A coupled rate-dependent/rate-independent system for adhesive contact in Kirchhoff–Love plates

Abstract We perform a dimension reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in the setting of visco-elastodynamic plates. We work with a weak solvability notion inspired by the theory of (purely) rate-independent processes, and accordingly term the related solutions “semistable energetic”. For semistable energetic solutions, the momentum balance holds in a variational sense, whereas the flow rule for the adhesion parameter is replaced by a semistability condition coupled with an energy-dissipation inequality. Prior to addressing the dimension reduction analysis, we show that semistable energetic solutions to the three-dimensional damped adhesive contact model converge, as the viscosity term tends to zero, to three-dimensional semistable energetic solutions for the undamped corresponding system. We then perform a dimension reduction analysis, both in the case of a vanishing viscosity tensor, and in the complementary setting in which the damping is assumed to go to infinity as the thickness of the plate tends to zero. In both regimes, the presence of adhesive contact yields a nontrivial coupling of the in-plane and out-of-plane contributions. In the vanishing-viscosity case, we additionally confine the analysis to the case in which also inertia is neglected: in the vanishing-thickness limit, we thus obtain purely rate-independent evolution for the adhesive contact phenomenon, still formulated in terms of the semistable energetic solution concept. In the second, undamped scenario, inertia is instead encompassed; thus the limiting evolution retains a mixed rate-dependent/rate-independent character, and is again given in terms of an energy-dissipation inequality and a semistability condition.