Statistical origins of fractional derivatives in viscoelasticity

Many linear viscoelastic materials show constitutive behavior involving fractional order derivatives. Linear, time invariant systems without memory have exponential decay in time but, contradictorily, not the power law decay associated with fractional derivatives. The physics literature has noted that apparently-non-exponential decays can be observed when several simultaneously decaying processes have closely spaced exponential decay rates. Many engineer-researchers interested in viscoelastic damping, however, seem unaware of these observations. In this letter I give an unoriginal explanation, but with a fresh engineering flavor, for the appearance of these fractional order derivatives. By this explanation, fractional order damping can be expected from many materials with sufficiently disordered dissipation mechanisms.