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On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach

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Abstract

In this paper, the self-adjointness of Eringen’s nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen’s model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever’s fundamental vibration frequency with respect to increasing Eringen’s small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen’s nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.

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References

  1. 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  Google Scholar 

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

    MATH  Google Scholar 

  3. 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 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Reddy J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010)

    Article  MATH  Google Scholar 

  7. Elishakoff I., Pentaras D., Dujat K., Versaci C., Muscolino G., Storch J., Bucas S., Challamel N., Natsuki T., Zhang Y.Y., Wang C.M., Ghyselinck G.: Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact. Wiley–ISTE, London (2012)

    Book  Google Scholar 

  8. Challamel N.: Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams. Compos. Struct. 105, 351–368 (2013)

    Article  Google Scholar 

  9. Maugin G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  10. 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, 1404–1421 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Challamel, N., Lerbet, J., Wang, C.M., Zhang Z.: Analytical length scale calibration of nonlocal continuum from a microstructured buckling model. Z. Angew. Math. Mech. 94(5), 402–413 (2014)

  12. Challamel, N., Zhang, Z. Wang, C.M.: Nonlocal equivalent continuum for the buckling and the vibrations of microstructured beams. ASCE J. Nanomech. Micromech. doi:10.1061/(ASCE)NM.2153-5477.0000062 (2014)

  13. Wang C.M., Zhang Z., Challamel N., Duan W.H.: Calibration of Eringen’s small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model. J. Phys. D: Appl. Phys. 46, 345501 (2013)

    Article  Google Scholar 

  14. Duan W., Challamel N., Wang C.M., Ding Z.: Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams. J. Appl. Phys. 114(104312), 1–11 (2013)

    Google Scholar 

  15. Zhang Z., Challamel N., Wang C.M.: Eringen’s small length scale coefficient for buckling of nonlocal Timoshenko beam based on a microstructured beam model. J. Appl. Phys. 114(114902), 1–6 (2013)

    Google Scholar 

  16. Adali S.: Variationalprinciples for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory. Phys. Lett. A 372, 5701–5705 (2008)

    Article  MATH  Google Scholar 

  17. Adali S.: Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam models. Nano Lett. 9(5), 1737–1741 (2009)

    Article  Google Scholar 

  18. Roach G.F.: Green’s Functions. Cambridge University Press, Cambridge (1999)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Phadikar J.K., Pradhan S.C.: Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comput. Mater. Sci. 49, 492–499 (2010)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  25. Silverman I.K.: Discussion on the paper of “Salvadori M.G., Numerical computation of buckling loads by finite differences. Trans. ASCE, 1951; 116, 590–636, 1951.”, Trans. ASCE, 1951; 116, 625–626

  26. Salvadori M.G.: Numerical computation of buckling loads by finite differences. Trans. ASCE, 1951; 116, 590–624 (590–636 with the discussion)

  27. Challamel N., Wang C.M., Elishakoff I.: Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis. Eur. J. Mech. A/Solids 44, 125–135 (2014)

    Article  MathSciNet  Google Scholar 

  28. Rosenau P.: Dynamics of nonlinear mass-spring chains near the continuum limit. Phys. Lett. A 118(5), 222–227 (1986)

    Article  MathSciNet  Google Scholar 

  29. Wattis J.A.D.: Quasi-continuum approximations to lattice equations arising from the discrete non-linear telegraph equation. J. Phys. A Math. Gen. 33, 5925–5944 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Andrianov I.V., Awrejcewicz J., Weichert D.: Improved continuous models for discrete media. Math. Probl. Eng. 986242, 1–35 (2010)

    Article  Google Scholar 

  31. Michelitsch, T.M., Collet, B., Wang, X.: Nonlocal constitutive laws generated by matrix functions: lattice dynamic models and their continuum limits. Int. J. Eng. Sci. 80, 106–123 (2014)

  32. Sheppard, W.F.: Central differences formulae. In: Proceedings, London Mathematical Society, 31 (1899)

  33. Richardson L.F.: The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam. Philos. Trans. R. Soc. Lond. 210, 307–357 (1911)

    Article  MATH  Google Scholar 

  34. Rosenau P.: Hamiltonian dynamics of dense chains and lattices: or how to correct the continuum. Phys. Lett. A 311, 39–52 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rosenau P., Schochet S.: Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit. Chaos 15, 015111 (2005)

    Article  MathSciNet  Google Scholar 

  36. Wong M.W.: An Introduction to Pseudo-Differential Operators. World Scientific Publishing, Singapore (1999)

    Book  MATH  Google Scholar 

  37. Blevins R.D.: Formulas for Natural Frequency and Mode Shape. Krieger Publishing Company, Malabar (2001)

    Google Scholar 

  38. Leckie F.A., Lindberg G.M.: The effect of lumped parameters on beam frequencies. Aeronaut. Q. 14, 224–240 (1963)

    Google Scholar 

  39. Santoro R., Elishakoff I.: Accuracy of the finite difference method in stochastic setting. J. Sound Vib. 291, 275–284 (2006)

    Article  MATH  Google Scholar 

  40. Luongo A., Zulli D.: Parametric, external and self-excitation of a tower under turbulent wind flow. J. Sound Vib. 330, 3057–3069 (2011)

    Article  Google Scholar 

  41. Wallis, R.F.: Effect of free ends on the vibration frequencies of one-dimensional lattice. Phys. Rev. 105(2), 540–545 (1957)

  42. Kivshar Y.S., Zhang F., Takeno S.: Nonlinear surface modes in monoatomic and diatomic lattices. Physica D 113, 248–260 (1998)

    Article  MATH  Google Scholar 

  43. dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids (2014), in press

  44. Alibert J.J., dell’Isola F., Seppecher P.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–74 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre de Recherche, UBS - LIMATB, Université Européenne de Bretagne, University of South Brittany UBS, Rue de Saint Maudé, BP 92116, 56321, Lorient Cedex, France

    Noël Challamel

  2. Department of Civil and Environmental Engineering, National University of Singapore, Kent Ridge Crescent, Singapore, 119260, Singapore

    Zhen Zhang & C. M. Wang

  3. Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77843-3123, USA

    J. N. Reddy

  4. Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada

    Q. Wang

  5. Institut Jean le Rond d’Alembert, CNRS UMR 7190, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, Tour 55-65, 75252, Paris Cedex 05, France

    Thomas Michelitsch & Bernard Collet

Authors
  1. Noël Challamel
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  2. Zhen Zhang
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  3. C. M. Wang
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  4. J. N. Reddy
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  5. Q. Wang
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  6. Thomas Michelitsch
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  7. Bernard Collet
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Correspondence to Noël Challamel.

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Challamel, N., Zhang, Z., Wang, C.M. et al. On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Arch Appl Mech 84, 1275–1292 (2014). https://doi.org/10.1007/s00419-014-0862-x

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  • DOI: https://doi.org/10.1007/s00419-014-0862-x

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