Skip to main content
SpringerLink
Log in

Cross-section effect on mechanics of nonlocal beams

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

In the current practice, the one-dimensional nonlocal constitutive relations are always employed to model beam-type structures, regardless of the inherent three-dimensional interactions between atoms, resulting in inaccurately predicted nonlocal structural behaviors. To improve modeling, the present work first reveals and establishes how the nonlocal interactions in beams’ width and height directions substantially affect the bending behaviors of nanobeams. Based on the new concept of a general three-dimensional nonlocal constitutive relation, a three-dimensional nonlocal Euler–Bernoulli beam model is developed for bending analysis. Closed-form solutions for the deflections of simply supported and cantilevered beams are derived. The cross-section effect on the nonlocal stress, the bending moment and the deflection is explored. Moreover, an effective calibration method is proposed for the determination of the intrinsic characteristic length based on molecular dynamics simulations. It has been found that the actually nonlocal physical dimension of beam-type structures is not in agreement with that from the one-dimensional geometric direction of classical beam mechanics, and the beams’ width and height must be taken into consideration, instead of just employing the beam’s length, a practice which has been somewhat blindly undertaken in the current practice without indeed much theoretical support and understanding. A beam problem at nanoscale has to be viewed and treated as a “three-dimensional” geometric and physical problem due to its geometric dimension in length and its physical dimensions in both the thickness and the width. When its intrinsic characteristic length is comparable to its width or thickness, a rigidity-softening effect of a beam can be observed. Further, it is established that the nonlocal effect is more sensitive in the thickness direction, as compared with the width direction.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Cowley, E.R.: Lattice dynamics of silicon with empirical many-body potentials. Phys. Rev. Lett. 60(23), 2379 (1988)

    Google Scholar 

  2. Yakobson, B.I., Brabec, C.J., Bernholc, J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76(14), 2511 (1996)

    Google Scholar 

  3. Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecular systems. J. Elast. 100(1–2), 63–143 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Duan, K., He, Y., Li, Y., Liu, J., Zhang, J., Hu, Y., Lin, R., Wang, X., Deng, W., Li, L.: Machine-learning assisted coarse-grained model for epoxies over wide ranges of temperatures and cross-linking degrees. Mater. Des. 183, 108130 (2019)

    Google Scholar 

  5. Li, L., Lin, R., Ng, T.Y.: A fractional nonlocal time-space viscoelasticity theory and its applications in structural dynamics. Appl. Math. Model. 84, 116–136 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Cosserat, E., Cosserat, F.: Théorie des corps déformables, A. Hermann et fils (1909)

  7. Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3(5), 731–742 (1967)

    MATH  Google Scholar 

  8. Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5), 425–435 (1972)

    MATH  Google Scholar 

  9. Edelen, D.G.B., Laws, N.: On the thermodynamics of systems with nonlocality. Arch. Ration. Mech. Anal. 43(1), 24–35 (1971)

    MathSciNet  MATH  Google Scholar 

  10. Nowinski, J.L.: On the nonlocal theory of wave propagation in elastic plates. Science 51, 608–613 (1984)

    Google Scholar 

  11. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, Berlin (2002)

    MATH  Google Scholar 

  12. Patnaik, S., Sidhardh, S., Semperlotti, F.: A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity. Int. J. Solids Struct. 202, 398–417 (2020)

    Google Scholar 

  13. de Sciarra, F.M.: Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua. Int. J. Solids Struct. 45(14–15), 4184–4202 (2008)

    MATH  Google Scholar 

  14. Pisano, A., Sofi, A., Fuschi, P.: Nonlocal integral elasticity: 2D finite element based solutions. Int. J. Solids Struct. 46(21), 3836–3849 (2009)

    MATH  Google Scholar 

  15. de Sciarra, F.M.: On non-local and non-homogeneous elastic continua. Int. J. Solids Struct. 46(3–4), 651–676 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Khodabakhshi, P., Reddy, J.N.: A unified integro-differential nonlocal model. Int. J. Eng. Sci. 95, 60–75 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Shaat, M.: Iterative nonlocal elasticity for kirchhoff plates. Int. J. Mech. Sci. 90, 162–170 (2015)

    MathSciNet  Google Scholar 

  18. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)

    Google Scholar 

  19. Srinivasa, A.R., Reddy, J.N.: An overview of theories of continuum mechanics with nonlocal elastic response and a general framework for conservative and dissipative systems. Appl. Mech. Rev. 69(3), 030802 (2017)

    Google Scholar 

  20. Barretta, R., Faghidian, S.A., de Sciarra, F.M.: Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. Int. J. Eng. Sci. 136, 38–52 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Ouakad, H.M., Valipour, A., Żur, K.K., Sedighi, H.M., Reddy, J.: On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity. Mech. Mater. 148, 103532 (2020)

