Monotonic, completely monotonic, and exponential relaxation functions in linear viscoelasticity

A priori restrictions on the relaxation function of linear viscoelasticity are studied under regularity assumptions weaker than those usually made in the literature. The new set of assumptions is sufficient to define, by a limit procedure, the work done in deformation processes in which some parts are subject either to extreme retardations or to extreme accelerations. The use of such processes results in a considerable simplification of the proofs of some classical results. Under the same assumptions, we give a characterization of the monotonicity of the relaxation function in terms of work. We also extend an earlier one-dimensional characterization of complete monotonicity due to Day, and prove that the work done in every closed path in stress-strain space is nonnegative if and only if the relaxation function is of exponential type.