Skip to main content
SpringerLink
Log in

Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model

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

Abstract

Several studies indicate that Eringen’s nonlocal model may lead to some inconsistencies for both Euler-Bernoulli and Timoshenko beams, such as cantilever beams subjected to an end point force and fixed-fixed beams subjected a uniform distributed load. In this paper, the elastic buckling behavior of nanobeams, including both Euler-Bernoulli and Timoshenko beams, is investigated on the basis of a stress-driven nonlocal integral model. The constitutive equations are the Fredholm-type integral equations of the first kind, which can be transformed to the Volterra integral equations of the first kind. With the application of the Laplace transformation, the general solutions of the deflections and bending moments for the Euler-Bernoulli and Timoshenko beams as well as the rotation and shear force for the Timoshenko beams are obtained explicitly with several unknown constants. Considering the boundary conditions and extra constitutive constraints, the characteristic equations are obtained explicitly for the Euler-Bernoulli and Timoshenko beams under different boundary conditions, from which one can determine the critical buckling loads of nanobeams. The effects of the nonlocal parameters and buckling order on the buckling loads of nanobeams are studied numerically, and a consistent toughening effect is obtained.

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

Access this article

Log in via an institution

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. 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)

    MATH  Google Scholar 

  2. GAVAN, K. B., WESTRA, H. J. R., VAN DER DRIFT, E. W. J. M., VENSTRA, W. J., and VAN DER ZANT, H. S. J. Size-dependent effective Young’s modulus of silicon nitride cantilevers. Applied Physics Letter, 94, 233108 (2009)

    Google Scholar 

  3. IDIART, M. I. and FLECK, N. A. Size effects in the torsion of thin metal wires. Modelling and Simulation in Materials Science and Engineering, 18, 015009 (2010)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. MINDLIN, R. D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 57–78 (1964)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. ERINGEN, A. C. and EDELEN, D. G. B. On nonlocal elasticity. International Journal of Engineering Science, 10, 233–248 (1972)

    MathSciNet  MATH  Google Scholar 

  10. 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)

    Google Scholar 

  11. ERINGEN, A. C. Theory of nonlocal elasticity and some applications. Res Mechanica, 21, 313–342 (1987)

    Google Scholar 

  12. ARASH, B. and WANG, Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Computational Materials Science, 51, 303–313 (2012)

    Google Scholar 

  13. 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)

    MathSciNet  Google Scholar 

  14. BEHERA, L. and CHAKRAVERTY, S. Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: a review. Archives of Computational Methods in Engineering, 24, 481–494 (2017)

    MathSciNet  MATH  Google Scholar 

  15. REDDY, J. N. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288–307 (2007)

    MATH  Google Scholar 

  16. AYDOGDU, M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-Dimensional Systems and Nanostructures, 41, 1651–1655 (2009)

    Google Scholar 

  17. BERRABAH, H. M., TOUNSI, A., SEMMAH, A., and BEDIA, E. A. A. Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams. Structural Engineering and Mechanics, 48, 351–365 (2013)

    Google Scholar 

  18. EMAM, S. A. A general nonlocal nonlinear model for buckling of nanobeams. Applied Mathematical Modelling, 37, 6929–6939 (2013)

    MathSciNet  MATH  Google Scholar 

  19. WANG, C. M., ZHANG, Y. Y., RAMESH, S. S., and KITIPORNCHAI, S. Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 39, 3904–3909 (2006)

    Google Scholar 

  20. THAI, H. T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Science, 52, 56–64 (2012)

    MathSciNet  MATH  Google Scholar 

  21. THAI, H. T. and VO, T. P. A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. International Journal of Engineering Science, 54, 58–66 (2012)

    MathSciNet  MATH  Google Scholar 

  22. ZHANG, Y. Y., WANG, C. M., and CHALLAMEL, N. Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. Journal of Engineering Mechanics-ASCE, 136, 562–574 (2010)

    Google Scholar 

  23. GHANNADPOUR, S. A. M., MOHAMMADI, B., and FAZILATI, J. Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Composite Structures, 96, 584–589 (2013)

