Acta Mechanica

, Volume 230, Issue 1, pp 243–264

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

• G. Y. Zhang
• X.-L. Gao
Original Paper

## 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.

## References

1. 1.
Antman, S.S.: Kirchhoff’s problem for nonlinearly elastic rods. Q. Appl. Math. 32, 221–240 (1974)
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)
3. 3.
Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44, 1–23 (1992)
4. 4.
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)
5. 5.
Nizette, M., Goriely, A.: Towards a classification of Euler–Kirchhoff filaments. J. Math. Phys. 40, 2830–2866 (1999)
6. 6.
Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33, 1–26 (1993)
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)
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)
9. 9.
da Fonseca, A.F., Galvão, D.S.: Mechanical properties of nanosprings. Phys. Rev. Lett. 92, 175502-1-4 (2004)
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)
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)
13. 13.
Till, J., Rucker, D.C.: Elastic stability of Cosserat rods and parallel continuum robots. IEEE Trans. Robot. 33, 718–733 (2017)
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)
15. 15.
Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Springer, Dordrecht (2000)
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)
17. 17.
Liu, D.S., Wang, C.H.T.: Variational principle for a special Cosserat rod. Appl. Math. Mech. 30, 1169–1176 (2009)
18. 18.
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
19. 19.
Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)
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)
23. 23.
Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)
24. 24.
Güven, U.: The investigation of the nonlocal longitudinal stress waves with modified couple stress theory. Acta Mech. 221, 321–325 (2011)
25. 25.
Güven, U.: A more general investigation for the longitudinal stress waves in microrods with initial stress. Acta Mech. 223, 2065–2074 (2012)
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)
27. 27.
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
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)
30. 30.
Hassanpour, S., Heppler, G.R.: Theory of micropolar gyroelastic continua. Acta Mech. 227, 1469–1491 (2016)
31. 31.
Lembo, M.: On nonlinear deformations of nonlocal elastic rods. Int. J. Solids Struct. 90, 215–227 (2016)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
47. 47.
Bîrsan, M., Altenbach, H.: On the theory of porous elastic rods. Int. J. Solids Struct. 48, 910–924 (2011)
48. 48.
Lembo, M.: On the stability of elastic annular rods. Int. J. Solids Struct. 40, 317–330 (2003)
49. 49.
Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)
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)
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)
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)
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)
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)
56. 56.
Frankel, S.: Complete approximate solutions of the equation $$\text{ x } = \tan \text{ x }$$. Natl. Math. Mag. 11(4), 177–182 (1937)
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)

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

## Authors and Affiliations

• G. Y. Zhang
• 1
• X.-L. Gao
• 1
1. 1.Department of Mechanical EngineeringSouthern Methodist UniversityDallasUSA