Experiment and the Constitutive Equation

In the Table are collected data from several experiments with different starting levels, but all with the same orifice area, and all done in the same cylindrical tank. Look at the data for some clues as to the behavior of this system. Note that it takes 60 sec to drain the tank from an initial level of 10 cm. When the initial level is set to 50 cm, it takes 130 sec for the tank to drain  

The relationships that we have just described in words are much easier to see if we plot the data as level versus time and do so all on one graph. We can use a few functions to pull this all together. The data in the table has been entered as a matrix of levels at each time called totdat  

The matrix form of the data needs to be transformed into data sets having each time and each level in pairs. We can do this in one small program as follows and the new data set will be called sepdat for the separated data  

For reference the dashed line across the data is set at the 10-cm level. With this we can see that once this level is reached, then independent of the starting point, it takes 50 sec to finish the process. The fluid moving out of the vessel then has no memory of the level at which the process was initiated. What we seek now is the relationship between the rate of change in level and the level in the tank, since both the material balance and the experimental data drive in this direction. We can get to this by computing the rate of change in level as a function of time for each experiment and then plotting this for comparison. 

The approximate rate of change can be computed from the data by taking the slope between successive data points and plotting this versus the time at that second point. We can write a function to do this and then plot the data. The algorithm for implementing this procedure on the set sepdat is  

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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... 

In the example given, the constitutive equation used is a multimode Phan Tien Tanner (PTT). It requires the evaluation of both linear and nonlinear material-response parameters. The linear parameters are a sufficient number of the discrete relaxation spectrum 2, and r]i pairs, which are evaluated from small-strain dynamic experiments. The values of the two nonlinear material-response parameters are evaluated as follows. Three semiarbitrary initial values of the two nonlinear parameters are chosen and the principal normal stress difference field is calculated for each of them using the equation of motion and the multimode PTT. They are compared at each field point (i, j) to the experimentally obtained normal stress difference and used in the following cost function F... 

Figure 13.36. These numerical methods also give results that are similar to experiments. Other approaches use the continuity equation and equation of motion developed for fluid flow (see Chapter 12) with a constitutive equation for the powder mass. The constitutive equation for powder flow is a problem that has no solution at this time. Several simple formulas in terms of tensor invariants and deviation tensors [83]...

One application of the solutions (4-55)-(4-61) is to evaluate the effect of viscous dissipation in the use of a shear rheometer to measure the viscosity of a Newtonian fluid. In this experiment, we subject the fluid in a thin gap between two plane walls to a shear flow by moving one of the walls in its own plane at a known velocity and then measuring the shear stress produced at either wall (by measuring the total tangential force and dividing by the area). In the absence of viscous dissipation, the velocity profile is linear and the shear rate is simply given by the tangential velocity U divided by the gap width d. Now, the constitutive equation, (2-87), for an incompressible Newtonian fluid applied to this simple flow situation takes the form... 

The relationship between the rate of a chemical reaction and the species concentrations cannot be predicted and -must be determined from experiment. Detailed discussions of the analysis of experimental reaction rate data to get the constitutive equation relating the reaction rate to species concentrations are given in Kinetics and Mechanism, 3rd ed., by J. W. Moore and R. G. Pearson, John Wiley Sons, New York (1981), Chaps. 2 and 3 Chemical Reaction Engineering, 2nd ed., by O. Levenspiel, John Wiley Sons, New York (1972), Chap. 3 and Introduction to Chemical Engineering Analysis, by T. W. F. Russell and M. M. Denn, John Wiley Sons, New York (1972), Chap. 5., ... 

In the FEBEX project blocks made of Ca-bentonite are tested for their suitability in radioactive waste isolation. This material has been tested in various laboratory experiment to identify the material parameters. The modelling of gas and water flow in engineered barriers demands beneath the intrinsic permeability and porosity constitutive equations for capillary pressure and relative permeabilities. 

In adherence of solids, surface effects, rheological effects and fracture mechanics are mixed in an intricate manner, and no experiment can be performed involving only the chemical aspects. The adherence of elastic solids is now clear that of viscoelastics is in progress for other materials, the constitutive equation must be known and taken into account. 

