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

TL;DR

This paper analyzes a coupled rate-dependent and rate-independent adhesive contact model in visco-elastodynamic plates, performing dimension reduction and convergence analysis to derive simplified models in various limiting regimes.

Contribution

It introduces the Semistable Energetic solution concept for the coupled adhesive contact system and performs a detailed dimension reduction analysis in different damping and viscosity regimes.

Findings

Convergence of 3D damped solutions to undamped solutions as viscosity tends to zero.

Derivation of purely rate-independent evolution in the vanishing-thickness limit.

Retention of mixed rate-dependent/rate-independent behavior in the undamped, thin-plate limit.

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 semi-stability 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.