Lodge Integral Equation

Recall from Chapter 3 that the linear Maxwell equation can be written in an integral form. The nonlinear upper-convected Maxwell equation can also be written in an integral form, namely 

Here y (r, t ) is the shear strain that accumulated between the times t and t. Let us illustrate the meaning of y (r, t ) with two examples. 

If t 0, then since the step occurred at time zero, no strain was added to the material between r andr and y(t, t) = 0. However, if t 0, then between t and t the strain yb was added to the material and y (t, t ) = yo. Thus, for a step shear strain at time zero 

If r 0, then the shearing has been continuous at a rate y from time t until the present time t, so the total strain accumulated between t and the present time f is y(f — t ). However, if t 0, then since no shearing occurred from time t up to time zero, the total accumulated strain between r and f is that which accumulated since time zero, namely yt. Thus 

We leave it as an exercise to the reader to calculate the shear stress at time t for the two strain histories given by eq.s 4.3.20 and 4.3.21. As one more example, let us consider an extensional flow. 

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It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

The connection between the double value of the slip parameter obtained from the viscometric functions and the violation of the Lodge-Meissner rule becomes more evident when the time-strain separability of the model is considered. For this purpose, the Johnson-Segalman model can be rewritten under the form of a single integral equation, cancelling the Cauchy term, which gives the following form in simple shear flows ... 

The K-BKZ and other integral constitutive equations discussed above can be regarded as generalizations of the Lodge integral, eq4.3.18. The upper-convected Maxwell (UCM) equation, which is the differential equivalent of the Lodge equation, can also be generalized to make possible more realistic predictions of nonlinear phenomena. 

Additional complexity can he brought to the constitutive equation in its integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in a large class of constitutive equations. In a perfect body, the strain energy W may be linked to strain and stress by ... 

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