Generalized linear Maxwell model

Christenson and McKinley [ 19] evaluated a generalized linear Maxwell model as well as the upper convected Maxwell model and the Giesekus model. These authors worked with the tensorial forms of these functions which are capable of correctly treating large strain deformations. 

For the generalized linear Maxwell model (GLM) and the upper convected model (UCM), the only material parameters needed are contained in the relaxation time spectrum of the material which can be obtained from simple linear viscoelastic measurements. For the Giesekus model, one needs in addition the mobility factors which Christensen and McKinley obtained by fitting the stress-strain curves of the adhesive. The advantage of the Giesekus model was that it provided them with a better description of the stress-strain curves. This, of course, is to be expected since those curves were used to deduce the parameters of the model. 

We have used the generalized phenomenological Maxwell model or Boltzmann s superposition principle to obtain the basic equation (Eq. (4.22) or (4.23)) for describing linear viscoelastic behavior. For the kind of polymeric liquid studied in this book, this basic equation has been well tested by experimental measurements of viscoelastic responses to different rate-of-strain histories in the linear region. There are several types of rate-of-strain functions A(t) which have often been used to evaluate the viscoelastic properties of the polymer. These different viscoelastic quantities, obtained from different kinds of measurements, are related through the relaxation modulus G t). In the following sections, we shall show how these different viscoelastic quantities are expressed in terms of G(t) by using Eq. (4.22). 

The only parameter in (11) having dimensions of time is C. Although this is not a relaxation time per se, it can be associated with a "Maxwell-type relaxation time, as follows. Although the linear Maxwell model predicts a constant (Newtonian) viscosity, it may be generalized by utilizing a co-rotational reference frame which follows the local rotation and translation of each fluid element [9]. When a term is added to account for the high shear limiting behavior, the result is the co-rotational form of the Jeffreys model ... 

There are several models to describe the viscoelastic behavior of different materials. Maxwell model, Kelvin-Voigt model, Standard Linear Solid model and Generalized Maxwell models are the most frequently applied. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

Both these models find their basis in network theories. The stress, as a response to flow, is assiimed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

In flow situations where the elastic properties play a role, viscoelastic fluid models are generally needed. Such models may be linear (e.g., Voigt, Maxwell) or nonlinear (e.g., Oldroyd). In general they are quite complex and will not be treated in this chapter. For further details, interested readers are referred to the textbooks by Bird et al. [6] and Barnes et al. [25],... 

Fig. 7. (a) General step-strain deformation history relevant to Boltzmann-type linear superposition, (b) Schematic of stress additivity of responses for a Maxwell model in a two-step strain history (see text for discussion). 

Fig. 8. (a) Stress-strain plot for a generalized Maxwell model to different strain rates, as depicted in figure. Plot shows nonlinear stress-strain behavior in spite of material model (Maxwell) following laws of linear viscoelasticity (see text), (b) Stress and strain data from different strain rates given in (a) divided by strain rate dy/dt, demonstrating that material model follows linear viscoelasticity (see text). 

Fig. 8 Linear elastic and viscous modulus functions G co, T) and G"(a>, T) of gum EPDM2504, drawn using the G of a six elements generalized Maxwell model and the respective Cl, C2 parameters of a WLF type equation with 100 °C as reference temperature experimeutal data from a frequency-temperature sweep protocol at 1 deg. strain amplitude with a closed-cavity torsional harmonic rheometer are displayed for comparison with the calculated maps...

While the exponential stress relaxation predicted by the viscoelastic analog of the Mawell element, ie., a single exponential, is qualitatively similar to the relaxation of polymeric liquids, it does not describe the detailed response of real materials. If, however, it is generalized by assembling a number of Maxwell elements in parallel, it is possible to fit the behavior of real materials to a level of accuracy limited only by the precision and time-range of the experimental data. This leads to the generalized, or multi-mode. Maxwell model for linear viscoelastic behavior, which is represented mathematically by a sum of exponentials as shown by Eq. 4.16. 

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