Skip to main content
SpringerLink
Log in

Nonlinear stress-driven nonlocal formulation of Timoshenko beams made of FGMs

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

Motivated by the paradoxical results obtained from the differential nonlocal elasticity theory in some cases (e.g., bending and vibration problems of cantilevers), several attempts have been recently made to develop nonlocal beam models based on the integral (original) formulation of Eringen’s nonlocal theory. These models can be classified into two main groups including strain- and stress-driven ones which have the capability of capturing the softening and hardening behaviors of material caused by nanoscale (nonlocal) effects, respectively. In the present paper, a novel stress-driven nonlocal formulation is developed for the nonlinear analysis of Timoshenko beams made of functionally graded materials. To this end, the governing equations are first derived in the context of integral form of stress-driven nonlocal model. The proposed model can be used for arbitrary kernel functions, and the paradox related to cantilever is resolved by it. The governing equations of stress-driven model in differential form together with corresponding constitutive boundary conditions are also derived. The Timoshenko beam under various end conditions is considered as the problem under study whose nonlinear static bending is analyzed. Furthermore, the generalized differential quadrature method is employed in the solution procedure. The effects of nonlocal parameter, FG index, length-to-thickness ratio and nonlinearity on the deflection of fully clamped, fully simply supported, clamped–simply supported and clamped–free beams are investigated. The presented formulation and results may be helpful in understanding nonlocal phenomena in nano-electro-mechanical systems.

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. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surface. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  3. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Article  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  7. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eringen, A.C.: A unified theory of thermomechanical materials. Int. J. Eng. Sci. 4, 179–202 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Eringen, A.C.: On nonlocal microfluid mechanics. Int. J. Eng. Sci. 11, 291–306 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eringen, A.C.: Theory of nonlocal electromagnetic elastic solids. J. Math. Phys. 14, 733–740 (1973)

    Article  ADS  MATH  Google Scholar 

  13. Eringen, A.C.: Theory of nonlocal thermoelasticity. Int. J. Eng. Sci. 12, 1063–1077 (1974)

    Article  MATH  Google Scholar 

  14. Eringen, A.C., Kim, B.S.: Stress concentration at the tip of crack. Mech. Res. Commun. 1, 233–237 (1974)

    Article  Google Scholar 

  15. Eringen, A.C.: Theory of nonlocal piezoelectricity. J. Math. Phys. 25, 717–727 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  19. Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)

    Article  ADS  Google Scholar 

  20. Wang, Q., Varadan, V.K.: Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater. Struct. 15, 659–666 (2006)

    Article  ADS  Google Scholar 

  21. Rouhi, H., Ansari, R.: Nonlocal analytical Flugge shell model for axial buckling of double-walled carbon nanotubes with different end conditions. NANO 7, 1250018 (2012)

    Article  Google Scholar 

  22. Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)

    Article  Google Scholar 

  23. Wang, K.F., Wang, B.L., Kitamura, T.: A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech. Sin. 32, 83–100 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Zhang, D., Lei, Y., Shen, Z.: Effect of longitudinal magnetic field on vibration response of double-walled carbon nanotubes embedded in viscoelastic medium. Acta Mech. Solida Sin. 31, 187–206 (2018)

    Article  Google Scholar 

  25. Eltaher, M.A., Khater, M.E., Emam, S.A.: A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Model. 40, 4109–4128 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ansari, R., Rouhi, H., Sahmani, S.: Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int. J. Mech. Sci. 53, 786–792 (2011)

    Article  Google Scholar 

  27. Shen, H.S., Xu, Y.M., Zhang, C.L.: Prediction of nonlinear vibration of bilayer graphene sheets in thermal environments via molecular dynamics simulations and nonlocal elasticity. Comput. Meth. Appl. Mech. Eng. 267, 458–470 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Liang, Y., Han, Q.: Prediction of the nonlocal scaling parameter for graphene sheet. Eur. J. Mech. A Solids 45, 153–160 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Ansari, R., Shahabodini, A., Rouhi, H.: A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions. Curr. Appl. Phys. 15, 1062–1069 (2015)

    Article  ADS  Google Scholar 

  30. Wang, Q.: Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, 124301 (2005)

    Article  ADS  Google Scholar 

  31. Lu, P., Lee, H.P., Lu, C.: Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys. 99, 073510 (2006)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  33. Wang, Q., Liew, K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)

