Regular and singular kernel problems in magneto-viscoelasticity

Abstract

Magneto-viscoelastic materials find their interest in a variety of applications in which mechanical properties are coupled with magnetic ones. In particular, new materials such as magneto-rheological elastomers or, in general, magneto-sensitive polymeric composites are more and more widely employed in new materials. The deformation evolution is assumed to be viscoelastic, that is, the stress–strain relation depends on the deformation history of the material further to on the deformation at the present time. This is a characteristic feature of all materials with memory, namely those materials whose mechanical and/or thermodynamical response depends on time not only via the present time, but also through the whole past history. To describe this behaviour integro-differential model equations are adopted subject to the fading memory assumption which corresponds to require that, asymptotically, effects of past deformation events become negligible. Magneto-viscoelastic materials are modelled aiming to describe viscoelastic materials whose mechanical response is also influenced by the presence of a magnetic field. Thus, the model system is obtained on coupling the viscoelasticity linear integro-differential equation with a nonlinear partial differential equation which describes magnetic effects. The attention is focussed on the kernel of the integro-differential equation: both regular as well as a singular kernel, at $$t=0$$ , problems are analysed. Indeed, singular kernel models allow to describe a wider class of materials and are also connected to materials modelled via kernels described via fractional derivatives.