Skip to main content
SpringerLink
Log in

Kirchhoff’s Rod Theory

  • Chapter
  • First Online:
Book cover Modeling Nonlinear Problems in the Mechanics of Strings and Rods

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

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The terminology “director” was introduced in a seminal paper [100] by Ericksen and Truesdell in 1958. For further discussion on the historical developments following the publication of this paper, see [12, 243, 245].

  2. 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. 3.

    Recall that the axial vector b of any skew-symmetric tensor B = −B T has the property that b ×c = Bc for any vector c. As a skew-symmetric tensor has three independent components, there is a one-to-one correspondence between B and its axial vector b (cf. Eqns. (1.17) and (1.18)).

  4. 4.

    The 3-2-3 set of Euler angles is also used in several texts, for example, Ginsberg [115, Section 4.2], Kelvin and Tait [180], Routh [305, 306], and Whittaker [362].

  5. 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. 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. 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. 8.

    For the reader comparing our treatment to Love’s classic work, the relations (5.28) are identical to [213, Eqn. (8), Page 386].

  9. 9.

    You may wish to look at our earlier discussion of the helix in Section 3.4

  10. 10.

    A derivation of these results was first presented in [137]. The developments used in [137] can also be found in the text [309].

  11. 11.

    As discussed by Naghdi [244], the freedom to make such a selection can be advantageous for formulating contact problems. It can also be advantageous when attempting to develop rod models for deformable bodies as in [286].

  12. 12.

    We discuss this matter in additional detail in Section 6.4.3 of Chapter 6

  13. 13.

    The prescription (5.68) is a generalization of Eqn. (56)1 in [264] to the case of an intrinsically curved rod.

  14. 14.

    See, e.g., [213, Eqn. (3), Section 260] or the more recent work [164, Eqn. (20)].

  15. 15.

    A derivation of these results was first presented in [137] and is based on the approximation (5.2): r ∗ = r(ξ, t) + ∑ α = 1 2 θ α d α (ξ, t). However, the derivation is difficult to follow. A recent treatment by [309] is far more transparent (see also [260] and Section 7.12 in Chapter 7).

  16. 16.

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

  17. 17.

    Some of the details and manipulations involved are outlined in Exercises 5.2 and 5.3.

  18. 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. 19.

    A key calculation used to arrive at the simplification is discussed in Exercise 5.7.

  20. 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. 21.

    The reader is referred to Kehrbaum and Maddocks [178] for a lucid discussion of this integration along with commentary on the works of Ilyukhin [170], Langer and Singer [196], and Shi and Hearst [319].

  22. 22.

    See the discussion pertaining to Exercise 4.2 on Page 183.

  23. 23.

    Lagrange’s original work is presented in [191, 192, Sections IX.34–IX.41]. Other treatments of this problem can be found in Arnold [15, Section 30], Lewis et al. [205], Marsden and Ratiu [229, Chapter 1], and Whittaker [362, Sections 71–73].

  24. 24.

    See Exercise 5.8 for further details on the Lagrangian and the specific form of Lagrange’s equations of interest here.

  25. 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. 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. 27.

    For details on this calculation, see Exercise 5.9.

  28. 28.

    Greenhill’s development of [141, Eqn. (1)] forms the basis for Exercise 5.12 at the end of this chapter. We note that the bifurcation at hand cannot be predicted as a limit of his criterion (see Eqn. (5.252)) when the length of the rod ℓ → ∞.

  29. 29.

    This observation on the change in handedness as Δ varies does not appear to have been previously recorded in the literature.

  30. 30.

    See the works of Goriely [222, 249], Maddocks [86, 169, 178], Thompson [164, 340], and their coworkers.

  31. 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. 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. 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. 34.

    For additional discussions on tendril perversion, the reader is referred to Domokos and Healey [90], Keller [179], Männer [223], and McMillen and Goriely [235].

  35. 35.

    For additional references on this topic, the reader is referred to [121, 273, 295, 323, 365] and references therein.

  36. 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. 37.

