formulae for composite Timoshenko beam", Compos. Struct., 51(4), 381-388. https://doi.org/10.1016/S0263-8223(00)00153-7
  11. Barquero-Cabrero, J.D., Luevanos-Rojas, A., Lopez-Chavarria, S., Medina-Elizondo, M., Velazquez-Santillan, F. and Sandoval-Rivas, R. (2018), "Deflections and rotations in rectangular beams with straight haunches under uniformly distributed load considering the shear deformations", Smart. Struct. Syst., Int. J., 22(6), 689-697. https://doi.org/10.12989/sss.2018.22.6.689
  12. Beni, Y.T., Mehralian, F. and Razavi, H. (2015), "Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory", Compos. Struct., 120, 65-78. https://doi.org/10.1016/j.compstruct.2014.09.065
  13. Canales, F.G. and Mantari, J.L. (2017), "Elasto-plastic vibrational analysis of tapered bars under uniform axial loading considering shear deformation and rotary inertia", Int. J. Lin. Mech., 95, 103-116. https://doi.org/10.1016/j.ijnonlinmec.2017.06.001
  14. Chau-Dinh, T., Zi, G., Lee, P., Rubczuk, T. and Song, J. (2012), "Phantom-node method for shell models with arbitrary cracks", Comput. Struct., 92-93, 242-256. https://doi.org/10.1016/j.compstruc.2011.10.021
  15. Chu, L., Dui, G. and Ju, C. (2018), "Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory", Compos. Struct., 186, 39-49. https://doi.org/10.1016/j.compstruct.2017.10.083
  16. Domagalski, L. (2018), "Free and forced large amplitude vibrations of periodically inhomogeneous slender beams", Arch. Civil. Mech. Eng., 18(4), 1506-1519. https://doi.org/10.1016/j.acme.2018.06.005
  17. Farzampour, A., Eatherton, M.R., Mansouri, I. and Hu, J.W. (2019), "Effect of flexural and shear stresses simultaneously for optimized design of butterfly-shaped dampers: Computational study", Smart Struct. Syst., Int. J., 23(4), 329-335. http://doi.org/10.12989/sss.2019.23.4.329
  18. Furtak, K. and Rodacki, K. (2018), "Experimental investigation of load-bearing capacity of composite timber-glass I-beams", Arch. Cvil. Mech. Engin., 18(3), 956-964. https://doi.org/10.1016/j.acme.2018.02.002
  19. Ganesh, S., Kumar, K.S. and Mahato, P.K. (2016), "Free vibration analysis of delaminated composite plates using finite element method", Proce. Eng., 144, 1067-1075. https://doi.org/10.1016/j.proeng.2016.05.061
  20. Ghasemi, H., Park, H.S. and Rabczuk, T. (2017), "A level-set based IGA formulation for topology optimization of flexoelectric materials", Comput. Method. Appl. Mech. Engin., 313, 239-258. https://doi.org/10.1016/j.cma.2016.09.029
  21. Ghasemi, H., Park, H.S. and Rabczuk, T. (2018), "A multi-material level set-based topology optimization of flexoelectric composites", Comput. Method. Appl. Mech. Eng., 332, 47-62. https://doi.org/10.1016/j.cma.2017.12.005
  22. Hamdia, K.M., Ghasemi, H., Zhuang, X., Alajlan, N. and Rabczuk, T. (2018), "Sensitivity and uncertainty analysis for flexoelectric nanostructures", Comput. Method. Appl. Mech. Engin., 337, 95-109. https://doi.org/10.1016/j.cma.2018.03.016
  23. Hsu, Y.S. (2016), "Enriched finite element methods for Timoshenko beam free vibration analysis", Appl. Math. Model., 40(15-16), 7012-7033. https://doi.org/10.1016/j.apm.2016.02.042
  24. Guo, H., Zhuang, X. and Rabczuk, T. (2019), "A deep collocation method for the bending analysis of Kirchhoff plate". Comput. Mater. Contin., 59(2), 433-456. https://doi.org/10.32604/cmc.2019.06660
  25. Jelodari, I. and Nilseresht, A.H. (2018), "Effects of Lorentz force and induced electrical field on the thermal performance of a magnetic nanofluid-filled cubic cavity", J. Molec. Liq., 252, 296-310. https://doi.org/10.1016/j.molliq.2017.12.143.
