Fractional time-temperature superposition model and its validation for dynamic mechanical properties of viscoelastic material

Abstract: The viscoelastic damping structure is widely used in the vibration and noise control of the agricultural engineering vehicle because of its high vibration dissipation capability, simple structure and lower maintenance cost. The mechanical behavior of the viscoelastic materials displays anelastic feature and temperature-frequency dependence, so the precise dynamic modeling is the key step in the viscoelastic structure design and its vibration damping analysis process. For anelasticity, the fractional derivative defined based on global definition can precisely represent the history dependence of the system function, and be extensively applied to the viscoelastic models with fewer parameters and better data fitting ability. For temperature-frequency dependence, the time-temperature superposition (TTS) principle was adopted, according to which the frequency spectrums at various temperatures can be collapsed into a master curve at the reference temperature by multiplying the conversion factors. The master curve covers a wide reduced frequency range up to many orders of magnitudes. In this research, the fractional time-temperature superposition model (FTTSM) for dynamic mechanical properties of viscoelastic materials was proposed based on the fractional order relationship, the Vogel-Fulcher-Tammann equation and the WLF (Williams-Landel-Ferry) equation. The frequency conversion factor from FTTSM was derived and its parameter identification process was developed based on tensile test and DMA (dynamic thermo mechanical analysis) test. In order to understand the parameter influence on the conversion factor, the variation of the conversion factor was studied under the material parameter of 0.2, 0.4, 0.6, 0.8 and 1 and the environment parameter of 0, 0.4, 0.6, 0.8 and 1. For the application and validation, the tensile test was conducted following GB/T 528-1998 on the M350-10 kN type precise elongation apparatus (Testometric, Britain), after preparing the I type dumbbell-shaped sample following GB/T 9865.1. In addition, the DMA test was also carried out according to ISO 6721-1 using the DMA 242C (Netzsch, Germany), in a 3-piont bending mode with a 40 mm span between the supports, in which the sample was supported on 2 supporting edges, while the probe edge applied load to the sample. On the base of the test data, the master curves at reference temperature of 5 ℃ from FTTSM and WLF equation, constructed through horizontally superposing the isothermals at various temperatures onto the isothermal at reference temperature, were comparatively studied. Furthermore, the theoretical prediction over the same frequency span was made through fractional Kelvin-Voigt constitutive model (KFVEO) to testify the master curves. The results indicated that the frequency conversion factors from FTTSM and WLF equation showed a good consistence with the maximum error of 0.984 4% within temperature scope (-80-80 ℃), and the master curves constructed by FTTSM and WLF equation greatly extended the frequency range up to 10 decades. The RMSE (root mean square error) between the master curves from FTTSM and WLF and the KFVEO prediction value was1.291 and 1.834 respectively, which manifested the FTTSM was more precise. Regarding the extended frequency, the minimum extended frequency by FTTSM was 2 orders of magnitudes less than that by WLF equation, while the maximum extended frequency stayed at the same level for these 2 models. This indicated a higher frequency extended capacity of FTTSM. This research can provide the theoretical reference for the investigation of viscoelastic material on dynamic behavior prediction, physical aging and mechanism of creep damage evolution, and so on.