Jump conditions for strings and sheets from an action principle

TL;DR

This paper derives conditions for the compatibility and conservation laws across moving discontinuities in strings and sheets using an action principle, with applications to fracture and elastica.

Contribution

It introduces a unified action-based framework for jump conditions in inextensible strings and sheets, including higher derivative actions and dynamic fracture modeling.

Findings

Derived jump conditions from an action principle for strings and sheets.

Provided solutions for conservative sheet motions near line discontinuities.

Extended the framework to include elastica and higher derivative actions.

Abstract

I present conditions for compatibility of velocities, conservation of mass, and balance of momentum and energy across moving discontinuities in inextensible strings and sheets of uniform mass density. The balances are derived from an action with a time-dependent, non-material boundary, and reduce to matching of material boundary conditions if the discontinuity is stationary with respect to the body. I first consider a point discontinuity in a string and a line discontinuity in a sheet, in the context of classical inertial motion in three Euclidean dimensions. I briefly comment on line discontinuities terminating in point discontinuities in a sheet, discontinuous line discontinuities in a sheet, and an approach to dynamic fracture that treats a crack tip in a sheet as a time-dependent boundary point. I provide two examples of general solutions for conservative sheet motions near a line discontinuity. The approach also enables treatment of actions depending on higher derivatives of position; I thus derive balances for an \emph{elastica} which are applicable to moving contact problems.