Bending and stretching of thin viscous sheets

Abstract

Thin viscous sheets occur frequently in situations ranging from polymer processing to global plate tectonics. Asympotic expansions in the sheet's dimensionless ‘slenderness’ ε [Lt ] 1 are used to derive two coupled equations that describe the deformation of a two-dimensional inertialess sheet with constant viscosity μ 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 L0 and thickness H0 ≡ εL0, inflated by a constant excess pressure ΔP applied to one side (‘film blowing’). The sheet deforms initially by bending on a time scale με4/ΔP ≡ τb, and thereafter by stretching except in bending boundary layers of width δ ∼ L0(t/τb−1/3 at the clamped ends. (ii) An initially horizontal ‘viscous beam’ with length L0 and thickness H0 ≡ εL0, clamped at one end, deforms by bending on a time scale τb = μH20/gδρL30 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 ε−2τb. (iii) A sheet extruded horizontally at speed U0 from a slit of width H0 in a gravitational field deforms primarily by bending on a time scale (μH20/U30gδρ)1/4. The sheet's ‘hinge point’ moves in the direction opposite to the extrusion velocity, which may explain the observed retrograde motion of subducting oceanic lithosphere (‘trench rollback’).