Viscoelastic extra stress

Figure 3.1 Subdivided element for the discretization of viscoelastic extra-stress...

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

In this paragraph, a Lesaint-Raviart method is presented. A Newton algorithm allowing fixed values of the viscoelastic extra-stress components outside the finite elements is used. A fixed-point algorithm on those exter extra-stress components is also involved. Tffis quasi-Newton method needs a storage requirement of the same size as that related to a classical decoupled method, but allows improved convergence [39]. 

Two equations have been selected for the viscoelastic extra-stress component a generalized Oldroyd-B model (GOB) and a multimode Phan-Thien Tanner model (mPTT). The veilues of the corresponding parameters are given in sub-section 3-2... 

The changes in the viscoelastic extra-stress components along the symmetry axis are markedly different (Fig. 22) Ty is much more important at the contraction for the GOB model on the contrary, Tvyy is more pronounced for the PTT model at the die exit, Ty x is equivalent for the two constitutive equations, but relaxes more quickly for the mPTT model. Tyyy is much more important for the GOB model, which is again consistent with the more important value of the swelling observed in Fig. 20a. Obviously, Ty y remains equal to zero along the syirunetry axis for the two constitutive equations. 

Figirre 22. Viscoelastic extra-stress components (a Tyxx, b Tvyy synunetry axis., . ... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

This difficulty can be overcome by the use of a viscoelastic model limiting the effect of the singularity in the transport equations. In the Modified Upper Convected Maxwell (MUCM) proposed by Apelian et al. (see [1]), the relaxation time X is a function of the trace of the deviatoric part of the extra stress tensor ... 

Normal stress is an extra stress which is developed in viscoelastic material under shear, in directions normal to the plane of shear. The normal stress is expressed by the first and second normal stress coefficients (Bird et al, 1987)... 

Equations (1.1) and (1.2) are not closed because of the presence of the extra stress tensor x. Therefore, one more equation is required, which is provided by a viscoelastic constitutive model [35, 49, 79). According to nonequilibritun thermodynamics, the most thermodynamically consistent way to describe the constitutive model is in terms of internal (structural) variables for which separate evolution equations are to be described [49]. The simplest case is when a single, second-order... 

In the absence of magnetic field, the theory exhibits viscoelastic transversally anisotropic behavior with symmetric stress tensor and orientation of director caused only by flow. Thus, this simplified approach has led to a closed set of two coupled anisotropic viscoelastic equations of quasilinear type for evolution of director and extra stress the anisotropic properties in the set being described by viscoelastic evolution equation for director. Although this theory has been developed for low enough value of Deborah number, it is still possible to compare the simulations with experimental data. 

The viscoelastic component of the extra-stress tensor can be computed with different differential viscoelastic material models. One model employed in this study is the Maxwell-model, which is defined as follows ... 

In this paper we present a constitutive relation for predicting the rheology of short glass fibers suspended in a polymeric matrix. The performance of the model is assessed through its ability to predict the steady-state and transient shear rheology as well as qualitatively predict the fiber orientation distribution of a short glass fiber (0.5 mm, L/D < 30) filled polypropylene. In this approach the total extra stress is equal to the sum of the contributions from the fibers (a special form of the Doi theory), the polymer and the rod-polymer interaction (multi-mode viscoelastic constitutive relation). 

In this approach the contribution to the extra stress of the matrix and the rod-matrix interaction is captured using a multi-mode viscoelastic constitutive relation. For the model predictions in the paper, we chose to use the Phan-Thien Tanner equation (PTT) [13]. The ability to indirectly capture the rod-matrix interaction is based on the presence of the fiber retarding the long relaxation... 

The dynamic mechanical thermal analyzer (DMTA) is an important tool for studying the structure-property relationships in polymer nanocomposites. DMTA essentially probes the relaxations in polymers, thereby providing a method to understand the mechanical behavior and the molecular structure of these materials under various conditions of stress and temperature. The dynamics of polymer chain relaxation or molecular mobility of polymer main chains and side chains is one of the factors that determine the viscoelastic properties of polymeric macromolecules. The temperature dependence of molecular mobility is characterized by different transitions in which a certain mode of chain motion occurs. A reduction of the tan 8 peak height, a shift of the peak position to higher temperatures, an extra hump or peak in the tan 8 curve above the glass transition temperature (Tg), and a relatively high value of the storage modulus often are reported in support of the dispersion process of the layered silicate. 

---