Isothermal physical aging of PEEK and PPS investigated by fractional Maxwell model

Sampling Points-Independent Identification of the Fractional Maxwell Model of Viscoelastic Materials Based on Stress Relaxation Experiment Data

Considerable development has been observed in the area of applying fractional-order rheological models to describe the viscoelastic properties of miscellaneous materials in the last few decades together with the increasingly stronger adoption of fractional calculus. The fractional Maxwell model is the best-known non-integer-order rheological model. A weighted least-square approximation problem of the relaxation modulus by the fractional Maxwell model is considered when only the time measurements of the relaxation modulus corrupted by additive noises are accessible for identification. This study was dedicated to the determination of the model, optimal in the sense of the integral square weighted model quality index, which does not depend on the particular sampling points applied in the stress relaxation experiment. It is proved that even when the real description of the material relaxation modulus is entirely unknown, the optimal fractional Maxwell model parameters can be recovered from the relaxation modulus measurements recorded for sampling time points selected randomly according to respective randomization. The identified model is a strongly consistent estimate of the desired optimal model. The exponential convergence rate is demonstrated both by the stochastic convergence analysis and by the numerical studies. A simple scheme for the optimal model identification is given. Numerical studies are presented for the materials described by the short relaxation times of the unimodal Gauss-like relaxation spectrum and the long relaxation times of the Baumgaertel, Schausberger and Winter spectrum. These studies have shown that the appropriate randomization introduced in the selection of sampling points guarantees that the sequence of the optimal fractional Maxwell model parameters asymptotically converge to parameters independent of these sampling points. The robustness of the identified model to the measurement disturbances was demonstrated by analytical analysis and numerical studies.

Phenomenological models for the isothermal physical aging of PEEK

Creep test is a useful tool to study thermal aging and deformation mechanisms of semi-crystalline polymers, such as polyether-ether-ketone (PEEK). Hou and Chen proposed a power law to fit creep data of PEEK aged at different temperatures and the master curve built from those data. This paper attempts to complement that analysis by introducing Kohlrausch function as an alternative to the fractional Maxwell's model associated to the power law. Although the fitting of experimental data and the mathematical conditions imposed to equations that describe curves that can be superimposed by translations, are obeyed by both models, this paper demonstrates that Kohlrausch function provides a better phenomenological description of the creep response of PEEK due to the physical interpretation of the fitting parameters and their dependence on the aging time and temperature.