Lodge integral

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 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 ... 

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 tip of the thermocouple well which broke off, the so-called lost part , was carried by the sodium flow to the superheater inlet distributor where it lodged and from which it was later recovered. The collision points of the lost part with the internals of the loop were deduced and the damage to the loop was estimated from the corresponding damage to the surface of the lost part. It was concluded that the integrity of the loop had not been impaired. 

Vlachopoulos (1981) and Tanner (1988a) have reviewed the various efforts to relate B to rheological material functions. The simplest and most widely used is Tanner s (1970a), which foilows from Lodge s work (1964) on recoverable strain. It assumes unconstrained recovery after steady shear and applies an integral model of the BKZ type (4.4.2) with one relaxation time. The result is... 

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