Modeling of the viscoelastic behavior of amorphous polymers by the differential and integration fractional method: the relaxation spectrum H( τ)

Abstract

The dynamic mechanical behavior of poly(methyl methacrylate) (PMMA) from deep in the glass to the glass transition region has been studied by DMTA and analyzed by using a phenomenological fractional model in which the dynamic stress appears as a non-integer-order derivative of the strain. In order for the model to accurately represent the experimental data, three non-integer values for the derivative order are required. These values are related to two relaxation mechanisms. In the low temperature region (i.e. the β relaxation of PMMA), the derivative order is smaller and near 0.2, which indicates behavior close to the ideal elastic solid (glassy). For higher temperatures (between the β and the α relaxations), the derivative order is higher, indicating more viscoelastic behavior. In this work, modeling of the viscoelastic behavior of polymers using the fractional calculus approach is presented and the extended fractional solid (EFS) model is used to fit the experimental data of PMMA. In addition, the continuous relaxation spectrum H( τ) of PMMA is calculated from the model using the inverse Stieljes integral transformation. Finally, the effect of thermal treatment on the non-integer model parameters and on the distribution of relaxation times is obtained.