    Article  ADS  Google Scholar 

  34. Abazari, A.M., Safavi, S.M., Rezazadeh, G., Villanueva, L.G.: Modelling the size effects on the mechanical properties of micro/nano structures. Sensors 15, 28543–28562 (2015)

    Article  Google Scholar 

  35. Challamel, N., Zhang, Z., Wang, C.M., Reddy, J.N., Wang, Q., Michelitsch, T., Collet, B.: On non-conservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from discrete-based approach. Arch. Appl. Mech. 84, 1275–1292 (2014)

    Article  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhu, X., Li, L.: Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Compos. Struct. 78, 87–96 (2017)

    Article  Google Scholar 

  39. Norouzzadeh, A., Ansari, R.: Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Phys. E 88, 194–200 (2017)

    Article  Google Scholar 

  40. Norouzzadeh, A., Ansari, R., Rouhi, H.: Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach. Appl. Phys. A 123, 330 (2017)

    Article  ADS  Google Scholar 

  41. 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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Koutsoumaris, C.C., Eptaimeros, K.G., Tsamasphyros, G.J.: A different approach to Eringen’s nonlocal integral stress model with applications for beams. Int. J. Solids Struct. 112, 222–238 (2017)

    Article  Google Scholar 

  43. Romano, G., Barretta, R.: Nonlocal elasticity in nanobeams: the stress-driven integral model. Int. J. Eng. Sci. 115, 14–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. Romano, G., Barretta, R.: Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Compos. Part B 114, 184–188 (2017)

    Article  Google Scholar 

  45. Romano, G., Barretta, R., Diaco, M.: On nonlocal integral models for elastic nano-beams. Int. J. Mech. Sci. 131–132, 490–499 (2017)

    Article  Google Scholar 

  46. Romano, G., Barretta, R., Diaco, M., Marotti de Sciarra, F.: Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. Int. J. Mech. Sci. 121, 151–156 (2017)

    Article  Google Scholar 

  47. Romano, G., Luciano, R., Barretta, R., Diaco, M.: Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Contin. Mech. Thermodyn. 30, 641–655 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Apuzzo, A., Barretta, R., Luciano, R., Marotti de Sciarra, F., Penna, R.: Free vibrations of Bernoulli–Euler nano-beams by the stress-driven nonlocal integral model. Compos. Part B Eng. 123, 105–111 (2017)

    Article  Google Scholar 

  49. Faraji Oskouie, M., Ansari, R., Rouhi, H.: Bending of Euler–Bernoulli nanobeams based on the strain- and stress-driven nonlocal integral models: a numerical approach. Acta Mech. Sin. 34, 871–882 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Faraji Oskouie, M., Ansari, R., Rouhi, H.: A numerical study on the buckling and vibration of nanobeams based on the strain- and stress-driven nonlocal integral models. Int. J. Comput. Mater. Sci. Eng. 7, 1850016 (2018)

    MATH  Google Scholar 

  51. Faraji Oskouie, M., Ansari, R., Rouhi, H.: Stress-driven nonlocal and strain gradient formulations of Timoshenko nanobeams. Eur. Phys. J. Plus 133, 336 (2018)

    Article  MATH  Google Scholar 

  52. Faraji Oskouie, M., Norouzzadeh, A., Ansari, R., Rouhi, H.: Bending of small-scale Timoshenko beams based on the integral/differential nonlocal–micropolar elasticity theory: a finite element approach. Appl. Math. Mech. Eng. Ed. 40, 767–782 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  53. Eltaher, M.A., Abdelrahman, A.A., Al-Nabawy, A., Khater, M., Mansour, A.: Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Appl. Math. Comput. 235, 512–529 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Barretta, R., Luciano, R., Marotti-de-Sciarra, F., Ruta, G.: Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. Eur. J. Mech. A Solids 72, 275–286 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis: With Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics. OUP, Oxford (2014)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    M. Roghani

  2. Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah, 44891-63157, Iran

    H. Rouhi

Authors
  1. M. Roghani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Rouhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Rouhi.

Additional information

Communicated by Andreas Öchsner.

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

Roghani, M., Rouhi, H. Nonlinear stress-driven nonlocal formulation of Timoshenko beams made of FGMs. Continuum Mech. Thermodyn. 33, 343–355 (2021). https://doi.org/10.1007/s00161-020-00906-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-020-00906-z

Keywords

Search

Navigation