Elastic buckling with viscous dissipation: Compression-induced buckling of an elastic film on a viscous foundation

When a thin, stiff film bound to a compliant foundation is compressed, it buckles into a variety of modes including sinusoidal wrinkles, tall well-spaced ridges, or deep folds. Buckling in “energy-conserving” systems has been studied heavily - examples include compression of an elastic film on an elastic foundation, compression of an elastic film floating on a dense liquid in gravity, or an elastic film under boundary tension. In all such cases, the buckling mechanics can be captured by finding the buckling mode that minimizes the total energy of the film-substrate system. But not all systems conserve energy. Here we consider the extreme case of an elastic film on a viscous foundation which fully dissipates energy.

 

Experiments are conducted by laying an elastic film on a viscous layer which is itself supported by a prestretched rubber strip. Unstretching the strip compresses the liquid layer, which transmits viscous stress to the film, which then buckles. Remarkably, buckles can appear either as uniform wrinkles, or as tall ridges separated by nearly-flat regions. For a finite-length film, we develop a shear lag model to describe the evolution of compressive stress in the film prior to buckling. A linear stability analysis of this base solution can describe the wrinkling mechanics, but the appearance of well-spaced ridges is mysterious. Finite element simulations under 2D plane strain conditions offer key insights on how three different mechanisms of energy release: relaxation from the film ends, wrinkle growth, or ridge growth compete. A key finding is that ridges initiate from a long-wavelength buckling mode which appears very close to the buckling threshold. Thus, unlike in energy-conserving systems where ridge formation is a post-buckling transition, here ridge formation appears as a near-threshold phenomenon. We map the state of the film as flat, wrinkled, or ridged in the parameter space of compression rate vs strain.