Study On Fractional Model Of Polymer Stress Relaxation And Loss Tangent

The convergence and numerical calculation of Mittag-Leffler function E_α,β(z) (0≤β<α<1)was discussed. The cause of calculation difficulty for large \z was pointed out. It was showed that the truncation with certain convergence condition for the asymptotic formula (-1/z) E_{-α,β-α} (1/z) was not always feasible, which resulted from the items of the series change with no regularity following the oscillation of (?) function between positive and negative.We proposed that the asymptotic formula was truncated at a set item. Following this algorithm, there is a range of z among which the two formulas can give consistent results. But to condition truncation, it did not exist for some parameters or change with them. It was showed that choosing the first item as the value of asymptotic formula for |z|>>1 which was used by many authors often has obvious errors comparing with results calculated by condition truncation or our algorithm.Based on genetic algorithm and conjugate gradient, a method for optimizing the model parameters was developed.The relaxation characteristics of the fractional Zener model were analyzed. The fractional model was used to simulate the stress relaxation process of polyurethane and VALOX, and the results show that this model can give good description to the relaxation process with different characteristics. Moreover, the relaxation process of VALOX displayed fractal structure.The loss tangent characteristics of the fractional Zener model were analyzed. The simulation to the unsaturated polyester resin gave good results. The logarithm of relaxation time log₁₀^Ï„and the relaxation indexαchange linearly with the temperature. A two process fractional model was used to simulate the HDPE with orientation structure. It was concluded that HDPE contain two relaxation process, one sharp and the other slow.