Limitations of the Newtonian time scale in relation to non-equilibrium rheological states and a theory of quasi-properties

Abstract

The behaviour of complex materials under stress is described in terms of entities which are not strictly 'physical properties'. These so-called 'quasi-properties' range from entities hardly distinguishable from dimensionally true physical properties to concepts which are much less clearly defined.Quasi-properties measure an ordered process towards equilibrium rather than a state of equilibrium. The Newtonian definition for equality of tune intervals which leads to the concepts velocity, acceleration, momentum and force having whole-number dimensional exponents, does not apply to 'quasi-equilibrium states'. In order to keep the Newtonian time scale, fractional differential equations are introduced.The simplest fractional differential equation relating stress, strain and time integrates to a series equation whose first term is a simple power law (Nutting's equation) already known to describe the behaviour of many complex materials under constant stress. The physical meaning of the fractional differential is considered.An apparatus is described for loading test-pieces of plastics and the like under tension or compression at constant stress to a preselected strain, and for following the subsequent stress dissipation; and the results of tests on thirty-eight materials are studied statistically.Introducing a second term from the series equation (and, very rarely, a third term) greatly improves the fit for materials for which Nutting's equation is inadequate and explains hitherto unaccountable anomalies when the Nutting plot is otherwise satisfactory. Constants derived by the equation from constant-stress and constant-strain conditions are compared.The form of the series equation suggests that the relative importance of the second term may sometimes disclose the presence of undissipated stresses in the materials.The accuracy of tests on individual test-pieces is high, but, on account of frequent lack of homogeneity in the samples available, the use of the unmodified Nutting equation is often adequate even when the addition of a second term would significantly improve individual curves.Some alternative treatments are discussed, but, both theoretically and practically, the fractional differential approach is preferred for most of the materials tested.

Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations

“…To demonstrate the ability of properly formulated fractional constitutive models and the resulting quasi-properties to compactly describe the complex time-dependent properties of real viscoelastic materials, we revisit Scott-Blair et al's [31] original stress relaxation data and fit the measurements with the FMM discussed in §2b. In figure 2, we re-plot representative data reported for the original stress relaxation and creep experiments performed by Scott-Blair et al [31].…”

“…However, it is in general not possible using standard procedures to find the underlying form of the constitutive model that could subsequently be used to predict the response of the material to another mode of excitation [3]. Scott-Blair et al [31] attempted to model measurements of anomalous stress relaxation in a range of materials using a higher-order Nutting equation of the form…”

“…The quasi-property V also reduces, respectively, to the limits of a modulus G (units: (Pa)) or a viscosity η (units: (Pa s)) in these limiting cases. The preferred term proposed by Scott-Blair et al [31] for a constitutive law exhibiting spring-pot-like behaviour was 'the principle of intermediacy'. We present a vectorial graphical representation of the spring-pot in the electronic supplementary material to aid in understanding this intermediate nature of the spring-pot.…”

“…For dimensional consistency, the constant V must have the units (Pa s α ) where 0 ≤ α ≤ 1, and can be equated to Scott-Blair's concept of a quasi-property [31]. The formulation of fractional constitutive equations in terms of quasi-properties has fallen out of use in the recent rheological literature.…”

“…We first briefly outline the basic definitions of fractional calculus in a form most useful for applications in rheology. We then connect the framework to the studies of Scott-Blair and co-workers [24,31], and show, using Scott-Blair et al's [31] original data on 'highly anomalous butyl rubber', how the use of the FMM to extract the quasi-properties of this material is superior to the use of conventional spring-dashpot models that characterize creep and stress relaxation. Next, we emphasize the utility of fractional constitutive models, and highlight the shortcomings of linear constitutive models for describing complex fluid interfaces using interfacial rheology data obtained from highly viscoelastic bovine serum albumin (BSA) and Acacia gum interfaces.…”