Structural Engineering and Mechanics

  • 제73권6호
  • /
  • Pages.685-698
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  • 2020
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Size-dependent dynamic stability of a FG polymer microbeam reinforced by graphene oxides

  • Wang, Yuewu (Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory of CO2 Utilization and Reduction Technology, Department of Energy and Power Engineering, Tsinghua University) ;
  • Xie, Ke (Institute of Systems Engineering, China Academy of Engineering Physics) ;
  • Fu, Tairan (Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Beijing Key Laboratory of CO2 Utilization and Reduction Technology, Department of Energy and Power Engineering, Tsinghua University)
  • 투고 : 2019.07.04
  • 심사 : 2019.11.12
  • 발행 : 2020.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

The dynamic stability of a functionally graded polymer microbeam reinforced by graphene oxides subjected to a periodic axial force is investigated. The microbeam is assumed to rest on an elastic substrate and is subjected to various immovable boundary restraints. The weight fraction of graphene oxides nanofillers is graded across the beam thickness. The effective Young's modulus of the functionally graded graphene oxides reinforced composite (FG-GORC) was determined using modified Halpin-Tsai model, with the mixture rule used to evaluate the effective Poisson's ratio and the mass density. An improved third order shear deformation theory (TSDT) is used in conjunction with the Chebyshev polynomial-based Ritz method to derive the Mathieu-Hill equations for dynamic stability of the FG-GORC microbeam, in which the scale effect is taken into account based on modified couple stress theory. Then, the Mathieu-Hill equation was solved using Bolotin's method to predict the principle unstable regions of the FG-GORC microbeams. The numerical results show the effects of the small scale, the graphene oxides nanofillers as well as the elastic substrate on the dynamic stability behaviors of the FG-GORC microbeams.

키워드

과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China, NSFC, China Postdoctoral Science Foundation

This study was supported by the National Key Research and Development Program of China (No. 2016YFC0802500), the National Natural Science Foundation of China (No. 51976097), the Science Fund for Creative Research of Groups of NSFC (No. 51621062), and the China Postdoctoral Science Foundation (2018M641333). We thank Prof. D.M. Christopher for editing the English

