Viscoelasticity, linear four-parameter model

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

An example for polytetrafluoroethylene is shown in Fig. 16-15. The experimental points are rather well matched by curves calculated from the simple mechanical model of Fig. 16-16 with suitable values of the four parameters and an added constant background loss of 0.1 X 10" cm /dyne for J". This model differs from those usually associated with linear viscoelasticity by the inclusion of the inertial unit w//, with dimensions of mass per unit length. The values of Co and G are determined by the limiting levels of J on the two sides of the dispersion, m/l by the resonance frequency and r) by the breadth of the dispersion. The magnitude of t] is reasonable, 10 poises. However, that of m/l is too large to be identifiable physically as a real mass per unit length. This result corresponds to the elementary calculation that to produce an audio-frequency resonance with an elastic member of the shape and modulus of the sample, an inertial member with mass considerably greater than that of the sample would be required. 

Spring and damper elements can be combined in a variety of arrangements to produce a simulated viscoelastic response. Early models due to Maxwell and Kelvin combine a linear spring in series or in parallel with a Newtonian damper as shown in Fig. 3.21. Other basic arrangements include the three-parameter solid and the four-parameter fluid as shown in Fig. 3.22. 

FIGURE 15.1 Linear viscoelastic models (a) linear elastic (b) linear viscous (c) Maxwell element (d) Voigt-Kelvin element (e) three-parameter (f) four-parameter. 

For each mode of the model, there are four fitting parameters Ty, T j, q, and Gj. The two parameter sets and G control the linear viscoelastic properties of the melt e.g., G and G"), and all the and G values can therefore be obtained empirically by fitting the linear viscoelastic data for a given melt. The remaining two parameter sets and [q ] are nonlinear... 

In another research, the thermo-mechanical behavior of SMPs was described by both linear and nonlinear viscoelastic theories [4]. In this woik four element mechanical units consisting of spring, dashpot and frictional device were used to derive a constitutive differential equation. In order to determine the material properties by a constitutive differential equation the modulus, viscosity and other parameters were assumed to decay exponentially with temperature. Liu et al. [5] developed a constitutive equation for SMPs based on thermodynamic concepts of entropy and internal energy. They adopted the concept of frozen strain and demonstrated the utility of the model by simulating the stress and strain... 

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