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On the Bending and Twisting of Rods with Misfit

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Abstract

We derive a one-dimensional variational problem representing the elastic energy of a rod with misfit, starting from a nonlinear, three-dimensional elastic energy with nontrivial preferred strain. Our approach to dimension reduction is to find a Gamma-limit as the thickness of the rod tends to 0. The limiting energy is a quadratic function of the rates at which the rod bends and twists, and we give explicit expressions for the preferred curvature and twist in the special case of isotropic elastic moduli.

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Notes

  1. By misfit we mean preferred elastic strain. For example, this may be caused by thermal stress or the tendency to expand (perhaps anisotropically) due to water absorption. Similar concepts in the literature sometimes go by the names prestrain, prestress, non-Euclidean elasticity, and incompatible elasticity.

  2. We gratefully acknowledge extensive input from Alexander Shtukenberg, with whom we had many discussions about twisted crystals. These discussions formed the critical nucleus for the investigation presented here.

  3. See, however, the final paragraph of this subsection, concerning the paper [8].

  4. Familiarity with \(\varGamma \)-convergence is not necessary to read this work if one takes on faith that Theorem 1 is an acceptable derivation of a rod energy. See [30] for a review of thin structures and \(\varGamma \)-convergence.

  5. Six dimensions are associated to rigid motions. Twisting and stretching contribute one mode each, and bending contributes two dimensions. The last two modes are due to shear.

  6. There is something counterintuitive about this approach. It looks like \(\boldsymbol{\beta }\) and \(\xi \) are of similar status, because we minimize over both of them, but instead our analysis treats \(\xi \), \(\omega \) and \(\boldsymbol{\kappa }\) similarly. The reason for this will become clear in the proof of Theorem 2. The key is that \(\boldsymbol{{F}}^{(j)}\) are orthogonal to symmetrized gradients, but are not in general orthogonal to each other.

  7. In this section we conflate the typical radius of curvature with \(L\). These are closely related quantities in the present work because the small-thickness limit of Theorem 1 holds the midline fixed as \(h\) tends to 0.

  8. Both results considered the \(\varGamma \)-limit of an inextensible plate energy in the small-width limit. Kirby and Fried [17] used the Frenet frame with some additional assumptions amounting to small twist, whereas Freddi et al. [12] used the material frame and derived a corrected Sadowsky functional that holds even with large twist.

  9. The words ‘wide’ and ‘narrow’ are used inconsistently in the literature. Here as in [3] and [4] we refer to \(w\) and \(\sqrt{tL}\), whereas in [12] the authors use ‘narrow’ in relation to \(w/L\). Additionally, we note that the analysis of the narrow ribbon in [3] is similar to that in [4] but with misfit.

  10. As in the introduction, \(\boldsymbol{x}\) is the reference coordinates after rescaling the cross section and \(\boldsymbol{z}= (x_{1}, hx_{2}, hx_{3})\) the reference coordinates with physically correct dimensions.

  11. A real bilayer is made of two different materials, so both the misfit and the Hooke’s law would depend on \({\boldsymbol{x}}_{\mathrm{cs}} \). In this paper, however, we have taken the Hooke’s law to be independent of \({\boldsymbol{x}}_{\mathrm{cs}} \) (since our goal is to explore the effect of misfit in the simplest possible setting). Thus our bilayer is made from two materials with the same Hooke’s law but different prestrains.

  12. The third eigenvector has eigenvalue 0.

  13. Rods are well-approximated by a theory allowing only finitely many degrees of freedom. Sheets are much less restricted. Of course the ribbon might do something more complex than twisting or bending: kelp leaves, for an example, sometimes wrinkle [18].

  14. Namely the substitution of silicon atoms for aluminum (and a monovalent cation). This can be precisely measured because \(\gamma \)-ray irradiation causes smoky discoloration proportional to the \(\operatorname{Al}\) concentration, and has a known effect on the lattice constants.

  15. These conditions should hold for all of the symmetries, not just \(\boldsymbol{{Q}}\). This would imply that \(\boldsymbol{{m}}\) is diagonal and \(m_{11} = m_{22}\), where the \(c\) axis aligns with \(x_{3}\). It would also yield more information about the elastic stiffness tensor. See [16] for measurements the elastic stiffness tensor of quartz.

  16. For compactness of notation, here and below we write \(\boldsymbol{{A}}{\boldsymbol{x}}_{\mathrm{cs}} \) to mean \(\boldsymbol{{A}}\) times \((0, x_{2}, x_{3})\).

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  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Robert V. Kohn & Ethan O’Brien

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  1. Robert V. Kohn
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Correspondence to Ethan O’Brien.

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This work was supported partially by the National Science Foundation grants OISE-0967140 and DMS-1311833. This work was also supported partially by the MRSEC Program of the National Science Foundation under Award Number DMR-1420073.

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Kohn, R.V., O’Brien, E. On the Bending and Twisting of Rods with Misfit. J Elast 130, 115–143 (2018). https://doi.org/10.1007/s10659-017-9635-4

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