Abstract
The theory of an elastic rod whose centerline is inextensible and whose cross sections remain plane and normal to the centerline is discussed. This theory, which is known as Kirchhoff rod theory, is presented in the modern context of a Cosserat rod theory. The governing equations for this widely used theory result in a set of equations to determine a rotation tensor P and a position vector r. This theory has a celebrated history in part because of Kirchhoff’s discovery that the equations governing static deformations of the rod are analogous to those for the rotational motion of a rigid body. A range of applications of the theory is also presented in this chapter. These examples include a terminally loaded rod which is bent and twisted and an initially curved rod which is straightened.
“A new idea, supple in application to a variety of mechanical theories and formalisms, was proposed by DUHEM [1893, 1, Ch. II]: A body is to be regarded as a collection not only of points but also of directions associated with the points: These vectors, which we shall call the directors of the body, are susceptible of rotations and stretches independent of the deformation of material elements.”
J. L. Ericksen and C. A. Truesdell [100, Page. 297] commenting on Pierre Duhem’s (1861–1916) contribution in [94] to the historical development of theories for rods and shells.
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Notes
- 1.
- 2.
Recall that an orthogonal tensor Q has the property that QQ T = I and so \(\det \left (\mathbf{Q}\right ) = \pm 1\). A proper-orthogonal tensor Q has a determinant of 1. From Euler’s theorem in rigid body dynamics, proper-orthogonal tensors and rotation tensors are synonymous.
- 3.
- 4.
- 5.
The 3-2-1 set of Euler angles are sometimes known as the Tait-Bryan angles and are prominent in aircraft and vehicle dynamics. The first instance of their development and use dates to the seminal work of Fick and Helmholtz [166, 167] on the kinematics of the eye (see [156, 253] and references therein) in the mid-1800s and Tait’s work [336] on rigid body dynamics in 1868.
- 6.
The operator skew(i) transforms i into a skew-symmetric tensor such that i ×b = skew(i)b for any vector b (cf. Eqn. (1.19)).
- 7.
The axial vectors in Eqns. (5.24) and (5.25) can be computed by direct differentiation of the tensor P or using the relative angular velocity vector proposed by Casey and Lam [51]. Applications of Casey and Lam’s relative angular velocity vector can be found in the textbook [265, Sections 6.7 and 6.8].
- 8.
- 9.
You may wish to look at our earlier discussion of the helix in Section 3.4
- 10.
- 11.
- 12.
- 13.
- 14.
- 15.
- 16.
For ease of exposition and without loss of generality, we assume that there is at most one such point.
- 17.
- 18.
As we saw earlier for a string, the jump condition (5.83) simplifies to \(\mathbf{M}_{O_{\gamma }} -\mathbf{r}\left (\gamma,t\right ) \times \mathbf{F}_{\gamma } = \mathbf{0}\) (i.e., M γ = 0 when the directors are ignored (cf. Eqn. (1.87))). As mentioned in an earlier chapter, the physical interpretation of this result is that a string can neither transmit nor resist a moment, so application of a nonzero M γ at ξ = γ is not feasible.
- 19.
A key calculation used to arrive at the simplification is discussed in Exercise 5.7.
- 20.
The jump conditions \(\left [\!\left [\mathbf{n}\right ]\!\right ]_{\gamma } + \mathbf{F}_{\gamma } = \mathbf{0}\) and \(\left [\!\left [\mathbf{m}\right ]\!\right ]_{\gamma } + \mathbf{M}_{\gamma } = \mathbf{0}\) where γ = 0 and γ = ℓ are used to compute the boundary conditions on n and m.
- 21.
- 22.
See the discussion pertaining to Exercise 4.2 on Page 183.
- 23.
- 24.
See Exercise 5.8 for further details on the Lagrangian and the specific form of Lagrange’s equations of interest here.
- 25.
The application here of the dual Euler basis vectors to explain the terminal loading on the rod is novel and, to the best of our knowledge, has not appeared previously in the vast literature on terminally loaded rods.
- 26.
This classic result can be used to show that the solutions to the second-order differential equation (5.144) conserve \(\frac{EI} {2} \left (\frac{\partial \alpha ^{2}} {\partial \xi } \right )^{2} + U\) and can be expressed in terms of Weierstrassian elliptic functions. For further details on this integration and the Routhian reduction procedure that can be used to establish Eqn. (5.144), the exposition in Whittaker [362, Section 71] is highly recommended.
- 27.
For details on this calculation, see Exercise 5.9.
- 28.
- 29.
This observation on the change in handedness as Δ varies does not appear to have been previously recorded in the literature.
- 30.
- 31.
It is useful to note that this choice implies that \(\frac{\partial \alpha ^{2}} {\partial x}\left (x = x_{1} = 0\right ) = \pm 2\left (1 -\sqrt{\varDelta }\right )\). To see this result, it suffices to set e = 1 in Eqn. (5.160) and then solve for \(\frac{\partial \alpha ^{2}} {\partial x}\) when \(\cos \left (\alpha _{0}^{2}\right ) = 2\sqrt{\varDelta }- 1\).
- 32.
Our discussion here, while also based on the behavior of s ℓ as F 0 varies, is different from Coyne’s. In particular, his discussion has statements on stability and instability that we have not been able to follow.
- 33.
The terminology is based on the use of the word perversion (or “Verkehrung” (= reversal)) by the famed topologist Johann B. Listing (1808–1882) for a transformation that changes the handedness of a helix (see [208, Page 22]).
- 34.
- 35.
- 36.
The interested reader is referred to the paper by Liu et al. [209] where an example of such an intrinsically curved body is used to help analyze perversions.
- 37.
- 38.
- 39.
This result was discussed in an earlier chapter of this text and the reader is referred to Eqn. (1.34) for details.
- 40.
In the sequel, our numerical results pertain to \(\phi _{0} = \frac{\pi } {2}\). We found that the solutions of Eqn. (5.180) for \(\phi _{0} = -\frac{\pi }{2}\) are qualitatively similar and so, in the interests of brevity, they are not mentioned.
- 41.
Whence our reference to the critical value \(f_{0} = \frac{1} {\mathcal{D}}\) in the caption for Figure 5.22.
- 42.
- 43.
Note that we can express each of the vector equations \(\mathbf{d}_{i}^{'} = \left (\mathbf{P}\left (\boldsymbol{\nu }+\boldsymbol{\nu }_{0}\right )\right ) \times \mathbf{d}_{i}\) in terms of their E k components to form three first-order differential equations. However, in the interests of brevity, we refrain from writing out the components.
- 44.
- 45.
- 46.
A translation of Euler’s comments on stability can be found in [254, Pages 102–103]. Timoshenko and Gere [345, Chapter 2] contains a comprehensive discussion of the buckling of a strut and the dependency of the critical loads on the boundary conditions. The text by Ziegler [375] is also highly recommended.
- 47.
- 48.
Our comments here are intimately related to the discussion on Q SF and \(\boldsymbol{\omega }_{\text{SF}}\) in Section 5.10.
- 49.
Further details on the procedure to establish equations of this form are discussed in Exercise 5.14.
- 50.
- 51.
- 52.
The terminology we use here is from the literature on DNA testing (cf. Ðuričković et al. [89]).
- 53.
That is, we are considering a set of three infinitesimal rotations superposed on a finite rotation.
- 54.
- 55.
- 56.
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O’Reilly, O.M. (2017). Kirchhoff’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_5
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