Constitutive equation Integral models

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 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)... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

Additional complexity can he brought to the constitutive equation in its integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in a large class of constitutive equations. In a perfect body, the strain energy W may be linked to strain and stress by ... 

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

In axisymmetric flow situations, the global pressure drop in a capillary rheometer is well described by the three constitutive equations. If one focuses on the entrance pressure drop, the numerical entrance pressure drop related to Bagley correction is foimd to be less important than the corresponding experimental data for the differential models for LDPE and LLDPE melts. For the Wagner integral constitutive equation, the computed entrance pressure drops are found to be lower for both fluids, but the computed values are closer to the experimental data for LLDPE than those related to the LDPE melt. This descrepancy, previously reported in the literature, needs further investigation. 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

Similar constitutive equations are used to approximate the integrals representing the interfacial heat transfer rates by convection and conduction through the stagnant films in the vicinity of a catalytic solid surface. Hence, the film model can be used to approximate the interfacial heat transport (3.167) by ... 

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. 

However expansions of the type of Eq. (20.4) would not be able to explain such phenomena as stress relaxation ( 10) or the overshoot phenomenon in stress growth ( 11). It might prove fruitful to explore alternate methods of obtaining constitutive equations which would not be of the form of Eq. (20.4), i.e. stress = function of y and its derivatives. After all, time derivatives of stress could appear [as in Eq. (4.23)], or stress could be given as an integral over the strain history. This question of the connection between dumbbell models and continuum mechanics has been studied extensively in a series of papers by Giesekus (3/). 

Just as Eqs. (6.58) and (6.59) were obtained in Chapter 6, the integral forms of Eq. (7.53) can be obtained. Then, with Eq. (7.49), the integral form of the constitutive equation for the Rouse chain model is given by (with u replaced by p)... 

Modeling of the coupled transfer phenomena (mass, heat, and momentiun) need to be solved considering appropriate constitutive equations for the evolving properties in order to optimize the integrity of the final material system and at the same time to control the final shape of produced composite element. 

Once the single-step data are known, then integration of equation 56 (or eq. 53) for different strain histories leads to predictions for the material response in any deformation history of interest. A very powerful method of evaluating constitutive equations is the double-step strain history in which the second-step response should be able to be predicted by the model in question. Figure 33 shows the set of two-step histories that is dealt with here and subsequently. Figures 34,... 

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. 

---