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Mechanics of a String

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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

We start this book with arguably the simplest theory of a deformable elastic body - that of a string or, as it is also known, a one-dimensional continuum. Our motivation is the development of a theory that can accommodate a range of effects such as gravity, spatial discontinuities in velocity, applied forces which are concentrated at a point, large displacements and stretches, and nonlinear material behavior. This theory is used to develop models for a variety of problems ranging from chains to conveyor belts and bungee cords to hanging cables. Initially a wide range of kinematical results are established. Then, the balance laws are presented and a methodology for using these laws to establish the equations governing the motions of both inextensible and elastic strings is presented.

“In a theory ideally worked out, the progress which we should be able to trace would be, in other particulars, one from less to more, but we may say that, in regard to the assumed physical principles, progress consists in passing from more to less.”

A. E. H. Love [213, Page 1] commenting on the historical development of theories for continuous media.

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Notes

  1. 1.

    For additional background on Euler angles and other parameterizations of a rotation, the interested reader is referred to the authoritative review [321] by Malcolm Shuster (1943–2012).

  2. 2.

    It is an interesting exercise to perform this integration for a space curve where \(\boldsymbol{\omega }_{\text{SF}}\) is constant. As can be seen from the developments in Section 1.3.4, the resulting curve will either be a circle or a circular helix.

  3. 3.

    The interested reader is referred to [252, 253, 267] for further details and references.

  4. 4.

    The representation for a rotation tensor Q in terms of the angle of rotation and axis of rotation is known as Euler’s representation and can be found in Eqn. (5.14) on Page 223.

  5. 5.

    As first shown by Green and Rivlin [139] some fifty years ago, superposed rigid body motions can be exploited to furnish elegant formulations of the balance laws. The paper [139] has influenced numerous researchers seeking to develop balance laws (governing equations) for deformable media.

  6. 6.

    See, for example, [56, 58, 201, 279, 316, 342, 343].

  7. 7.

    Details on the continuity and boundedness assumptions on these fields can be inferred from our discussion following (1.68).

  8. 8.

    Eshelby’s collected works were recently assembled and published in [104]. The recent collection [332] of articles by researchers in this area provides an interesting panorama and a variety of perspectives on the legacy of Eshelby’s work.

  9. 9.

    We refer the reader to the example considered in Section 1.8, Exercises 1.7 and 2.7, and Section 8.8.2 of Chapter 8 for further details on these correspondences.

  10. 10.

    These identities have counterparts in the continuum mechanics of a three-dimensional body that can be found in the papers [230, 263]. The counterpart to Eqn. (1.88) is also discussed in Section 8.8.2 of Chapter 8

  11. 11.

    This identity was first established in [264]. An outline of the derivation of the identity (1.88) is presented in Exercises 1.3 and 1.4.

  12. 12.

    This parallels the methodology used in establishing constitutive relations for a three-dimensional continuum. For the reader’s convenience, a rapid summary of this procedure is presented in Section 8.6 of Chapter 8 Additional developments are presented in Exercise 8.4.

  13. 13.

    Restrictions of this type are common in continuum mechanics and the interested reader is referred to [351, Section 87] for a review of these restrictions. In the context of one-dimensional continua, Antman’s masterful discussion in [12, Chapter III, Section 3] has greatly influenced our exposition.

  14. 14.

    For further details on the Legendre transformation, we refer the reader to the lucid discussion of this transformation in Lanczos [195].

  15. 15.

    For an inextensible string μ = 1 and, after computing \(\dot{\mu }\), one finds that r ′ ⋅ v ′ = 0.

  16. 16.

    We follow Casey and Carroll [49] and do not presume that p  ⊥  = p. Related invariance requirements for F γ and \(\varPhi _{\text{E}_{\gamma }}\) are discussed in [277].

  17. 17.

    A detailed description of this procedure can be found in Section 2.3.2 of Chapter 2

  18. 18.

    Alternatively, we could use the Leibnitz rule and the constancy of C to establish the sought-after expression.

  19. 19.

    Observe that Eqn. (1.129) is simply a restatement of the identity \(\varPhi _{\text{E}_{\gamma }} = \mathsf{B}_{\gamma }\dot{\gamma } + \mathbf{F}_{\gamma } \cdot \mathbf{v}_{\gamma }\) (where F γ  = 0) applied to the present problem.

  20. 20.

    The strain energy function \(\rho _{0}\psi = \frac{EA} {2} \left (\mu -1\right )^{2}\) is insufficient to examine the phase transformation. Instead, what is required is a strain energy function such as the one shown in Figure 2.5 (cf. Eqn. (2.10)). The computation of C and \(\mathsf{B}_{\ell_{1}}\) for this strain energy function follows our previous developments, but the algebraic details are more complicated and are presented in Exercise 2.7.

  21. 21.

    We refer the reader to Eqn. (1.144) in Exercise 1.7 and Abeyaratne and Knowles [5, Chapter 2].

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    Oliver M. O’Reilly

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O’Reilly, O.M. (2017). Mechanics of a String. 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_1

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