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On the modeling of the dynamic‐elastic modulus for polymer materials under isochronal conditions
Author(s) -
ReyesMelo E.,
MartínezVega J.,
GuerreroSalazar C.,
OrtizMéndez U.
Publication year - 2004
Publication title -
journal of applied polymer science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.575
H-Index - 166
eISSN - 1097-4628
pISSN - 0021-8995
DOI - 10.1002/app.20898
Subject(s) - viscoelasticity , relaxation (psychology) , fractional calculus , polymer , thermodynamics , materials science , dynamic modulus , diagram , modulus , dynamic mechanical analysis , physics , mathematical analysis , mathematics , composite material , psychology , social psychology , statistics
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