DOI QR코드

DOI QR Code

Vibration analysis of micro composite thin beam based on modified couple stress

  • Ehyaei, Javad (Faculty of Engineering, Department of Mechanics, Imam Khomeini International University) ;
  • Akbarizadeh, M. Reza (Faculty of Engineering, Department of Mechanics, Imam Khomeini International University)
  • 투고 : 2016.10.15
  • 심사 : 2017.05.18
  • 발행 : 2017.11.25

초록

In this article, analytical solution for free vibration of micro composite laminated beam on elastic medium based on modified couple stress are presented. The surrounding elastic medium is modeled as the Winkler elastic foundation. The governing equations and boundary conditions are obtained by using the principle of minimum potential energy for EulerBernoulli beam. For investigating the effect of different parameters including material length scale, beam thickness, some numerical results on different cross ply laminated beams such as (90,0,90), (0,90,0), (90,90,90) and (0,0,0) are presented on elastic medium. Free vibration analysis of a simply supported beam is considered utilizing the Fourier series. Also, the fundamental frequency is obtained using the principle of Hamilton for four types of cross ply laminations with hinged-hinged boundary conditions and different beam theories. The fundamental frequency for different thin beam theories are investigated by increasing the slenderness ratio and various foundation coefficients. The results prove that the modified couple stress theory increases the natural frequency under the various foundation for free vibration of composite laminated micro beams.

