Analysis of properties of creep curves generated by the linear viscoelasticity theory under arbitrary loading programs at initial stage

A. V. Khokhlov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 1, pp. 65-95.

Résumé

The general equation of creep curves family generated by the linear integral constitutive relation of viscoelasticity (with an arbitrary creep compliance function) under arbitrary non-decreasing stress histories at initial stage of loading up to a given stress level is derived and analyzed. Basic qualitative properties of the theoretic creep curves and their dependence on a rise time magnitude, on a loading program shape at initial stage and on creep function characteristics are studied analytically in the uni-axial case assuming creep compliance is an increasing convex-up function of time. Monotonicity and convexity intervals of creep curves, their asymptotic behavior at infinity and conditions for convergence to zero of the deviation from the creep curve under instantaneous (step) loading to a constant stress with time tending to infinity are examined. Two-sided bounds have been obtained for such creep curves and for deviation from the creep curve under step loading and for differences of creep curves with different initial programs of loading up to a given stress level. The uniform convergence of the theoretic creep curves family (with fixed loading law at initial stage) to the creep curve under step loading with the rise time tending to zero has been proved. The analysis revealed the importance of convexity restriction imposed on a creep compliance and the key role of its derivative limit value at infinity. It is proved that the derivative limit value equality to zero is the criterion for memory fading. General properties and peculiarities of the theoretic creep curves and their dependence on loading program shape at initial stage are illustrated by the examination of the classical rheological models (consisting of two or three spring and dashpot elements), fractional models and hybrid models (with piecewise creep function). The main classes of linear models are considered and specific features of their theoretic creep curves are marked. The results of the analysis are helpful to examine the linear viscoelasticity theory abilities to provide an adequate description of basic rheological phenomena related to creep and to indicate the field of applicability or non-applicability of the linear theory considering creep test data for a given material. The results constitutes the analytical foundation for obtaining precise two-sided bounds and correction formulas for creep compliance via theoretic or experimental creep curves with initial stage of loading (ramp loading, in particular) and for development of identification, fitting and verification techniques.

Détail

Keywords: linear viscoelasticity, creep compliance, theoretic creep curves, initial loading stage influence, loading program shape, ramp loading, two-sided bounds, deviation asymptotics, fading memory, regular and singular models, fractional models.
Mots-clés : rise time, convergence

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