    In the interests of brevity, the developments of these two representations of the balance laws are discussed in Exercises 5.14 and 5.15.

  38. 38.

    A complementary perspective on solutions of this type can be found in the discussion of Ericksen’s uniform states in Section 6.6 in Chapter 6

  39. 39.

    This result was discussed in an earlier chapter of this text and the reader is referred to Eqn. (1.34) for details.

  40. 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. 41.

    Whence our reference to the critical value \(f_{0} = \frac{1} {\mathcal{D}}\) in the caption for Figure 5.22.

  42. 42.

    The derivation of this form of m ′ + r ′ ×n = 0 is similar to the derivation of Eqn. (5.119) and is discussed in Exercise 5.14.

  43. 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. 44.

    We are grateful to Tyler McMillen and Alain Goriely for generously supplying the code needed to construct the solutions shown in Figures 5.23 and 5.24.

    Fig. 5.24
    figure 24

    The centerline r(ξ) and tangent indicatrix c t for the approximation \(\mathcal{P}\) to a perversion that is shown in Figure 5.23. The arrows indicate the direction of increasing ξ.

    Full size image
  45. 45.

    See Figure 3.7 on Page 115 in Chapter 3

  46. 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. 47.

    See [213, Section 264] or [345, Chapter 2].

  48. 48.

    Our comments here are intimately related to the discussion on Q SF and \(\boldsymbol{\omega }_{\text{SF}}\) in Section 5.10.

  49. 49.

    Further details on the procedure to establish equations of this form are discussed in Exercise 5.14.

  50. 50.

    The identities (5.209) are particularly helpful in establishing several of the results which follow and enabling comparisons with [89, Eqns. (9) and (10)].

  51. 51.

    The representation (5.228) is a dimensionless form of [213, Eqn. (42), Section 271].

  52. 52.

    The terminology we use here is from the literature on DNA testing (cf. Ðuričković et al. [89]).

  53. 53.

    That is, we are considering a set of three infinitesimal rotations superposed on a finite rotation.

  54. 54.

    Additional discussions of this criterion can be found in [213, Section 272] and [345, Pages 156–157]. Ziegler’s delightful introductory text [375, Chapter 5, Section 3] has an informative discussion of the case where P = 0.

  55. 55.

    This extension is presented in far greater detail in Love [213, Section 261] than in Larmor’s original work [197, Section 8].

  56. 56.

    Parameterizations of this type are sometimes used in studies on the dynamics of satellites in orbit. See, for example, Rumyantsev [311] and articles such as [272, 333] citing this work.

References

  1. Antman, S.S.: Kirchhoff’s problem for nonlinearly elastic rods. Quarterly of Applied Mathematics 32, 221–240 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antman, S.S.: Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107, second edn. Springer-Verlag, New York (2005)

    Google Scholar 

  3. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, second edn. Springer-Verlag, New York (1989). Translated from the Russian by K. Vogtmann and A. Weinstein

    Google Scholar 

  4. 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)

    Google Scholar 

  5. Casey, J.: A treatment of rigid body dynamics. ASME Journal of Applied Mechanics 50, 905–907 and 51 227 (1983). URL http://dx.doi.org/10.1115/1.3167171

  6. Casey, J., Lam, V.C.: On the relative angular velocity tensor. ASME Journal of Mechanisms, Transmissions, and Automation in Design 108, 399–400 (1986). URL http://dx.doi.org/10.1115/1.3258746

  7. Chouaieb, N., Goriely, A., Maddocks, J.H.: Helices. Proceedings of the National Academy of Sciences 103 (25), 9398–9403 (2006). URL http://dx.doi.org/10.1073/pnas.0508370103

  8. Clauvelin, N., Audoly, B., Neukirch, S.: Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology. Journal of the Mechanics and Physics of Solids 57 (9), 1623–1656 (2009). URL http://dx.doi.org/10.1016/j.jmps.2009.05.004

  9. Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. Journal of Elasticity 60 (3), 173–221 (2000). URL http://dx.doi.org/10.1023/A:1010911113919

