Structural Engineering and Mechanics

  • 제65권5호
  • /
  • Pages.573-585
  • /
  • 2018
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Modelling of graded rectangular micro-plates with variable length scale parameters

  • Aghazadeh, Reza (Department of Mechanical Engineering, Middle East Technical University) ;
  • Dag, Serkan (Department of Mechanical Engineering, Middle East Technical University) ;
  • Cigeroglu, Ender (Department of Mechanical Engineering, Middle East Technical University)
  • 투고 : 2017.06.02
  • 심사 : 2018.01.05
  • 발행 : 2018.03.10
⟨ 이전 논문 다음 논문 ⟩

초록

This article presents strain gradient elasticity-based procedures for static bending, free vibration and buckling analyses of functionally graded rectangular micro-plates. The developed method allows consideration of smooth spatial variations of length scale parameters of strain gradient elasticity. Governing partial differential equations and boundary conditions are derived by following the variational approach and applying Hamilton's principle. Displacement field is expressed in a unified way to produce numerical results in accordance with Kirchhoff, Mindlin, and third order shear deformation theories. All material properties, including the length scale parameters, are assumed to be functions of the plate thickness coordinate in the derivations. Developed equations are solved numerically by means of differential quadrature method. Proposed procedures are verified through comparisons made to the results available in the literature for certain limiting cases. Further numerical results are provided to illustrate the effects of material and geometric parameters on bending, free vibrations, and buckling. The results generated by Kirchhoff and third order shear deformation theories are in very good agreement, whereas Mindlin plate theory slightly overestimates static deflection and underestimates natural frequency. A rise in the length scale parameter ratio, which identifies the degree of spatial variations, leads to a drop in dimensionless maximum deflection, and increases in dimensionless vibration frequency and buckling load. Size effect is shown to play a more significant role as the plate thickness becomes smaller compared to the length scale parameter. Numerical results indicate that consideration of length scale parameter variation is required for accurate modelling of graded rectangular micro-plates.

키워드

과제정보

연구 과제 주관 기관 : Scientific and Technological Research Council of Turkey (TUBITAK)

