Partial Constraint Singularities in Elastic Rods

Abstract

We present a unified classical treatment of partially constrained elastic rods. Partial constraints often entail singularities in both shapes and reactions. Our approach encompasses both sleeve and adhesion problems, and provides simple and unambiguous derivations of counterintuitive results in the literature. Relationships between reaction forces and moments, geometry, and adhesion energies follow from the balance of energy during quasistatic motion. We also relate our approach to the balance of material momentum and the concept of a driving traction. The theory is generalizable and can be applied to a wide array of contact, adhesion, gripping, and locomotion problems.

Pseudomomentum: origins and consequences

“…where clearly ( 62) is just a specific case of (63). The prescription ( 62) is consistent with O'Reilly [78] and with our prior work [80], while the prescription ( 63) is consistent with Kienzler and G. Herrmann [75].…”

“…We demonstrate that the propulsive material force on the confined rod can be obtained directly by integrating the pseudomomentum balance, and obtain reaction forces at points of geometric and material discontinuity from the singular pseudomomentum balance without any appeal to micromechanical arguments [49]. As in O'Reilly's analysis [79,80] of Bigoni's sleeve constraint [76], this approach requires a prescription for singular sources of pseudomomentum, but once this leap has been taken the results follow immediately.…”

“…This group has also offered a theoretical analysis that, we believe, incorrectly conflates forces and material forces, lumping them both under a general heading of configurational or "Eshelby-like" effects. In this section, we present our perspective, which is heavily influenced by the analyses of Kienzler and G. Herrmann [75] and O'Reilly [19,78,79], and partially laid out in prior publications [80,81]. After presenting the bulk and singular balance laws for momentum and pseudomomentum for a non-uniform planar Euler elastica, we apply our approach specifically to the problem of planar serpentine locomotion of a rod through a curved channel [48,49], and attempt to delineate which forces appearing in the problem are actual forces and which are configurational forces.…”

“…with allowance for singular supplies of momentum R and pseudomomentum Y that do not explicitly appear in the action [80]. The quantity R represents such things as reaction forces from external constraints at the discontinuity, as might arise at the edge of a sleeve [76].…”

“…The quantity R represents such things as reaction forces from external constraints at the discontinuity, as might arise at the edge of a sleeve [76]. The interpretation of Y is still a matter of discussion in the literature [19,78,80] and will be considered for our specific problem in the following section.…”