  26. Kashani, B.K. and Sani, A.A. (2016), "Free vibration analysis of horizontal cylindrical shells including sloshing effect utilizing polar finite element", Europ. J. Mech. A/Solids., 58, 187-201. https://doi.org/10.1016/j.euromechsol.2016.02.002
  27. Ke, L.L. and Wang, Y.S. (2011), "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Compos. Struct., 93(2), 342-350. https://doi.org/10.1016/j.compstruct.2010.09.008
  28. Kutlu, A. and Omurtag, M.H. (2012), "Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method", Int. J. Mech. Sci., 65(1), 64-74. https://doi.org/10.1016/j.ijmecsci.2012.09.004
  29. Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  30. Lee, J. and Schultz, W.W. (2004), "Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method", J. Sound. Vib., 269(3-5), 609-621. https://doi.org/10.1016/S0022-460X(03)00047-6
  31. Lei, J., He, Y., Zhang, B., Gan, Z. and Zeng, P. (2013a), "Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory", Int. J. Eng. Sci., 72, 36-52. https://doi.org/10.1016/j.ijengsci.2013.06.012
  32. Lei, Z.X., Liew, K.M. and Yu, J.L. (2013b), "Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment", Compos. Struct., 106, 128-138. https://doi.org/10.1016/j.compstruct.2013.06.003
  33. Lei, Z.X., Zhang, L.W. and Liew, K.M. (2015), "Elastodynamic analysis of carbon nanotube-reinforced functionally graded plates", Int. J. Mech. Sci., 99, 208-217. https://doi.org/10.1016/j.ijmecsci.2015.05.014
  34. Lenci, S., Clementi, F. and Rega, G. (2017), "Comparing Nonlinear Free Vibrations of Timoshenko Beams with Mechanical or geometric curvature definition", Defin. Proce. IUTAM., 20, 34-41. https://doi.org/10.1016/j.piutam.2017.03.006
  35. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  36. Mirsalehi, M., Azhari, M. and Amoushahi, H. (2017), "Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method", Europ. J. Mech. A/Solid., 61, 1-13. https://doi.org/10.1016/j.euromechsol.2016.08.008
  37. Mohammadimehr, M. and Mehrabi, M. (2017), "Stability and free vibration analysis of double-bonded micro composite sandwich cylindrical shells conveying fluid flow", Appl. Math. Model., 47, 685-709. https://doi.org/10.1016/j.apm.2017.03.054.
  38. Mohammadimehr, M. and Mehrabi, M. (2018), "Electro-thermomechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow", Appl. Math. Model., 60, 255-272. https://doi.org/10.1016/j.apm.2018.03.008
  39. Mohammadimehr, M., Salemi, M. and Navi, B.R. (2016a), "Bending, buckling, and free vibration analysis of MSGT microcomposite Reddy plate reinforced by FG-SWCNTs with temperature-dependent material properties under hydro-thermomechanical loadings using DQM", Compos. Struct., 138, 361-380. https://doi.org/10.1016/j.compstruct.2015.11.055
  40. Mohammadimehr, M., Mohammadimehr, M.A. and Dashti, P. (2016b), "Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic microplate based on surface stress and modified couple stress theories using differential quadrature method", Appl. Mathe. Mech., 37(4), 529-554. https://doi.org/10.1007/s10483-016-2045-9
  41. Mohammadimehr, M., Mehrabi, M., Hadizadeh, H. and Hadizadeh, H. (2018a), "Surface and size dependent effects on static, buckling, and vibration of micro composite beam under thermo-magnetic fields based on strain gradient theory", Steel Compos. Struct., Int. J., 26(4), 513-531. http://doi.org/10.12989/scs.2018.26.4.513
  42. Mohammadimehr, M., Nejad, E.S. and Mehrabi, M. (2018b), "Buckling and vibration analyses of MGSGT double-bonded micro composite sandwich SSDT plates reinforced by CNTs and BNNTs with isotropic foam & flexible transversely orthotropic cores", Struct. Eng. Mech., Int. J., 65(4), 491-504. http://doi.org/10.12989/sem.2018.65.4.491
  43. Mohammadimehr, M., Mohammadi-Dehabadi, A.A., Akhavan Alavi, S.M., Alambeigi, K., Bamdad, M., Yazdani, R. and Hanifehlou, S. (2018c), "Bending, buckling, and free vibration analyses of carbon nanotube reinforced composite beams and experimental tensile test to obtain the mechanical properties of nanocompositem", Steel Compos. Struct., Int. J., 29(3), 405-422. https://doi.org/12989/scs.2018.29.3.405 https://doi.org/10.12989/scs.2018.29.3.405
