On the modeling of the dynamic‐elastic modulus for polymer materials under isochronal conditions

A model based on the concept of fractional calculus is proposed for the description of the dynamic elastic modulus, E* = E′ + iE″, of polymer materials. This model takes into account three relaxation phenomena (α, β, and γ) under isochronal conditions. The differential equations obtained for this model have derivatives of fractional order between 0 and 1. Applying the Fourier transform to the fractional differential equations and associating each relaxation mode to cooperative or noncooperative movements, E*(iω,T) was evaluated. The isochronal diagrams of E′ and E″ clearly show three relaxation phenomena, each of them is manifested by a decrease of E′ when temperature increases. This decrease is associated with a maximum in E″(T) diagram for each relaxation mode. The shape of the three peaks (three maxima in E″(T) diagrams) depends of the fractional orders of this new fractional model. The mathematical description obtained of E* corresponds to a nonexponential relaxation behavior often encountered in the dynamics of polymer systems having three relaxation phenomena. This model will enable us to analyze the viscoelastic behavior of polymers.