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

- 61 Downloads

## Abstract

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.

## Preview

Unable to display preview. Download preview PDF.

## Notes

## References

- 1.Antman, S.S.: Kirchhoff’s problem for nonlinearly elastic rods. Q. Appl. Math.
**32**, 221–240 (1974)MathSciNetCrossRefGoogle Scholar - 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.Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci.
**44**, 1–23 (1992)MathSciNetCrossRefGoogle Scholar - 4.Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)zbMATHGoogle Scholar
- 5.Nizette, M., Goriely, A.: Towards a classification of Euler–Kirchhoff filaments. J. Math. Phys.
**40**, 2830–2866 (1999)MathSciNetCrossRefGoogle Scholar - 6.Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast.
**33**, 1–26 (1993)MathSciNetCrossRefGoogle Scholar - 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.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.da Fonseca, A.F., Galvão, D.S.: Mechanical properties of nanosprings. Phys. Rev. Lett.
**92**, 175502-1-4 (2004)CrossRefGoogle Scholar - 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.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.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.Till, J., Rucker, D.C.: Elastic stability of Cosserat rods and parallel continuum robots. IEEE Trans. Robot.
**33**, 718–733 (2017)CrossRefGoogle Scholar - 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.Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Springer, Dordrecht (2000)CrossRefGoogle Scholar
- 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.Liu, D.S., Wang, C.H.T.: Variational principle for a special Cosserat rod. Appl. Math. Mech.
**30**, 1169–1176 (2009)MathSciNetCrossRefGoogle Scholar - 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.Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct.
**14**, 431–440 (1978)CrossRefGoogle Scholar - 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.Zhang, R.J.: Size effects in Kirchhoff flexible rods. Phys. Rev. E
**81**, 056601-1-5 (2010)Google Scholar - 22.Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech.
**3**, 1–7 (1963)CrossRefGoogle Scholar - 23.Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal.
**17**, 85–112 (1964)MathSciNetCrossRefGoogle Scholar - 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.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.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.Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
- 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.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.Hassanpour, S., Heppler, G.R.: Theory of micropolar gyroelastic continua. Acta Mech.
**227**, 1469–1491 (2016)MathSciNetCrossRefGoogle Scholar - 31.Lembo, M.: On nonlinear deformations of nonlocal elastic rods. Int. J. Solids Struct.
**90**, 215–227 (2016)CrossRefGoogle Scholar - 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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.Bîrsan, M., Altenbach, H.: On the theory of porous elastic rods. Int. J. Solids Struct.
**48**, 910–924 (2011)CrossRefGoogle Scholar - 48.Lembo, M.: On the stability of elastic annular rods. Int. J. Solids Struct.
**40**, 317–330 (2003)MathSciNetCrossRefGoogle Scholar - 49.Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
- 50.Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, New York (2002)Google Scholar
- 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.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.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.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.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.Frankel, S.: Complete approximate solutions of the equation \(\text{ x } = \tan \text{ x }\). Natl. Math. Mag.
**11**(4), 177–182 (1937)MathSciNetCrossRefGoogle Scholar - 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