On pressing of a buckled film

Abstract

The problem of smoothing a locally debonded and distorted thin film structure is considered. The structure consists of a thick rigid but deformable substrate and a thin elastic film, where the film is buckled in the debonded region as a result of compression of the substrate. A transverse point load is applied at the center of the buckled segment of the film simulating the action of a press. The film is modeled as a von Karman's thin plate. The problem is formulated in terms of the variational Principle of Stationary Potential Energy coupled with Griffith's Energy Criterion governing delamination growth. A self-consistent system of nonlinear equations describing deformation and delamination of the structure is derived. The associated transversality condition is found as a consequence of the variation of the debond edges, with the energy release rate seen to be comprised of mode I and mode II fracture. The problem is recast into a mixed formulation in terms of the transverse displacement and the membrane strain of the film, which renders a linear boundary value problem coupled with a nonlinear integrability condition associated with the prescribed in plane displacements at the debond edges. A special technique is introduced to modify the integrability condition by indirect evaluation of the corresponding integral. A closed form analytical solution is obtained and analyzed. Numerical simulations reveal the typical pressing scenario which appears to involve a snap-through behavior. Analysis of the growth condition shows that further delamination does not occur for the present type of loading, suggesting that successful pressing is possible under appropriate conditions.