  10. Coleman, B.D., Swigon, D.: Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 362 (1820), 1281–1299 (2004). URL http://dx.doi.org/10.1098/rsta.2004.1393

  11. Cosserat, E., Cosserat, F.: Sur la statique de la ligne déformable. Compte Rendus de l’Académie des Sciences, Paris 145, 1409–1412 (1907)

    MATH  Google Scholar 

  12. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. A. Hermann et Fils, Paris (1909)

    MATH  Google Scholar 

  13. Coyne, J.: Analysis of the formation and elimination of loops in twisted cable. IEEE Journal of Oceanic Engineering 15 (2), 72–83 (1990). URL http://dx.doi.org/10.1109/48.50692

  14. Cusumano, J.P., Moon, F.C.: Chaotic non-planar vibrations of the thin elastica: Part I: Experimental observation of planar instability. Journal of Sound and Vibration 179 (2), 185–208 (1995). URL http://dx.doi.org/10.1006/jsvi.1995.0013

  15. Cusumano, J.P., Moon, F.C.: Chaotic non-planar vibrations of the thin elastica: Part II: Derivation and analysis of a low-dimensional model. Journal of Sound and Vibration 179 (2), 209–226 (1995). URL http://dx.doi.org/10.1006/jsvi.1995.0014

  16. Darwin, C.R.: On the movements and habits of climbing plants. Journal of the Linnean Society of London, Botany 9 (33–34), 1–118 (1865). URL 10.1111/j.1095-8339.1865.tb00011.x. Second edition printed in London in 1875 by J. Murray.

  17. Davies, M.A., Moon, F.C.: 3-D spatial chaos in the elastica and the spinning top: Kirchhoff analogy. Chaos 3 (1), 93–99 (1993). URL http://dx.doi.org/10.1063/1.165969

  18. Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: J.P. Mesirov, K. Schulten, D. Sumners (eds.) Mathematical approaches to biomolecular structure and dynamics (Minneapolis, MN, 1994), The IMA Volumes in Mathematics and its Applications, vol. 82, pp. 71–113. Springer-Verlag, New York (1996). URL http://dx.doi.org/10.1007/978-1-4612-4066-2_6

  19. Ðuričković, B., Goriely, A., Maddocks, J.H.: Twist and stretch of helices explained via the Kirchhoff-Love rod model of elastic filaments. Physical Review Letters 111, 108,103 (2013). URL http://link.aps.org/doi/10.1103/PhysRevLett.111.108103

  20. Domokos, G., Healey, T.J.: Multiple helical perversions of finite, intrinsically curved rods. International Journal of Bifurcation and Chaos 15 (3), 871–890 (2005). URL http://dx.doi.org/10.1142/S0218127405012430

  21. Duhem, P.: Le potentiel thermodynamique et la pression hydrostatique. Annales Scientifiques de l’École Normale Supérieure 10, 183–230 (1893). URL http://eudml.org/doc/81051

  22. Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Archive for Rational Mechanics and Analysis 1 (1), 295–323 (1958). URL http://dx.doi.org/10.1007/BF00298012

  23. Euler, L.: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti: Additamentum 1 De Curvis Elasticis. Leonhardi Euleri Opera Omnia, Series prima (Opera mathematica), Vol. XXIV, Auctoritate et impensis Societatis Scientiarum Naturalium Helveticae. Orell Füssli, Zürich (1952). An English translation of this work can be found in [254].

    Google Scholar 

  24. Ginsberg, J.H.: Advanced Engineering Dynamics, second edn. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  25. Gore, J., Bryant, Z., Nöllmann, M., Le, M.U., Cozzarelli, N.R., Bustamante, C.: DNA overwinds when stretched. Nature 442, 836–839 (2006). URL http://dx.doi.org/10.1038/nature04974

  26. Goriely, A.: The Mechanics and Mathematics of Biological Growth. Springer-Verlag, New York (2017)

    MATH  Google Scholar 

  27. Goriely, A., Neukirch, S.: The mechanics of attachment in twining plants. Physical Review Letters 97, 184,302 (2006). URL http://dx.doi.org/10.1103/PhysRevLett.97.184302