참고문헌

  1. Arani, A.G. and Kiani, F. (2018), "Nonlinear free and forced vibration analysis of microbeams resting on the nonlinear orthotropic visco-Pasternak foundation with different boundary conditions", Steel Compos. Struct., 28(2), 149-165. http://dx.doi.org/10.12989/scs.2018.28.2.149.
  2. Arefi, M., Bidgoli, E.M.R., Dimitri, R. and Tornabene, F. (2018), "Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", Aerosp. Sci. Technol., 81, 108-117. https://doi.org/10.1016/j.ast.2018.07.036.
  3. Ashrafi, B., Hubert, P. and Vengallatore S. (2006), "Carbon nanotube-reinforced composites as structural materials for microactuators in microelectromechanical systems", Nanotechnology, 17(19), 4895. https://doi.org/10.1088/0957-4484/17/19/019.
  4. Bolotin, V.V. (1964), The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, CA, USA.
  5. Chen, L. and Zhang, W.P. (2017), "Chebyshev polynomials and their some interesting applications", Adv. Differ. Equ. 2017, 303. https://doi.org/10.1186/s13662-017-1365-1.
  6. Chen, X. Lu, Y. and Li, Y. (2019), "Free vibration, buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium", Appl. Math. Model., 67, 430-448. https://doi.org/10.1016/j.apm.2018.11.004.
  7. Chen,W.Q., Lu, C.F. and Bian, Z.G. (2004), "A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation", Appl. Math. Model., 28(10), 877-890. https://doi.org/10.1016/j.apm.2004.04.001.
  8. Chong, A.C.M., Yang, F. and Lam, D.C.C. (2001), "Torsion and bending of micron-scaled structures", J. Mater. Res., 16, 1052-1058. https://doi.org/10.1557/JMR.2001.0146.
  9. Ebrahimi, F. and Barati, M.R. (2016a), "Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams", Mech. Adv. Mater. Struct., 24(11), 924-936. https://doi.org/10.1080/15376494.2016.1196795.
  10. Ebrahimi, F. and Barati, M.R. (2016b), "Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field", Appl. Phys. A, 122, 451. https://doi.org/10.1007/s00339-016-0001-3.
  11. Ebrahimi, F. and Barati, M.R. (2017a), "Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory", Mech. Adv. Mater. Struct., 25(4), 350-359. https://doi.org/10.1080/15376494.2016.1255830.
  12. Ebrahimi, F. and Barati, M.R. (2017b), "Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium", J. Brazilian Soc. Mech. Sci. Eng., 39(3), 937-952. https://doi.org/10.1007/s40430-016-0551-5.
  13. Ebrahimi, F. and Barati, M.R. (2017c), "Through-the-length temperature distribution effects on thermal vibration analysis of nonlocal strain-gradient axially graded nanobeams subjected to nonuniform magnetic field", J. Therm. Stresses, 40(5), 548-563. https://doi.org/10.1080/01495739.2016.1254076.
  14. Ebrahimi, F., Barati, M.R. and Haghi, P. (2017d), "Thermal effects on wave propagation characteristics of rotating strain gradient temperature-dependent functionally graded nanoscale beams", J. Therm. Stresses, 40(5), 535-547. https://doi.org/10.1080/01495739.2016.1230483.
  15. Ebrahimi, F. and Barati, M.R. (2017e), "Porosity-dependent vibration analysis of piezo-magnetically actuated heterogeneous nanobeams", Mech. Syst. Signal Pr., 93, 445-459. https://doi.org/10.1016/j.ymssp.2017.02.021.
  16. Ebrahimi, F. and Barati, M.R. (2018a), "Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment", J. Vib. Control, 24(3), 549-564. https://doi.org/10.1177/1077546316646239.
  17. Ebrahimi, F. and Barati, M.R. (2018b), "Stability analysis of porous multi-phase nanocrystalline nonlocal beams based on a general higher-order couple-stress beam model", Struct. Eng. Mech., 65(4), 465-476. https://doi.org/10.12989/sem.2018.65.4.465.
  18. Ebrahimi, F. and Dabbagh, A. (2017), "Nonlocal strain gradient based wave dispersion behavior of smart rotating magneto-electro-elastic nanoplates", Mater. Res. Express, 4(2), 025003. https://doi.org/10.1088/2053-1591/aa55b5.
  19. Ebrahimi, F. and Salari, E. (2015a), "Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams", Smart Mater. Struct., 24, 125007. https://doi.org/10.1088/0964-1726/24/12/125007.
  20. Ebrahimi, F., Salari, E. and Hosseini, S.A.H. (2015b), "Thermomechanical vibration behavior of FG nanobeams subjected to linear and non-linear temperature distributions", J. Therm. Stresses, 38(12), 1360-1386. https://doi.org/10.1080/01495739.2015.1073980.
  21. Ebrahimi, F. and Salari, E. (2015c), "A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position", CMES: Computer Modeling in Engineering & Sciences, 105(2), 151-181. https://doi.org/10.3970/cmes.2015.105.151.
