Aging of Viscoelastic Materials: A Mathematical Model

The aim of this note is to propose a mathematical model to describe aging effects occurring in viscoelastic materials. In the classical formulation, viscoelasticity is modeled through an integro-differential equation, where the convolution kernel takes into account the delay effects induced by viscosity. Usually, such a kernel is a given function, related to the rheological properties of the material. On the other hand, it is reasonable to conjecture that, as the material changes/deteriorates over time (and these changes are referred to as aging), the shape of the kernel should change as well. The key idea is then to consider an integro-differential equation where the convolution kernel is a function of time itself. This allows a more realistic description of the evolution, requiring the development of a new mathematical theory apt to treat dynamical systems acting on time-dependent spaces.