We consider the problem of debonding of a thin gel domain from a rigid substrate. Starting with a variational approach involving the total energy of a gel, we formulate the boundary value problem of the governing equations in two-space dimensions. We consider the case that the aspect ratio, η, the quotient of the thickness of the film with respect to its length, is very small. We assume that the gel is partially debonded at the dimensionless horizontal location denoted by 0 < δ< 1. The appropriate limiting problem with respect to η, with fixed δ, yields an approximate solution corresponding to a deformation that is homogeneous both on the bonded part and on the debonded part of the gel, but whose gradient and vertical component are discontinuous across the interface x= δ. This approximate solution determines, up to first order, the energy release rate on δ, giving the critical value for the gel thickness at which it becomes unstable against debonding.
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We are grateful to M. Barchiesi, M. Doi, D. Golovaty, O. Lavrentovich, R. Siegel, S. Walker, N. Walkington & W. Zhang for their insight and valuable comments. Also, the careful attention and in-depth remarks from the referees is much appreciated. In particular, we thank them for pointing us out the possibility of an overall equilibration of the normal traction at the interface by means of a shear force concentrated at the crack tip, and a plausible explanation for the likely breaking of vertical symmetry in terms of the difference between the top and bottom boundary conditions at the bonded part of the gel. We are specially grateful for the detailed and enriching comments and feedback from the referee Dr. Thomas Pence. M. C. Calderer acknowledges the support of the National Science Foundation, grant N DMS-1616866. D. Henao is indebted to the Institute for Mathematics and its Applications of the University of Minnesota and its program ‘Multiscale Mathematics and Computing in Science and Engineering’, where he was received as a long-term visitor, for greatly facilitating this research. He also acknowledges the School of Mathematics of the University of Minnesota and the City Library of San Fernando, VI Región, Chile for their warm hospitality, as well as the Chilean Ministries of Economy and of Education for partially funding his research through Millennium Nucleus NC130017 CAPDE and FONDECYT projects 1150038 and 1190018. ∘
- Calculus of variations
- Nonlinear elasticity
- Thin film