A Magneto-Viscoelasticity Problem with Aging

This study addresses a magneto-viscoelasticity problem, considering the one-dimensional case. The system under investigation is given by the coupling a non-linear partial differential equation with a linear integro-differential equation. The system models a viscoelastic body whose mechanical behavior, described by the linear integro-differential equation, is also influenced by an external magnetic field. The model here investigated aims to consider the concomitance of three different effects: viscoelasticity, aging and magnetization. In particular, the viscoelastic behavior is represented via an integro-differential equation whose kernel characterizes the properties of the material. In a viscoelastic material subject to the effects of aging, all changes in the response to deformation are due not only to the intrinsic memory of the material but also to deterioration with the age of the material itself. Thus, the relaxation function is not assumed to depend on the two times, present and past, via their difference, but to depend on both the present and past times as two independent variables. The sensibility to an external magnetic field is modeled by a non-linear partial differential equation taking its origin in the Landau–Lifschitz magnetic model. This investigation is part of a long-term research project aiming to provide new insight in the study of materials with memory and, in particular, viscoelastic materials. Specifically, the classical model of viscoelastic body introduced by Boltzmann represents the fundamental base from which a variety of generalizations have been considered in the literature. In particular, the effects on the viscoelastic body due to interaction with an external magnetic field are studied. The new aspect under investigation is the combined presence of the external magnetic field with the effect of aging. Indeed, the coupling of viscoelasticity, which takes into account the deterioration of the material with time, with the presence of an external magnetic field, was never considered in previous research. An existence and uniqueness result is proved under suitable regularity assumptions.