Stress retardation versus stress relaxation in linear viscoelasticity

TL;DR

This paper explores the relationship between stress relaxation and retardation in linear viscoelasticity, revealing their asymptotic equivalence at small Deborah numbers and addressing discrepancies between theory and experiments.

Contribution

It introduces a new perspective on the stress-retardation versus stress-relaxation dichotomy, suggesting a way to reconcile theoretical models with experimental observations.

Findings

Relaxation and retardation are asymptotically equivalent for small Deborah numbers.

Pure relaxation models are necessarily ill-posed pure retardation models.

The dichotomy offers a potential reconciliation between theory and experiments on viscoelastic liquids.

Abstract

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.