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Acta Mechanica

, Volume 230, Issue 3, pp 749–769 | Cite as

Infinitesimal deformations and stability of rods made of nonlocal elastic materials

  • Marzio LemboEmail author
Original Paper

Abstract

Aim of the paper is the formulation of a criterion of infinitesimal stability for a class of rods made of nonlocal elastic materials. To that end, the nonlinear equilibrium equations of naturally straight, inextensible rods subject to terminal loads are written, and the constitutive equation assumed to represent the material response in rods of finite length is discussed. Then, the equations describing the infinitesimal deformations superimposed upon a finite one are deduced. The expression of the work done by the increments of the external loads associated with an infinitesimal deformation is employed to formulate, for the considered rods, the criterion of infinitesimal stability which does not require the existence of a stored-energy function. The criterion is applied to study the stability of simply supported rods subject to axial forces, of rods with one end clamped and the other one constrained to have the tangent parallel to the undeformed rod axis, subject to axial forces and twisting couples, and of annular rods formed from naturally straight rods with the addition of twist. The results show that rods made of nonlocal elastic materials exhibit a reduction in rigidity with respect to rods having the same geometry and made of usual elastic materials with the same tensile and shear moduli.

Mathematics Subject Classification

74B20 74A60 74K10 74H55 

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References

  1. 1.
    Xia, Y., Yang, P., Sun, Y., Wu, Y., Mayers, B., Gates, B., Yin, Y., Kim, F., Yan, H.: One-dimensional nanostructures: synthesis, characterizations, and applications. Adv. Mater. 15, 353–389 (2003)CrossRefGoogle Scholar
  2. 2.
    Akita, S.: Nanomechanical applications of CNT. In: Matsumoto, Z. (ed.) Frontiers of Graphene and Carbon Nanotubes, Devices and Applications, pp. 187–200. Springer, Tokyo (2015)Google Scholar
  3. 3.
    Kaushik, B.K., Majumder, M.K.: Carbon Nanotube: Properties and Application. In: Kaushik, B.K., Majumder, M.K. (eds.) Carbon Nanotube Based VLSI Interconnects, pp. 17–37. Springer, Berlin (2015)Google Scholar
  4. 4.
    Eringen, A.C.: Vistas of nonlocal continuum physics. Int. J. Eng. Sci. 30, 1551–1565 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)CrossRefGoogle Scholar
  6. 6.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  7. 7.
    Thostenson, E.T., Ren, Z., Chou, T.-W.: Advances in the science and technology of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 61, 1899–1912 (2001)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Iijima, S., Brabec, C., Maiti, A., Bernholc, J.: Structural flexibility of carbon nanotubes. J. Chem. Phys. 104, 2089–2092 (1996)CrossRefGoogle Scholar
  12. 12.
    Wong, E.W., Sheehan, P.E., Lieber, C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277, 1971–1975 (1997)CrossRefGoogle Scholar
  13. 13.
    Falvo, M.R., Clary, G.J., Taylor II, R.M., Chi, V., Brooks Jr., F.P., Washburn, S., Superfine, R.: Bending and buckling of carbon nanotubes under large strain. Nature 389, 582–584 (1997)CrossRefGoogle Scholar
  14. 14.
    Yakobson, B.I., Avouris, P.: Mechanical properties of carbon nanotubes. In: Dresselhaus, M.S., Dresselhaus, G., Avouris, P. (eds.) Carbon Nanotubes. Topis Appl. Phys., vol. 80, pp. 287–327. Spinger, Berlin (2001)CrossRefGoogle Scholar
  15. 15.
    Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)CrossRefzbMATHGoogle Scholar
  16. 16.
    Sinir, B.G., Özhan, B.B., Reddy, J.N.: Buckling configurations and dynamic response of buckled Euler–Bernoulli beams with non-classical supports. Lat. Am. J. Solids Struct. 11, 2516–2536 (2014)CrossRefGoogle Scholar
  17. 17.
    Challamel, N., Camotin, D., Wang, C.M., Zhang, Z.: On lateral-torsional buckling of discrete elastic systems: a nonlocal approach. Eur. J. Mech. A/Solids 49, 106–113 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Challamel, N., Kocsis, A., Wang, C.M.: Discrete and non-local elastica. Int. J. Non-linear Mech. 77, 128–140 (2015)CrossRefGoogle Scholar
