I Nonlinear Viscoelasticity

In Chapter I we saw that nonlinear normal stresses can arise in simple shear (eq 1.5.15) and in torsion (Example 1.7.1) of an elastic solid. In this chapter our immediate goal is to develop constitutive equations that can predict normal stresses and other nonlinear phenomena in flowing viscoelastic liquids. The eventual goal of this effort is to use these equations to predict and control viscoelastic 

Venous experiments used by Weissenbeig (1947) and his co-workers to demonstrate normal stress effects in liquids. 

The breadth of the scope of nonlinear phenomena can be grasped in part by considering the various time-dependent probes of linear viscoelasticity cited in Table 3.3.2 sinusoidal oscillation, creep, constrained recoil, stress relaxation after step strain, stress relaxation after steady shearing, and stress growth after start-up of steady shearing. In the linear regime— that is, at small strains or small strain rates—the experimental results of any one of these probes (in simple shear, for example) can be used to predict results for any of the other probes, not only for simple shearing defor- 

Schematic diagram showing the behavior of viscoelastic fluids in the limits of low strain rates, low amplitude deformations, and high strain rates. Adapted from Pipkin (1972). 

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 

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