Skip to main content
SpringerLink
Log in

A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

There are exhaustive reports revolving around the nonlocal differential beam models of micro- and nano-electromechanical systems (MEMS–NEMS), carbon nanotubes (CNTs), nanomaterials, etc., since the nonlocal continuum theory is considered a viable option for shedding light on size effect phenomena. A constitutive equation with a second-order differential operator, Helmholtz type, is employed in these studies. However, these models do not produce quadratic, self-adjoint energy functionals and give rise to paradoxes in static and dynamical problems. The transformation of Eringen’s integral constitutive equation into a differential one is not an injective process in a finite domain and that is probably responsible for the inconsistencies and paradoxes raised. This work researches into the adequacy of a higher-order differential operator, bi-Helmholtz type, applied to engineering problems. The bi-Helmholtz-type operator is more effective for describing wave dispersion of atomic models than the Helmholtz type. Eringen’s nonlocal integral stress model is also looked into beams for both types of kernels. The nonlocal integral model’s key benefit is the energy consistent formulas produced, whereas its main drawback lies in the handling of the 1st kind Fredholm governing equation. Exploiting the modified nonlocal attenuation function (kernel) normalized in a finite domain, the 1st kind Fredholm integral equation is directly transformed into a 2nd kind one. Furthermore, the modified kernel successfully handles the physical inconsistencies in a finite domain, such as the reversion of the nonlocal integral stress model to the classic-local model when the nonlocal parameter tends to zero. Our research objective is to explore bi-Helmholtz operator’s application and the corresponding kernel in the nonlocal integral model. The results deduced and the conclusions reached can be considered adequate for the nonlocal integral form’s suitability for dealing with problems in nanoscale.

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. Kröner, E.: Interrelations between various branches of continuum mechanics. In: Kröner, E. (ed.) Mechanics of Generalized Continua: Proceedings of the IUTAM-Symposium on The Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart (Germany), pp. 330–340. Springer, Berlin (1968)

    Chapter  Google Scholar 

  2. Krumhansl, J.A.: Some considerations of the relation between solid state physics and generalized continuum mechanics. In: Kröner, E. (ed.) Mechanics of Generalized Continua, pp. 298–311. Springer, Berlin (1968)

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Kröner, E., Datta, B.K.: Nichtlokale Elastostatik: Ableitung aus der Gittertheorie. Zeitschrift für Phys. 196(3), 203–211 (1966)

    Article  Google Scholar 

  5. Krumhansl, J.A.: Generalized continuum field representations for lattice vibrations. In: Wallis, R.F. (ed.) Lattice Dynamics. Pergamon Press, Oxford (1965)

    Google Scholar 

  6. Kunin, I.A.: Model of elastic medium with simple structure and space dispersion. Prikl. Mat. Mekh. 30, 9 (1966). (in Russian)

    Google Scholar 

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

    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)

    Article  MATH  Google Scholar 

  9. Eringen, A.C.: On nonlocal fluid mechanics. Int. J. Eng. Sci. 10(6), 561–575 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eringen, A.C.: On nonlocal microfluid mechanics. Int. J. Eng. Sci. 11(2), 291–306 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eringen, A.C.: Theory of nonlocal electromagnetic elastic solids. J. Math. Phys. 14(6), 733–740 (1973)

    Article  MATH  Google Scholar 

  12. Eringen, A.C.: Theory of nonlocal piezoelectricity. J. Math. Phys. 25(3), 717–727 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eringen, A.C.: Nonlocal continuum mechanics based on distributions. Int. J. Eng. Sci. 44(3–4), 141–147 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Eringen, A.C.: A unified theory of thermomechanical materials. Int. J. Eng. Sci. 4(2), 179–202 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cemal Eringen, A., Kim, B.S.: Stress concentration at the tip of crack. Mech. Res. Commun. 1(4), 233–237 (1974)

    Article  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)

    Article  Google Scholar 

  19. Edelen, D.G.B., Green, A.E., Laws, N.: Nonlocal continuum mechanics. Arch. Ration. Mech. Anal. 43(1), 36–44 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kress, R.: Linear Integral Equations. Springer, New York (2014)

    Book  MATH  Google Scholar 

  21. 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)

    Article  Google Scholar 

  22. Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94(11), 7281–7287 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Wang, C.M., Zhang, Y.Y., He, X.Q.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10), 105401 (2007)