  28. Goriely, A., Tabor, M.: Spontaneous helix hand reversal and tendril perversion in climbing plants. Physical Review Letters 80, 1564–1567 (1998). URL http://dx.doi.org/10.1103/PhysRevLett.80.1564

  29. Goyal, S., Perkins, N.C.: Looping mechanics of rods and DNA with non-homogeneous and discontinuous stiffness. International Journal of Non-Linear Mechanics 43 (10), 1121–1129 (2008). URL http://dx.doi.org/10.1016/j.ijnonlinmec.2008.06.013

  30. Goyal, S., Perkins, N.C., Lee, C.L.: Non-linear dynamic intertwining of rods with self-contact. International Journal of Non-Linear Mechanics 43 (1), 65–73 (2008). URL http://dx.doi.org/10.1016/j.ijnonlinmec.2007.10.004

  31. Green, A.E., Laws, N.: Remarks on the theory of rods. Journal of Elasticity 3 (3), 179–184 (1973). URL http://dx.doi.org/10.1007/BF00052892

  32. Green, A.E., Laws, N., Naghdi, P.M.: A linear theory of straight elastic rods. Archive for Rational Mechanics and Analysis 25 (4), 285–298 (1967). URL http://dx.doi.org/10.1007/BF00250931

  33. Green, A.E., Naghdi, P.M.: A derivation of jump condition for entropy in thermomechanics. Journal of Elasticity 8 (2), 119–182 (1978). URL http://dx.doi.org/10.1007/BF00052481

  34. 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

  35. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods. II Developments by direct approach. Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences 337 (1611), 485–507 (1974). URL http://dx.doi.org/10.1098/rspa.1974.0062

  36. Green, A.E., Zerna, W.T.: Theoretical Elasticity, second edn. Clarendon Press, Oxford (1968)

    Google Scholar 

  37. Greenhill, A.G.: On the strength of shafting when exposed both to torsion and to end thrust. Proceedings of the Institution of Mechanical Engineers 34 (1), 182–225 (1883). URL http://dx.doi.org/10.1243/PIME_PROC_1883_034_013_02

  38. Haslwanter, T.: Mathematics of three-dimensional eye rotations. Vision Research 35 (12), 1727–1739 (1995). URL http://dx.doi.org/10.1016/0042-6989(94)00257-M

  39. 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

  40. Healey, T.J., Mehta, P.G.: Straightforward computation of spatial equilibria of geometrically exact Cosserat rods. International Journal of Bifurcation and Chaos 15 (3), 949–965 (2005). URL http://dx.doi.org/10.1142/S0218127405012387

  41. van der Heijden, G.H.M., Neukirch, S., Goss, V.G.A., Thompson, J.M.T.: Instability and self-contact phenomena in the writhing of clamped rods. International Journal of Mechanical Sciences 45 (1), 161–196 (2003). URL http://dx.doi.org/10.1016/S0020-7403(02)00183-2

  42. van der Heijden, G.H.M., Thompson, J.M.T.: Helical and localised buckling in twisted rods: A unified analysis of the symmetric case. Nonlinear Dynamics 21 (1), 71–99 (2000). URL http://dx.doi.org/10.1023/A:1008310425967

  43. van der Heijden, G.H.M., Yagasaki, K.: Horseshoes for the nearly symmetric heavy top. Zeitschrift für Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathématiques et de Physique Appliquées 65 (2), 221–240 (2014). URL http://dx.doi.org/10.1007/s00033-013-0319-z

  44. Helmholtz, H.: Ueber die normalen Bewegungen des menschlichen Auges. Archiv für Ophthalmologie 9 (2), 153–214 (1863). URL http://dx.doi.org/10.1007/BF02720895

  45. von Helmholtz, H.: A Treatise on Physiological Optics, vol. III. Dover Publications, New York (1962). Translated from the (1910) third German edition and edited by J.P.C. Southall