참고문헌

  1. Aghazadeh, R., Cigeroglu, E. and Dag, S. (2014), "Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories", Eur. J. Mech. A. Sol., 46, 1-11.
  2. Aifantis, E.C. (1999), "Strain gradient interpretation of size effects", J. Fract., 95(1-4), 299-314. https://doi.org/10.1023/A:1018625006804
  3. Akgoz, B. and Civalek, O. (2013), "Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory", Meccan., 48(4), 863-873. https://doi.org/10.1007/s11012-012-9639-x
  4. Akgoz, B. and Civalek, O. (2015), "A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory", Acta Mech., 226(7), 2277-2294. https://doi.org/10.1007/s00707-015-1308-4
  5. Alipour Ghassabi, A., Dag, S. and Cigeroglu, E. (2017), "Free vibration analysis of functionally graded rectangular nano-plates considering spatial variation of the nonlocal parameter", Arch. Mech., 69(2), 105-103.
  6. Andrew, W.M. and Jonathan, S.C. (2005), "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", J. Micromech. Microeng., 15(5), 1060. https://doi.org/10.1088/0960-1317/15/5/024
  7. Ansari, R., Faghih Shojaei, M., Mohammadi, V., Bazdid-Vahdati, M. and Rouhi, H. (2015), "Triangular mindlin microplate element", Comput. Meth. Appl. Mech. Eng., 295, 56-76. https://doi.org/10.1016/j.cma.2015.06.004
  8. Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V. and Darabi, M.A. (2013), "Thermal buckling analysis of a mindlin rectangular FGM microplate based on the strain gradient theory", J. Therm. Stress., 36(5), 446-465. https://doi.org/10.1080/01495739.2013.770657
  9. Asghari, M. (2012), "Geometrically nonlinear micro-plate formulation based on the modified couple stress theory", J. Eng. Sci., 51, 292-309. https://doi.org/10.1016/j.ijengsci.2011.08.013
  10. Bellman, R.E. and Casti, J. (1971), "Differential quadrature and long-term integration", J. Math. Anal. Appl., 34, 235-238. https://doi.org/10.1016/0022-247X(71)90110-7
  11. Eringen, A.C. (1972), "Nonlocal polar elastic continua", J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  12. Eshraghi, I., Dag, S. and Soltani, N. (2015), "Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates", Compos. Part B, 78, 338-348. https://doi.org/10.1016/j.compositesb.2015.03.095
  13. Eshraghi, I., Dag, S. and Soltani, N. (2016), "Bending and free vibrations of functionally graded annular and circular microplates under thermal loading", Compos. Struct., 137, 196-207. https://doi.org/10.1016/j.compstruct.2015.11.024
  14. Farahmand , H., Mohammadi, M., Iranmanesh, A. and Naseralavi, S.S. (2015), "Exact solution for free vibration analysis of functionally graded microplates based on the strain gradient theory", J. Multisc. Comput. Eng., 13(6), 463-474. https://doi.org/10.1615/IntJMultCompEng.2015014164
  15. Farokhi, H. and Ghayesh, M.H. (2015), "Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory", J. Mech. Sci., 90, 133-144. https://doi.org/10.1016/j.ijmecsci.2014.11.002
  16. Farokhi, H. and Ghayesh, M.H. (2016), "Nonlinear size-dependent dynamics of an imperfect shear deformable microplate", J. Sound Vibr., 361, 226-242. https://doi.org/10.1016/j.jsv.2015.09.025
  17. Fu, Y., Du, H. and Zhang, S. (2003), "Functionally graded TiN/TiNi shape memory alloy films", Mater. Lett., 57(20), 2995-2999. https://doi.org/10.1016/S0167-577X(02)01419-2
  18. Gholami, R. and Ansari, R. (2016), "A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates", Nonlin. Dyn., 84(4), 2403-2422. https://doi.org/10.1007/s11071-016-2653-0
  19. Gurtin, M.E., Weissmuller, J. and Larche, F. (1998), "The general theory of curved deformable interfaces in solids at equilibrium", Philos. Mag. A, 178, 1093-1109.
  20. Hassanin, H. and Jiang, K. (2014), "Net shape manufacturing of ceramic micro parts with tailored graded layers", J. Micromech. Microeng., 24(1), 015018. https://doi.org/10.1088/0960-1317/24/1/015018
  21. Jomehzadeh, E., Noori, H.R. and Saidi, A.R. (2011), "The sizedependent vibration analysis of micro-plates based on a modified couple stress theory", Phys. E, 43(4), 877-883. https://doi.org/10.1016/j.physe.2010.11.005
  22. Kahrobaiyan, M.H., Rahaeifard, M., Tajalli, S.A. and Ahmadian, M.T. (2012), "A strain gradient functionally graded eulerbernoulli beam formulation", J. Eng. Sci., 52, 65-76. https://doi.org/10.1016/j.ijengsci.2011.11.010
  23. Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2012), "Free vibration of size-dependent mindlin microplates based on the modified couple stress theory", J. Sound Vibr., 331(1), 94-106. https://doi.org/10.1016/j.jsv.2011.08.020
  24. Kim, J. and Reddy, J.N. (2013), "Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stressbased third-order theory", Compos. Struct., 103, 86-98. https://doi.org/10.1016/j.compstruct.2013.03.007
  25. Kim, J. and Reddy, J.N. (2015), "A general third-order theory of functionally graded plates with modified couple stress effect and the von Karman nonlinearity: Theory and finite element analysis", Acta Mech., 226(9), 2973-2998. https://doi.org/10.1007/s00707-015-1370-y
  26. 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. Sol., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  27. Lazopoulos, K.A. (2009), "On bending of strain gradient elastic micro-plates", Mech. Res. Commun., 36(7), 777-783. https://doi.org/10.1016/j.mechrescom.2009.05.005
  28. Lou, J. and He, L. (2015), "Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory", Compos. Struct., 131, 810-820. https://doi.org/10.1016/j.compstruct.2015.06.031
  29. Ma, H.M., Gao, X.L. and Reddy, J.N. (2011), "A non-classical Mindlin plate model based on a modified couple stress theory", Acta Mech., 220(1-4), 217-235. https://doi.org/10.1007/s00707-011-0480-4
