Skip to main content
Log in

Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton’s principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. RAFII-TABAR, H., GHAVANLOO, E., and FAZELZADEH, S. A. Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Physics Reports, 638, 1–97 (2016)

    Article  MathSciNet  Google Scholar 

  2. LIM, C.W., ZHANG, G., and REDDY, J. N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. IMBODEN, M. and MOHANTY, P. Dissipation in nanoelectromechanical systems. Physics Reports, 534, 89–146 (2014)

    Article  MathSciNet  Google Scholar 

  4. ARASH, B. and WANG, Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Modeling of Carbon Nanotubes, Graphene and Their Composites, Springer, Berlin, 57–82 (2014)

    Chapter  Google Scholar 

  5. ELTAHER, M. A., KHATER, M. E., and EMAM, S. A. A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Applied Mathematical Modelling, 40, 4109–4128 (2016)

    Article  MathSciNet  Google Scholar 

  6. WANG, K., WANG, B., and KITAMURA, T. A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mechanica Sinica, 32, 83–100 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. CORDERO, N. M., FOREST, S., and BUSSO, E. P. Second strain gradient elasticity of nano-objects. Journal of the Mechanics and Physics of Solids, 97, 92–124 (2016)

    Article  MathSciNet  Google Scholar 

  8. KRISHNAN, A., DUJARDIN, E., EBBESEN, T., YIANILOS, P., and TREACY, M. Young’s modulus of single-walled nanotubes. Physical Review B, 58, 14013 (1998)

    Article  Google Scholar 

  9. WANG, L., ZHENG, Q., LIU, J. Z., and JIANG, Q. Size dependence of the thin-shell model for carbon nanotubes. Physical Review Letters, 95, 105501 (2005)

    Article  Google Scholar 

  10. DIAO, J., GALL, K., and DUNN, M. L. Atomistic simulation of the structure and elastic prop- erties of gold nanowires. Journal of the Mechanics and Physics of Solids, 52, 1935–1962 (2004)

    Article  MATH  Google Scholar 

  11. LI, C. and CHOU, T. W. A structural mechanics approach for the analysis of carbon nanotubes. International Journal of Solids and Structures, 40, 2487–2499 (2003)

    Article  MATH  Google Scholar 

  12. LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  13. LEI, J., HE, Y., GUO, S., LI, Z., and LIU, D. Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Advances, 6, 105202 (2016)

    Article  Google Scholar 

  14. TREACY, M. J., EBBESEN, T., and GIBSON, J. Exceptionally high Young’s modulus observed for individual carbon nanotubes. nature, 381, 678–680 (1996)

    Article  Google Scholar 

  15. AGRAWAL, R., PENG, B., GDOUTOS, E. E., and ESPINOSA, H. D. Elasticity size effects in ZnO nanowires—a combined experimental-computational approach. Nano Letters, 8, 3668–3674 (2008)

    Article  Google Scholar 

  16. NATSUKI, T., TANTRAKARN, K., and ENDO, M. Effects of carbon nanotube structures on mechanical properties. Applied Physics A: Materials Science and Processing, 79, 117–124 (2004)

    Article  Google Scholar 

  17. TANG, C. and ALICI, G. Evaluation of length-scale effects for mechanical behaviour of micro- and nanocantilevers: I, experimental determination of length-scale factors. Journal of Physics D: Applied Physics, 44, 335501 (2011)

    Article  Google Scholar 

  18. RU, C. Q. Effective bending stiffness of carbon nanotubes. Physical Review B, 62, 9973 (2000)

    Article  Google Scholar 

  19. WANG, Q. and VARADAN, V. Wave characteristics of carbon nanotubes. International Journal of Solids and Structures, 43, 254–265 (2006)

    Article  MATH  Google Scholar 

  20. KULATHUNGA, D. D. T. K., ANG, K. K., and REDDY, J. N. Accurate modeling of buckling of single- and double-walled carbon nanotubes based on shell theories. Journal of Physics: Condensed Matter, 21, 435301 (2009)

    Google Scholar 

  21. ERINGEN, A. C. Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710 (1983)

    Article  Google Scholar 

  23. MINDLIN, R. D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  24. MINDLIN, R. D. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1, 417–438 (1965)

    Article  Google Scholar 

  25. TOUPIN, R. A. Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, 11, 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  26. MINDLIN, R. D. and TIERSTEN, H. F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 11, 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  27. ZHOU, S., LI, A., and WANG, B. A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials. International Journal of Solids and Structures, 80, 28–37 (2016)