  44. Mohammadimehr, M., Afshari, H., Salemi, M., Torabi, K. and Mehrabi, M. (2019), "Free vibration and buckling analyses of functionally graded annular thin sector plate in-plane loads using GDQM", Struct. Eng. Mech., Int. J., 71(5), 525-544. http://doi.org/10.12989/.2019.71.5.525
  45. Mohammadimehr, M., Mehrabi, M. and Mousavinejad, F.S. (2020), "Magneto-mechanical vibration analysis of single-/three-layered micro-Timoshenko porous beam and graphene platelet as reinforcement based on modified strain gradient theory and differential quadrature method", J. Vib. Control. https://doi.org/10.1177%2F1077546320949083 https://doi.org/10.1177%2F1077546320949083
  46. Nanthakumar, S.S., Lahmer, T., Zhuang, X., Zi, G. and Rabczuk, T. (2016), "Detection of material interfaces using a regularized level set method in piezoelectric structures", Inverse Prob. Sci. Eng., 24(1), 153-176. https://doi.org/10.1080/17415977.2015.1017485
  47. Nguyen-Thanh, N., Valizadeh, N., Nguyen, M.N., Nguyen-Xuan, H., Zhuang, X., Areias, P., Zi, G., Bazilevs, Y.D.L.L.R.T., De Lorenzis, L. and Rabczuk, T. (2015), "An extended isogeometric thin shell analysis based on Kirchhof-Love theory", Comput. Method. Appl. Mech. Eng., 284, 265-291. https://doi.org/10.1016/j.cma.2014.08.025
  48. Ozdemir, Y.I. (2018), "Using fourth order element for free vibration parametric analysis of thick plate resting on elastic foundation", Struct. Eng. Mech., Int. J., 65(3), 213-222. http://doi.org/10.12989/sem.2018.65.3.213
  49. Paluszny, A., Tang, X.H. and Zimmerman, R.W. (2013), "Fracture and impulse based finite-discrete element modeling of fragmentation", Comput. Mech., 52(5), 1071-1084. http://doi.org/10.1007/s00466-013-0864-5
  50. Pandey, S. and Pradyumna, S. (2015), "A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells", Compos. Struct., 133, 438-450. https://doi.org/10.1016/j.compstruct.2015.07.087
  51. Plagianakos, T.S. and Papadopoulos, E.G. (2015), "Coupled higher-order layerwise mechanics and finite element for cylindrical composite and sandwich shells with piezoelectric transducers", Europ. J. Mech. A/Solids., 54, 11-23. https://doi.org/10.1016/j.euromechsol.2015.06.003
  52. Pradhan, S.C. and Mandal, U. (2013), "Finite element analysis of CNTs based on nonlocal elasticity and Timoshenko beam theory including thermal effect", Physica E., 53, 223-232. https://doi.org/10.1016/j.physe.2013.04.029
  53. Qing, G., Qiu, J. and Liu, Y. (2018), "Free vibration analysis of Stiffened Laminated Plate using FEM", Mater. Today. Proce., 5(2), 5313-5321. https://doi.org/10.1016/j.matpr.2017.12.115
  54. Rabczuk, T., Ren, H. and Zhuang, X. (2019), "A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem", Comput. Mater. Contin., 59(1), 31-55. https://doi.org/10.32604/cmc.2019.04567
  55. Rajabi, J. and Mohammadimehr, M. (2019), "Bending analysis of a micro sandwich skew plate using extended Kantorovich method based on Eshelby-Mori-Tanaka approach", Comput. Concrete, Int. J., 23(5), 361-376. http://doi.org/10.12989/cac.2019.23.5.361
  56. Romanoff, J., Reddy, J.N. and Jelovica, J. (2016), "Using nonlocal Timoshenko beam theories for prediction of micro- and macro-structural responses", Compos. Struct., 156, 410-420. https://doi.org/10.1016/j.compstruct.2015.07.010
  57. Sapountzakis, E.J., Tsipiras, V.J. and Argyridi, A.K. (2015), "Torsional vibration analysis of bars including secondary torsional shear deformation effect by the boundary element method", J. Sound. Vib., 355, 208-231. https://doi.org/10.1016/j.jsv.2015.04.032
  58. Setoodeh, A.R., Shojaee, M. and Malekzadeh, P. (2018), "Application of transformed differential quadrature to free vibration analysis of FG-CNTRC quadrilateral spherical panel with piezoelectric layers", Comput. Method. Appl. Mech. Eng., 335, 510-537. https://doi.org/10.1016/j.cma.2018.02.022
