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A one-dimensional model of 3-D structure for large deformation: a general higher-order rod theory

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Abstract

A higher-order one-dimensional model in a curvilinear cylindrical coordinate system and associated finite element model are presented. A general displacement field of the cross section of the rod or the structures in the polar coordinate system is assumed, and the associated governing equation of motion in the Lagrangian frame of reference is derived. Since the displacement field considered is very general, the theory is not limited to rods but can be used to analyze thick, solid, or hollow arbitrary cross section members or a shell structure whose axis can be given as a space curve. A nonlinear finite element model of the theory which can model large deformation is developed. Numerical examples are presented to illustrate the usefulness and accuracy of the model in analyzing shell structures (e.g., spiral duct), whose central axis is a space curve, subjected to point loads and internal or external pressure.

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  • 31 August 2018

    The correction in Eq. (B.1).

References

  1. Euler, L.: Genuina principia doctrinae de statu aequilibrii et motu corporum tam perfecte flexibilium quam elasticorum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 15(1770), 381–413 (1771)

    Google Scholar 

  2. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann, Paris (1909)

    MATH  Google Scholar 

  3. Cosserat, E., Cosserat, F.: Sur la statique de la ligne deformable. CR Acad. Sci. Paris 145, 1409–1412 (1907)

    MATH  Google Scholar 

  4. Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  5. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods. I. Derivations from the three-dimensional equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 337(1611), 451–483 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods. II. Developments by direct approach. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 337(1611), 485–507 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Antman, S.S.: Nonlinear problems of elasticity, volume 107 of applied mathematical sciences. Springer, New York (2005)

    Google Scholar 

  8. Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49(1), 55–70 (1985)

    Article  MATH  Google Scholar 

  9. Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. Part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58(1), 79–116 (1986)

    Article  MATH  Google Scholar 

  10. Goyal, S., Perkins, N.C., Lee, C.L.: Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of dna and cables. J. Comput. Phys. 209(1), 371–389 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goyal, S., Perkins, N.C., Lee, C.L.: Non-linear dynamic intertwining of rods with self-contact. Int. J. Non Linear Mech. 43(1), 65–73 (2008)

    Article  MATH  Google Scholar 

  12. Kumar, A., Mukherjee, S.: A geometrically exact rod model including in-plane cross-sectional deformation. J. Appl. Mech. 78(1), 011010 (2011)

    Article  Google Scholar 

  13. Fang, C., Kumar, A., Mukherjee, S.: A finite element analysis of single-walled carbon nanotube deformation. J. Appl. Mech. 78(3), 034502 (2011)

    Article  Google Scholar 

  14. 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(20), 2849–2858 (2011)

    Article  Google Scholar 

  15. Kumar, A., Healey, T.J.: A generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints. Comput. Methods Appl. Mech. Eng. 199(25), 1805–1815 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Arbind, A., Reddy, J.N.: Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model. Int. J. Non Linear Mech. 79, 38–47 (2016)

    Article  Google Scholar 

  17. Simo, J.C., Vu-Quoc, L.: A geometrically-exact rod model incorporating shear and torsion-warping deformation. Int. J. Solids Struct. 27(3), 371–393 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Arbind, A., Reddy, J.N.: A general higher order one-dimensional theory for analysis of solid body in cylindrical co-ordinate system and it’s nonlinear finite element model for large deformation. Comput. Methods Appl. Mech. Eng 328, 99–121 (2018)

    Article  Google Scholar 

  19. Reddy, J.N.: An Introduction to Continuum Mechanics, 2nd edn. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  20. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis: With Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics, 2nd edn. OUP Oxford, Oxford (2015)

    Google Scholar 

  21. Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 3rd edn. Wiley, New York (2017)

    Google Scholar 

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Authors and Affiliations

  1. Advanced Computational Mechanics Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA

    A. Arbind & J. N. Reddy

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  1. A. Arbind
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  2. J. N. Reddy
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Correspondence to J. N. Reddy.

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Arbind, A., Reddy, J.N. A one-dimensional model of 3-D structure for large deformation: a general higher-order rod theory. Acta Mech 229, 1803–1831 (2018). https://doi.org/10.1007/s00707-017-2048-4

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