Other Integral Constitutive Equations

Wagner s Eq. 10.10 is special case of a more general integral model of Rivlin and Sawyer, which is, in turn, a generalization called the K-BKZ model [7]. In the K-BKZ model, the strain is described by a linear combination of the Finger and Cauchy tensors, and as a result it is possible to accommodate a non-zero second normal stress difference.  

Another special case of the Rivlin-Sawyer model that is a generalization of Eq. 10.10 was proposed by Wagner and Demarmels [12] who added a dependency on the Cauchy tensor to Eq. 10.10 to provide for a non-zero second normal stress difference and a better fit to data for planar extension, which is defined in Section 10.9. Their model is shown as Eq. 10.11. 

The K BKZ model differs from the Rivim Sawyer model in that in the former the coefficients of the strain tensors are derivatives of a strain energy function like that describing free energy in a deformed rubber. In the Rivlin-Sawyer and tteigner models, there is no general strain energy function. 

They assumed that the parameter j8 is constant and found that a value of -0.27 fitted their planar extension data. While it is now understood that the normal stress ratio is a function of strain, it is shown in Section 10.4.5 that it does approach a specific limiting value as y 0. Based on data for one melt in step shear and start-up of steady uniaxial extension, Wagner and Demarmels proposed the following empirical relationship for the damping function with 

The first constitutive equation to be derived from a tube model was that of Doi and Edwards [1]. It can be obtained from Eq. 10.5 by first using Eq. 10.13 for the memory function  

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The K-BKZ and other integral constitutive equations discussed above can be regarded as generalizations of the Lodge integral, eq4.3.18. The upper-convected Maxwell (UCM) equation, which is the differential equivalent of the Lodge equation, can also be generalized to make possible more realistic predictions of nonlinear phenomena. 

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. 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

This identity is useful for relating integral and differential constitutive equations, as we shall see in Section 3.4.4. A thorough discussion of this and other relationships among kinematic tensors can be found in Astarita and Marrucci (1974). 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

Equation (26) is what is known as a rate-type constitutive equation, and it gives the extra stress in implicit form. Integral equations, on the other hand, are explicit in the stress, and the simplest of these can be derived based on the Boltzmann superposition principle. According to this principle, the stress in a material at any time can be obtained by adding stress tbatjndivjduallv result from earh Vio... 

There are numerous other constitutive equations of both differential and integral type for polymer melts, and some do a better job of matching data from a variety of experiments than does the PTT equation. The overall structure of the differential equations is usually of the form employed here The total stress is a sum of individual stress modes, each associated with one term in the linear viscoelastic spectrum, and there is an invariant derivative similar in structure to the one in the PTT equation, but with different quadratic nonlinearities in t and Vv. The Giesekus model, for example, which is also widely used, has the following form ... 

In applied viscoelasticity not all the constitutive equations are formulated by an a-priori defined internal energy y/, but the constitutive model is expressed directly by the functional relation between the stress and the strain through an hereditary integral. In rheology this class of constitutive models is called Rivlin-Sawyers models Fxmg s [164], Fosdick and Yu s [165] and many other models currently used belong to this constitutive class. 

This integral model has been used in numerical flow simulations for a number of flow problems more or less successfully (see Refs [28,31-34]). A recent review [35] on the subject gives a list of problems solved with this model through numerical simulation, including many flows from polymer-processing operations. Other flows solved with a number of different constitutive equations can be found in a recent book on computational rheology [36]. 

In some special cases it is possible to solve the equations of motion [Eq. (11)] entirely independently of any knowledge of the constitutive relation and to obtain a universal shear stress distribution that applies to all fluids. In other cases it is not possible to do this because the evaluation of certain integration constants requires knowledge of the specific constitutive relation. Because of space limitations, we illustrate only one case of each type here. 

Undoubtedly, this new kind of integrated approach is well representative of what should be membrane engineering, with final objectives clearly defined, the right hypothesis and choice of simple equations for modeling, a realistic representation of real complex solutions and the set-up of efficient simulation tools involving successive intra- and extrapolation steps. It appears to be easily extended to other membrane operations, in other fields of applications. It should provide stakeholders with information needed to make their decision costs, safety, product quality, environment impact, and so on of new process. Coupled with the need to check the robustness of the new plant and the quality of final output, it should constitute the right way to develop the use of membranes as essential instruments for process intensification with industrial units at work. 

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