Abstract
In the discrete elastic rod formulation, expressions for the variations, gradients, and Hessians of kinematic variables induced by changes to the vertices are required. The present chapter provides the background and intermediate computations that are needed to establish the desired representations for these gradients and Hessians. Some of resulting expressions are employed to compute the elastic restoring forces in a discrete elastic rod in a later chapter.
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Notes
- 1.
It is tempting to assume (in error) that the material vectors are unaltered by the change in the vertices. However, as the variations in \(\mathbf {m}^{k-1}_1\), \(\mathbf {m}^{k-1}_2\), \(\mathbf {m}^k_1\), and \(\mathbf {m}^k_2\) induced by variations in the vertices are orthogonal to \(\left ( \kappa \mathbf {b}\right )_k\), the variations \(\delta \mathbf {m}^{k-1}_1\), \(\delta \mathbf {m}^{k-1}_2\), \(\delta \mathbf {m}^k_1\), and \(\delta \mathbf {m}^k_2\) are absent in the final expressions for \(\delta \kappa _{k_1}\) and \(\delta \kappa _{k_2}\).
References
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Jawed, M.K., Novelia, A., O’Reilly, O.M. (2018). Variations, Gradients, and Hessians. In: A Primer on the Kinematics of Discrete Elastic Rods. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-76965-3_6
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DOI: https://doi.org/10.1007/978-3-319-76965-3_6
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