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Green and Naghdi’s Rod Theory

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Modeling Nonlinear Problems in the Mechanics of Strings and Rods

Part of the book series: Interaction of Mechanics and Mathematics ((IMM))

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Abstract

The rod theory discussed in this chapter originated in a paper by Green and Laws in 1966 [128]. In this theory, the material curve is extensible, and the directors d α can change their length and relative orientation. This theory was further developed in a series of papers by Green, Naghdi, and several of their coworkers. They also showed how it could be established by integrating the three-dimensional equations of continuum mechanics. Here, the rod theory is presented in the most general form for an elastic rod with two directors and possible discontinuities. The linearized theory is discussed in a series of exercises and well-known rod theories that can be considered as constrained theories are presented.

“As often happens in the history of science, the simple ideas are the hardest to achieve; simplicity does not come of itself but must be created.”

C. A. Truesdell [350, Page 251].

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Notes

  1. 1.

    For more information on this matter, see [127, 131, 137, 186, 247, 309].

  2. 2.

    See Exercise 7.1.

  3. 3.

    This problem is discussed in further detail in Exercise 7.3.

  4. 4.

    When the transverse shears γ 13 and γ 23 are constrained to be identically zero, the equations governing flexure often reduce to those for the classic Bernoulli-Euler beam.

  5. 5.

    Cauchy’s representation theorem for a function \(f = f\left (\mathbf{a},\mathbf{b},\mathbf{c}\right )\) states that if \(f\left (\mathbf{a},\mathbf{b},\mathbf{c}\right ) = f\left (\mathbf{Q}\mathbf{a},\mathbf{Q}\mathbf{b},\mathbf{Q}\mathbf{c}\right )\) for all possible rotations Q, then f can be expressed as a function of the inner products a ⋅ a, b ⋅ b, c ⋅ c, a ⋅ b, b ⋅ c, and a ⋅ c, and the triple product \(\left [\mathbf{a},\mathbf{b},\mathbf{c}\right ]\). Discussions of Cauchy’s theorem can be found in Antman [12, Section 7.8] and Truesdell and Noll [351, Section 11].

  6. 6.

    This series of works was summarized and critiqued in the paper [260].

  7. 7.

    Some of the consequences of selecting a different centerline can be seen in Figure 5.8 on Page 201.

  8. 8.

    As usual, for ease of exposition and without loss of generality, we assume that there is at most one such point.

  9. 9.

    As summarized in Section 8.6, a parallel situation arises in the three-dimensional theory. That is, the constitutive relations for the stress tensor of a hyperelastic continuum automatically satisfy the local form of the balance of angular momentum provided the strain energy function is properly invariant under superposed rigid body motions.

  10. 10.

    The developments for three-dimensional continua subject to internal constraints are discussed in Section 8.6 of Chapter 8 and in Exercise 8.4 that can be found on Page 373.

  11. 11.

    Equations (7.76)1, 3 were first presented in [137] and, for the reasons discussed in [260], are only useful on the end surfaces. For a linear theory, it is shown in Exercise 7.2 that the identifications (7.76) simplify considerably.

  12. 12.

    The local form of the balance of angular momentum (7.50)6 is used to establish this identity.

  13. 13.

    See, for example, the derivation of the balance laws for Kirchhoff’s rod theory presented in [285].

  14. 14.

    See Eqn. (5.36).

  15. 15.

    We refer the reader to Eqn. (8.46) on Page 356 for further details on the divergence of P. Our development of Eqn. (7.106) was inspired by the treatment in Rubin [307].

  16. 16.

    The intrinsic director forces k α can be attributed to the terms \(\sum _{r=1}^{3}\mathbf{T}^{r} \frac{\partial \theta ^{\alpha }} {\partial \theta ^{r}} = \mathbf{T}^{\alpha }\) in Eqn. (7.106).

  17. 17.

    Details on this calculation can be found in [260, Section 4.1].

  18. 18.

    The general functional form of the solutions for the resulting ordinary differential equations can be found in [251, Eqns. (A4)–(A10)]. For a circular rod, the general solutions for δ 11(ξ) = δ 22(ξ) and \(\frac{\partial u_{3}} {\partial \xi } (\xi )\) are presented in [138, Eqns. (9.16)–(9.18)].

  19. 19.

    As emphasized in Antman [12], such deformations can be prevented by ensuring that the strain energy function has terms that prohibit unrealistic strains. For the linearized theory under consideration, the strain energy function is such that finite forces can produce physically meaningless results.

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  1. Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA, USA

    Oliver M. O’Reilly

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O’Reilly, O.M. (2017). Green and Naghdi’s Rod Theory. In: Modeling Nonlinear Problems in the Mechanics of Strings and Rods. Interaction of Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-50598-5_7

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