Lodge-Meissner relationship

Transient birefringence measurements were used by Larson et al. [112] to test the validity of the Lodge-Meissner relationship for entangled polymer solutions. This relationship states that the ratio of the first normal stress difference to the shear stress following a step strain is simply Nx/%xy - y, where y is the strain. Those authors found the relationship was valid, except for ultrahigh molecular weight materials. 

This relation, which also results from the K-BKZ model, is referred to as the Lodge-Meissner relationship (124) and results for materials with a finite elastic modulus at zero time. 

At small strains y 0, so G(y, t) must reduce to the linear viscoelastic modulus G(/), and h y) must approach unity for small y. Note also in Figure 4.4.1 that the normal stress modulus N /y equals the shear stress modulus xn/y this implies that melt I obeys the Lodge-Meissner relationship, eq 4.2.8. Note that this relation follows directly from eqs. 4.4.4 and 4.4.5. 

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. 

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 ... 

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]. 

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