    Google Scholar 

  46. Hoffman, K.A., Manning, R.S., Maddocks, J.H.: Link, twist, energy, and the stability of DNA minicircles. Biopolymers 70 (2), 145–157 (2003). URL http://dx.doi.org/10.1002/bip.10430

  47. Ilyukhin, A.A.: Three-Dimensional Problems of the Nonlinear Theory of Elastic Rods. Naukova Dumka, Kiev, Ukraine (1979)

    Google Scholar 

  48. Jawed, M.K., Dieleman, P., Audoly, B., Reis, P.M.: Untangling the mechanics and topology in the frictional response of long overhand elastic knots. Physical Review Letters 115 (11), 118,302 (2015). URL http://dx.doi.org/10.1103/PhysRevLett.115.118302

  49. Kamien, R.D., Lubensky, T.C., Nelson, P., O’Hern, C.S.: Twist-stretch elasticity of DNA. In: Symposia EE - Statistical Mechanics in Physics and Biology, MRS Online Proceedings Library Archive, vol. 463, pp. 43–48 (1996). URL http://dx.doi.org/10.1557/PROC-463-43

  50. Kehrbaum, S., Maddocks, J.H.: Elastic rods, rigid bodies, quaternions and the last quadrature. Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical Sciences and Engineering 355 (1732), 2117–2136 (1997). URL http://dx.doi.org/10.1098/rsta.1997.0113

  51. Keller, J.B.: Tendril shape and lichen growth. In: G.F. Oster (ed.) Some Mathematical Questions in Biology; Symposium On Mathematical Biology At The Annual Meeting of The American Association for the Advancement Of Science, San Francisco, California, USA, Jan. 6, 1980, Lectures On Mathematics in the Life Sciences, pp. 257–274. American Mathematical Society, Providence, Rhode Island (1980)

    Google Scholar 

  52. Kelvin, L., Tait, P.G.: A Treatise on Natural Philosophy, Reprinted edn. Cambridge University Press, Cambridge (1912)

    Google Scholar 

  53. Kirchhoff, G.: Über des gleichgewicht und die Bewegung eines unendlich dünnen elastichen Stabes. Crelles Journal für die reine und angewandte Mathematik 56, 285–313 (1859). URL http://dx.doi.org/10.1515/crll.1859.56.285

  54. Kreyszig, E.: Differential Geometry, Revised and reprinted edn. Toronto University Press, Toronto (1964)

    MATH  Google Scholar 

  55. Kumar, A., Healey, T.J.: A generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints. Computer Methods in Applied Mechanics and Engineering 199 (25), 1805–1815 (2010). URL http://dx.doi.org/10.1016/j.cma.2010.02.007

  56. Lagrange, J.L.: Mécanique Analytique. In: J.A. Serret, G. Darboux (eds.) Joseph Louis de Lagrange Oeuvres, vol. 11/12, fourth edn. Georg Olms Verlag, Heidelberg (1973)

    Google Scholar 

  57. Lagrange, J.L.: Analytical Mechanics, Boston Studies in the Philosophy of Science, vol. 191. Kluwer Academic Publishers Group, Dordrecht (1997). Translated from the 1811 French original, with an introduction and edited by A. Boissonnade and V. N. Vagliente, and with a preface by C. G. Fraser

    Google Scholar 

  58. Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review 38 (4), 605–618 (1996). URL http://dx.doi.org/10.1137/S0036144593253290

  59. Larmor, J.: On the direct application of the principle of least action to the dynamics of solid and fluid systems, and analogous elastic problems. Proceedings of the London Mathematical Society s1-15 (1), 170–185 (1883). URL http://dx.doi.org/10.1112/plms/s1-15.1.170

  60. 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

  61. Lauderdale, T.A., O’Reilly, O.M.: On the restrictions imposed by non-affine material symmetry groups for elastic rods: Application to helical substructures. European Journal of Mechanics. A. Solids 26 (4), 701–711 (2007). URL http://dx.doi.org/10.1016/j.euromechsol.2006.10.003

