Lodge-Meissner rule

It requires that the principal stress axes should coincide with the principal strain axes. This rrile has been experimentally checked hy many authors [24, 56] Actually, the use of the Gordon-Schowalter derivative involves the violation of the Lodge - Meissner rule, indeed when a equals 0 or 2, either the upper or the lower convected derivatives implies that the relationship is respected. In the general case, the double value of the slip parameter is a natural way to accommodate this rule. 

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 significance of the double value of a is clearly shown in equations (50) since respect of the Lodge-Meissner rule or of material objectivity requires the identity of equations (50a) and (50b) which can only be achieved by two different values for a ... 

At least, using the complete Phan Thien Tanner equation, with non-affine motion and modified kinetics enables a correct description of the data in shear and in elongation. However, the parameters that can be determined for this model are bound to be some compromise. This is necessary in order to minimize the deviation to the Lodge-Meissner rule, due to the use of the Gordon-Schowalter derivative. This is also required to give adequate description of both the shear and uniaxial elongational behaviour. Additional undesirable phenomena in some flows have also been pointed out such as oscillations in transient flows. 

Lodge Meissner rule unsatisfied (2 slip parameters). Oscillations in transient stress growth. Negative relaxation modulus in large step shear strain. 

Lodge Meissner rule unsatisfied. Oscillations in transient stress growth. 

According to Eq. (18.B.15), the extinction angle of the birefringence, x> is solely determined by the step strain applied initially and remains the same (independent of time) as long as the Lodge-Meissner rule holds. 

This suggests that at sufficiently small strains, the stress ratio (N la) should become equal to the strain. This relationship is known as the Lodge-Meissner rule [36]. 

This implies that the Lodge-Meissner rule (Eq. 10.35) continues to be valid at large strains. We note that if time-strain superposability is valid, this implies that the transient first normal... 

It was pointed out in Section 10.4.3 that wall slip can cause a large error in the determination of the strain in step-strain experiments, and the true strain maybe much less than the nominal strain inferred from the displacement of a rheometer surface. The observation that N /a is independent of time does not, by itself, imply that there is no slip unless this ratio is also equal to the nominal strain applied. And when the Lodge-Meissner rule is not obeyed, it is often taken as evidence that slip is occurring, and the stress ratio Nj/cris used in place of the nominal strain as the independent variable in reporting shear stress and normal stress difference data [40]. 

A rule that can be derived theoretically from general premises likely to be valid for polymer melts and solutions is the Lodge-AfeicSfrier relationship (L ge and Meissner, 1972) between the shear stress and the first nomml stress difference after a step shear strain ... 

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