A Material Momentum Balance Law for Rods

Abstract

A material momentum balance law is presented in this paper where it is also specialized for a variety of rod and string theories. The local form of the law is assumed to be identically satisfied, while the jump condition provides an extra equation which is often needed to solve problems involving the application of rod and string theories. The balance law is also related to several existing conservation laws for strings and rods, including Kelvin's circulation theorem. A novel identity for the singular sources at a discontinuity is also established.

Some perspectives on Eshelby-like forces in the elastica arm scale

“…We have examined Bosi et al's [1] recently developed elastica arm scale using a material force balance from O'Reilly [4]. This balance law features a contact material force C which is the one-dimensional counterpart of Eshelby's energy-momentum tensor and is a necessary Our analysis has shown how continuity of C enables a solution of the tangential components of the reaction forces F a 1 and F a 2 , and we were also able to use the conservation of C − ρgĝ · r to establish the governing equation (4.2) for the arm scale when the self-weight of the rod was included.…”

Chain paradoxes

“…In a different approach proposed in [22], the localized energy balance such as (2.30) is taken to be identically satisfied, which essentially amounts at using it to read off the dissipation D. The place of (2.30) is there taken by a jump condition for the material momentum, which usually determines the front's evolution law. As shown in [22], this approach, which is closely related to the theory of configurational forces, is also amenable to a variational formulation.…”

“…As shown in [22], this approach, which is closely related to the theory of configurational forces, is also amenable to a variational formulation.…”

“…22 A further limitation of the model is the strict one-dimensionality assumed for the motion, which is likely to be violated as y → 0. At the external shock, by (3.9c) and (5.4a), the dissipative power W * + is given by…”

On Adhesive and Buckling Instabilities in the Mechanics of Carbon Nanotubes Bundles

“…XIX ]. The discussion in Love's classic text [23] is supplemented by material on branching, adhesion and material momentum from recent works (see [19,22,24,25] and references therein). Referring to Figure 4, the centerline of the rod is parameterized by an arc-length coordinate s ∈ [0, ℓ] and the position of a point on the centerline is denoted by the vector-valued function r(s).…”