  62. Lewis, D., Ratiu, T., Simo, J.C., Marsden, J.E.: The heavy top: A geometric treatment. Nonlinearity 5 (1), 1–48 (1992). URL http://dx.doi.org/10.1088/0951-7715/5/1/001

  63. Lionnet, T., Joubaud, S., Lavery, R., Bensimon, D., Croquette, V.: Wringing out DNA. Physical Review Letters 96, 178,102 (2006). URL http://link.aps.org/doi/10.1103/PhysRevLett.96.178102

  64. Listing, J.B.: Vorstudien zur Topologie. Göttinger Studien. Vandenboeck & Ruprecht, Göttingen (1848). URL https://archive.org/details/vorstudienzurto00listgoog

  65. Liu, J., Huang, J., Su, T., Bertoldi, K., Clarke, D.R.: Structural transition from helices to hemihelices. PLoS ONE 9 (4), e93,183 (2014). URL http://dx.doi.org/10.1371/journal.pone.0093183

  66. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, fourth edn. Cambridge University Press, Cambridge (1927)

    Google Scholar 

  67. Majumdar, A., Prior, C., Goriely, A.: Stability estimates for a twisted rod under terminal loads: A three-dimensional study. Journal of Elasticity 109 (1), 75–93 (2012). URL http://dx.doi.org/10.1007/s10659-012-9371-8

  68. Majumdar, A., Raisch, A.: Stability of twisted rods, helices and buckling solutions in three dimensions. Nonlinearity 27 (12), 2841–2867 (2014). URL http://dx.doi.org/10.1088/0951-7715/27/12/2841

  69. Männer, J.: On the form problem of embryonic heart loops, its geometrical solutions, and a new biophysical concept of cardiac looping. Annals of Anatomy - Anatomischer Anzeiger 195 (4), 312–323 (2013). URL http://dx.doi.org/10.1016/j.aanat.2013.02.008

  70. Manning, R.S.: Conjugate points revisited and Neumann-Neumann problems. SIAM Review 51 (1), 193–212 (2009). URL http://dx.doi.org/10.1137/060668547

  71. Manning, R.S., Rogers, K.A., Maddocks, J.H.: Isoperimetric conjugate points with application to the stability of DNA minicircles. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454 (1980), 3047–3074 (1998). URL http://dx.doi.org/10.1098/rspa.1998.0291

  72. Marko, J.F.: Stretching must twist DNA. Europhysics Letters 38 (3), 183–188 (1997). URL http://stacks.iop.org/0295-5075/38/i=3/a=183

  73. Marko, J.F., Siggia, E.D.: Bending and twisting elasticity of DNA. Macromolecules 27 (4), 981–988 (1994). URL http://dx.doi.org/10.1021/ma00082a015. Errata for this paper were published on page 4820 of Volume 29

  74. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, vol. 17, second edn. Springer-Verlag, New York (1999). URL http://dx.doi.org/10.1007/978-0-387-21792-5

  75. Marshall, J.S., Naghdi, P.M.: A thermodynamical theory of turbulence. I. Basic developments. Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 327 (1595), 415–448 (1989). URL http://dx.doi.org/10.1098/rsta.1989.0001

  76. McMillen, T., Goriely, A.: Tendril perversion in intrinsically curved rods. Journal of Nonlinear Science 12 (3), 241–281 (2002). URL http://dx.doi.org/10.1007/s00332-002-0493-1

  77. Mielke, A., Holmes, P.: Spatially complex equilibria of buckled rods. Archive for Rational Mechanics and Analysis 101 (4), 319–348 (1988). URL http://dx.doi.org/10.1007/BF00251491

  78. Naghdi, P.M.: The theory of shells and plates. In: C. Truesdell (ed.) Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, pp. 425–640. Springer-Verlag, Berlin, Heidelberg (1973). URL http://dx.doi.org/10.1007/978-3-662-39776-3_5

  79. Naghdi, P.M.: On the formulation of contact problems of shells and plates. Journal of Elasticity 5 (3–4), 379–398 (1975). URL http://dx.doi.org/10.1007/BF00126998

