Jump conditions for strings and sheets from an action principle

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 elastica which are applicable to moving contact problems.