Viscoelastic Maxwell Model

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. 

One feature of the Maxwell model is that it allows the complete relaxation of any applied strain, i.e. we do not observe any energy stored in the sample, and all the energy stored in the springs is dissipated in flow. Such a material is termed a viscoelastic fluid or viscoelastic liquid. However, it is feasible for a material to show an apparent yield stress at low shear rates or stresses (Section 6.2). We can think of this as an elastic response at low stresses or strains regardless of the application time (over all practical timescales). We can only obtain such a response by removing one of the dashpots from the viscoelastic model in Figure 4.8. When a... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Figure 4. The imaginary part of the calculated viscosity is plotted as a function of the Fourier frequency at the triple point (solid line). Also shown is the prediction of the Maxwell viscoelastic model (dashed line), given by Eq. (239). The viscosity is scaled by a2/ /(mkBT) and the frequency is scaled by x l, where xsc = [ma2/kBT l/2. For more details see the text.

Together with Eq. 3.3-17, Eq. 3.3-16 is the White-Metzner constitutive equation, which has been used frequently as a nonlinear viscoelastic model. Of course, for small deformations, X(i) = dx/dt, and the single Maxwell fluid equation (Eq. 3.3-9) is obtained. 

Viscoelasticity Models For characterization with viscoelasticity models, simulation models have been developed on the basis of Kelvin, Maxwell, and Voigt elements. These elements come from continuum mechanics and can be used to describe compression. 

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 ... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

David and Augsburger (63) studied the decay of compressional forces for a variety of excipients, compressed with flat-faced punches on a Stokes rotary press. They found that initial compressive force could be subject to a fairly rapid decay and that this rate was dependent on the deformation behavior of the excipient for the materials studied, they found that maximum loss in compression force was for compressible starch and MCC, which was followed by compressible sugar and DCP. This was attributed to differences in the extent of plastic flow. The decay curves were analyzed using the Maxwell model of viscoelastic behavior. Maxwell model implies first order decay of compression force. 

The material properties (moduli and relaxation times) are then calculated from knowledge of network structure and free volume. Strains imparted by processing and reaction are determined. These inputs are then applied to a viscoelastic bi-Maxwell model, whereby stress in the polymer is determined. Time is then incremented and the procedure repeated until the cure profile is complete. 

In 1874, Boltzmann formulated the theory of viscoelasticity, giving the foundation to the modem rheology. The concept of the relaxation spectmm was introduced by Thompson in 1888. The spring-and-dashpot analogy of the viscoelastic behavior (Maxwell and Voigt models) appeared in 1906. The statistical approach to polymer problems was introduced by Kuhn [1930]. 

The Kelvin (or Voigt) model therefore gives an acceptable first approximation to creep and recovery behavior but does not predict relaxation. By comparison, the previous model (Maxwell model) could account for relaxation but was poor in relation to creep and recovery. It is evident therefore that a better simulation of viscoelastic materials may be achieved by combining tbe two models. 

Figure 6.5 Maxwell viscoelastic model and its prediction of stress relaxation in a melt after the cessation of steady shear flow. t<, is the relaxation time.

FIGURE 20.2 Maxwell and Kelvin-Voigt viscoelastic model. 

Real materials exhibit a much more complex behavior compared to these simplified linear viscoelastic models. One way of simulating increased complexity is by combining several models. If, for instance, one combines in series a Maxwell and a Voigt model, a new body is created, called the Burger model (Figure 4-15). 

For a typical solid that exhibits linear viscoelasticity, the Maxwell model applies as... 

Maxwell model (Maxwell element) n. A concept useful in modeling the deformation behavior of viscoelastic materials. It consists of an elastic spring in series with a viscous dashpot. When the ends are pulled apart with a definite force, the spring deflects instantaneously to its stretched position then motion is steady as the dashpot opens. A simple combination of these two types provides a fair analogic representation of real viscoelastic behavior under stress. 

Voigt element n. This is a Voight model which is a component, together with other Voight or Maxwell components, of a more complex viscoelastic model system, such as the standard linear solid. 

Fig. 19 Mechanical-viscoelastic model of Lin and Chen (1999) with two Maxwell models to describe SME in segmented PUs. (a) General model, (b) Change of the model in the shape-memory cycle, (c) Shape-memory behavior for two PU samples. Solid lines indicate the recoverable ration curves of the model. Taken from ref. [36], Copyright 1999. Reprinted with permission of John WUey Sons, Inc.

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). 

The two basic viscoelastic models include the Kelvin-Voigt (K-V) and the Maxwell elements. The K-V element behaves as a solid when sheared, since the deformed material regains its initial state after the applied stress is relaxed. The components of equivalent shear modulus and equivalent viscosity, respectively, are... 

FIGURE 2. Two-element models for linear viscoelasticity (a) Maxwell model (b) Voigt (Kelvin) model. 

Fig. 3 Common viscoelastic models (a) Maxwell, (b) Kelvin-Voigt, (c) Wiechert, (d) Kelvin, and (e) standard linear solid...

They presented a theoretical approach to predict the behavior of silicone rubber under uniaxial stress. The model is based on the concept of the classical Maxwell treatment of viscoelasticity and stress relaxation behavior, and the Hookean spring component was replaced by an ideal elastomer component. From the test data, the substitution permits the new model estimation of the cross-link density of the silicone elastomer and allows a stress level to be predicted as a complex function of extension, cross-link density, absolute temperature, and relaxation time. Tock and co-workersh" ] found quite good agreementbetweenthe experimental behavior based on the new viscoelastic model. By using dynamic mechanical analysis (DMA), the authors would have been able to obtain similar information on the silicone elastomer. 

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