Abstract
Post-buckling configurations of nanorods with various end conditions are described by means of exact analytical solutions of the nonlinear differential equations governing equilibrium configurations of rods that are made of an Eringen’s nonlocal material and deform according to the kinematics of Kirchhoff’s theory. The general solutions in terms of Weierstrass elliptic functions of the equilibrium equations for planar flexural deformations of naturally straight rods are deduced; then, these solutions are specialized to rods subject to compressive axial forces and applied to the study of post-buckling behavior. Comparison of the results for nonlocal and classical rods having the same geometry and tensile modulus shows that the former exhibit, with respect to the latter, a reduction in rigidity that becomes more significant for greater values of the material parameter which accounts for small-scale effects in the response of an Eringen’s nonlocal material.
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References
Kirchhoff, G.: Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. J. f. Reine. angew. Math. (Crelle) 56, 285–313 (1859)
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, Reprint of the 4th edn. Dover Publications, New York (1944)
Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44, 1–23 (1992)
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
Peddison, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
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)
Wang, C.M., Zhang, Y.Y., Xiang, Y., Reddy, J.N.: Recent studies on buckling of carbon nanotubes. Appl. Mech. Rev. 63, 030804 (2010)
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)
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)
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)
Yakobson, B.I., Brabec, C.J., Bernholc, J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 2511–2514 (2013)
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)
Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)
Sinir, B.G., Özhan, B.B., Reddy, J.N.: Buckling configurations and dynamic response of buckled Euler–Bernoulli beams with non-classical supports. Latin Am. J. Solids Struct. 11, 2516–2536 (2014)
Wang, C.M., Xiang, Y., Kitipornchai, S.: Postbuckling of nano rods/tubes based on nonlocal beam theory. Int. J. Appl. Mech. 1, 259 (2009)
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)
Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R., Sahmani, S.: Postbuckling characteristics of nanobeams based on the surface elasticity theory. Compos. B 55, 240–246 (2013)
Xu, S.P., Xu, M.R., Wang, C.M.: Stability analysis of nonlocal elastic columns with initial imperfections. Mathematical problems in Engineering 341232 (2013)
Challamel, N., Kocsis, A., Wang, C.M.: Discrete and non-local elastica. Int. J. Non-linear Mech. 77, 128–140 (2015)
Lembo, M.: On nonlinear deformations of nonlocal elastic rods. Int. J. Solids Struct. 90, 215–227 (2016)
Coleman, B.C., 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)
Binet, J.: Mémoire sur l’intégration des équations de la courbe élastique à double courbure. Compte Rendu de l’Académie des Sciences T. XVIII, 1115–1119 (1844)
Wantzel, P.-L.: Note sur l’intégration des équations de la courbe élastique à double courbure. Compte Rendu de l’Académie des Sciences T. XVIII, 1197–1201 (1844)
Hermite, C.: Sur quelques applications des fonctions elliptiques. Gauthier-Villars, Paris (1885)
Coleman, B.C., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elast. 60, 173–221 (2000)
Lembo, M.: On the stability of elastic annular rods. Int. J. Solids Struct. 40, 317–330 (2003)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Bianchi, L.: Lezioni sulla teoria delle funzioni di variabile complessa e delle funzioni ellittiche. Zanichelli, Bologna (1930)
Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Springer, New York (1988)
Kumar, D., Heinrich, C., Waas, A.M.: Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories. J. Appl. Phys. 103, 073521 (2008)
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Lembo, M. Exact solutions for post-buckling deformations of nanorods. Acta Mech 228, 2283–2298 (2017). https://doi.org/10.1007/s00707-017-1834-3
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DOI: https://doi.org/10.1007/s00707-017-1834-3