    Article  Google Scholar 

  28. YANG, F., CHONG, A. C. M., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. {iItnternational Journal of Solids and Structures}, 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  29. HADJESFANDIARI, A. R. and DARGUSH, G. F. Couple stress theory for solids. International Journal of Solids and Structures, 48, 2496–2510 (2011)

    Article  Google Scholar 

  30. PEDDIESON, J., BUCHANAN, G. R., and MCNITT, R. P. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41, 305–312 (2003)

    Article  Google Scholar 

  31. WANG, C. M., ZHANG, Y. Y., and HE, X. Q. Vibration of nonlocal Timoshenko beams. Nanotechnology, 18, 105401 (2007)

    Article  Google Scholar 

  32. LIM, C. W. and WANG, C. M. Exact variational nonlocal stress modeling with asymptotic higher- order strain gradients for nanobeams. Journal of Applied Physics, 101, 054312 (2007)

    Article  Google Scholar 

  33. HU, Y. G., LIEW, K. M., WANG, Q., HE, X. Q., and YAKOBSON, B. I. Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. Journal of the Mechanics and Physics of Solids, 56, 3475–3485 (2008)

    Article  MATH  Google Scholar 

  34. REDDY, J. N. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. International Journal of Engineering Science, 48, 1507–1518 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. SHEN, H. S. and ZHANG, C. L. Nonlocal shear deformable shell model for post-buckling of axially compressed double-walled carbon nanotubes embedded in an elastic matrix. Journal of Applied Mechanics, 77, 041006 (2010)

    Article  Google Scholar 

  36. SHEN, H. S. and ZHANG, C. L. Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates. Computational Materials Science, 50, 1022–1029 (2011)

    Article  Google Scholar 

  37. PENG, X. W., GUO, X. M., LIU, L., and WU, B. J. Scale effects on nonlocal buckling analysis of bilayer composite plates under non-uniform uniaxial loads. Applied Mathematics and Mechanics (English Edition), 36(1), 1–10 (2015) https://doi.org/10.1007/s10483-015-1900-7

    Article  MathSciNet  MATH  Google Scholar 

  38. GHORBANPOUR-ARANI, A., KOLAHDOUZAN, F., and ABDOLLAHIAN, M. Nonlocal buck- ling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory. Applied Mathematics and Mechanics (English Edition), 39(4), 529–546 (2018) https://doi.org/ 10.1007/s10483-018-2319-8

    Article  MathSciNet  Google Scholar 

  39. PARK, S. K. and GAO, X. L. Bernoulli-Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16, 2355–2359 (2006)

    Article  Google Scholar 

  40. MA, H. M., GAO, X. L., and REDDY, J. N. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids, 56, 3379–3391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. ŞIMŞEK, M. and REDDY, J. N. Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. International Journal of Engineering Science, 64, 37–53 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. REDDY, J. N. and KIM, J. A nonlinear modified couple stress-based third-order theory of func- tionally graded plates. Composite Structures, 94, 1128–1143 (2012)

    Article  Google Scholar 

  43. KOMIJANI, M., REDDY, J. N., and ESLAMI, M. R. Nonlinear analysis of microstructure- dependent functionally graded piezoelectric material actuators. Journal of the Mechanics and Physics of Solids, 63, 214–227 (2014)

    Article  MathSciNet  Google Scholar 

  44. KONG. S., ZHOU, S., NIE, Z., and WANG, K. Static and dynamic analysis of micro beams based on strain gradient elasticity theory. International Journal of Engineering Science, 47, 487–498 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. WANG, B., ZHAO, J., and ZHOU, S. A micro scale Timoshenko beam model based on strain gradient elasticity theory. European Journal of Mechanics-A/Solids, 29, 591–599 (2010)

    Article  Google Scholar 

  46. WANG, B., DENG, Z. C., and ZHANG, K. Nonlinear vibration of embedded single-walled car- bon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory. Applied Mathematics and Mechanics (English Edition), 34(3), 269–280 (2013) https://doi.org/10.1007/s10483-013-1669-8

    Article  MathSciNet  MATH  Google Scholar 

  47. MOHAMMADIMEHR, M., FARAHI, M. J., and ALIMIRZAEI, S. Vibration and wave propaga- tion analysis of twisted micro-beam using strain gradient theory. Applied Mathematics and Mechanics (English Edition), 37(10), 1375–1392 (2016) https://doi.org/10.1007/s10483-016-2138-9

    Article  MathSciNet  MATH  Google Scholar 

  48. ZHANG, B., HE, Y., LIU, D., GAN, Z., and SHEN, L. Size-dependent functionally graded beam model based on an improved third-order shear deformation theory. European Journal of Mechanics-A/Solids, 47, 211–230 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. LI, L. and HU, Y. Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. International Journal of Engineering Science, 97, 84–94 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. LI, L. and HU, Y. Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects. International Journal of Mechanical Sciences, 120, 159–170 (2017)