The effects of shear flow on the PS/LLDPE morphology were investigated by observing the structure of quenched samples under the scanning electron microscope, SEM. Predictions based on the constitutive equations were compared with observations from the dynamic shear experiments at 200 °C (see Fig. 7.22). The frequency variations of Gb, Gb", and t b were found to be in good agreement with computations based on Eqs. 7.101, 7.102, and 7.103. However, to get such agreement, seven parameters (viz., Vi2, < ), initial value of the anisotropy parameter, qijo, initial size of the dispersion, and three dimensionless equation parameters) were required (Lee and Park 1994). 

The DB was determined at 10 cylindrical rock salt specimens with both methods. The results are presented in Fig. 11.42. In conclusion, the results of all experiments are within an acceptable deviation range. The continuous line in Fig. 11.42 represents the calculated dilatancy boundary due to the constitutive equations as proposed by Pusch and Alkan [2002]. 

Despite their highly successful record, MD or MC simulations are still hardly extended to the direct interpretation of complex set-ups, typical of most rheological experiments. In such cases it is preferable to employ mean-field or continuum descriptions, based of the numerical solution of the constitutive equations describing hydrodynamic properties. Such techniques were for instance applied to the prediction of transient director patterns of liquid crystalline nematic samples [11-14]. Hydrodynamic treatments are algebraically complex and computationally intensive, and their implementation is limited mostly to nematic phases. 

In mathematical terms, for elastic solids and viscous liquids, the constitutive equation, which relates the components of the stress tensor or matrix to the components of the strain tensor or the rate of strain tensor, involves only one material constant which can be determined in a single experiment. Needless to say, one picks this single test based on the ease of experi-... 

In contrast to simple elastic solids and viscous liquids, the situation with polymeric fluids is somewhat more complicated. Polymer melts (and most adhesives are composed of polymers) display elements of both Newtonian fluid behavior and elastic solid behavior, depending on the temperature and the rate at which deformation takes place. One therefore characterizes polymers as viscoelastic materials. Furthermore, if either the total strain or the rate of strain is low, the behavior may be described as one of linear or infinitesimal viscoelasticity. In such a case, the stress-deformation relationship (the constitutive equation) involves not just a single time-independent constant but a set of constants called the relaxation spectrum,(2) and this, too, may be determined from a single stress relaxation experiment, or an experiment involving small-amplitude oscillatory motion. 

Quantum mechanics is based on several statements called postulates. These postulates are assumed, not proven. It may seem difficult to understand why an entire model of electrons, atoms, and molecules is based on assumptions, but the reason is simply because the statements based on these assumptions lead to predictions about atoms and molecules that agree with our observations. Not just a few isolated observations Over decades, millions of measurements on atoms and molecules have yielded data that agree with the conclusions based on the few postulates of quantum mechanics. With agreement between theory and experiment so abundant, the unproven postulates are accepted and no longer questioned. In the following discussion of the fundamentals of quantum mechanics, some of the statements may seem unusual or even contrary. However questionable they may seem at first, realize that statements and equations based on these postulates agree with experiment and so constitute an appropriate model for the description of subatomic matter, especially electrons. 

Experimental measurements of mechanical properties are usually made by observing external forces and changes in external dimensions of a body with a certain shape—a cube, disc, rod, or fiber. The connection between forces and deformations in a specific experiment depends not only on the stress-strain relations (the constitutive equation) but also on two other relations. These are the equation of continuity, expressing conservation of mass ... 

Here p is the density, t the time, Xi the three Cartesian coordinates, and o,- the components of velocity in the respective directions of these coordinates. In equation 2, the index j may assume successively the values 1, 2, 3 gj is the component of gravitational acceleration in the j direction, and atj the appropriate component of the stress tensor (see below). (A third equation, describing the law of conservation of energy, can be omitted for a process at constant temperature the discussion in this chapter is limited to isothermal conditions.) Now, many experiments are purposely designed so that both sides of equation 1 are zero, and so that in equation 2 the inertial and gravitational forces represented by the first and last terms are negligible. In this case, the internal states of stress and strain can be calculated from observable quantities by the constitutive equation alone. For infinitesimal deformations, the appropriate relations for viscoelastic materials involve the same geometrical form factors as in the classical theory of equilibrium elasticity they are described in connection with experimental methods in Chapters 5-8 and are summarized in Appendix C. 