  59. Shahedi, S. and Mohammadimehr, M. (2019), "Vibration analysis of rotating fully-bonded and delaminated sandwich beam with CNTRC face sheets and AL-foam flexible core in thermal and moisture environments", Mech. Based Des. Struct. Mach., 48(5), 584-614. https://doi.org/10.1080/15397734.2019.1646661
  60. Soleimani, I. and Beni, Y.T. (2018), "Vibration analysis of nanotubes based on two-node size dependent axisymmetric shell element", Arch. Civil. Mech. Eng., 18(4), 1345-1358. https://doi.org/10.1016/j.acme.2018.04.009
  61. Song, Z.G., Zhang, L.W. and Liew, K.M. (2016), "Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments", Int. J. Mech. Sci., 115-116, 339-347. https://doi.org/10.1016/j.ijmecsci.2016.06.020
  62. Soni, S., Jain, N.K. and Joshi, P.V. (2018), "Vibration analysis of partially cracked plate submerged in fluid", J. Sound. Vib., 412, 28-57. https://doi.org/10.1016/j.jsv.2017.09.016
  63. Tabbakh, M. and Nasihatgozar, M. (2018), "Buckling analysis of nanocomposite plates coated by magnetostrictive layer", Smart Struct. Syst., Int. J., 22(6), 743-751. http://.doi.org/10.12989/sss.2018.22.6.743
  64. Tang, X.H., Paluszny, A. and Zimmerman, R.W. (2013), "Energy conservative property of impulse-based methods for collision resolution", Int. J. Numer. Method. Eng., 95(6), 529-540. https://doi.org/10.1002/nme.4537
  65. Tang, X., Paluszny, A. and Zimmerman, R.W. (2014), "An impulse-based energy tracking method for collision resolution", Comput. Method. Appl. Mech. Eng., 278, 160-185. https://doi.org/10.1016/j.cma.2014.05.004
  66. Thai, S., Thai, H.T., Vo, T.P. and Nguyen-Xuan, H. (2017), "Nonlinear static and transient isogeometric analysis of functionally graded microplates based on the modified strain gradient theory", Eng. Struct., 153, 598-612. https://doi.org/10.1016/j.engstruct.2017.10.002
  67. Thai, C.H., Ferreira, A.J.M. and Nguyen-Xuan, H. (2018), "Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory", Compos. Struct., 192, 274-288. https://doi.org/10.1016/j.compstruct.2018.02.060
  68. Vu-Bac, N., Lahmer, T., Zhuang, X., Nguyen-Thoi, T. and Rabczuk, T. (2016), "A software framework for probabilistic sensitivity analysis for computationally expensive models", Comput. Method. Appl. Mech. Eng., 100, 19-31. https://doi.org/10.1016/j.advengsoft.2016.06.005
  69. Xu, Y., Qian, Y. and Song, G. (2016), "Stochastic finite element method for free vibration characteristics of random FGM", Appl. Math. Model., 40(23-24), 10238-10253. https://doi.org/10.1016/j.apm.2016.07.025
  70. Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solid. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  71. Yang, J., Xiong, J., Ma, L., Zhang, G., Wang, X. and Wu, L. (2014), "Study on vibration damping of composite sandwich cylindrical shell with pyramidal truss-like cores", Compos. Struct., 117, 362-372. https://doi.org/10.1016/j.compstruct.2014.06.042
  72. Yang, Y., Chen, L., Xu, D. and Zheng, H. (2016), "Free and forced vibration analyses using the four-node quadrilateral element with continuous nodal stress", Eng. Anal. Bound. Element., 70, 1-11. https://doi.org/10.1016/j.enganabound.2016.05.005
  73. Yue, Y.M., Xu, K.Y. and Chen, T. (2016), "A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects", Compos. Struct., 136, 278-286. https://doi.org/10.1016/j.compstruct.2015.09.046
  74. Zeinkiewicz, O.C. and Taylor, R.L. (2000), The Finite Element Method, (5th edition), Oxford: Butterworth- Heinemann, USA.
  75. Zhang, H., Wang, C.M., Ruocco, E. and Challamel, N. (2016a), "Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation", Eng. Struct., 126, 252-263. https://doi.org/10.1016/j.engstruct.2016.07.062
  76. Zhang, L.W., Zhang, Y. and Liew, K.M. (2016b), "Free vibration analysis of triangular nanocomposite plates subjected to in-plane stresses using an element-free method", Compos. Struct., 149, 247-260. https://doi.org/10.1016/j.compstruct.2016.04.019
  77. Zhu, P., Lei, Z.X. and Liew, K.M. (2012), "Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory", Compos. Struct., 94(4), 1450-1460. https://doi.org/10.1016/j.compstruct.2011.11.010