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Theory of an Elastic Rod with Extension and Shear

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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 now consider a generalization of Kirchhoff’s rod theory to a theory which can accommodate extensibility of the centerline and transverse shear of the cross sections. The theory is constructed by allowing the rod to extend and shear. Consequently, the only significant change to the balance laws lies in the strain energy function. Theories of this type were developed in the early 1970s by various authors including Antman [10], Green and Laws [130], and Reissner [299, 300]. Antman’s papers discussing this theory have been hugely influential. This theory has several aspects which has made it popular in the applied mathematics, biophysics, computational mechanics, and mechanics communities (see, e.g., [12, 86, 158, 199, 327]). In addition to discussing applications of the theory to the stretching of DNA molecules, the linearized version of the theory is shown to include Timoshenko’s beam theory. We also take this opportunity to discuss a treatment of material symmetry for rods.

“In this way, one arrives at the kinematical model of a rod consisting of a one- dimensional continuum M 1 and a set of two vector fields \(\left (\mathop{\mathbf{d}}\limits_{1}^{i}\left (\xi,t\right ),\mathop{\mathbf{d}}\limits_{2}^{i}\left (\xi,t\right )\right )\) in M 1 whose values fix a homogeneous deformation of the cross section of the rod through the point ξ.”

R. A. Toupin’s discussion in [348, Page 90] of a model for a rod as a material curve with a set of directors.

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Notes

  1. 1.

    By way of additional background, we also note that if a function is invariant only under proper-orthogonal transformations (i.e., rotations) then it is said to be hemitropic. The adjective isotropic pertains to the case where the function is invariant under orthogonal transformations.

  2. 2.

    Necessary conditions for the positive definiteness of the strain energy function (6.57) include a 33 > 0, b 33 > 0, and a 33 b 33 − c 33 2 > 0.

  3. 3.

    Our use of the symbol Q to denote the rotation tensor associated with the uniform state should not be confused with the use of the same symbol to denote an orthogonal transformation in an earlier section of this chapter.

  4. 4.

    A proof of this result can be found in [267].

  5. 5.

    The shear correction factor is a constant in beam theory that is used to match static and dynamic solutions of the three-dimensional theory to those for the rod theory. The factor depends on the geometry of the cross section and the type of comparisons used (cf. [74, 93, 143, 310] and references therein). For a square cross section of a linearly elastic isotropic rod-like body with ν = 0. 3, k ≈ 0. 85 (0. 822) if a comparison based on a static (dynamic) solution is employed (cf. [93, Table 3]).

  6. 6.

    An analysis of the resulting equations can be found in Antman and Liu [13]. We also refer the reader to the paper by Coleman et al. [66] for a related analysis for a planar rod theory.

  7. 7.

    The reader is also referred to the closely related Exercises 5.6 and 7.4

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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). Theory of an Elastic Rod with Extension and Shear. 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_6

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