키워드

참고문헌

  1. Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  2. Akgoz, B. and Civalek, O. (2015), "A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory", Acta Mechanica, 226(7), 2277-2294. https://doi.org/10.1007/s00707-015-1308-4
  3. Akgoz, B. and Civalek, O. (2015), "A novel microstructuredependent shear deformable beam model", Int. J. Mech. Sci., 99, 10-20. https://doi.org/10.1016/j.ijmecsci.2015.05.003
  4. Akgoz, B. and Civalek, O. (2015), "Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity", Compos. Struct., 134, 294-301. https://doi.org/10.1016/j.compstruct.2015.08.095
  5. Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070
  6. Attia, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187
  7. Au, F.T.K., Zheng, D.Y. and Cheung, Y.K., (1999), "Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C 1 modified beam vibration functions", Appl. Math. Model., 23(1), 19-34. https://doi.org/10.1016/S0307-904X(98)10045-8
  8. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B, 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  9. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct. Syst., 18(4), 755-786. https://doi.org/10.12989/sss.2016.18.4.755
  10. Bellifa, H., Benrahou, K.H., Hadji, L., Houari, M.S.A. and Tounsi, A. (2016), "Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position", J Braz. Soc. Mech. Sci. Eng., 38(1), 265-275. https://doi.org/10.1007/s40430-015-0354-0
  11. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  12. Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Adda Bedia, E.A. (2013), "A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets", J. Sandw. Struct. Mater, 15(6), 671-703. https://doi.org/10.1177/1099636213498888
  13. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., 14(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  14. Bouderba, B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2016), "Thermal stability of functionally graded sandwich plates using a simple shear deformation theory", Struct. Eng. Mech., 58(3), 397-422. https://doi.org/10.12989/sem.2016.58.3.397
  15. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  16. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  17. Bousahla, A.A., Benyoucef, S., Tounsi, A. and Mahmoud, S.R. (2016), "On thermal stability of plates with functionally graded coefficient of thermal expansion", Struct. Eng. Mech., 60(2), 313-335. https://doi.org/10.12989/sem.2016.60.2.313
  18. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Meth., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  19. Chen, W., Li, L. and Xu, M., (2011), "A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation", Compos. Struct., 93(11), 2723-2732. https://doi.org/10.1016/j.compstruct.2011.05.032
  20. Chen, W.J. and Li, X.P. (2013), "Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory", Arch. Appl. Mech., 83(3), 431-444. https://doi.org/10.1007/s00419-012-0689-2
  21. Chikh, A., Tounsi, A., Hebali, H. and Mahmoud, S.R. (2017), "Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT", Smart Struct. Syst., 19(3), 289-297. https://doi.org/10.12989/sss.2017.19.3.289
  22. Draiche, K., Tounsi, A. and Mahmoud, S.R. (2016), "A refined theory with stretching effect for the flexure analysis of laminated composite plates", Geomech. Eng., 11(5), 671-690. https://doi.org/10.12989/gae.2016.11.5.671
  23. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  24. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248 https://doi.org/10.1016/0020-7225(72)90039-0
  25. Ghadiri, M., Zajkani, A. and Akbarizadeh, M.R. (2016), "Thermal effect on dynamics of thin and thick composite laminated microbeams by modified couple stress theory for different boundary conditions", Appl. Phys. A, 122(12), 1023. https://doi.org/10.1007/s00339-016-0534-5
  26. Gurses, M., Akgoz, B. and Civalek, O. (2012), "Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation", Appl. Math. Comput., 219(6), 3226-3240 https://doi.org/10.1016/j.amc.2012.09.062
  27. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  28. Herakovich, C.T. (2012), "Mechanics of composites: a historical review", Mech. Res. Commun., 41, 1-20. https://doi.org/10.1016/j.mechrescom.2012.01.006
  29. Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257
  30. Jahangiri, R., Jahangiri, H. and Khezerloo, H., (2015), "FGM micro-gripper under electrostatic and intermolecular Van-der Waals forces using modified couple stress theory", Steel Compos. Struct., 18(6), 1541-1555. https://doi.org/10.12989/scs.2015.18.6.1541
  31. Kapania, R.K. and Raciti, S. (1989), "Recent advances in analysis of laminated beams and plates, Part I-Shear effects and buckling", AIAA J., 27(7), 923-935. https://doi.org/10.2514/3.10202
  32. Kocaturk, T. and Akbas, S.D. (2013), "Wave propagation in a microbeam based on the modified couple stress theory", Struct. Eng. Mech., 46(3), 417-431. https://doi.org/10.12989/sem.2013.46.3.417
  33. Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39, 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  34. Matsunaga, H. (1999), "Vibration and buckling of deep beam-columns on two-parameter elastic foundations", J. Sound Vib., 228(2), 359-376. https://doi.org/10.1006/jsvi.1999.2415
  35. Mindlin, R.D. (1963), "Influence of couple-stresses on stress concentrations", Exper. Mech., 3(1), 1-7. https://doi.org/10.1007/BF02327219
  36. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Rat. Mech. Anal., 16(1), 51-78. https://doi.org/10.1007/BF00248490
  37. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in linear elasticity", Arch. Rat. Mech. Anal., 11(1), 415-448. https://doi.org/10.1007/BF00253946
  38. Mohammad-Abadi, M. and Daneshmehr, A.R. (2015), "Modified couple stress theory applied to dynamic analysis of composite laminated beams by considering different beam theories",, Int. J. Eng. Sci., 87, 83-102. https://doi.org/10.1016/j.ijengsci.2014.11.003
  39. Nix, W.D. and Gao, H. (1998), "Indentation size effects in crystalline materials: a law for strain gradient plasticity", J. Mech. Phys. Solid., 46(3), 411-425. https://doi.org/10.1016/S0022-5096(97)00086-0
  40. Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16(11), 2355. https://doi.org/10.1088/0960-1317/16/11/015
  41. Pradhan, S.C. and T. Murmu, (2009), "Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method", J. Sound Vib., 321(1), 342-362. https://doi.org/10.1016/j.jsv.2008.09.018
  42. Thambiratnam, D. and Zhuge, Y. (1996), "Free vibration analysis of beams on elastic foundation", Comput. Struct., 60(6), 971-980. https://doi.org/10.1016/0045-7949(96)00053-3
  43. Tounsi, A., Houari, M.S.A. and Bessaim, A. (2016), "A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate", Struct. Eng. Mech., 60(4), 547-565. https://doi.org/10.12989/sem.2016.60.4.547
  44. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Rat. Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945
  45. Wanji, C., Chen, W. and Sze, K.Y. (2012), "A model of composite laminated Reddy beam based on a modified couple-stress theory", Compos. Struct., 94(8), 2599-2609. https://doi.org/10.1016/j.compstruct.2012.02.020
  46. 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
  47. Ying, J., Lu, C.F. and Chen, W.Q. (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Compos. Struct., 84(3), 209-219. https://doi.org/10.1016/j.compstruct.2007.07.004
  48. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693
  49. Zhou, D. (1993), "A general solution to vibrations of beams on variable Winkler elastic foundation", Comput. Struct., 47(1), 83-90. https://doi.org/10.1016/0045-7949(93)90281-H