Acta Mechanica

, Volume 230, Issue 1, pp 243–264 | Cite as

A non-classical Kirchhoff rod model based on the modified couple stress theory

  • G. Y. Zhang
  • X.-L. GaoEmail author
Original Paper


A new non-classical Kirchhoff rod model is developed using the modified couple stress theory, which contains one material length scale parameter and can account for microstructure-dependent size effects. The governing equations and boundary conditions are determined simultaneously by a variational formulation based on the principle of minimum total potential energy. The newly developed model recovers its classical elasticity-based counterpart as a special case when the microstructure effect is not considered. To illustrate the new non-classical Kirchhoff rod model, two sample problems are analytically solved by directly applying the general formulas derived. One problem is the equilibrium analysis of a helical rod of circular cross section deformed from a straight rod, and the other is the buckling of a straight rod of circular cross section induced by an axial compressive force. In the former, the rod undergoes a twisting-dominated deformation, while in the latter the rod deformation is bending dominated. Two closed-form expressions are obtained for the force and couple needed in deforming the helical rod, and an analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, with the microstructure effect incorporated in each case. These formulas reduce to those based on classical elasticity when the microstructure effect is suppressed. For the helical rod problem, the numerical results show that the couple predicted by the current non-classical rod model is significantly larger than that predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase in the rod radius. For the buckling problem, it is found that the critical buckling load based on the new non-classical Kirchhoff rod model is always higher than that given by the classical elasticity-based model, with the difference being significant for a very thin rod.


