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 time 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.