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 \emph{elastica} which are applicable to moving contact problems.Comment: Added figure with picture of dancing tube guy, small clarifications, and term that was missing from an equatio

Junctions in shell structures: A review

“…The jump conditions were also generalized in several papers by Eremeyev and Pietraszkiewicz [69][70][71] and Pietraszkiewicz et al [72] to account for the phase transition phenomena in shells. Hanna [73] presented 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.…”

Chain paradoxes

“…As a consequence, by (5.18), P is predicted to diverge to +∞ as t → t f . By (5.19), equation (5.18) can be given an easier expression, 21) which shows how P diverges as η → 0, unless f = 1 or f = 0. The former is the case of free fall, when (5.21) reduces to P = 3P 0 (1 − η).…”

“…The latter is the case of no dissipation at the internal shock, when (5.21) reduces to P = P 0 (1 − η). 21 More generally, figure 6a illustrates a phase portrait of equation (5.17) on the plane (η, η ) in the range of η that interests us. The unboundedness ofẏ,ÿ and P could be regarded as paradoxical.…”

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“…is identically satisfied, and expression(35) becomesn · x + m·m 2B = c .A rod x(s, t) partially constrained by a frictionless sleeve and subject to end loads S and P. A reaction force R and moment M are present at the sleeve edge s = s0(t).…”