    Google Scholar 

  24. CHALLAMEL, N., LERBET, J., WANG, C. M., and ZHANG, Z. Analytical length scale calibration of nonlocal continuum from a microstructured buckling model. Zeitschrift für Angewandte Mathematik und Mechanik, 94, 402–413 (2014)

    MathSciNet  MATH  Google Scholar 

  25. XU, X. J. and ZHENG, M. L. Analytical solutions for buckling of size-dependent Timoshenko beams. Applied Mathematics and Mechanics (English Edition), 40(7), 953–976 (2019) https://doi.org/10.1007/s10483-019-2494-8

    MathSciNet  MATH  Google Scholar 

  26. MENG, L. C., ZOU, D. J., LAI, H., GUO, Z. L., HE, X. Z., XIE, Z. J., and GAO, C. F. Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force. Applied Mathematics and Mechanics (English Edition), 39(12), 1805–1824 (2018) https://doi.org/10.1007/s10483-018-2395-9

    MathSciNet  MATH  Google Scholar 

  27. YU, Y. J., XUE, Z. N., LI, C. L., and TIAN, X. G. Buckling of nanobeams under nonuniform temperature based on nonlocal thermoelasticity. Composite Structures, 146, 108–113 (2016)

    Google Scholar 

  28. TOUNSI, A., SEMMAH, A., and BOUSAHLA, A. A. Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. Journal of Nanomechanics and Micromechanics, 3, 20–25 (2013)

    Google Scholar 

  29. EBRAHIMI, F. and SALARI, E. Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams. Mechanics of Advanced Materials and Structures, 23, 1379–1397 (2016)

    Google Scholar 

  30. LEI, J., HE, Y., LI, Z., GUO, S., and LIU, D. Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams. Journal of Thermal Stresses, 42, 526–539 (2019)

    Google Scholar 

  31. REFAEINEJAD, V., RAHMANI, O., and HOSSEINI, S. A. H. An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories. Scientia Iranica, 24, 1635–1653 (2017)

    Google Scholar 

  32. SARI, M. E. S., AL-KOUZ, W. G., and ATIEH, A. Buckling analysis of axially functionally graded tapered nanobeams resting on elastic foundations, based on nonlocal elasticity theory. Strojniski Vestnik-Journal of Mechanical Engineering, 64, 772–782 (2018)

    Google Scholar 

  33. ROBINSON, M. T. A. and ADALI, S. Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation. Composite Structures, 206, 95–103 (2018)

    Google Scholar 

  34. EBRAHIMI, F. and BARATI, M. R. Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. Journal of the Brazilian Society of Mechanical Sciences, 39, 937–952 (2017)

    Google Scholar 

  35. NIKNAM, H. and AGHDAM, M. M. A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Composite Structures, 119, 452–462 (2015)

    Google Scholar 

  36. RAHMANI, O. and JANDAGHIAN, A. A. Buckling analysis of functionally graded nanobeams based on a nonlocal third-order shear deformation theory. Applied Physics A-Materials Science & Processing, 119, 1019–1032 (2015)

    Google Scholar 

  37. SIMSEK, M. and YURTCU, H. H. Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Composite Structures, 97, 378–386 (2013)

    Google Scholar 

  38. NEJAD, M. Z., HADI, A., and RASTGOO, A. Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory. International Journal of Engineering Science, 103, 1–10 (2016)

    MathSciNet  MATH  Google Scholar 

  39. YIN, G. S., DENG, Q. T., and YANG, Z. C. Bending and buckling of functionally graded Poisson’s ratio nanoscale beam based on nonlocal theory. Iranian Journal of Science and Technology Transaction A-Science, 39, 559–565 (2015)

    Google Scholar 

  40. BENAHMED, A., FAHSI, B., BENZAIR, A., ZIDOUR, M., BOURADA, F., and TOUNSI, A. Critical buckling of functionally graded nanoscale beam with porosities using nonlocal higher-order shear deformation. Structural Engineering and Mechanics, 69, 457–466 (2019)

    Google Scholar 

  41. 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

    MathSciNet  MATH  Google Scholar 

  42. HOSSEINI, S. A. H. and RAHMANI, O. Surface effects on buckling of double nanobeam system based on nonlocal Timoshenko model. International Journal of Structural Stability and Dynamics, 16, 1550077 (2016)

