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 *( i ω, 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. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 94: 657–670, 2004