Bending and stretching of thin viscous sheets

Thin viscous sheets occur frequently in situations ranging from polymer processing to global plate tectonics. Asympotic expansions in the sheet's dimensionless 'slenderness' epsilon much less than 1 are used to derive two coupled equations that describe the deformation of a two-dimensional inertialess sheet with constant viscosity mu and variable thickness and curvature in response to arbitrary loading. Three model problems illustrate the partitioning of thin-sheet deformation between stretching and bending modes: (i) A sheet with fixed (hinged or clamped) ends, initially flat and of length Lo and thickness H-0 equivalent to epsilonL(0) inflated by a constant excess pressure DeltaP applied to one side ('film blowing'). The sheet deforms initially by bending nn a time scale mu epsilon (4)/DeltaP, and thereafter stretching except in bending boundary layers of width delta similar to L-0(t/taub())(-1/3) at the clamped ends. (ii) An initially horizontal 'viscous beam' with length L-0 and thickness H-0 equivalent to epsilonL(0), clamped at one end, deforms by bending on a time scale tau (b) = muH(0)(2)/g delta rhoL(0)(3) until it hangs nearly vertically. Thereafter it deforms by bending in a thin boundary layer at the clamped end, and elsewhere by stretching on a slow time scale epsilon (-2)tau (b). (iii) A sheet extruded horizontally at speed U-0 from a slit of width Ho in a gravitational fiels deforms primarily by bending on a time scale (muH(0)(2)/U(0)(3)g delta rho)(1/4) point' moves in the direction opposite to the extrusion velocity, which map explain the observed retrograde motion of subducting oceanic lithosphere ('trench rollback').