  80. 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, Bethlehem PA 1980, pp. 47–104. Martinus Nijhoff, The Hague (1982)

    Google Scholar 

  81. Naghdi, P.M., Rubin, M.B.: Constrained theories of rods. Journal of Elasticity 14, 343–361 (1984). URL http://dx.doi.org/10.1007/BF00125605

  82. Naghdi, P.M., Rubin, M.B.: On the significance of normal cross-sectional extension in beam theory with application to contact problems. International Journal of Solids and Structures 25 (3), 249–265 (1989). URL http://dx.doi.org/10.1016/0020-7683(89)90047-4

  83. Nizette, M., Goriely, A.: Towards a classification of Euler-Kirchhoff filaments. Journal of Mathematical Physics 40 (6), 2830–2866 (1999). URL http://dx.doi.org/10.1063/1.532731

  84. Nordenholz, T.R., O’Reilly, O.M.: On steady motions of an elastic rod with application to contact problems. International Journal of Solids and Structures 34 (9), 1123–1143 & 3211–3212 (1997). URL http://dx.doi.org/10.1016/S0020-7683(96)00054-6

  85. Novelia, A., O’Reilly, O.M.: On the dynamics of the eye: geodesics on a configuration manifold, motions of the gaze direction and Helmholtz’s theorem. Nonlinear Dynamics 80 (3), 1303–1327 (2015). URL http://dx.doi.org/10.1007/s11071-015-1945-0

  86. Oldfather, W.A., Ellis, C.A., Brown, D.M.: Leonhard Euler’s elastic curves. Isis 20 (1), 72–160 (1933). URL http://www.jstor.org/stable/224885

  87. 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

  88. O’Reilly, O.M.: The dual Euler basis: Constraints, potentials, and Lagrange’s equations in rigid body dynamics. ASME Journal of Applied Mechanics 74 (2), 256–258 (2007). URL http://dx.doi.org/10.1115/1.2190231

  89. O’Reilly, O.M.: A material momentum balance law for rods. Journal of Elasticity 86 (2), 155–172 (2007). URL http://dx.doi.org/10.1007/s10659-006-9089-6

  90. O’Reilly, O.M.: Intermediate Engineering Dynamics: A Unified Treatment of Newton-Euler and Lagrangian Mechanics. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  91. O’Reilly, O.M., Sena, M., Feeley, B.T., Lotz, J.C.: On representations for joint moments using a joint coordinate system. ASME Journal of Biomechanical Engineering 135 (11), 114,504–114,504 (2013). URL http://dx.doi.org/10.1115/1.4025327

  92. O’Reilly, O.M., Srinivasa, A.R.: A simple treatment of constraint forces and constraint moments in the dynamics of rigid bodies. ASME Applied Mechanics Reviews 67 (1), 014,801–014,801–8 (2014). URL http://dx.doi.org/10.1115/1.4028099

  93. O’Reilly, O.M., Tan, M.H.: Steady motions of an axisymmetric satellite: An atlas of their bifurcations. International Journal of Non-Linear Mechanics 39 (6), 921–940 (2004). URL http://dx.doi.org/10.1016/S0020-7462(03)00075-1

  94. O’Reilly, O.M., Tresierras, T.N.: On the evolution of intrinsic curvature in rod-based models of growth in long slender plant stems. International Journal of Solids and Structures 48 (9), 1239–1247 (2011). URL http://dx.doi.org/10.1016/j.ijsolstr.2010.12.006

  95. Pai, D.K.: Strands: Interactive simulation of thin solids using Cosserat models. Computer Graphics Forum 21 (3), 347–352 (2002). URL http://dx.doi.org/10.1111/1467-8659.00594

  96. de Payrebrune, K.M., O’Reilly, O.M.: On constitutive relations for a rod-based model of a pneu-net bending actuator. Extreme Mechanics Letters 8 (C), 38–46 (2016). URL http://dx.doi.org/10.1016/j.eml.2016.02.007

