Properties of the strain rate sensitivity function produced by the linear viscoelasticity theory and existence of its maximum with respect to strain and strain rate

Strain rate sensitivity of stress-strain curves family generated by the Boltzmann–Volterra linear viscoelasticity constitutive equation (with an arbitrary relaxation modulus) under uni-axial loadings at constant strain rates is studied analytically as the function of strain and strain rate. The general expression for strain rate sensitivity index is derived and analyzed assuming relaxation modulus being arbitrary. Dependence of the strain rate sensitivity index on strain and strain rate and on relaxation modulus qualitative characteristics is examined, conditions for its monotonicity and for existence of extrema, the lower and the upper bounds and the limit values of the strain rate sensitivity as strain rate tends to zero or to infinity are studied. It is found out that (within the framework of the linear viscoelasticity) the strain rate sensitivity index which is, generally speaking, the function of two independent variables (namely strain and strain rate), depends on the single argument only that is the ratio of strain to strain rate. So defined function of one real variable is termed the strain rate sensitivity function and it may be regarded as a material function. The explicit integral expression (and the two-sided bound) for relaxation modulus in terms of strain rate sensitivity function is derived which enables one to restore relaxation modulus assuming a strain rate sensitivity function is given. The strain rate sensitivity function is represented as a linear function of ratio of tangent modulus to secant modulus of a stress-strain curve at any fixed constant strain rate and can be evaluated in such a way using experimental data. It is proved that the strain rate sensitivity value is confined in the interval from zero to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) whatever strain and strain rate magnitudes. It is found out that the linear theory can reproduce increasing or decreasing or non-monotone dependences of strain rate sensitivity on strain rate (for any fixed strain) and it can provide existence of local maximum or minimum or several extrema as well without any complex restrictions on the relaxation modulus. General properties and peculiarities of the theoretic strain rate sensitivity function are illustrated by the examination of the classical regular and singular rheological models (consisting of two, three or four spring and dashpot elements) and fractional models. Namely, the Maxwell, Kelvin–Voigt, standard linear solid, Zener, anti-Zener, Burgers, anti-Burgers, Scott–Blair, fractional Kelvin–Voigt models and their parallel connections are considered. The carried out analysis let us to conclude that the linear viscoelasticity theory (supplied with common relaxation function which are non-exotic from any point of view) is able to produce high values of strain rate sensitivity index close to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) and to provide existence of the strain rate sensitivity index maximum with respect to strain rate. Thus, it is able to simulate qualitatively existence of a flexure point on log-log graph of stress dependence on strain rate and its sigmoid shape which is one of the most distinctive features of superplastic deformation regime observed in numerous materials tests.

viscoelasticity, stress-strain curves at constant strain rates, strain hardening, strain rate sensitivity index (function), pseudoplastic media, fractional models, fractional differential equations, superplasticity, sigmoid curve, titanium and aluminum alloys, ceramics