  22. Ebrahimi, F., Ghasemi, F. and Salari, E. (2016), “Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities”. Meccanica, 51(1), 223-249. https://doi.org/10.1007/s11012-015-0208-y.
  23. Ebrahimi, F. and Salari, E. (2016), "Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams", Mech. Adv. Mater. Struct., 23(12), 1379-1397. https://doi.org/10.1080/15376494.2015.1091524.
  24. Ebrahimi, F. and Hosseini, S.H.S. (2016), "Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates", J. Therm. Stresses, 39(5), 606-625. https://doi.org/10.1080/01495739.2016.1160684.
  25. Ebrahimi, F. and Mokhtari, M. (2015), "Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method", J. Brazilian Soc. Mech. Sci. Eng., 37(4), 1435-1444. https://doi.org/10.1007/s40430-014-0255-7.
  26. Fox, L. and Parker, I.B. (1968), Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, UK.
  27. Graphene-info. (2019), Graphene Oxide: Introduction and Market News; Metalgrass LTD, Herzerlia, Israel, https://www.graphene-info.com/graphene-oxide, (accessed: 01-June-2019)
  28. Harris, B. (1986), Engineering Composite Materials, Institute of Metals, London, UK.
  29. Javani, R., Bidgoli R.M. and Kolahchi R. (2019), "Buckling analysis of plates reinforced by Graphene platelet based on Halpin-Tsai and Reddy theories", Steel Compos. Struct., 31(4), 419-427. http://dx.doi.org/10.12989/scs.2019.31.4.419.
  30. Karami, B., Shahsavari, D., Janghorban, M. and Tounsi, A. (2019), "Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", Int. J. Mech. Sci., 156, 94-105. https://doi.org/10.1016/j.ijmecsci.2019.03.036.
  31. Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y. (2009), "Flexural vibration and elastic buckling of a cracked timoshenko beam made of functionally graded materials", Mech. Adv. Mater. Struct., 16(6), 488-502. https://doi.org/10.1080/15376490902781175.
  32. Ke, L.L., Yang, J. and Kitipornchai, S. (2013), "Dynamic stability of functionally graded carbon nanotube-reinforced composite beams", Mech. Adv. Mater. Struct., 20(1), 28-37. https://doi.org/10.1080/15376494.2011.581412.
  33. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids. 51, 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  34. Liang, K., Sun, Q. and Liu, X.R. (2018), "Investigation on imperfection sensitivity of composite cylindrical shells using the nonlinearity reduction technique and the polynomial chaos method", Acta Astronaut, 146, 349-358. https://doi.org/10.1016/j.actaastro.2018.03.018.
  35. Li, C., Thostenson, E.T. and Chou, T.W. (2008), "Sensors and actuators based on carbon nanotubes and their composites: A review", Compos. Sci. Technol., 68(6), 1227-1249. https://doi.org/10.1016/j.compscitech.2008.01.006.
  36. Li, X., Bhushan, B., Takashima, K., Baek, C.W. and Kim, Y.K.. (2003), "Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques", Ultramicroscopy, 97(1), 481-494. https://doi.org/10.1016/S0304-3991(03)00077-9.
  37. Mao, J.J. and Zhang, W. (2019), "Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces", Compos. Struct., 216(5), 392-405. https://doi.org/10.1016/j.compstruct.2019.02.095.
  38. Mahkam, M., Rafi, A.A., Faraji, L. and Zakerzadeh, E. (2015), "Preparation of poly (methacrylic acid)-graphene oxide nanocomposite as a pH-Sensitive drug carrier through in-situ copolymerization of methacrylic acid with polymerizable graphene", Polymer-Plastics Technology and Engineering, 54(9), 916-922. https://doi.org/10.1080/03602559.2014.961081.
  39. Mohammed, A. and Cagri, M. (2018), "Dynamic stability of sandwich functionally graded micro-beam based on the nonlocal strain gradient theory with thermal effect", Compos. Struct., 201, 1018-1030. https://doi.org/10.1016/j.compstruct.2018.06.035.
  40. Mindlin, R.D. (1963), "Influence of couple-stresses on stress concentrations", Exp. Mech., 3, 1-7. https://doi.org/10.1007/BF02327219.
  41. Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16, 2355-2359. https://doi.org/10.1088/0960-1317/16/11/015.
  42. Potts, R.J., Dreyer, R.D., Bielawski, W.C. and Ruoff, S.R. (2011), "Graphene-based polymer nanocomposites", Polymer, 52(1), 5-25. https://doi.org/10.1016/j.polymer.2010.11.042.
  43. Ramaratnam, A. and Jalili, N. (2006), "Reinforcement of piezoelectric polymers with carbon nanotubes: pathway to next-generation sensors", J. Intel. Mat. Syst. Str., 17(3), 199-208. https://doi.org/10.1177/1045389X06055282.
  44. Reddy, J.N. (2003), Mechanics of Laminated Composite Plates and Shells: Theory and Application, (2nd Edition), CRC Press, New York, Washington D.C., USA.
