Equation constitutive

A constitutive equation is a relation between the extra stress (t) and the rate of deformation that a fluid experiences as it flows. Therefore, theoretically, the constitutive equation of a fluid characterises its macroscopic deformation behaviour under different flow conditions. It is reasonable to assume that the macroscopic behaviour of a fluid mainly depends on its microscopic structure. However, it is extremely difficult, if not impossible, to establish exact quantitative 

Depending on their constitutive behaviour, polymeric liquids are classified as  

For purely viscous fluids, the rheological constitutive equation that relates the stresses x to the velocity gradients is the generalized Newtonian model [5,6,21] and is written as 

In the above equations, t]o is the zero shear rate viscosity, t)oo is the infinite shear rate viscosity, Xc is a time constant, and n is again the power-law index. The magnitude y of the rate-of-strain tensor is given by 

The effect of temperature on the viscosity is of primordial importance in polymer processing, where tight control of temperatures is required for a successful 

Another expression for the temperature-dependence of the viscosity is the Arrhenius law [21]  

Non-Newtonian fluids (polymer solutions and melts) are rheologically complex materials, which exhibit both viscous and elastic effects, and are therefore called viscoelastic [6]. Regarding viscoelasticity, a plethora of constitutive equations exist with varying degrees of success and popularity. Standard textbooks on the 

In order to develop the theory, it is necessary to set up constitutive equations for the quantities F, tj(, and (Leslie )- We assume that these quantities are single-valued functions of 

We now invoke the fundamental principle of classical physics that material properties are indifferent to the frame of reference or the observer. Hence the constitutive equations should be invariant under proper orthogonal transformations. It is seen that rif and do not transform as tensors. The parameters (3.1.12) must therefore be replaced by 

In view of the constitutive assumptions, it is clear that h j, Nf, IV, and /, can be varied arbitrarily and independently of all other quantities and hence their coefficients must vanish, i.e.. 

Let us write the stress and the intrinsic director body force as 

Since rfy and Ni can be chosen arbitrarily and independently of the static parts t and gl 

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

The laws of mechanics alone are not sufficient to determine the relationships between forces and motions in a body. It is necessary to know the evolution with time of functions that depend on the nature of the material, that are in 

The causality indifference principle states that the physical behavior of a material at a time t is independent of all future events. Thus, the stress of material at a point P at time t depends only on motions at times 0 /. 

The local action principle establishes that the behavior of a particular element of a material is determined by the motion properties of that element and is independent of the behavior of any other element. The causality indifference principle together with the local action principle lead to the principle of determinism, which states that the stress of a given element of a material at time t depends only on the deformation of that element at times 0 t. 

The frame indifference principle states that constitutive equations must exhibit coordinate indifference, that is, the properties of a material must be independent of the reference frame. 

A review of the basic definitions of stress and strain was given in Chapter 2. It was noted that a linear elastic solid in uniaxial tension or pure shear obeys Hooke s laws given by, 

For polymers, the torsion test is often the test of choice because, as discussed in Chapter 2, the time dependent (viscoelastic) behavior of polymers is principally due to the deviatoric (shear or shape change) stress components rather than the dilatoric (volume change) stress components. Typically, constant strain rate tests are often used for either tension, compression or torsion as discussed in Chapter 3. If the material is linear elastic, the stress rate is proportional to the strain rate as the modulus is time independent. That is. 

On the other hand, if the modulus is time dependent a term must be added for the time derivative of the modulus. In fact, in Chapter 3 it was found that the differential equation for the elementary Maxwell model (where p, is viscosity) was given by 

Elementary creep and relaxation tests as a means to experimentally characterize polymers were discussed in Chapter 3. Further, elementary mechanical models and the related differential equations were discussed as 

The most fundamental approach to calculating the stresses arising in a given deformation is to use a molecular dynamics model based on the principles governing the behavior of individual polymer molecules. Such a model generally consists of the following components  

A rule for computing the stresses from the segmental orientation distribution function. 

Suen et al. [5] have reviewed recent developments in this field, and Kroger [6] has summarized in some detail work on micro- and mesoscopic models for the nonlinear rheological behavior of complex fluids, in particular those that cannot be solved by analytical methods. The calculation of the stresses arising in flow fields using molecular dynamics is a very intensive operation from a computational point of view. 

The computational requirements of a molecular dynamics model can be greatly reduced if we make use of averaging to produce a mean-field model such as those based on the concept of a molecule in a tube. Here, instead of starting from a detailed picture of the interactions between individual molecules, we focus attention on a single molecule, a test chain, and represent the effect of all the surrounding molecules by an average field of constraints. Such models can be used to calculate the response to homogeneous deformations such as step shear and steady-simple shear. 

For the simulation of more complex flows, one needs a constitutive equation or a rheological equation of state. Nearly all of the many equations that have been proposed over the past fifty years are basically empirical in nature, and only in the last twenty-five years have such models been developed on the basis of mean field molecular theories, e.g., tube models. Although the early models were often developed with a molecular viewpoint in mind, it is best to think of them as continuum models or semi-empirical models. The relaxation mechanisms invoked were crude, involving concepts such as network rupture or anisotropic friction without the molecular detail required to predict a priori the dependence of viscoelastic behavior on molecular structure. While these lack a firm molecular basis and thus do not have universal validity or predictive capability, they have been useful in the interpretation of experimental data. In more recent times, constitutive equations have been derived from mean field models of molecular behavior, and these are described in Chapter 11. We describe in this section a few constitutive equations that have proven useful in one or another way. More complete treatments of this subject are given by Larson [7] and by Bird et al. [8]. 

To proceed further we have to make more specific assumptions about the dynamic contributions Uj and Uj to the stress and couple stress tensors. This means that some relations between these stresses and the motion of the material will have 

In a rigid body motion the dynamic variables Wi and Vij are not necessarily zero and therefore we require combinations of these quantities that do vanish in such types of motion. In a rigid body motion [24, p.81] 

Material frame-indifference of the viscous stress iij means that we require it to be a hemitropic function of the above named variables, that is. 

The experiments of Miesowicz [201] and Zwetkoff [288] suggest that Uj has a linear dependence upon N and A, and accepting such a linear dependence we can write 

In order to progress in the derivation of iij the inequality (4.61) can now be developed further by employing the notation introduced in Section 4.2.1 and introducing some terminology which will be valuable later. We can write, using the expression (4.74) and noting that CijkAkj = 0 because A is symmetric, 

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Leslie F M 1968 Some constitutive equations for liquid crystals Arch. Ration. Mech. Analysis 28 265-83... 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Model (material) parameters used in viscoelastic constitutive equations... 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

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... 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

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. 

Single-integral constitutive equations for viscoelastic fluids... 

Some of the integral or differential constitutive equations presented in this and the previous section have an exact equivalent in the other group. There are, however, equations in both groups that have no equivalent in the other category. 

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

Phan-Thien, N. and Tanner, R.T., 1977. A new constitutive equation derived from network theory, Non-Newtonian Fluid Mech. 2, 353-365. 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

Equations of continuity and motion in a flow model are intrinsically connected and their solution should be described simultaneously. Solution of the energy and viscoelastic constitutive equations can be considered independently. 

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. 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

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 the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

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... 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

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