  19. 19.
    Wang, C.M., Xiang, Y., Kitipornchai, S.: Postbuckling of nano rods/tubes based on nonlocal beam theory. Int. J. Appl. Mech. 1, 259–266 (2009)CrossRefGoogle Scholar
  20. 20.
    Xu, S.P.: Elastica type buckling analysis of micro/nano-rods using nonlocal elasticity theory. In: Proceedings of Second Asian Conference on Mechanics of Functional Materials and Structures, Nanjing, pp. 219–222 (2010)Google Scholar
  21. 21.
    Xu, S.P., Xu, M.R., Wang, C.M.: Stability analysis of nonlocal elastic columns with initial imperfections. Math. Probl. Eng. 2013, 341232 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lembo, M.: On nonlinear deformations of nonlocal elastic rods. Int. J. Solids Struct. 90, 215–227 (2016)CrossRefGoogle Scholar
  23. 23.
    Lembo, M.: Exact solutions for post-buckling deformations of nanorods. Acta Mech. 228, 2283–2298 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lembo, M.: Exact equilibrium solutions for nonlinear spatial deformations of nanorods with application to buckling under terminal force and couple. Int. J. Solids Struct. 135, 274–288 (2018)CrossRefGoogle Scholar
  25. 25.
    Kirchhoff, G.: Über das Gleichgewicht und die Bewegug eines unendlich dünnen elastichen Stabes. J. f. reine. angew. Math. (Crelle) 56, 285–313 (1859)Google Scholar
  26. 26.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, Reprint of the Fourth Edition. Dover Publications, New York (1944)Google Scholar
  27. 27.
    Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44, 1–23 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Coleman, B.D., Dill, E.H., Lembo, M., Lu, Z., Tobias, I.: On the dynamics of rods in the theory of Kirchhoff and Clebsch. Arch. Ration. Mech. Anal. 121, 339–359 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Poisson, S.D.: Sur les lignes élastiques à double courbure. Correspondence sur l’École Royale Polytechnique Tome Troisième, 355–360 (1816)Google Scholar
  30. 30.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, New York (1937)zbMATHGoogle Scholar
  31. 31.
    Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)CrossRefGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Romano, G., Barretta, R., Diaco, M., Marotti-deSciarra, F.: Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int. J. Mech. Sci. 121, 151–156 (2017)CrossRefGoogle Scholar
  34. 34.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, New York (1965)zbMATHGoogle Scholar
  35. 35.
    Wang, C.C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoff, Leyden (1973)zbMATHGoogle Scholar
  36. 36.
    Gurtin, M., Spector, S.J.: On stability and uniqueness in finite elasticity. Arch. Ration. Mech. Anal. 70, 153–165 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Spector, S.J.: On uniqueness in finite elasticity with general loading. J. Elast. 10, 145–161 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Spector, S.J.: On uniqueness for the traction problem in finite elasticity. J. Elast. 12, 367–383 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hadamard, J.: Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique. Hermann, Paris (1903)zbMATHGoogle Scholar
  40. 40.
    Knops, R.J., Wilkes, E.W.: In: Truesdell, C. (ed.) Theory of Elastic Stability, Mechanics of Solids, vol. III. Springer, Berlin (1984)Google Scholar
  41. 41.
    Lembo, M.: On the stability of elastic annular rods. Int. J. Solids Struct. 40, 317–330 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)CrossRefGoogle Scholar
  43. 43.
    Challamel, N., Wang, C.M.: On lateral-torsional buckling of non-local beams. Adv. Appl. Math. Mech. 3, 389–398 (2010)MathSciNetGoogle Scholar
  44. 44.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  45. 45.
    Greenhill, A.G.: On the strength of shafting when exposed both to torsion and to end thrust. In: Proceedings of the Institution of Mechanical Engineers, pp. 182–209 (1883)Google Scholar
  46. 46.
    Timoshenko, S.T., Gere, J.M.: Theory of Elastic Stability, 2nd edn. McGraw-Hill, London (1961)Google Scholar
  47. 47.
    Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di IngegneriaUniversità di Roma TreRomeItaly

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