  97. Prusinkiewicz, P.: Modeling plant growth and development. Current Opinion in Plant Biology 7 (1), 79–83 (2004). URL http://dx.doi.org/10.1016/j.pbi.2003.11.007

  98. Routh, E.J.: The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, sixth, revised and enlarged edn. MacMillan & Co., London (1905)

    Google Scholar 

  99. Routh, E.J.: The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies, seventh edn. Macmillan, London (1905)

    Google Scholar 

  100. 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

  101. Rumyantsev, V.V.: Stability of steady-state satellite motion. Cosmic Research 6, 135–138 (1968)

    Google Scholar 

  102. Schlick, T.: Modeling superhelical DNA: recent analytical and dynamic approaches. Current Opinion in Structural Biology 5 (2), 245–262 (1995). URL http://dx.doi.org/10.1016/0959-440X(95)80083-2

  103. Shi, Y., Hearst, J.E.: The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling. The Journal of Chemical Physics 101 (6), 5186–5200 (1994). URL http://dx.doi.org/10.1063/1.468506

  104. Shi, Y., Hearst, J.E., Bishop, T.C., Halvorson, H.R.: Erratum: “The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling” [J. Chem. Phys. 101, 5186 (1994)]. The Journal of Chemical Physics 109 (7), 2959–2961 (1998). URL http://dx.doi.org/10.1063/1.476848

  105. Shuster, M.D.: A survey of attitude representations. American Astronautical Society. Journal of the Astronautical Sciences 41 (4), 439–517 (1993)

    MathSciNet  Google Scholar 

  106. Silk, W., Erickson, R.O.: Kinematics of plant growth. Journal of Theoretical Botany 76 (4), 481–501 (1979). URL http://dx.doi.org/10.1016/0022-5193(79)90014-6

  107. Simmonds, J.G.: A Brief on Tensor Analysis, second edn. Springer-Verlag, New York (1994). URL http://dx.doi.org/10.1007/978-1-4419-8522-4

  108. Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions - a geometrically exact approach. Computer Methods in Applied Mechanics and Engineering 66 (2), 125–161 (1988). URL http://dx.doi.org/10.1016/0045-7825(88)90073-4

  109. Sokolnikoff, I.S.: Tensor Analysis: Theory and Applications. John Wiley & Sons, Inc., London (1951)

    Google Scholar 

  110. Stepanov, S.I.: On the steady motions of a gyrostat satellite. Journal of Applied Mathematics and Mechanics 33 (1), 121–126 (1969). URL http://dx.doi.org/10.1016/0021-8928(69)90120-8

  111. Tait, P.G.: On the rotation of a rigid body about a fixed point. Proceedings of the Royal Society of Edinburgh 25 (1868). Reprinted in pages 86–127 of [338]

    Google Scholar 

  112. Thompson, J.M.T., Champneys, A.R.: From helix to localized writhing in the torsional post-buckling of elastic rods. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 452 (1944), 117–138 (1996). URL http://dx.doi.org/10.1098/rspa.1996.0007

  113. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability, second edn. McGraw-Hill, New York (1961)

    Google Scholar 

  114. Umetani, N., Schmidt, R., Stam, J.: Position-based elastic rods. In: K. Erleben, J. Pettré (eds.) Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 21–30. Eurographics Association (2014)

    Google Scholar 

  115. Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, fourth edn. Cambridge University Press, Cambridge (1937)

    Google Scholar 

  116. Wilson, B.F.: Apical control of branch growth and angle in woody plants. American Journal of Botany 87 (5), 601–607 (2000). URL http://www.amjbot.org/content/87/5/601.abstract

  117. Ziegler, H.: Principles of Structural Stability, second edn. Birkhaüser, Basel (1977). URL http://dx.doi.org/10.1007/978-3-0348-5912-7

Download references

Author information

Authors and Affiliations

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

    Oliver M. O’Reilly

Authors
  1. Oliver M. O’Reilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

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

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50598-5_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50596-1

  • Online ISBN: 978-3-319-50598-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation