Stress Relaxation And Creep Of Polymers Studied By The Fractional Model

Polymer has been used widely in industry, agriculture, advanced science and ourdaily life, Knowing mechanical properties of polymers is very important. Mechanicalproperties of many polymers depends intensively on the temperature and time, it isviscoelastic problem. The classical models to describe the viscoelastic behavior aremade up of the elastic and viscous elements, the corresponding constitutive equationsare in the form of integer order derivatives. The simple classical model can representspecific viscoelastic process qualitatively. Sufficient accuracy for quantitativesimulation of the experiment data could be obtained when complex models are used.As there are too many adjustable parameters, however, such models often lackgenerality.Viscoelasticity theory made a great development because fractional calulus isintroduced to viscoelastic constitutive equations. Constitutive equations are obtainedby replacing integer order by fractional order that is not always stability. A fractionalviscoelastic element, called “spring-pot”, was proposed by Koeller to overcome thisshortcoming. When the springs and dashpots in the classic models are replaced by aspring-pots, the constitutive equations obtained can automatically guaranteemechanical and thermodynamical stability. Comparing with the classical models ofinteger order derivatives, the fractional models involve fewer parameters with quiteclear interpretations and they provide generally appropriate description to the creep,relaxation and dynamical data over a broad time/frequency range.In this paper, the generalized fractional models are used to study quasi-static anddynamical moduli of three samples, the nature-rubber gum vulcanizetes GR-S andHeave and the polyisobutylene. The H-Fox function plays a dominant role in theapplication of fractional theories. The convergence and the numerical calculation ofthe function are discussed. Genetic algorithm and the conjugated gradient method arecombined to optimize the model parameters. The results show that Zener model candescribe the static and dynamic moduli of GR-S and Hevea perfectly over the entiretime/frequency range. The parallel Maxwell model well presents the static relaxationand dynamic storage moduli of polyisobutylene.The creep curves of high-density polyethylene at different stress levels and aging time were fitted using generalized fractional Maxwell model. The curves of creepcompliance at different elapsed time and stress levels are constructed using thetime-aging time superposition principle. The results show that factional Maxwellmodel can describe the creep compliance of high-density polyethylene perfectly.When the time was scaled by the relaxation time, the creep curve at the same stresslevels and different aging time superpose naturally. It is to say that the time-agingtime superposition is naturally satisfied.