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  1. 1.
    Antman, S.S.: Kirchhoff’s problem for nonlinearly elastic rods. Q. Appl. Math. 32, 221–240 (1974)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44, 1–23 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)zbMATHGoogle Scholar
  5. 5.
    Nizette, M., Goriely, A.: Towards a classification of Euler–Kirchhoff filaments. J. Math. Phys. 40, 2830–2866 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33, 1–26 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Liangruksa, M., Laomettachit, T., Wongwises, S.: Theoretical study of DNA’s deformation and instability subjected to mechanical stress. Int. J. Mech. Sci. 130, 324–330 (2017)CrossRefGoogle Scholar
  8. 8.
    Westcott, T.P., Tobias, I., Olson, W.K.: Elasticity theory and numerical analysis of DNA supercoiling: an application to DNA looping. J. Phys. Chem. 99, 17926–17935 (1995)CrossRefGoogle Scholar
  9. 9.
    da Fonseca, A.F., Galvão, D.S.: Mechanical properties of nanosprings. Phys. Rev. Lett. 92, 175502-1-4 (2004)CrossRefGoogle Scholar
  10. 10.
    Kumar, A., Mukherjee, S., Paci, J.T., Chandraseker, K., Schatz, G.C.: A rod model for three dimensional deformations of single-walled carbon nanotubes. Int. J. Solids Struct. 48, 2849–2858 (2011)CrossRefGoogle Scholar
  11. 11.
    Zhang, P., Parnell, W.J.: Band gap formation and tunability in stretchable serpentine interconnects. ASME J. Appl. Mech. 84, 091007-1-7 (2017)Google Scholar
  12. 12.
    Burgner-Kahrs, J., Rucker, D.C., Choset, H.: Continuum robots for medical applications: a survey. IEEE Trans. Robot. 31, 1261–1280 (2015)CrossRefGoogle Scholar
  13. 13.
    Till, J., Rucker, D.C.: Elastic stability of Cosserat rods and parallel continuum robots. IEEE Trans. Robot. 33, 718–733 (2017)CrossRefGoogle Scholar
  14. 14.
    Hassanpour, S., Heppler, G.R.: Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Math. Mech. Solids 22, 224–242 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Springer, Dordrecht (2000)CrossRefGoogle Scholar
  16. 16.
    Cao, D.Q., Liu, D.S., Wang, C.H.T.: Nonlinear dynamic modelling for MEMS components via the Cosserat rod element approach. J. Micromech. Microeng. 15, 1334–1343 (2005)CrossRefGoogle Scholar
  17. 17.
    Liu, D.S., Wang, C.H.T.: Variational principle for a special Cosserat rod. Appl. Math. Mech. 30, 1169–1176 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)CrossRefGoogle Scholar
  20. 20.
    Wang, J.S., Cui, Y.H., Feng, X.Q., Wang, G.F., Qin, Q.H.: Surface effects on the elasticity of nanosprings. Europhys. Lett. 92, 16002-1-6 (2010)Google Scholar
  21. 21.
    Zhang, R.J.: Size effects in Kirchhoff flexible rods. Phys. Rev. E 81, 056601-1-5 (2010)Google Scholar
  22. 22.
    Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)CrossRefGoogle Scholar
  23. 23.
    Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Güven, U.: The investigation of the nonlocal longitudinal stress waves with modified couple stress theory. Acta Mech. 221, 321–325 (2011)CrossRefGoogle Scholar
  25. 25.
    Güven, U.: A more general investigation for the longitudinal stress waves in microrods with initial stress. Acta Mech. 223, 2065–2074 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)CrossRefGoogle Scholar
  27. 27.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  28. 28.
    Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rods. In: Altenbach, H., Eremeyev, V.A. (eds.), Generalized Continua from the Theory to Engineering Applications, pp. 179–248. Springer, Wien (2013)Google Scholar
  29. 29.
    Güven, U.: Two mode Mindlin–Herrmann rod solution based on modified couple stress theory. Z. Angew. Math. Mech. 94, 1011–1016 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hassanpour, S., Heppler, G.R.: Theory of micropolar gyroelastic continua. Acta Mech. 227, 1469–1491 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lembo, M.: On nonlinear deformations of nonlocal elastic rods. Int. J. Solids Struct. 90, 215–227 (2016)CrossRefGoogle Scholar
  32. 32.
    Arefi, M.: Analysis of wave in a functionally graded magneto-electro-elastic nano-rod using nonlocal elasticity model subjected to electric and magnetic potentials. Acta Mech. 227, 2529–2542 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Park, S.K., Gao, X.-L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59, 904–917 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gao, X.-L., Huang, J.X., Reddy, J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. 224, 2699–2718 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Gao, X.-L., Mahmoud, F.F.: A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects. Z. Angew. Math. Phys. 65, 393–404 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Gao, X.-L., Zhang, G.Y.: A microstructure- and surface energy-dependent third-order shear deformation beam model. Z. Angew. Math. Phys. 66, 1871–1894 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gao, X.-L., Zhang, G.Y.: A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects. Proc. R. Soc. A 472, 20160275-1-25 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Ma, H.M., Gao, X.-L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)CrossRefGoogle Scholar
  40. 40.
    Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)CrossRefGoogle Scholar
  41. 41.
    Park, S.K., Gao, X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)CrossRefGoogle Scholar
  42. 42.
    Roque, C.M.C., Ferreira, A.J.M., Reddy, J.N.: Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method. Appl. Math. Model. 37, 4626–4633 (2013)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Şimşek, M., Reddy, J.N.: A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos. Struct. 101, 47–58 (2013)CrossRefGoogle Scholar
  44. 44.
    Zhang, G.Y., Gao, X.-L., Guo, Z.Y.: A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium. Acta Mech. 228, 3811–3825 (2017)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Zhou, S.-S., Gao, X.-L.: A nonclassical model for circular Mindlin plates based on a modified couple stress theory. ASME J. Appl. Mech. 81, 051014-1-8 (2014)Google Scholar
  46. 46.
    Zhou, X., Wang, L., Qin, P.: Free vibration of micro- and nano-shells based on modified couple stress theory. J. Comput. Theor. Nanosci. 9, 814–818 (2012)CrossRefGoogle Scholar
  47. 47.
    Bîrsan, M., Altenbach, H.: On the theory of porous elastic rods. Int. J. Solids Struct. 48, 910–924 (2011)CrossRefGoogle Scholar
  48. 48.
    Lembo, M.: On the stability of elastic annular rods. Int. J. Solids Struct. 40, 317–330 (2003)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
  50. 50.
    Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, New York (2002)Google Scholar
  51. 51.
    Gao, X.-L., Mall, S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 38, 855–874 (2001)CrossRefGoogle Scholar
  52. 52.
    Gao, X.-L., Park, S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)CrossRefGoogle Scholar
  53. 53.
    Liu, Y.Z., Zu, J.W.: Stability and bifurcation of helical equilibrium of a thin elastic rod. Acta Mech. 167, 29–39 (2004)CrossRefGoogle Scholar
  54. 54.
    Chong, A.C.M., Yang, F., Lam, D.C.C., Tong, P.: Torsion and bending of micron-scaled structures. J. Mater. Res. 16, 1052–1058 (2001)CrossRefGoogle Scholar
  55. 55.
    Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)CrossRefGoogle Scholar
  56. 56.
    Frankel, S.: Complete approximate solutions of the equation \(\text{ x } = \tan \text{ x }\). Natl. Math. Mag. 11(4), 177–182 (1937)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Ugural, A.C., Fenster, S.K.: Advanced Mechanics of Materials and Applied Elasticity, 5th edn. Prentice-Hall, Upper Saddle River, New Jersey (2012)zbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringSouthern Methodist UniversityDallasUSA

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