Spatial energy decay estimate in dynamical problems for a micropolar viscoelastic solid

Abstract

We consider a linear micropolar viscoelastic solid occupying a domainB in dynamical conditions. First, on assuming thatB is of the kindB={∈R:x’ =(x1,x2)∈D(x3);x3∈R++}, and that the body is subjected to boundary data different from zero only onD(0), we estimate for any fixedt>0, in terms of the initial and boundary data, the «energy» of the portions of the solid at distance greater thanz fromD(0)(gt(z)) and its norm inL1(0,t) (Gt(z)). Moreover we show that, if there exists somez0≥0, such that past histories vanish onD(z) withz≥z0, then for any fixedt>0 the points (x’’, z) withz−z0≥Vt are at rest, while forz−z0≤Vt, Gt(z) decays withz−z0, the decay rate being described by the factor \(1 - \frac{{z - z_0 }}{{Vt}}\).V is a computable positive constant depending on the relaxation functions, the mass density and the microinertial tensor. Finally these last results are extended to more general domains under the hypothesis that the initial and boundary data have a bounded support. In our analysis we make use of a Maximal Free Energy which allows us to impose very mild restrictions on the relaxation functions.