Linear viscoelastic models dynamic moduli

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

In Chapter 4, we studied the fundamental importance of the relaxation modulus G t) in linear viscoelasticity. Here, we shall show how the theoretical form of G t) in the Doi-Edwards model is derived in terms of molecular structural and dynamic parameters. In the Doi-Edwards theory the study of G t) includes the nonlinear region. However, we shall postpone full discussion of G t) in the nonlinear region until Chapter 12. 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

The investigations of model compositions, based on linear elastomers and various fillers, have shown that the yield stress also may be characterized by the value of the complex shear modulus measured at various frequencies. The dependence of the dynamic modulus on the filler concentration characterizes critical concentrations of the filler, above which the viscoelastic behavior of composition drastically changes. Dynamic modulus corresponding to the yield stress does not depend on the matrix viscosity or its nature. This fact indicates a predominant role of the structural frame for rheological properties of filled polymers. 

Therefore, with the exception of the Giesekus model, the parameters for all of these constitutive equations can be deduced from the relaxation time spectrum of the material which can be obtained from the small strain linear viscoelasticity measurements alone. There are various numerical methods in the literature which allow the determination of this spectrum from measured viscoelastic master curves, such as dynamic modulus, relaxation modulus, and creep compliance. 

Within the scope of this paper the influence of the strain rate and temperature on mechanical properties like the modulus of semi-crystalline materials in both static and dynamic load situations are investigated with regard to modelling the non-linear viscoelastic deformation behavior of polymers. 

The described models are mostly calibrated according to static or dynamic experiments. While static experiments result in common stress-strain diagrams for mostly isothermal load situations, dynamic experiments allow the determination of the temperature dependent modulus. These values are usually measured using a dynamic mechanical analysis (DMA), which operates in the range of linear viscoelasticity at low stress- and strain-amplitudes. 

This square-root dependence on tw is a fundamental featme of linear chains in the Rouse model. The shear modulus at intermediate frequencies is a signature of the internal, intra-chain dynamics, which is determined by the topology of the GGS. As stressed before, the viscoelastic relaxation forms can be expressed through the relaxation spectrum H r), see Eq. 27. Here one finds [3] ... 

Nonlinear soil behavior can be approximated by an equivalent linear characterization of soil dynamic properties. The method makes use of the exact continuum solution of wave propagation in horizontally layered viscoelastic materials subjected to vertically propagating transient motions (e.g., Roesset 1977). It models the nonlinear variation of soil shear modulus and... 

Graessley and Stmglinski (1986) developed a theory, that predicts the linear dynamic viscoelastic properties of binary blends of monodisperse flexible homopolymers, based on the tube model, by incorporating constraint release and path fluctuations into the reptation motion (see Chapter 4). They assumed that the stress relaxation modulus of a binary blend, Gb(0, may be represented by... 

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