To supplement the above transient i.e., nonperiodic) experiments and provide information corresponding to very short times, the stress may be varied periodically, usually with a sinusoidal alternation at a frequency v in cycles/sec (Hz) or o(= 270 ) in radians/sec. A periodic experiment at frequency to is qualitatively equivalent to a transient experiment at time f = 1 /oj, as will be evident in the examples shown in Chapter 2. If the viscoelastic behavior is linear, it is found that the strain will also alternate sinusoidally but will be out of phase with the stress (Fig. 1-7). This can be shown from the constitutive equation as follows. Let... 

As pointed out in Chapter 1, the forces and displacements which are measured in a mechanical experiment are related to the states of stress and strain by the constitutive equation which describes the viscoelastic properties sought, as well as the equations of motion and continuity (equations 1 and 2 of Chapter 1). Ordinarily, there is considerable simplification because there is no change in density with time, and because gravitational forces can be neglected. In transient experiments (creep and stress relaxation), inertial forces can also be neglected by suitable restriction of the time scale, eliminating short times. In periodic (oscillatory) experiments, inertial forces may or may not play an important role depending on the frequency, sample dimensions, and mechanical consistency as described in Section D below. 

Excitation spectra arise from transitions between different quantum states of the system, corresponding to different nondegenerate solutions of the Schrodinger equation. In quantum chemistry it is common practice to treat the solution of the Schrodinger equation within the Bom-Oppenheimer approximation [1] and separate the electronic and nuclear degrees of freedom. Consequently the excitation spectra are also separated into an electronic and a roto-vibrational spectrum. The former is studied mainly in optical (UV/vis) spectroscopy experiments and will constitute the main subject of this chapter the latter, which can be investigated by infrared, microwave or Raman spectroscopy measurements, provides fine-structure corrections to the electronic spectrum. 

In spite of many efforts, attempts to model complicated properties of liquid crystalline polymers (LCPs) are far from being complete. The constitutive equations of continuum type for thermotropic LCPs were proposed only last year. Multipara-metric character of these equations is the challenging problem for LCP simulations. The chapter by Chen and Leonov ( Liquid Crystalline Polymers Theories, Experiments, and Nematodynamic Simulations of Shearing Flows ) reviews the major findings in this field, describes new continuum theory valid for thermotropic LCPs, and illustrates simulations of their shearing flows. 

Another key issue for the application of this theory is the determination of the phenomenological coefficients appearing in the constitutive equations. Of course, the ultimate determination must come from careful viscometric experiments. As these are apparently not yet available, we have had to resort to experience and separate theoretical analysis to provide estimates of these parameters for our calculations. For the viscous type coefficients, i.e., those appearing in (7) and (8), we have relied on prior experience with micropolar flows [7,9,14]. In particular, we have focused on those values which will produce plug-shaped profiles with appropriate boundary conditions. The values selected here reduce to the molecular value of the viscosity of water, under standard conditions, when the solid volume fraction goes to zero. As to the... 

Oscillatory shear flow has long been used to characterize the linear viscoelastic properties of polymer solutions and melts. In Chapter 5 we describe the basic principles of such experiments. In this section we present the material functions for small-amplitude oscillatory shear flow using the constitutive equations presented in the preceding section. 

In this chapter, we have presented the fundamentals of molecular theory for the viscoelasticity of flexible homogeneous polymers, namely the Rouse/Zimm theory for dilute polymer solutions and unentangled polymer melts, and the Doi-Edwards theory for concentrated polymer solutions and entangled polymer melts. In doing so, we have shown how the constitutive equations from each theory have been derived and then have compared theoretical prediction with experiment. The material presented in this chapter is very important for understanding how the molecular parameters of polymers are related to the rheological properties of homopolymers. 

Although the experiments described in the foregoing section are very helpful for developing our intuition about the behavior of viscoelastic fluids such as polymers, they are not suitable for obtaining and cataloging information about specific polymeric materials. For the characterization of polymers it is necessary to make careful measurements of stresses in systems where the velocity or displacement field is known within rather strict limits. These rheometric experiments provide information about one or more of the stress components as functions of shear rate, frequency, or of other controllable variables these functions are generally referred to as material functions , since they are different for each material. Once these material functions have been measured, they can be used to test various empirical or molecular expressions for the stress tensor (that is, the constitutive equation), or they can be used to establish the values of the parameters that appear in these stress-tensor expressions. 

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