This paper is concerned with the analysis of the axial deformation of a viscoelastic rod whose heredity is modelled in terms of fractional derivatives. We prove the existence, the uniqueness, and the strong asymptotic stability of the solution. Our approach (which can be extended to several other types of physical models with fractional damping) is based on the input-output behaviour of a suitable diffusion system and allows the transformation of the original model into an equivalent augmented system whose analysis is less complicated. The advantage of this transformation lies in the fact that the system in its augmented form admits an easily identifiable Lyapunov's functional