Viscoelasticity Maxwell equations

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

Willhite, 1998). The polymer viscoelastic properties must be measured in an oscillation flow meter and included in the Maxwell equation. 

Equation (4.11) is called the Maxwell equation and r, as given by Eq. (4.12), is the relaxation time of the viscoelastic system. The relaxation time plays an important role in determining whether the system behaves more as a Newtonian fluid or as a Hookean solid. If r is very large and... 

In summary, if G t), which is contained in Eqs. (4.30), (4.34)-(4.37), (4.49)-(4.51), (4.63) and (4.73), is known, all the linear viscoelastic quantities can be calculated. In other words, all the various viscoelastic properties of the polymer are related to each other through the relaxation modulus G t). This result is of course the consequence of the generalized Maxwell equation or equivalently Boltzmann s superposition principle. The experimental results of linear viscoelastic properties of various polymers support the phenomenological principle. Some viscoelastic properties play more important roles than the others in certain rheological processes related to... 

When the solid feature dominates the mechanical response of a shear deformation, the shear stress cr is proportional to the shear strain y, and the proportionality coefficient is the shear modulus E. On the other hand, when the liquid feature dominates the response, the shear stress cr is proportional to the shear rate y, the proportionality coefficient is the shear viscosity 77. Maxwell equation of linear viscoelasticity can be applied to describe the continuous switching between the solid and the liquid (Maxwell 1867),... 

To study the kinetics of temperature stresses in polymers, to analyse the influence of various factors on the flow of the examined processes, and to model the relaxation behaviour in polymers, a nonlinear constitutive differential equation is used in the paper. This equation was proposed by G.I. Gurevich [1], who called it the nonlinear generalized Maxwell equation out of respect for J. Maxwell s ideas [2] that served as a partial basis for deducing the equation. Total deformation is regarded as the sum of elastic, viscoelastic and temperature deformations ... 

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. 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

In Chapter 1 it was pointed out that the Maxwell fluid is a very simple model of the first order effects observed with viscoelastic liquids. The equation of a Maxwell fluid is... 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

In the Maxwell model for viscoelastic deformation, it is assumed that the total strain is equal to the elastic strain plus the viscous strain. This is expressed in the two following differential equations from Equations 14.2 and 14.3. 

Using a Maxwell model as a constitutive equation for a viscoelastic fluid, one can show that the instantaneous shear stress at the wall is smaller in the viscoelastic fluid than in the corresponding Newtonian fluid. 

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. 

A simple equation that shows much of viscoelastic behaviour is due to Maxwell (Figure C4-5). You see that it is the same as the Newtonian equation, but with one additional stress term containing the relaxation time. We can see two extreme cases ... 

Constitutive equations of the Maxwell-Wiechert tjq)e have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of pol3maer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple spring-dashpot mechanical analogy leading to the linear equation ... 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts. 

M. Renardy, A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell-type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 181-191. 

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 ... 

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]. 

Three equations are basic to viscoelasticity (1) Newton s law of viscosity, a = ijy, (2) Hooke s law of elasticity. Equation 1.15, and (3) Newton s second law of motion, F = ma, where m is the mass and a is the acceleration. One can combine the three equations to obtain a basic differential equation. In linear viscoelasticity, the conditions are such that the contributions of the viscous, elastic, and the inertial elements are additive. The Maxwell model is ... 

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