Maxwell rheological model

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

According to Maxwell s rheological model of long-term action of constant loading, the stress o0 and viscous deformations ev are characterized by the equations ... 

The paradox of Maxwell s model. A popular representation of models in rheology mimics the equivalent electrical circuits with dipolar components. The elastic component is naturally symbolized by a spring and the viscous component by a damper or dashpot (a piston filled with a viscous fluid able to circulate). The viscoelastic relaxation is thus represented with these two components mounted in series, as shown in Figure 11.12a and is known as Maxwell s model (Oswald 2005). (In this representation, the customary notation is used for facilitating comparison with the literature.)... 

FIGURE 11.12 Maxwell s rheological model (a) and equivalent circuit model (b). 

The data is taken from Zhang and Soong (1992). Two rheological models describing the d5mamic behaviour of dampers were applied in the calculations the Kelvin fractional model and the Maxwell fractional model. 

The origin of the theory of viscoelasticity may be traced to various isolated researchers in the last decades of the nineteenth Century. This early stage of development is essentially due to the work of Maxwell, Kelvin and Voigt who independently studied the one dimensional response of such materials. The linear constitutive relationships introduced therein are the base of rheological models which are still used in many applications [121]. Their works led to Boltzmann s [122] first formulation of three dimensional theory for the isotropic medium, which... 

Another way to introduce fractional derivatives is through rheological models of fractional order. In particular, the fractional Maxwell element corresponds to a spring in series with a fractional damper. The one-dimensional linear stress, <7, versus strain, e, relation of a spring in parallel with the fractional Maxwell element can expressed in terms of fractional derivatives [171], e.g.,... 

There have been numerous studies on the film-blowing process. Since the initial thin-shell approximation proposed by Pearson and Petrie [125, 126] with the Newtonian model assumed for deformation, various rheological models have been incorporated in simulations, such as the power-law model [127,128], a crystallization model [129], the Maxwell model [130-133], the Leonov model [133], a viscoplasti-c-elastic model [134], the K-BKZ/PSM model [135-137], and a nonisothermal viscosity model [138]. A complete set of experimental data was reported by Gupta [139] for the Styron 666 polystyrene and by Tas [140] for three different grades of LDPE. 

Earlier in the theory of viscoelasticity many rheological models with combinations of the Maxwell and the Voigt - Kelvin bodies were considered (see Freudental and Geiringer [1]). These models have constitutive laws for stresses o j and strains eij which include time derivatives of arbitrary order. 

Physics-Based Ground-Motion Simulation, Fig. 5 Examples of rheological models used to incorporate the effect of attenuation, (a) Generalized Maxwell model (Emmerich and Kom 1987), (b) generalized... 

On the other hand, the Maxwell fluid model explains the response of complex fluids to an oscillatory shear rate. The frequency-dependent behavior of this model, displayed into linear responses to applied shear rates has been found to be applicable to a variety of complex fluid systems. Although the linear viscoelasticity is useful for understanding the relationship between the microstructure and the rheological properties of complex fluids, it is important to bear in mind that the linear viscoelasticity theory is only valid when the total deformation is quite small. Therefore, its ability to distinguish complex fluids with similar micro- and nanostructure or molecular structures (e.g. linear or branched polymer topology) is limited. However, complex fluids with similar linear viscoelastic properties may show different non-linear viscoelastic properties [31]. 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

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