Single-Mode Differential-Type Constitutive Equations

1 Single-Mode Differential-Type Constitutive Equations 

Therefore, the total strain of the spring and dashpot at any time t is the sum of that due to the spring (reversible) and that due to the dashpot (irreversible). Combining Eqs. (3.1) and (3.2) and generalizing the resulting expression to a three-dimensional form (i.e., in tensor form), we obtain 

However, Eq. (3.3) is valid only for extremely small strain rates because the spring and dashpot mechanical model is based on the premise that the Hookean material is subjected to an infinitesimally small displacement gradient. In order to overcome this limitation, Oldroyd (1950) proposed a generalization of Eq. (3.3) by replacing the partial derivative d/dt with the convected derivative b/bt (see Chapter 2), yielding 

It can easily be shown that Eq. (3.4) with the covariant components of a (see Eq. (2.104) for the definition of the covariant convected derivative) yields the following material 

It is seen that the material functions obtained from the covariant convected derivative of a are different from those obtained from the contravariant convected derivative of a. Experimental results reported to date indicate that the magnitude of N2 is much smaller than that of (say -A 2/ i 0.2-0.3). Therefore, the rheology community uses only the contravariant convected derivative of a when using Eq. (3.4), which is referred to as the upper convected Maxwell model. However, the limitations of the upper convected Maxwell model lie in that, as shown in Eq. (3.6), (1) it predicts shear-rate independent viscosity (i.e., Newtonian viscosity, t]q), (2) is proportional to over the entire range of shear rate, and (3) N2 = 0. There is experimental evidence (Baek et al. 1993 Christiansen and Miller 1971 Ginn and Metzner 1969 Olabisi and Williams 1972) that suggests Nj is negative. Also, as will be shown later in this chapter, and also in Chapter 5, in steady-state shear flow for many polymeric liquids, (1) l (k) follows Newtonian behavior at low y and then decreases as y increases above a certain critical value, and (2) increases with at low y and then increases with y (l n 2) as y increases further above a certain critical value. 

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