Abstract
An analytical expression for the strain energy of a constrained extensible Cosserat elastica is developed for general planar shapes and deformations of the rod. This strain energy function naturally couples tangential stretch and reference and current curvatures of the centroidal curve. The model considers a rigid rectangular cross-section of the rod which remains normal to the centroidal curve. The constitutive equations for the tangential force, shear force and bending moment are consistent with a restriction based on the balance of angular momentum that requires a stress-like tensor to be symmetric in a similar manner to the symmetry of the Cauchy stress in a three-dimensional continuum. Examples show that coupling of tangential stretch and reference and current curvatures of the centroidal curve in the new strain energy function can significantly influence predictions of tangential force, shear force and bending moments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antman, S.: Equilibrium states of nonlinearly elastic rods. J. Math. Anal. Appl. 23, 459–470 (1968)
Antman, S.: General solutions for plane extensible elasticae having nonlinear stress-strain laws. Q. Appl. Math. 26, 35–47 (1968)
Antman, S.S.: The theory of rods. In: Linear Theories of Elasticity and Thermoelasticity, pp. 641–703. Springer, Berlin (1973)
Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107. Springer, Berlin (1991)
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, 451–483 (1974)
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, 485–507 (1974)
Naghdi, P.M.: The Theory of Shells and Plates. Springer, Berlin (1973)
Naghdi, P.M., Rubin, M.B.: Restrictions on nonlinear constitutive equations for elastic shells. J. Elast. 39, 33–163 (1995)
Rubin, M.B.: Restrictions on nonlinear constitutive equations for elastic rods. J. Elast. 44, 9–36 (1996)
Rubin, M.B.: Cosserat theories: Shells, rods and points. Springer Science & Business Media 79 (2000)
Rubin, M.B.: Angular momentum in a special nonlinear elastic rod. J. Elast. (2024) Submitted
Tadjbakhsh, I.: The variational theory of the plane motion of the extensible elastica. Int. J. Eng. Sci. 4, 433–450 (1966)
Taloni, A., Vilone, D., Ruta, G.: General theory for plane extensible elastica with arbitrary undeformed shape. Int. J. Eng. Sci. 193, 103941 (2023)
Funding
No funding was received for conducting this study.
Author information
Authors and Affiliations
Contributions
I am the sole author and am responsible for all aspects of this paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rubin, M.B. A Strain Energy Function for Planar Response of a Constrained Cosserat Extensible Elastica with a General Reference Planar Shape. J Elast (2024). https://doi.org/10.1007/s10659-024-10053-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10659-024-10053-0
Keywords
- Constrained Cosserat rod
- Coupled strain energy function
- Elastica
- Planar motion
- Symmetric stress-like tensor