QUARTERLY OF APPLIED MATHEMATICS VOLUME LIV, NUMBER 4 DECEMBER 1996, PAGES 687-696 SOME QUASI-STATIC PROBLEMS WITH ELASTIC AND VISCOUS BOUNDARY CONDITIONS IN LINEAR VISCOELASTICITY By CARLO ALBERTO BOSELLO and GIORGIO GENTILI Dipartimento di Matematica, Universita degli Studi di Bologna, Italy Abstract. We study the quasi-static behaviour of a linearly viscoelastic body which is subject to boundary forces respectively of elastic type and of viscous type. The ensu- ing problems exhibit dynamic boundary conditions. We impose on the memory kernel only those restrictions deriving from thermodynamics and, making use of the Fourier transform method, we show existence and uniqueness of the solution to each problem. 1. Introduction. A linearly viscoelastic body is described by the constitutive equa- tion: pOO T(x, t) — Go(x)E(x, t) + / G(x, s)E(x, t —s) ds (1.1) Jo where T is the Cauchy stress (second-order) tensor, E = |(Vu + VuT), u is the dis- placement vector, and G(x, t) = Go(x) + / G(x, s)ds, t> 0 (1.2) Jo is a symmetric fourth-order tensor representing the relaxation function of the viscoelastic material. The quasi-static behaviour of a continuum medium is described by the equation V • T(x, t) + f(x, t) = 0, (x, t)€fixR, (1-3) together with suitable boundary conditions (here Q is an open and bounded region of R3 with sufficiently regular boundary). For materials of type (1.1), (1.3) turns out to be an integro-differential equation of elliptic type, depending on time, whose integral kernel is G. We shall assume that G satisfies the fading memory principle, at least in its weak form (see for instance [1]). Furthermore, we shall impose on G the restriction dictated by the Second Law of Ther- modynamics in the Clausius form ([7]). In particular, as Fabrizio and Morro pointed out ([3] and [4]), we shall distinguish reversible from irreversible processes in the sense that, Received May 25, 1994. 1991 Mathematics Subject Classification. Primary 73F99, 45K05, 35J25. ©1996 Brown University 687