Maxwell-Type Differential Constitutive Equations

Many differential constitutive equations of the Maxwell type have been proposed most of them ate of the form 

Authors Constitutive Models f. u Fits to Data for Polymer Melts 

Johnson and Segalman (1977) (D T-l-T D) 0 Predicts negative shear stress in step shear. Spurious oscillations in stait-up of steady shearing. Singularities in steady extensional flows. 

White and Metzner (1963,1977) a(2D D) /2 0 Poor fits in step shears. N2 — 0. Singularities in steady extensional flows. 

Larson (1984) D-.t(t + Gl) 0 Fits data reasonably well for a variety of different types of deformation, except it predicts N2 — O 

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As our second major topic, we present the simplest equations from each of the three important classes of constitutive equations, namely the differential equations from the retarded-motion expansion, the Maxwell-type differential equations, and the integral equations. Third and finally, we summarize the more accurate constitutive equations that we feel are the most promising for simply and realistically describing viscoelastic fluids and for modeling viscoelastic flows. More complete treatments of nonlinear constitutive equations are available elsewhere (Tanner, 1985 Bird et al., 1987 Larson, 1988 Joseph, 1990). Throughout this chapter, our examples are drawn from the literature on polymeric... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Finally, more accurate differential and integral constitutive equations were presented, and their successes and failures in de-st bing experimental data, were discussed. No single nonlinear constitutive equation is best for all purposes, and thus one s choice of an appropriate constitutive equation must be guided by the problem at himd, the accuracy with which one wishes to solve the problem, and the effort one is willing to expend to solve it. Generally differential models of the Maxwell type are easier to implement numerically, and some are available in fluid mechanics codes. Also, some cmistitutive equations are better founded in molecular theory, as discussed in Chiqpter 11. 

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