Stress retardation versus stress relaxation in linear viscoelasticity

• Preliminary examination of an approach to relaxation and retardation in viscoelasticity.
• Relaxation and retardation can be asymptotically equivalent for small Deborah number.
• Causal relaxation is shown to correspond to ill-posed retardation through this equivalence.

We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for “relaxation” and “retardation.” Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation.