Rheological implications of completely monotone fading memory

In the constitutive equation modeling of a (linear) viscoelastic material, the “fading memory” of the relaxation modulus G(t) is a fundamental concept that dates back to Boltzmann [Ann. Phys. Chem. 7, 624 (1876)]. There have been various proposals that range from the experimental and pragmatic to the theoretical about how fading memory should be defined. However, if, as is common in the rheological literature, one assumes that G(t) has the following relaxation spectrum representation: G(t)=∫0∞ exp(−t/τ)[H(τ)/τ]dτ, t > 0, then it follows automatically that G(t) is a completely monotone function. Such functions have quite deep mathematical properties, that, in a rheological context, spawn interesting and novel implications. For example, because the set of completely monotone functions is closed under positive linear combinations and products, it follows that the dynamics of a linear viscoelastic material, under appropriate stress–strain stimuli, will involve a simultaneous mixture of different molecular int...