Material Functions for Oscillatory Shear Flow

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. 

When a small-amplitude oscillatory (sinusoidal) shear strain is imposed on a linear viscoelastic fluid, we expect to observe an oscillatory response in shear stress, which can be represented by 

Let us consider the upper convected Maxwell model given by Eq. (3.4). Since we are only interested in small-amplitude oscillations with Uj = Vi(t,x2), all nonlinear terms appearing in the convected derivative of stress tensor a (see Eq. (2.107)) can be neglected and thus Eq. (3.4) reduces to the classical Maxwell equation, Eq. (3.3). Applying Eq. (3.79) to (3.3) we obtain  

G o)) is called the dynamic storage modulus and G ((o) is called the dynamic loss modulus. Comparison of Eq. (3.83) with Eq. (3.86) establishes the following 

Small-amplitude oscillatory analysis can readily be applied to any nonlinear constitutive equation. For instance, applying Eq. (3.79) to the Oldroyd three-constant model, Eq. (3.21), we obtain 

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