    MathSciNet  MATH  Google Scholar 

  43. EBRAHIMI, F. and BARATI, M. R. Magnetic field effects on buckling characteristics of smart flexoelectrically actuated piezoelectric nanobeams based on nonlocal and surface elasticity theories. Microsystem Technologies, 24, 2147–2157 (2018)

    Google Scholar 

  44. EBRAHIMI, F. and KARIMIASL, M. Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams. Mechanics of Advanced Materials and Structures, 25, 943–952 (2018)

    Google Scholar 

  45. EBRAHIMI-NEJAD, S. and BOREIRY, M. Comprehensive nonlocal analysis of piezoelectric nanobeams with surface effects in bending, buckling and vibrations under magneto-electro-thermomechanical loading. Materials Research Express, 5, 035028 (2018)

    Google Scholar 

  46. 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)

    Google Scholar 

  47. WANG, Q. and LIEW, K. M. Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Physics Letters A, 363, 236–242 (2007)

    Google Scholar 

  48. CHALLAMEL, N. and WANG, C. M. The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology, 19, 345703 (2008)

    Google Scholar 

  49. REDDY, J. N. and PANG, S. D. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics, 103, 023511 (2008)

    Google Scholar 

  50. LI, C., YAO, L. Q., CHEN, W. Q., and LI, S. Comments on nonlocal effects in nano-cantilever beams. International Journal of Engineering Science, 87, 47–57 (2015)

    Google Scholar 

  51. POLIZZOTTO, C. Nonlocal elasticity and related variational principles. International Journal of Solids and Structures, 38, 7359–7380 (2001)

    MathSciNet  MATH  Google Scholar 

  52. BURHANETTIN ALTAN, S. Uniqueness of initial-boundary value problems in nonlocal elasticity. International Journal of Solids and Structures, 25, 1271–1278 (1989)

    MathSciNet  MATH  Google Scholar 

  53. TUNA, M. and KIRCA, M. Exact solution of Eringen’s nonlocal integral model for vibration and buckling of Euler-Bernoulli beam. International Journal of Mechanical Sciences, 107, 54–67 (2016)

    MATH  Google Scholar 

  54. ROMANO, G. and BARRETTA, R. Comment on the paper “Exact solution of Eringen’s nonlocal integral model for bending of Euler Bernoulli and Timoshenko beams” by Meral Tuna & Mesut Itirca. International Journal of Mechanical Sciences, 109, 240–242 (2016)

    MATH  Google Scholar 

  55. ROMANO, G., BARRETTA, R., DIACO, M., and DE SCIARRA, F. M. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Mechanical Sciences, 121, 151–156 (2017)

    Google Scholar 

  56. ZHU, X., WANG, Y., and DAI, H. H. Buckling analysis of Euler-Bernoulli beams using Eringen’s two-phase nonlocal model. International Journal of Engineering Science, 116, 130–140 (2017)

    MathSciNet  MATH  Google Scholar 

  57. ROMANO, G. and BARRETTAI, R. Nonlocal elasticity in nanobeams: the stress-driven integral model. International Journal of Engineering Science, 115, 14–27 (2017)

    MathSciNet  MATH  Google Scholar 

  58. BARRETTA, R., LUCIANO, R., DE SCIARRA, F. M., and RUTA, G. Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. European Journal of Mechanics A-Solids, 72, 275–286 (2018)

    MathSciNet  MATH  Google Scholar 

  59. ZHANG, J. Q., QING, H., and GAO, C. F. Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress-derived nonlocal integral model. Zeitschrift für Angewandte Mathematik und Mechanik (2019) https://doi.org/10.1002/zamm.201900148

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Peng Jiang, Hai Qing & Cunfa Gao

Authors
  1. Peng Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hai Qing
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cunfa Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hai Qing.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11672131), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures of China (No. MCMS-0217G02), and the Priority Academic Program Development of Jiangsu Higher Education Institutions of China (No. 11672131)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, P., Qing, H. & Gao, C. Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model. Appl. Math. Mech.-Engl. Ed. 41, 207–232 (2020). https://doi.org/10.1007/s10483-020-2569-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-020-2569-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation