Monotone increase of the strain rate sensitivity value of any parallel connection of the fractional Kelvin–Voigt models

A. V. Khokhlov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 3, pp. 56-67

Résumé

We continue to analyze the properties of the strain rate sensitivity value of the stress-strain curves at constant strain rates generated by the Boltzmann–Volterra linear viscoelasticity constitutive equation with an arbitrary relaxation modulus (in uni-axial case) and its dependence on strain, strain rate and relaxation modulus characteristics. The expression for the strain rate sensitivity value of the parallel connection of any number of the fractional Kelvin–Voigt models (each one governed by three parameters) is derived and analytically studied. In particular, arbitrary connections of the Scott Blair fractional elements (specified by power relaxation modulus) are considered. We prove that the strain rate sensitivity takes the values in the range from zero to the maximal exponent of the models connected whatever strain and strain rate magnitudes are; and in case only “fractal elements” are connected, the lower bound (and the limit value as the strain rate tends to zero) is non-zero and is equal to the minimal exponent of the models connected. The main result of the article is that we prove that strain rate sensitivity value of the studied models increases with the growth of the strain rate for any fixed strain (it has no peak value). This result is similar to the one obtained earlier for any parallel connections on non-linear power-law viscous elements and to its generalization on parallel connections of viscoplastic Herschel–Bulkley models (and the Shvedov–Bingham models as well) accounting for threshold stress. It means that there is no inflection point on the log-log graph of stress dependence on strain rate generated by any model of the class under consideration. This implies that such fractal models are not able to produce sigmoid shape of stress dependence on strain rate (in logarithmic scales) which is the distinctive feature of superplastic deformation regime and so they aren’t suitable for modeling superplasticity of materials. This result supplements and elaborates the capability of the linear viscoelasticity theory to provide existence of the strain rate sensitivity index maximum as well as its high values close to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) which have been discovered in previous contribution.

Détail

Keywords: viscoelasticity, stress-strain curves, strain rate sensitivity value, superplasticity, sigmoid curve, Voigt fractional models, fractional differential equations, power non-linear viscous element.
Mots-clés : fractal element

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