Relations from Nonlinear Constitutive Equations

The interrelationships for linear viscoelasticity in Sections B to F are accepted with almost the confidence given deductions from the laws of thermodynamics. Relations from nonlinear viscoelasticity theory are less well established. Many nonlinear constitutive equations have been proposed. Some predict certain relations which are in close accord with experiment and can be accepted with confidence but fail in other respects. A very thorough analysis with emphasis on viscoelastic liquids is provided by the treatise of Bird, Armstrong, and Hassager. ° 

The first manifestation of nonlinear behavior with increasing strain or strain rate is the appearance of normal stress differences in shearing deformation. For steady-state shear flow at small shear rates, several nonlinear models ° predict the relation for the primary normal stress difference given as equation 62 of Chapter 1, which, combined with equation 54, gives 

Thus the primary normal stress difference in steady flow at small y gives the same information as G at low frequencies. At higher values of the respective arguments, 

A sinusoidally varying shear strain rate with small amplitude, such that 721 = 721 cos cor, evokes a sinusoidally varying normal stress difference T — (T22 but at twice the frequency of the strain. This result, which would be expected from symmetry considerations because of the proportionality to 721, is predicted by the phenomenological models previously quoted.5 -54 jj,g oscillatory stress difference is superposed on a constant stress and both are proportional to 721 if 721 is small. The coefficient is now defined as the ratio (ai i — ff22)/(72i) - It is the sum of a constant term and two oscillating terms  

Thus oscillatory measurements of the primary normal stress difference give the same information as oscillatory shear stress measurements. Oscillatory measurements of the secondary normal stress difference, however, would provide additional 

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This equation is valid only in the linear region, which may be rare in biology. Equation (11.194) may be used for the evolution of all biological networks, which can be characterized by thermodynamic considerations. Equation (11.194) is valid for both linear and nonlinear constitutive relations, and can be used for quasi-equilibrium and far-from-equilibrium regions of the thermodynamic branch. 

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

Analytical redundancy relations (ARRs) Are mathematical equations that relate known system inputs, known parameters and quantities obtained by measurements from a real system. Their evaluation results in so-called ARR residuals that are identical to zero or close to zero in narrow limits as long as the system is healthy. Residuals that deviate distinguishably from zero serve as fault indicators. If nonlinear constitutive element equations do not permit to eliminate unknown variables in a candidate for an ARR in closed symbolic form then residuals are given implicitly and can be determined by numerically solving a set of equations. As inputs into ARRs may be time derivatives of measured quantities, measurement noise is to be filtered appropriately. The differentiation is carried out in discrete time. 

If the deformations are not kept small, but are carried to the point where the elastic behavior is nonlinear, equations 38 and 39 do not hold. For soft polymeric solids, deviations from linearity appear sooner i.e., at smaller strains) in extension than in shear, because of the geometrical effects of Hnite deformations. At substantial deformations, the relations between creep and recovery are much more complicated than those given above, and require formulation by nonlinear constitutive relationships. 

The linear viscoelastic phenomena described in the preceding chapter are all interrelated. From a single quite simple constitutive equation, equation 7 of Chapter 1, it is possible to derive exact relations for calculating any one of the viscoelastic functions in shear from any other provided the latter is known over a sufficiently wide range of time or frequency. The relations for other types of linear deformation (bulk, simple extension, etc.) are analogous. Procedures for such calculations are summarized in this chapter, together with a few remarks about relations among nonlinear phenomena. 

This relation can be derived from several phenomenological models and also from molecular theory. According to the constitutive equation of Bird and Carreau, it holds for finite 7 even though equations 67 and 74 are limited to small 7. With large deformations or large strain rates, other nonlinear phenomena will... 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

Transient Response Creep. The creep behavior of the polymeric fluid in the nonlinear viscoelastic regime has some different features from what were foimd with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation behavior (or vice versa). For most cases, the material properties that appear in the constitutive equations are written in terms of the relaxation response. We discuss this subsequently in the context of the K-BKZ model. 

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

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