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 ≡ τ_b, and thereafter by stretching except in bending boundary layers of width δ ∼ L₀(t/τ_b^−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 τ_b = 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⁻²τ_b. (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’).