Viscous control of shallow elastic fracture: peeling without precursors

Abstract

We consider peeling of an elastic sheet away from an elastic substrate through propagation of a fluid-filled crack along the interface between the two. The peeling is driven by a bending moment applied to the sheet and is resisted by viscous flow towards the crack tip and by the toughness of any bonding between the sheet and the substrate. Travelling-wave solutions are determined using lubrication theory coupled to the full equations of elasticity and fracture. The propagation speed $v$

scales like

$M^{3}/\unicode[STIX]{x1D707}\bar{E}^{2}d^{5}=Bd\unicode[STIX]{x1D705}^{3}/144\unicode[STIX]{x1D707}$

, where

$d$

is the sheet’s thickness,

$B=\bar{E}d^{3}/12$

its stiffness,

$\bar{E}=E/(1-\unicode[STIX]{x1D708}^{2})$

its plane-strain modulus,

$\unicode[STIX]{x1D707}$

the fluid viscosity,

$M$

the applied bending moment and

$\unicode[STIX]{x1D705}=M/B$

the sheet’s curvature due to bending; and the prefactor depends on the dimensionless toughness. If the toughness is small then there is a region of dry shear failure ahead of the fluid-filled region. The expressions for the propagation speed have been used to derive new similarity solutions for the spread of an axisymmetric fluid-filled blister in a variety of regimes: constant-flux injection resisted by elastohydrodynamics in the tip leads to spread proportional to

$t^{4/13}$

,

$t^{4/17}$

and

$t^{7/19}$

for peeling-by-bending, gravitational spreading and peeling-by-pulling, respectively.