  30. Mahmoud, F.F. and Shaat, M. (2015), "A new Mindlin FG plate model incorporating microstructure and surface energy effects", Struct. Eng. Mech., 53(1), 105-130. https://doi.org/10.12989/sem.2015.53.1.105
  31. Mindlin, R.D. (1963), "Influence of couple-stresses on stress concentrations", Exp. Mech., 3(1), 1-7. https://doi.org/10.1007/BF02327219
  32. Mindlin, R.D. (1965), "Second gradient of strain and surfacetension in linear elasticity", J. Sol. Struct., 1(4), 417-438. https://doi.org/10.1016/0020-7683(65)90006-5
  33. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couplestresses in linear elasticity", Arch. Rat. Mech. Anal., 11(1), 415-448. https://doi.org/10.1007/BF00253946
  34. Mohammadimehr, M., Rousta Navi, B. and Ghorbanpour Arani, A. (2016), "Modified strain gradient reddy rectangular plate model for biaxial buckling and bending analysis of double-coupled piezoelectric polymeric nanocomposite reinforced by FGSWNT", Compos. Part B, 87, 132-148. https://doi.org/10.1016/j.compositesb.2015.10.007
  35. Ng, C.H.W., Zhao, Y.B., Xiang, Y. and Wei, G.W. (2009), "On the accuracy and stability of a variety of differential quadrature formulations for the vibration analysis of beams", J. Eng. Appl. Sci., 1, 1-25.
  36. Nikolov, S., Han, C.S. and Raabe, D. (2007), "On the origin of size effects in small-strain elasticity of solid polymers", J. Sol. Struct., 44(5), 1582-1592. https://doi.org/10.1016/j.ijsolstr.2006.06.039
  37. 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
  38. Ramezani, S. (2012), "A shear deformation micro-plate model based on the most general form of strain gradient elasticity", J. Mech. Sci., 57(1), 34-42. https://doi.org/10.1016/j.ijmecsci.2012.01.012
  39. Reddy, J.N. and Kim, J. (2012), "A nonlinear modified couple stress-based third-order theory of functionally graded plates", Compos. Struct., 94(3), 1128-1143. https://doi.org/10.1016/j.compstruct.2011.10.006
  40. Sahmani, S. and Ansari, R. (2013), "On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory", Compos. Struct., 95, 430-442. https://doi.org/10.1016/j.compstruct.2012.07.025
  41. Salehipour, H., Nahvi, H. and Shahidi, A.R. (2015), "Exact closedform free vibration analysis for functionally graded micro/nano plates based on modified couple stress and three-dimensional elasticity theories", Compos. Struct., 124, 283-291. https://doi.org/10.1016/j.compstruct.2015.01.015
  42. Setoodeh, A. and Rezaei, M. (2017), "Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation", Struct. Eng. Mech., 61(2), 209-220. https://doi.org/10.12989/sem.2017.61.2.209
  43. Shenas, A.G. and Malekzadeh, P. (2016), "Free vibration of functionally graded quadrilateral microplates in thermal environment", Thin Wall. Struct., 106, 294-315. https://doi.org/10.1016/j.tws.2016.05.001
  44. Shu, C. (2000), Differential Quadrature and its Application in Engineering, Springer, London, U.K.
  45. Simsek, M., Aydin, M., Yurtcu, H.H. and Reddy, J.N. (2015), "Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory", Acta Mech., 226(11), 3807-3822. https://doi.org/10.1007/s00707-015-1437-9
  46. Stolken, J.S. and Evans, A.G. (1998), "A microbend test method for measuring the plasticity length scale", Acta Mater., 46(14), 5109-5115. https://doi.org/10.1016/S1359-6454(98)00153-0
  47. Thai, H.T. and Choi, D.H. (2013), "Size-dependent functionally graded kirchhoff and mindlin plate models based on a modified couple stress theory", Compos. Struct., 95, 142-153. https://doi.org/10.1016/j.compstruct.2012.08.023
  48. Thai, H.T. and Kim, S.E. (2013), "A size-dependent functionally graded Reddy plate model based on a modified couple stress theory", Compos. Part B, 45(1), 1636-1645. https://doi.org/10.1016/j.compositesb.2012.09.065
  49. Thai, H.T. and Vo, T.P. (2013), "A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory", Compos. Struct., 96, 376-383. https://doi.org/10.1016/j.compstruct.2012.09.025
  50. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Rat. Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945
  51. Tsiatas, G.C. (2009), "A new kirchhoff plate model based on a modified couple stress theory", J. Sol. Struct., 46(13), 2757-2764. https://doi.org/10.1016/j.ijsolstr.2009.03.004
  52. Wang, B., Zhou, S., Zhao, J. and Chen, X. (2011), "A sizedependent kirchhoff micro-plate model based on strain gradient elasticity theory", Eur. J. Mech. A. Sol., 30(4), 517-524. https://doi.org/10.1016/j.euromechsol.2011.04.001
  53. Wang, Y.G., Lin, W.H. and Liu, N. (2013), "Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory", J. Mech. Sci., 71, 51-57. https://doi.org/10.1016/j.ijmecsci.2013.03.008
  54. Witvrouw, A. and Mehta, A. (2005), "The use of functionally graded poly-SiGe layers for MEMS applications", Mater. Sci. For., 492-493, 255-260.
  55. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", J. Sol. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  56. Yin, L., Qian, Q., Wang, L. and Xia, W. (2010), "Vibration analysis of microscale plates based on modified couple stress theory", Acta Mech. Sol. Sin., 23(5), 386-393. https://doi.org/10.1016/S0894-9166(10)60040-7
  57. Zhong, Z.Y., Zhang, W.M., Meng, G. and Wang, M.Y. (2015), "Thermoelastic damping in the size-dependent microplate resonators based on modified couple stress theory", J. Microelectr. Syst., 24(2), 431-445. https://doi.org/10.1109/JMEMS.2014.2332757

피인용 문헌

  1. Buckling treatment of piezoelectric functionally graded graphene platelets micro plates vol.38, pp.3, 2021, https://doi.org/10.12989/scs.2021.38.3.337