Dynamics of Two Linearly Elastic Bodies Connected by a Heavy Thin Soft Viscoelastic Layer

In this paper, we extend the asymptotic analysis in (Licht et al. in J. Math. Pures Appl. 99:685–703, 2013) performed, in the framework of small strains, 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 (Licht et al. in J. Math. Pures Appl. 99:685–703, 2013), 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.