    Google Scholar 

  22. Li, L., Hu, Y.: Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Malikan, M., Krasheninnikov, M., Eremeyev, V.A.: Torsional stability capacity of a nano-composite shell based on a nonlocal strain gradient shell model under a three-dimensional magnetic field. Int. J. Eng. Sci. 148, 103210 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Malikan, M., Uglov, N.S., Eremeyev, V.A.: On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures. Int. J. Eng. Sci. 157, 103395 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Sedighi, H.M., Malikan, M., Valipour, A., Żur, K.K.: Nonlocal vibration of carbon/boron-nitride nano-hetero-structure in thermal and magnetic fields by means of nonlinear finite element method. J. Comput. Des. Eng. 7, 591–602 (2020)

    Google Scholar 

  26. Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41(3–5), 305–312 (2003)

    Google Scholar 

  28. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2), 288–307 (2007)

    MATH  Google Scholar 

  29. Lu, P., Lee, H.P., Lu, C., Zhang, P.Q.: Application of nonlocal beam models for carbon nanotubes. Int. J. Solids Struct. 44(16), 5289–5300 (2007)

    MATH  Google Scholar 

  30. Kiani, K., Mehri, B.: Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J. Sound Vib. 329(11), 2241–2264 (2010)

    Google Scholar 

  31. Thai, H.-T.: A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Li, L., Li, X., Hu, Y.: Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 102, 77–92 (2016)

    MATH  Google Scholar 

  33. Nejad, M.Z., Hadi, A.: Non-local analysis of free vibration of bi-directional functionally graded euler-bernoulli nano-beams. Int. J. Eng. Sci. 105, 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Ebrahimi, F., Barati, M.R.: A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures. Int. J. Eng. Sci. 107, 183–196 (2016)

    Google Scholar 

  35. Lu, L., Guo, X., Zhao, J.: Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Li, L., Hu, Y.: Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects. Int. J. Mech. Sci. 120, 159–170 (2017)

    Google Scholar 

  37. Numanoğlu, H.M., Akgöz, B., Civalek, Ö.: On dynamic analysis of nanorods. Int. J. Eng. Sci. 130, 33–50 (2018)

    Google Scholar 

  38. Apuzzo, A., Barretta, R., Faghidian, S., Luciano, R., de Sciarra, F.M.: Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int. J. Eng. Sci. 133, 99–108 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Ghayesh, M.H., Farajpour, A.: Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory. Int. J. Eng. Sci. 129, 84–95 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Guo, S., He, Y., Liu, D., Lei, J., Li, Z.: Dynamic transverse vibration characteristics and vibro-buckling analyses of axially moving and rotating nanobeams based on nonlocal strain gradient theory. Microsyst. Technol. 24(2), 963–977 (2018)

    Google Scholar 

  41. Karami, B., Janghorban, M., Rabczuk, T.: Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Compos. B Eng. 182, 107622 (2020)

    Google Scholar 

  42. She, G.-L., Yuan, F.-G., Karami, B., Ren, Y.-R., Xiao, W.-S.: On nonlinear bending behavior of FG porous curved nanotubes. Int. J. Eng. Sci. 135, 58–74 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Li, C., Tian, X., He, T.: Size-dependent buckling analysis of Euler-Bernoulli nanobeam under non-uniform concentration. Arch. Appl. Mech. 90, 1845–1860 (2020)

    Google Scholar 

  44. Gholipour, A., Ghayesh, M.H.: Nonlinear coupled mechanics of functionally graded nanobeams. Int. J. Eng. Sci. 150, 103221 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Jankowski, P., Żur, K.K., Kim, J., Reddy, J.: On the bifurcation buckling and vibration of porous nanobeams. Compos. Struct. 250, 112632 (2020)

    Google Scholar 

  46. Sourani, P., Hashemian, M., Pirmoradian, M., Toghraie, D.: A comparison of the bolotin and incremental harmonic balance methods in the dynamic stability analysis of an euler-bernoulli nanobeam based on the nonlocal strain gradient theory and surface effects. Mech. Mater. 103403, 52 (2020)

    Google Scholar 

  47. Sahmani, S., Safaei, B.: Influence of homogenization models on size-dependent nonlinear bending and postbuckling of bi-directional functionally graded micro/nano-beams. Appl. Math. Model. 82, 336–358 (2020)

    MathSciNet  MATH  Google Scholar 

  48. Kiani, K., Kamil, K.: Vibrations of double-nanorod-systems with defects using nonlocal-integral-surface energy-based formulations. Compos. Struct. 113028, 21 (2020)

    Google Scholar 

  49. Farajpour, A., Ghayesh, M.H., Farokhi, H.: A review on the mechanics of nanostructures. Int. J. Eng. Sci. 133, 231–263 (2018)

    MathSciNet  MATH  Google Scholar 

  50. Wu, C.-P., Yu, J.-J.: A review of mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes using eringen’s nonlocal elasticity theory. Arch. Appl. Mech. 89(9), 1761–1792 (2019)

    Google Scholar 

  51. Zhu, X., Li, L.: Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. Int. J. Mech. Sci. 133, 639–650 (2017)

    Google Scholar 

  52. Zhu, X., Li, L.: Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Compos. Struct. 178, 87–96 (2017)

    Google Scholar 

  53. Eptaimeros, K., Koutsoumaris, C.C., Karyofyllis, I.: Eigenfrequencies of microtubules embedded in the cytoplasm by means of the nonlocal integral elasticity. Acta Mech. 231, 1669–1684 (2020)

