Fractional Calculus Description of Non-Linear Viscoelastic Behaviour of Polymers

Abstract

In recent decades, constitutive equations for polymers involving fractional calculus have been the object of ever increasing interest, due to their special suitability for describing self-similarity and memory effects, which are typical of viscoelastic behaviour in polymers. Thermodynamic validity of these equations can be ensured by obtaining them from analog models containing spring-pots with positive front factors. Failure of self-similarity in real polymers at short (local) and long (whole chain) scales has been addressed previously. In the past, interest in fractional differential descriptions of polymer viscoelasticity has been mainly concerned with linear viscoelasticity, despite the fact that in processing and end use conditions are largely in the non-linear range. In this paper, extension of fractional calculus models to the non-linear range of viscoelasticity is attempted, by accounting for stress activation of deformation and strain acceleration of annealing. Calculated stress-strain curves are compared with experimental results on an amorphous polymer (polycarbonate). The model adequately describes the general trends of yield and post-yield behaviour, but does not properly describe the gentle approach to yield observed experimentally.

Time-fractional KdV equation: formulation and solution using variational methods

“…As a further step, Riewe [1,2] formulated a version of the Euler-Lagrange equation for problems of calculus of variation with fractional derivatives. Recently, further studies concerning the fractional Euler-Lagrange equations can be found in the works of Agrawal and coworkers [5][6][7][8], Baleanu and coworkers [9][10][11][12][13][14][15], Tarasov and Zaslavsky [16,17] and others [18][19][20][21][22].…”

“…In last decades, much interest was devoted to apply fractional calculus to almost every field of science, engineering and mathematics [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The awareness of the importance of this type of equation has grown continuously include for viscoelasticity and rheology, image processing, mechanics, mechatronics, physics, and control theory, see for instance [23][24][25].…”

A new identification method of viscoelastic behavior: Application to the generalized Maxwell model

“…Lion [14,15] has investigated rheological models incorporating "fractional damping elements", called spring-pots, from the point of view of thermodynamics. Heymans [16] has extended the fractional calculus to non-linear viscoelasticity. Koeller [10] has defined the springpot as a rheological element whose stress is proportional to the fractional derivative of the strain.…”

Stability analysis of fractional-order systems with double noncommensurate orders for matrix case

“…436-453 , DOI: 10.2478/s13540-011-0027-3 study of real physical systems dynamical described by fractional-order calculus equations [11], it is found that fractional calculus is an adequate tool for the study of so called "anomalous" social and physical behaviors, in reflecting the non-local, frequency-and history-dependent properties of these phenomena [7,20]. For more knowledge of theory and applications on fractional calculus, please refer to [4,2,12,15,30,31].…”