    Article  Google Scholar 

  51. LI, X., LI, L., HU, Y., DING, Z. C., and DENG, W. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Composite Structures, 165, 250–265 (2017)

    Article  Google Scholar 

  52. LI, L. and HU, Y. Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 107, 77–97 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  53. LI, L., LI, X., and HU, Y. Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 102, 77–92 (2016)

    Article  MATH  Google Scholar 

  54. LI, L., HU, Y., and LING, L. Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Composite Structures, 133, 1079–1092 (2015)

    Article  Google Scholar 

  55. SAHMANI, S. and FATTAHI, A. M. Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory. Applied Mathematics and Mechanics (English Edition), 39(4), 561–580 (2018) https://doi.org/10.1007/s10483-018-2321-8

    Article  MathSciNet  MATH  Google Scholar 

  56. LU, L., GUO, X., and ZHAO, J. Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  57. LU, L., GUO, X., and ZHAO, J. A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms. International Journal of Engineering Science, 119, 265–277 (2017)

    Article  MATH  Google Scholar 

  58. LU, L., GUO, X., and ZHAO, J. On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy. International Journal of Engineering Science, 124, 24–40 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  59. SHEN, H. S. A novel technique for nonlinear analysis of beams on two-parameter elastic founda- tions. International Journal of Structural Stability and Dynamics, 11, 999–1014 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  60. LI, Z. M. and QIAO, P. On an exact bending curvature model for nonlinear free vibration analysis shear deformable anisotropic laminated beams. Composite Structures, 108, 243–258 (2014)

    Article  Google Scholar 

  61. WICKERT, J. Non-linear vibration of a traveling tensioned beam. International Journal of Nonlinear Mechanics, 27, 503–517 (1992)

    Article  MATH  Google Scholar 

  62. SHEN, H. S. and ZHANG, J. W. Perturbation analyses for the postbuckling of simply supported rectangular plates under uniaxial compression. Applied Mathematics and Mechanics (English Edition), 9(8), 793–804 (1988) https://doi.org/10.1007/BF02465403

    Article  MathSciNet  MATH  Google Scholar 

  63. SHEN, H. S., XIANG, Y., and LIN, F. Nonlinear vibration of functionally graded graphene- reinforced composite laminated plates in thermal environments. Computer Methods in Applied Mechanics and Engineering, 319, 175–193 (2017)

    Article  MathSciNet  Google Scholar 

  64. SHEN, H. S., LIN, F., and XIANG, Y. Nonlinear bending and thermal postbuckling of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations. Engineering Structures, 140, 89–97 (2017)

    Article  Google Scholar 

  65. SHEN, H. S., HE, X. Q., and YANG, D. Q. Vibration of thermally postbuckled carbon nanotube- reinforced composite beams resting on elastic foundations. International Journal of Non-Linear Mechanics, 91, 69–75 (2017)

    Article  Google Scholar 

  66. KIEN, D. K. Postbuckling behavior of beams on two-parameter elastic foundation. International Journal of Structural Stability and Dynamics, 4, 21–43 (2004)

    Article  MATH  Google Scholar 

  67. NAIDU, N. R. and RAO, G. V. Stability behaviour of uniform columns on a class of two parameter elastic foundation. Computers and Structures, 57, 551–553 (1995)

    Article  Google Scholar 

  68. TIMOSHENKO, S. P. and GERE, J. M. Theory of Elastic Stability, McGraw-Hill Book Company, New York (1961)

    Google Scholar 

  69. HORIBE, T. and ASANO, N. Large deflection analysis of beams on two-parameter elastic foun- dation using the boundary integral equation method. JSME International Journal Series A Solid Mechanics and Material Engineering, 44, 231–236 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bo Zhang.

Additional information

Citation: ZHANG, B., SHEN, H. M., LIU, J., WANG, Y. X., and ZHANG, Y. R. Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects. Applied Mathematics and Mechanics (English Edition), 40(4), 515–548 (2019) https://doi.org/10.1007/s10483-019-2482-9

Project supported by the National Natural Science Foundation of China (Nos. 11672252 and 11602204) and the Fundamental Research Funds for the Central Universities, Southwest Jiaotong University (No. 2682016CX096)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, B., Shen, H., Liu, J. et al. Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects. Appl. Math. Mech.-Engl. Ed. 40, 515–548 (2019). https://doi.org/10.1007/s10483-019-2482-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-019-2482-9

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Navigation