  45. Rokni, H., Milani, A.S. and Seethaler, R.J. (2012a), "Improvement in dynamic properties of laminated MWCNT-polystyrene composite beams via an integrated numerical-experimental approach", Compos. Struct., 94(8), 2538-2547. https://doi.org/10.1016/j.compstruct.2012.03.028.
  46. Rokni, H., Milani, A.S. and Seethaler, R.J. (2012b), "2D optimum distribution of carbon nanotubes to maximize fundamental natural frequency of polymer composite micro-beams", Compos. Part B: Eng., 43(2), 779-785. https://doi.org/10.1016/j.compositesb.2011.07.012.
  47. Saemul, S. and Ganesan, R. (2018), "Dynamic instability of rotating doubly-tapered laminated composite beams under periodic rotational speeds", Compos. Struct., 200, 711-728. https://doi.org/10.1016/j.compstruct.2018.05.133.
  48. Setoodeh, A. and Rezae, M. (2017), "Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation", Struct. Eng. Mech., 61(2), 209-220. http://dx.doi.org/10.12989/sem.2017.61.2.209.
  49. Shi, G. (2007), "A new simple third-order shear deformation theory of plates", Int. J. Solids Struct., 44(13), 4399-4417. https://doi.org/10.1016/j.ijsolstr.2006.11.031.
  50. Song, M., Kitipornchai, S. and Yang, J. (2016), "Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets", Compos. Struct., 159, 579-588. https://doi.org/10.1016/j.compstruct.2016.09.070.
  51. Trinh, C.L., Vo, P.T., Thai, H.T. and Nguyen, T.K. (2018), "Size-dependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions", Compos. Part B: Eng., 134, 225-245. https://doi.org/10.1016/j.compositesb.2017.09.054.
  52. Van Es, M.A. (2001), "Polymer-clay nanocomposites: the importance of particle dimensions", Ph.D. Dissertation, Delft University of Technology, Delft, Netherlands.
  53. Wang, A., Chen, H., Hao, Y. and Zhang, W. (2016), "Vibration and bending behavior of functionally graded nanocomposite doubly-curved shallow shells reinforced by graphene nanoplatelets", Results in Physics, 9, 550-559. https://doi.org/10.1016/j.rinp.2018.02.062.
  54. Wang, Y., Xie, K., Fu, T. and Shi, C. (2019a), "Vibration response of a functionally graded graphene nanoplatelet reinforced composite beam under two successive moving masses", Compos. Struct., 209, 928-939. https://doi.org/10.1016/j.compstruct.2018.11.014.
  55. Wang, Y., Xie, K., Shi, C. and Fu, T. (2019b), "Nonlinear bending of axially functionally graded microbeams reinforced by graphene nanoplatelets in thermal environments", Mater. Res. Express, 6, 085615. https://doi.org/10.1088/2053-1591/ab1eef.
  56. Wattanasakulpong, N., Gangadhara, B.P. and Donald, W.K. (2011), "Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams", Int. J. Mech. Sci., 53(9), 734-743. https://doi.org/10.1016/j.ijmecsci.2011.06.005.
  57. Wattanasakulpong, N. and Bui, T.Q. (2018), "Vibration analysis of third-order shear deformable FGM beams with elastic support by Chebyshev collocation method", Int. J. Struct. Stability Dynam., 18(5), 1850071. https://doi.org/10.1142/S0219455418500712.
  58. Weon, J.I. (2009), "Mechanical and thermal behavior of polyamide-6/clay nanocomposite using continuum-based micromechanical modeling", Macromol. Res., 17(10), 797-806. https://doi.org/10.1007/BF03218617.
  59. Wu, H., Yang, J. and Kitipornchai, S. (2017), "Dynamic instability of functionally graded multilayer graphene nanocomposite beams in thermal environment", Compos. Structu., 162(15), 244-254. https://doi.org/10.1016/j.compstruct.2016.12.001.
  60. Xin, J. Wang, J. and Yao, J. (2011), "Vibration, buckling and dynamic stability of a cracked cylindrical shell with time-varying rotating speed", Mech. Based. Des. Struct. Mach. Int. J., 39(4), 461-490. https://doi.org/10.1080/15397734.2011.569301.
  61. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  62. Yang, S.Y., Lin, W.N., Huang, Y.L., Tien, H.W., Wang, J.Y., Ma, M.C.C, Li, S.M. and Wang Y.S. (2010), "Synergetic effects of graphene platelets and carbon nanotubes on the mechanical and thermal properties of epoxy composites", Carbon, 49(3), 793-803. https://doi.org/10.1016/j.carbon.2010.10.014.
  63. Zhang, Z., et al. (2018), "Mechanical analysis of functionally graded graphene oxide-reinforced composite beams based on the first-order shear deformation theory", Mech. Adv. Mater. Struct., (In press). https://doi.org/10.1080/15376494.2018.1444216.
  64. Zhou, D., Lo, S.H., Au, F.T.K., Cheung, Y.K. and Liu, W.Q. (2006), "3-D vibration analysis of skew thick plates using Chebyshev-Ritz method", Int. J. Mech. Sci., 48(12), 1481-1493. https://doi.org/10.1016/j.ijmecsci.2006.06.015.