    Article  Google Scholar 

  25. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007)

    Article  MATH  Google Scholar 

  26. Wang, Q., Liew, K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363(3), 236–242 (2007)

    Article  Google Scholar 

  27. Wang, Q., Varadan, V.K.: Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater. Struct. 16(1), 178 (2007)

    Article  Google Scholar 

  28. Wang, Q., Varadan, V.K.: Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater. Struct. 15(2), 659 (2006)

    Article  Google Scholar 

  29. Lu, P., et al.: Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys. 99(7), 073510 (2006)

    Article  Google Scholar 

  30. Wang, Q.: Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98(12), 124301 (2005)

    Article  Google Scholar 

  31. Harik, V.M.: Mechanics of carbon nanotubes: applicability of the continuum-beam models. Comput. Mater. Sci. 24(3), 328–342 (2002)

    Article  Google Scholar 

  32. Harik, V.M.: Ranges of applicability for the continuum beam model in the mechanics of carbon nanotubes and nanorods. Solid State Commun. 120(7–8), 331–335 (2001)

    Article  Google Scholar 

  33. Krishnan, A., et al.: Young’s modulus of single-walled nanotubes. Phys. Rev. B 58(20), 14013–14019 (1998)

    Article  Google Scholar 

  34. Wong, E.W., Sheehan, P.E., Lieber, C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277(5334), 1971–1975 (1997)

    Article  Google Scholar 

  35. Treacy, M.M.J., Ebbesen, T.W., Gibson, J.M.: Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381(6584), 678–680 (1996)

    Article  Google Scholar 

  36. Challamel, N., et al.: On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Arch. Appl. Mech. 84(9), 1275–1292 (2014)

    Article  MATH  Google Scholar 

  37. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19(34), 345703 (2008)

    Article  Google Scholar 

  38. Fernández-Sáez, J., et al.: Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int. J. Eng. Sci. 99, 107–116 (2016)

    Article  MathSciNet  Google Scholar 

  39. Polizzotto, C.: Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38(42–43), 7359–7380 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mahmoud, F.F.: On the nonexistence of a feasible solution in the context of the differential form of Eringen’s constitutive model: a proposed iterative model based on a residual nonlocality formulation. Int. J. Appl. Mech. 9(7), 1750094 (2017)

    Article  MathSciNet  Google Scholar 

  41. Benvenuti, E., Simone, A.: One-dimensional nonlocal and gradient elasticity: Closed-form solution and size effect. Mech. Res. Commun. 48, 46–51 (2013)

    Article  Google Scholar 

  42. Polyanin, A., Manzhirov, A.: Handbook of Integral Equations. CRC Press, New York (2008)

    Book  MATH  Google Scholar 

  43. Chang, I.L.: Molecular dynamics investigation of carbon nanotube resonance. Modell. Simul. Mater. Sci. Eng. 21(4), 045011 (2013)

    Article  Google Scholar 

  44. Chang, I.L., Chang-Ming, H.: Vibrational behavior of single-walled carbon nanotubes: atomistic simulations. Jpn. J. Appl. Phys. 52(10R), 105101 (2013)

    Article  Google Scholar 

  45. Zhang, Y.Y., Wang, C.M., Tan, V.B.C.: Assessment of Timoshenko Beam models for vibrational behavior of single walled carbon nanotubes using molecular dynamics. Adv. Appl. Math. Mech. 1, 89–106 (2009)

    MathSciNet  Google Scholar 

  46. Duan, W.H., Wang, C.M., Zhang, Y.Y.: Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J. Appl. Phys. 101(2), 024305 (2007)

    Article  Google Scholar 

  47. Koutsoumaris, C.C., et al.: Application of bi-Helmholtz nonlocal elasticity and molecular simulations to the dynamical response of carbon nanotubes. AIP Conf. Proc. 1702, 190011 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  49. Ghavanloo, E., Fazelzadeh, S.A.: Nonlocal shell model for predicting axisymmetric vibration of spherical shell-Like nanostructures. Mech. Adv. Mater. Struct. 22(7), 597–603 (2015)

