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Consideriamo un solido viscoelastico micropolare lineare occupante un dominioB in condizioni dinamiche. Dapprima assumendo cheB sia del tipoB={∈R:x’ =(x 1,x 2)∈D(x 3);x 3∈R++} e che il corpo sia soggetto a dati al contorno diversi da zero solo suD(0), stimiamo ad ogni istantet>0 fissato, mediante i dati iniziali e al contorno, l’«energia» della porzione di solido a distanza maggiore diz daD(0) (g t(z)) e la sua norma inL 1(0, t) (Gt(z)). Inoltre mostriamo che se le storie passate sono nulle suD(z),αz≥z 0≥0, allora ad ogni istantet>0 fissato, i punti (x’’,z) conz−z 0≥Vt sono in quiete, mentre perz−z 0≤Vt, Gt (z) decade come \(1 - \frac{{z - z_0 }}{{Vt}}\).V è una costante positiva calcolabile che dipende dalle funzioni di rilassamento, dalla densità di massa, dal tensore di microinerzia. Infine estendiamo i risultati a domini più generali sotto l’ipotesi che i dati iniziali e al contorno abbiano supporto compatto. Nel nostro studio utilizziamo un’energia libera massimale che ci consente di imporre restrizioni molto deboli sulle funzioni di rilassamento.
Abstract
We consider a linear micropolar viscoelastic solid occupying a domainB in dynamical conditions. First, on assuming thatB is of the kindB={∈R:x’ =(x 1,x 2)∈D(x 3);x 3∈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)(g t(z)) and its norm inL 1(0,t) (Gt(z)). Moreover we show that, if there exists somez 0≥0, such that past histories vanish onD(z) withz≥z 0, then for any fixedt>0 the points (x’’, z) withz−z 0≥Vt are at rest, while forz−z 0≤Vt, Gt(z) decays withz−z 0, 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.
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Borrelli, A., Piras, E. Spatial energy decay estimate in dynamical problems for a micropolar viscoelastic solid. Ann. Univ. Ferrara 48, 275–306 (2002). https://doi.org/10.1007/BF02824750
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DOI: https://doi.org/10.1007/BF02824750