Ricerc@Sapienza

Materials whose mechanical and/or thermodynamical behaviour is determined not only by their present status but also by their past history can be termed materials with memory. A well known example of material with memory is a rigid heat conductor with memory. This model, according to [8] and [1], describes a body, assumed rigid, in which the memory affects its thermodynamical behaviour. Specifically, the heat flux relaxation function depends only on the time variable through the present time as well as the whole past history.

The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in vis- coelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials.

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.

The existence of solutions to a one-dimensional problem arising in magneto- viscoelasticity is here considered. Specifically, a non-linear system of integro- dierential equations is analysed; it is obtained coupling an integro-dierential equation modelling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial dierential equations modelling the presence of a magnetic field.