    MathSciNet  MATH  Google Scholar 

  54. Eptaimeros, K., Koutsoumaris, C.C., Tsamasphyros, G.: Nonlocal integral approach to the dynamical response of nanobeams. Int. J. Mech. Sci. 115, 68–80 (2016)

    Google Scholar 

  55. Fernández-Sáez, J., Zaera, R., Loya, J., Reddy, J.: Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int. J. Eng. Sci. 99, 107–116 (2016)

    MathSciNet  MATH  Google Scholar 

  56. Tuna, M., Kirca, M.: Exact solution of Eringen’s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams. Int. J. Eng. Sci. 105, 80–92 (2016)

    MathSciNet  MATH  Google Scholar 

  57. Faghidian, S.A.: Higher-order nonlocal gradient elasticity: A consistent variational theory. Int. J. Eng. Sci. 154, 103337 (2020)

    MathSciNet  MATH  Google Scholar 

  58. Fazlali, M., Faghidian, S.A., Asghari, M., Shodja, H.M.: Nonlinear flexure of Timoshenko-Ehrenfest nano-beams via nonlocal integral elasticity. Eur. Phys. J. Plus 135(8), 1–20 (2020)

    Google Scholar 

  59. Zhu, X., Wang, Y., Dai, H.H.: Buckling analysis of Euler-Bernoulli beams using Eringen’s two-phase nonlocal model. Int. J. Eng. Sci. 116, 130–140 (2017)

    MathSciNet  MATH  Google Scholar 

  60. Li, L., Lin, R., Ng, T.Y.: Contribution of nonlocality to surface elasticity. Int. J. Eng. Sci. 152, 103311 (2020)

    MathSciNet  MATH  Google Scholar 

  61. Bažant, Z.P., Jirásek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149 (2002)

    Google Scholar 

  62. Glaisher, J.W.L.: Liv. On a class of definite integrals-part ii, The London, Edinburgh, and Dublin. Philos. Mag. J. Sci. 42(282), 421–436 (1871)

    Google Scholar 

  63. Reddy, J.N., Pang, S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2), 023511 (2008)

    Google Scholar 

  64. Challamel, N., Reddy, J.N., Wang, C.M.: Eringen’s stress gradient model for bending of nonlocal beams. J. Eng. Mech. 142(12), 04016095 (2016)

    Google Scholar 

  65. Li, C., Yao, L., Chen, W., Li, S.: Comments on nonlocal effects in nano-cantilever beams. Int. J. Eng. Sci. 87, 47–57 (2015)

    Google Scholar 

  66. Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1–19 (1995)

    MATH  Google Scholar 

  67. Stillinger, F.H., Weber, T.A.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31(8), 5262 (1985)

    Google Scholar 

  68. Kiani, K., Efazati, M.: Nonlocal vibrations and instability of three-dimensionally accelerated moving nanocables. Phys. Scr. 95(10), 105005 (2020)

    Google Scholar 

  69. Shariati, A., Mohammad-Sedighi, H., Żur, K.K., Habibi, M., Safa, M., et al.: On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials 13(7), 1707 (2020)

    Google Scholar 

  70. Li, X., Li, L., Hu, Y., Ding, Z., Deng, W.: Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos. Struct. 165, 250–265 (2017)

    Google Scholar 

  71. Li, L., Hu, Y.: Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 107, 77–97 (2016)

    MathSciNet  MATH  Google Scholar 

  72. Norouzzadeh, A., Ansari, R., Rouhi, H.: An analytical study on wave propagation in functionally graded nano-beams/tubes based on the integral formulation of nonlocal elasticity. Waves Random Complex Media 30(3), 562–580 (2020)

    MathSciNet  Google Scholar 

  73. Fakher, M., Behdad, S., Naderi, A., Hosseini-Hashemi, S.: Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium. Int. J. Mech. Sci. 171, 105381 (2020)

    Google Scholar 

  74. Hosseini-Hashemi, S., Behdad, S., Fakher, M.: Vibration analysis of two-phase local/nonlocal viscoelastic nanobeams with surface effects. Eur. Phys. J. Plus 135(2), 190 (2020)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant no. 51605172), the Natural Science Foundation of Hubei Province (Grant no. 2016CFB191) and the Fundamental Research Funds for the Central Universities (Grant no. 2015MS014).

Author information

Authors and Affiliations

  1. State Key Lab of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Li Li

  2. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore, 639798, Republic of Singapore

    Rongming Lin

  3. State Key Lab of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Yujin Hu

Authors
  1. Li Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rongming Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yujin Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Lin, R. & Hu, Y. Cross-section effect on mechanics of nonlocal beams. Arch Appl Mech 91, 1541–1556 (2021). https://doi.org/10.1007/s00419-020-01839-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01839-4

Keywords

Search

Navigation