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Postbuckling analysis of microscale beams based on a strain gradient finite element approach

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Abstract

The postbuckling problem of Euler–Bernoulli (EB) microbeams under different boundary conditions is addressed in this paper using a non-classical finite element (FE) approach with a novel beam element. The proposed element is capable of considering the strain gradient effects and hence is appropriate to use at microscale. First, based on Mindlin’s strain gradient theory (SGT), a size-dependent EB beam model including strain gradient effects is developed. Then, by a FE approach, the non-classical beam element is constructed which needs two additional degrees of freedom per node as compared to the classical EB beam element. The new element is based upon Mindlin’s SGT, and can be easily reduced to that based on various higher-order elasticity theories such as the modified versions of strain gradient and couple stress theories. Selected numerical results are presented to show the reliability of the developed FE formulation, and also to study the small scale influences on the bifurcation diagrams of microbeams.

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References

  1. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487

    Article  Google Scholar 

  2. Ma Q, Clarke DR (1995) Size dependent hardness of silver single crystals. J Mater Res 10:853–863

    Article  ADS  Google Scholar 

  3. Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115

    Article  Google Scholar 

  4. Abu Al-Rub RK, Voyiadjis GZ (2004) Analytical and experimental determination of material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. Int J Plasticity 20:1139–1182

    Article  Google Scholar 

  5. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78

    Article  MathSciNet  MATH  Google Scholar 

  6. Mindlin RD (1965) Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct 1:417–438

    Article  Google Scholar 

  7. Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4:109–124

    Article  MATH  Google Scholar 

  8. Bakogianni DG, Lazopoulos KA (2007) Stability of strain gradient elastic bars in tension. Opt Lett 1:407–420

    Article  MathSciNet  MATH  Google Scholar 

  9. Lazopoulos KA, Lazopoulos AK (2010) Bending and buckling of thin strain gradient elastic beams. Eur J Mech A Solids 29:837–843

    Article  MATH  Google Scholar 

  10. Lazopoulos KA, Lazopoulos AK (2012) On the torsion problem of strain gradient elastic bars. Mech Res Commun 45:42–47

    Article  MATH  Google Scholar 

  11. Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ratoin Mech Anal 11:415–448

    Article  MathSciNet  MATH  Google Scholar 

  12. Koiter WT (1964) Couple stresses in the theory of elasticity I and II. In: Proceedings of K Ned Akad Wetensc (B), vol 67, pp 17–44

  13. Toupin RA (1962) Elastic materials with couple-stresses. Arch Ratoin Mech Anal 11:385–414

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang F, Chong ACM, Lam DCC (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    Article  MATH  Google Scholar 

  15. Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508

    Article  ADS  MATH  Google Scholar 

  16. Park SK, Gao X-L (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 16:2355

    Article  Google Scholar 

  17. Tsiatas GC (2009) A new Kirchhoff plate model based on a modified couple stress theory. Int J Solids Struct 46:2757–2764

    Article  MATH  Google Scholar 

  18. Şimşek M (2010) Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Int J Eng Sci 48:1721–1732

    Article  MATH  Google Scholar 

  19. Yin L, Qian Q, Wang L, Xia W (2010) Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech Solida Sinica 23:386–393

    Article  Google Scholar 

  20. Akgöz B, Civalek Ö (2011) Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 49:1268–1280

    Article  MathSciNet  Google Scholar 

  21. Chen W, Li L, Xu M (2011) A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Compos Struct 93:2723–2732

    Article  Google Scholar 

  22. Ke LL, Wang YS, Wang ZD (2011) Thermal effect on free vibration and buckling of size-dependent microbeams. Phys E 43:1387–1393

    Article  Google Scholar 

  23. Zhao JF, Zhou SJ, Wang BL (2011) Size dependent pull-in phenomena in electro-statically actuated micro-beam based on the modified couple stress theory. Adv Mater Res 335–336:633–640

    Google Scholar 

  24. Zhang J, Fu Y (2012) Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory. Meccanica 47:1649–1658

    Article  MathSciNet  MATH  Google Scholar 

  25. Baghani M (2012) Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. Int J Eng Sci 54:99–105

    Article  MathSciNet  Google Scholar 

  26. Ke LL, Yang J, Kitipornchai S, Bradford MA (2012) Bending, buckling and vibration of size-dependent functionally graded annular microplates. Compos Struct 94:3250–3257

    Article  Google Scholar 

  27. Mustapha KB, Zhong ZW (2012) Wave propagation characteristics of a twisted micro scale beam. Int J Eng Sci 53:46–57

    Article  MathSciNet  Google Scholar 

  28. Thai HT, Kim SE (2013) A size-dependent functionally graded reddy plate model based on a modified couple stress theory. Compos Part B 45:1636–1645

    Article  Google Scholar 

  29. Roque CMC, Fidalgo DS, Ferreira AJM, Reddy JN (2013) A study of a microstructure-dependent composite laminated timoshenko beam using a modified couple stress theory and a meshless method. Compos Struct 96:532–537

    Article  Google Scholar 

  30. Şimşek M, Reddy JN (2013) Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int J Eng Sci 64:37–53

    Article  MathSciNet  Google Scholar 

  31. Roque CMC, Ferreira AJM, Reddy JN (2013) Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method. Appl Math Model 37:4626–4633

    Article  MathSciNet  Google Scholar 

  32. Zhang B, He Y, Liu D, Gan Z, Shen L (2013) A non-classical Mindlin plate finite element based on a modified couple stress theory. Eur. J. Mech. A/Solids 42:63–80

    Article  ADS  MathSciNet  Google Scholar 

  33. Thai H-T, 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

    Article  Google Scholar 

  34. Gao X-L, Huang JX, Reddy JN (2013) A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech 224:2699–2718

    Article  MathSciNet  MATH  Google Scholar 

  35. Akgöz B, Civalek Ö (2013) Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica 48:863–873

    Article  MathSciNet  MATH  Google Scholar 

  36. Şimşek M, Reddy JN (2013) A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos Struct 101:47–58

    Article  Google Scholar 

  37. Simsek M, Kocatürk T, Akbas SD (2013) Static bending of a functionally graded microscale timoshenko beam based on the modified couple stress theory. Compos Struct 95:740–747

    Article  Google Scholar 

  38. Lazopoulos KA (2009) On bending of strain gradient elastic micro-plates. Mech Res Commun 36:777–783

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang B, Zhao J, Zhou S (2010) A micro scale timoshenko beam model based on strain gradient elasticity theory. Eur J Mech A/Solids 29:591–599

    Article  Google Scholar 

  40. Wang B, Zhou S, Zhao J, Chen X (2011) A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Eur J Mech A/Solids 30:517–524

    Article  MATH  Google Scholar 

  41. Zhao J, Zhou S, Wang B, Wang X (2012) Nonlinear microbeam model based on strain gradient theory. Appl Math Model 36:2674–2686

    Article  MathSciNet  MATH  Google Scholar 

  42. Akgöz B, Civalek Ö (2012) Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. J Vib Control 20:606–616

    Article  MathSciNet  MATH  Google Scholar 

  43. Akgöz B, Civalek Ö (2013) A Size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int J Eng Sci 70:1–14

    Article  MathSciNet  MATH  Google Scholar 

  44. Akgöz B, Civalek Ö (2013) Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech 224:2185–2201

    Article  MathSciNet  MATH  Google Scholar 

  45. Liang X, Hu S, Shen S (2013) Bernoulli–Euler dielectric beam model based on strain-gradient effect. ASME J Appl Mech 80:044502

    Article  Google Scholar 

  46. Xu L, Shen S (2013) Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity. Int J Appl Mech 05:1350015

    Article  Google Scholar 

  47. Lei J, He Y, Zhang B, Gan Z, Zeng P (2013) Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory. Int J Eng Sci 72:36–52

    Article  Google Scholar 

  48. Zhang B, He Y, Liu D, Gan Z, Shen L (2013) A novel size-dependent functionally graded curved mircobeam model based on the strain gradient elasticity theory. Compos Struct 106:374–392

    Article  Google Scholar 

  49. Li A, Zhou S, Zhou S, Wang B (2014) A size-dependent bilayered microbeam model based on strain gradient elasticity theory. Compos Struct 108:259–266

    Article  Google Scholar 

  50. Kahrobaiyan MH, Asghari M, Ahmadian MT (2013) Strain gradient beam element. Finite Elem Anal Des 68:63–75

    Article  MathSciNet  MATH  Google Scholar 

  51. Ansari R, Faghih Shojaei M, Mohammadi V, Rouhi H, Bazdid-Vahdati M (2015) Triangular Mindlin microplate element. Comput Meth Appl Mech Eng 295:56–76

    Article  MathSciNet  Google Scholar 

  52. Ansari R, Faghih Shojaei M, Rouhi H (2015) Small-scale timoshenko beam element. Eur J Mech/A Solids 53:19–33

    Article  MathSciNet  MATH  Google Scholar 

  53. Ansari R, Faghih Shojaei M, Ebrahimi F, Rouhi H, Bazdid-Vahdati M (2015) A novel size-dependent microbeam element based on Mindlin’s strain gradient theory. Comput, Eng. doi:10.1007/s00366-015-0406-1

    MATH  Google Scholar 

  54. Keller HB (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems, applications of bifurcation theory. In: Proceedings of advanced seminar, University of Wisconsin, Madison, Wisconsin, 1976. Academic Press, New York, pp 359–384

  55. Gholami R, Darvizeh A, Ansari R, Hosseinzadeh M (2014) Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory. Meccanica 49:1679–1695

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    F. Ebrahimi, R. Ansari, M. Faghih Shojaei & H. Rouhi

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  1. F. Ebrahimi
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  2. R. Ansari
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  3. M. Faghih Shojaei
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  4. H. Rouhi
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Correspondence to R. Ansari.

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Ebrahimi, F., Ansari, R., Faghih Shojaei, M. et al. Postbuckling analysis of microscale beams based on a strain gradient finite element approach. Meccanica 51, 2493–2507 (2016). https://doi.org/10.1007/s11012-016-0383-5

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  • DOI: https://doi.org/10.1007/s11012-016-0383-5

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