  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.
