Evaluation of classic and fractional models as constitutive relations for carbon black–filled rubber

Many constitutive relationships have been derived to model the viscoelastic behaviors of materials, but limited works are done to point out which model is the most suitable one in certain conditions. In this article, we present the detailed comparisons of the classic rheological and fractional derivative models to make up for the deficiency. First, creep and stress relaxation tests of carbon black (CB)–filled rubber are carried out at the room temperature for 12 h, and frequency-sweep tests are conducted at a temperature ranging from Tg ？ 10°C to Tg + 100°C under tension loading. The master curve of storage modulus over a wide range of frequency covering about 21 decades is constructed according to the time/frequency–temperature superposition (TTS/fTS). And then the classic models (Maxwell, Kelvin, and their generalized models) and fractional derivative models (4, 5, 6, and 10 parameters Zener models) are used to fit the test data. The fitting results show that the classic generalized models and fractional Zener models are quantitatively equivalent. The classic generalized models with several parameters are much simpler and are able to predict the viscoelastic behaviors for short-term or narrow frequency conditions, while the fractional derivative models are suitable for wide frequency conditions