Dynamics of two linearly elastic bodies connected by a heavy thin soft viscoelastic layer

TL;DR

This paper extends asymptotic analysis of two elastic bodies connected by a thin viscoelastic layer to include cases where the layer's mass remains positive, revealing persistent dynamic effects at the contact surface.

Contribution

It introduces a new asymptotic model accounting for the finite mass of the viscoelastic layer, capturing dynamic effects that previous models overlooked.

Findings

Dynamic effects at the contact surface persist in the limit.

The limiting behavior involves an additional variable for the adhesive layer dynamics.

Convergence results are established using nonlinear Trotter's theory.

Abstract

In this paper we extend the asymptotic analysis in [LLOO], performed on a structure consisting of two linearly elastic bodies connected by a thin soft nonlinear Kelvin-Voigt viscoelastic adhesive layer, to the case in which the total mass of the layer remains strictly positive as its thickness tends to zero. We obtain convergence results by means of a nonlinear version of Trotter's theory of approximation of semigroups acting on variable Hilbert spaces. Differently from the limit models derived in [LLOO], in the present analysis the dynamic effects on the surface to which the layer shrinks do not disappear. Thus, the limiting behavior of the remaining bodies is described not only in terms of their displacements on the contact surface, but also by an additional variable that keeps track of the dynamics in the adhesive layer.