Abstract
The purpose of this final chapter is to illuminate how the kinematic results presented in the earlier chapters are used to formulate the governing equations for the discrete elastic rod. In particular, representations of the kinetic and potential energies for the discrete elastic rod formulation are discussed. The gradients of the elastic energies associated with bending, stretching, and torsion are used to establish expressions for the internal forces in the rod. Assigned forces, including non-conservative forces, can also be incorporated into the discrete elastic rod formulation. To this end, we include examples, such as a force couple and an applied non-conservative force, to illuminate how applied forces and applied moments can be accommodated.
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Notes
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Given a vector-valued function \(\mathbf { a}\left (\mathbf {b}\right )\) where b is a vector, we recall from Sect. 6.2 that \(\frac {\partial \mathbf {a}}{\partial \mathbf {b}} = \sum _{r=1}^3 \sum _{s=1}^3 \frac {\partial a_r}{\partial b_s} \mathbf {E}_r\otimes \mathbf {E}_s\) where \(\mathbf {a} = \sum _{r=1}^3 a_r \mathbf {E}_r\) and \(\mathbf {b} = \sum _{s=1}^3 b_s \mathbf {E}_s\).
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These computations were also invaluable to us while attempting to eliminate typographical errors from the lengthy expressions for the Hessians.
References
Antman, S.S.: Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107, second edn. Springer-Verlag, New York (2005). URL http://dx.doi.org/10.1007/0-387-27649-1
Bergou, M., Audoly, B., Vouga, E., Wardetzky, M., Grinspun, E.: Discrete viscous threads. ACM Transactions on Graphics (SIGGRAPH) 29(4), 116:1–116:10 (2010). URL http://dx.doi.org/10.1145/1778765.1778853
Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Transactions on Graphics (SIGGRAPH) 27(3), 63:1–63:12 (2008). URL http://dx.doi.org/10.1145/1360612.1360662
Chadwick, P.: Continuum Mechanics, second corrected and enlarged edn. Dover Publications, New York (1999)
Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods. I Derivations from three-dimensional equations. Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences 337(1611), 451–483 (1974). URL http://dx.doi.org/10.1098/rspa.1974.0061
Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)
Healey, T.J.: Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7(4), 405–420 (2002). URL http://dx.doi.org/10.1177/108128028482
Kaldor, J.M., James, D.L., Marschner, S.: Efficient yarn-based cloth with adaptive contact linearization. In: ACM SIGGRAPH 2010 Papers, SIGGRAPH ’10, pp. 105:1–105:10. ACM, New York, NY, USA (2010). URL http://doi.acm.org/10.1145/1833349.1778842
Lang, H., Linn, J., Arnold, M.: Multi-body dynamics simulation of geometrically exact Cosserat rods. Multibody System Dynamics 25(3), 285–312 (2011). URL https://doi.org/10.1007/s11044-010-9223-x
Lauderdale, T.A., O’Reilly, O.M.: On transverse and rotational symmetries in elastic rods. Journal of Elasticity 82(1), 31–47 (2006). URL http://dx.doi.org/10.1007/s10659-005-9022-4
Loock, A., Schömer, E., Stadtwald, I.: A virtual environment for interactive assembly simulation: From rigid bodies to deformable cables. In: 5th World Multiconference on Systemics, Cybernetics and Informatics, pp. 325–332 (2001)
Lv, N., Liu, J., Ding, X., Liu, J., Lin, H., Ma, J.: Physically based real-time interactive assembly simulation of cable harness. Journal of Manufacturing Systems 43(Part 3), 385–399 (2017). URL https://doi.org/10.1016/j.jmsy.2017.02.001. Special Issue on the 12th International Conference on Frontiers of Design and Manufacturing
Naghdi, P.M.: Finite deformation of elastic rods and shells. In: D.E. Carlson, R.T. Shield (eds.) Proceedings of the IUTAM Symposium on Finite Elasticity: Held at Lehigh University, Bethlehem, PA, USA August 10–15, 1980, pp. 47–104. Springer Netherlands, Dordrecht (1982). URL https://doi.org/10.1007/978-94-009-7538-5_4
O’Reilly, O.M.: On constitutive relations for elastic rods. International Journal of Solids and Structures 35(11), 1009–1024 (1998). URL http://dx.doi.org/10.1016/S0020-7683(97)00100-5
O’Reilly, O.M.: Modeling Nonlinear Problems in the Mechanics of Strings and Rods: The Role of the Balance Laws. Interaction of Mechanics and Mathematics. Springer International Publishing, New York (2017). URL http://dx.doi.org/10.1007/978-3-319-50598-5
Rega, G.: Nonlinear vibrations of suspended cables - Part I: Modeling and analysis. ASME Applied Mechanics Reviews 57(6), 443–478 (2005). URL http://dx.doi.org/10.1115/1.1777224
Rega, G.: Nonlinear vibrations of suspended cables - Part II: Deterministic phenomena. ASME Applied Mechanics Reviews 57(6), 479–514 (2005). URL http://dx.doi.org/10.1115/1.1777225
Rubin, M.B.: Cosserat Theories: Shells, Rods, and Points. Kluwer Academic Press, Dordrecht (2000). URL http://dx.doi.org/10.1007/978-94-015-9379-3
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Jawed, M.K., Novelia, A., O’Reilly, O.M. (2018). Equations of Motion and Energetic Considerations. In: A Primer on the Kinematics of Discrete Elastic Rods. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-76965-3_8
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