Ricerc@Sapienza

We take advantage of the approximation of the stretched exponential function with a general Prony series in glass relaxation to give some results about the spectral analysis for the equation of viscoelasticity. Moreover, in the case of the Burgers model we carry out a complete investigation that leads to the representation of the solution.

Materials with memory, namely those materials whose mechanical and/or thermodynamical behavior depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behavior is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation.

The model of a viscoelastic body is considered focussing the attention on the the kernel of the integro- differential model equation. It represents the relaxation modulus which characterises the response of the material with memory to deformation. An overview on the classical viscoelasticity model is followed by different generalisations. Two different cases of relaxation functions, whose physical admissibility is guarantied by appropriate assumptions, are listed. The first one concerns a relaxation modulus at the initial time t = 0.

We consider an anisotropic hyperbolic equation with memory term: ?t2u(x,t)=?i,j=1n?i(aij(x)?ju)+?0t?|?|?2b?(x,t,?)?x?u(x,?)d?+R(x,t)f(x) for $x \in \Omega$ and $t\in (0, T)$ , which is a simplified model equation for viscoelasticity. The main result is a both-sided Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor $f(x)$ of the force term $R(x, t)\,f(x)$ .