Completely monotone fading memory relaxation modulii

In linear viscoelasticity, the fundamental model is the Boltzmann causal integral equation σ(t) = ∫_-∞^t G(t - τ)γ̇(τ)dτ which defines how the stress σ(t) at time t depends on the earlier history of the shear rate γ̇(τ) via the relaxation modulus (kernel) G(t). Physical reality is achieved by requiring that the form of the relaxation modulus G(t) gives the Boltzmann equation fading memory, so that changes in the distant past have less effect now than the same changes in the more recent past. A popular choice, though others have previously been proposed and investigated, is the assumption that G(t) be a completely monotone function. This assumption has much deeper ramifications than have been identified, discussed or exploited in the theological literature. The purpose of this paper is to review the key mathematical properties of completely monotone functions, and to illustrate how these properties impact on the theory and application of linear viscoelasticity and polymer dynamics. A more general representation of a completely monotone function, known in the mathematical literature, but not the theological, is formulated and discussed. This representation is used to derive new rheological relationships. In particular, explicit inversion formulas are derived for the relationships that are obtained when the relaxation spectrum model and a mixing rule are linked through a common relaxation modulus.