Abstract
The constitutive equations of chemically and physically ageing rubber in the audible frequency range are modelled as a function of ageing temperature, ageing time, actual temperature, time and frequency. The constitutive equations are derived by assuming nearly incompressible material with elastic spherical response and viscoelastic deviatoric response, using Mittag-Leffler relaxation function of fractional derivative type, the main advantage being the minimum material parameters needed to successfully fit experimental data over a broad frequency range. The material is furthermore assumed essentially entropic and thermo-mechanically simple while using a modified William–Landel–Ferry shift function to take into account temperature dependence and physical ageing, with fractional free volume evolution modelled by a nonlinear, fractional differential equation with relaxation time identical to that of the stress response and related to the fractional free volume by Doolittle equation. Physical ageing is a reversible ageing process, including trapping and freeing of polymer chain ends, polymer chain reorganizations and free volume changes. In contrast, chemical ageing is an irreversible process, mainly attributed to oxygen reaction with polymer network either damaging the network by scission or reformation of new polymer links. The chemical ageing is modelled by inner variables that are determined by inner fractional evolution equations. Finally, the model parameters are fitted to measurements results of natural rubber over a broad audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations.
Highlights
Rubber components such as sealing, damper and vibration isolator are frequently used for a long time under harsh environmental conditions including oxygen, water, ozone, oil, contaminants and radioactive radiation exposures, usually at various temperatures
The developed physical and chemical ageing models are implemented into constitutive stress and strain relations for rubber in the audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations
Relaxation spectrum at each physical ageing time shows a broad, continuous spectrum enabling a wider time scale found in real molecular networks to be taken into account as compared to classical models embodying discrete spectra, providing a plausible explanation of the successful fit of Mittag-Leffler function to measurement results while explaining the broader times scales involved in the fractional free volume and relaxation time evolution results using Caputo fractional time derivatives in the differential evolution equations
Summary
Rubber components such as sealing, damper and vibration isolator are frequently used for a long time under harsh environmental conditions including oxygen, water, ozone, oil, contaminants and radioactive radiation exposures, usually at various temperatures Their mechanical properties are commonly modelled as if they were at their virgin state, disregarding any ageing. Free volume brought out of thermodynamical equilibrium by a sudden temperature alteration spontaneously advances to regenerate its thermodynamical equilibrium free volume state This process is present throughout the temperature range including glassy, transition and rubbery regions. The developed physical and chemical ageing models are implemented into constitutive stress and strain relations for rubber in the audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations. This is performed in part 2 of this paper [23]
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