Damage and fatigue described by a fractional derivative model

As in [1], damage is associated with fatigue that a material undergoes. In this paper, because we work with viscoelastic solids represented by a fractional model, damage is described by the order of the fractional derivative, which represents the phase field satisfying Ginzburg–Landau equation, which describes the evolution of damage.Finally, in our model, damage is caused, not only by fatigue, but also directly by a source related to environmental factors and described by a positive time function.