Lodge equation

Lodge and Meissner have recently examined in detail the stresses during start-up of steady elongational and shear flows (374). The Lodge equation [Eq.(6.15)], with its flow-independent memory function, described the build-up of stress rather well (even for rapid deformations) until a critical strain was reached. Beyond the critical strain (which differed somewhat in shear and... 

A thorough solution of this equation, making use of the Lodge equation, was again presented by Janeschitz-Kriegl (General references, 1983, Appendix A3), with as the result... 

The Lodge equation can also be obtained in a differential form known as the Upper Convected Maxwell equation (UCM) ... 

Lodge was the first to formulate such an equation by a combination of Eq. (7.74) with the Boltzmann superposition principle as expressed by Eq. (5.111). Explicitly, the Lodge equation of state of rubber-like liquids , when written for homogeneous deformations, has the following form... 

Relaxation After a Step Strain for the Lodge Equation... 

Growth of the shear viscosity /+ after onset of steady shearing for the UCM/Lodge equation, compared to a Newtonian and a second-order fluid. 

Here r is given by the UCM equation 4.3.7 (or equivalently by the Lodge equation, eq.4.3.18), and is usually just a Newtonian term 2r7jD, where rj, is the solvent viscosity. The combination of these two terms is the Oldroyd-B constitutive equation (Oldroyd, 1950 see Exercise 4.6.4). Figure 4.3.4 compares the storage mod-... 

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. 

Relaxation After a Step Strain for the Lodge Equation Calculate the relaxation of the shear stress and the first normal stress... 

Validity of the linear stress-optical rule points at the dominant role of the network forces in pol3mier melts. The Lodge equation of state can be interpreted on this basis. We introduced the equation empirically, as an ap>-propriate combination of properties of rubbers with those of viscous liquids. It is possible to associate the equation with a microscopic model. Since the entanglement network, although temporary in its microscopic structure, leads under steady state conditions to stationary viscoelastic properties, we have to assume a continuous destruction and creation of stress-bearing chain sequences. This implies that at any time the network will consist of sequences of different ages. As long as a sequence exists, it can follow all imposed deformations. 

Here the contributions of all active chain sequences with different ages are superimposed. To show the agreement, we rewrite the Lodge equation... 

This relation, which Venerus et al. [22] call the Osaki-Lodge equation, has been found to be valid for entangled polystyrene solutions [22,42,43]. Another quantity of interest is the normal stress ratio defined as ... 

The Lodge equation satisfactorily passes the objectivity test using the Finger tensor U as strain measure [9] ... 

Unfortunately, the Lodge equation does not predict shear-thinning phenomena or nonzero normal stress coefficients. 

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