    Article  Google Scholar 

  50. Hu, Y.G., et al.: Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J. Mech. Phys. Solids 56(12), 3475–3485 (2008)

    Article  MATH  Google Scholar 

  51. Ansari, R., Arash, B.: Nonlocal flügge shell model for vibrations of double-walled carbon nanotubes with different boundary conditions. J. Appl. Mech. Trans. ASME 80(2), 021006 (2013)

    Article  Google Scholar 

  52. Ke, L.L., Liu, C., Wang, Y.S.: Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Phys. E 66, 93–106 (2015)

    Article  Google Scholar 

  53. Lazar, M., Maugin, G.A., Aifantis, E.C.: On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43(6), 1404–1421 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  54. Fafalis, D.A., Filopoulos, S.P., Tsamasphyros, G.J.: On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale. Eur. J. Mech. A. Solids 36, 25–37 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Koutsoumaris, C.C., Tsamasphyros, G.J.: Application of the bi-Helmholtz type nonlocal elasticity on the free vibration problem of carbon nanotubes. In: The 2014 International Conference on Theoretical Mechanics and Applied Mechanics. Venice, Italy (2014)

  56. Koutsoumaris, C.C., et al.: Exploring the Applicability of bi-Helmholtz type nonlocal elasticity to the dynamical response of carbon nanotubes. In: The 2nd International Conference on Mechanics, Fluids, Heat, Elasticity and Electromagnetic Fields. Athens, Greece (2014)

  57. Eringen, A.C.: Theory of nonlocal elasticity and some applications. Res Mech. 21(4), 313–342 (1987)

    Google Scholar 

  58. Altan, S.B.: Uniqueness of initial-boundary value problems in nonlocal elasticity. Int. J. Solids Struct. 25(11), 1271–1278 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  59. Angela Pisano, A., Fuschi, P.: Closed form solution for a nonlocal elastic bar in tension. Int. J. Solids Struct. 40(1), 13–23 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  61. Abdollahi, R., Boroomand, B.: Benchmarks in nonlocal elasticity defined by Eringen’s integral model. Int. J. Solids Struct. 50(18), 2758–2771 (2013)

    Article  Google Scholar 

  62. Fuschi, P., Pisano, A.A., De Domenico, D.: Plane stress problems in nonlocal elasticity: finite element solutions with a strain-difference-based formulation. J. Math. Anal. Appl. 431(2), 714–736 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wang, Y.B., Zhu, X.W., Dai, H.H.: Exact solutions for the static bending of Euler–Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Adv. 6(8), 085114 (2016)

    Article  Google Scholar 

  64. Wang, Y., et al.: Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model. Math. Mech. Solids.: 1081286517750008. https://doi.org/10.1177/1081286517750008

  65. 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)

    Article  MathSciNet  Google Scholar 

  66. Challamel, N.: Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models. Comptes Rendus Mécanique. 346(4), 320–335 (2018)

    Article  Google Scholar 

  67. Eptaimeros, K.G., Koutsoumaris, C.C., Tsamasphyros, G.J.: Nonlocal integral approach to the dynamical response of nanobeams. Int. J. Mech. Sci. 115–116, 68–80 (2016)

    Article  Google Scholar 

  68. Fernández-Sáez, J., Zaera, R.: Vibrations of Bernoulli–Euler beams using the two-phase nonlocal elasticity theory. Int. J. Eng. Sci. 119, 232–248 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  70. Zhu, X., Li, L.: On longitudinal dynamics of nanorods. Int. J. Eng. Sci. 120, 129–145 (2017)

    Article  Google Scholar 

  71. 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 

  72. Borino, G., Failla, B., Parrinello, F.: A symmetric nonlocal damage theory. Int. J. Solids Struct. 40(13–14), 3621–3645 (2003)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  74. Atkinson, K.E.: The numerical solution of integral equations of the second kind. In: Cambridge Monographs on Applied and Computational Mathematics. 1997, Cambridge: Cambridge University Press (1997)

Download references

Author information

Authors and Affiliations

  1. Division of Mechanics School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    C. Chr. Koutsoumaris & K. G. Eptaimeros

Authors
  1. C. Chr. Koutsoumaris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. G. Eptaimeros
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to C. Chr. Koutsoumaris or K. G. Eptaimeros.

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

Koutsoumaris, C.C., Eptaimeros, K.G. A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body. Acta Mech 229, 3629–3649 (2018). https://doi.org/10.1007/s00707-018-2180-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2180-9

Search

Navigation