Lodge-Meissner relation

Keeping in mind the previous remarks, it must be recognized that the lower the value of the slip parameter, the smaller the deviation to the Lodge-Meissner relation. This may be especially interesting, since in such a case, one can expect a single value of a that enables some kind of compromise for an acceptable depiction of the rheological shear functions. This can be expected,... 

FIGURE 6.17 (Bottom) Birefringence for uncrosslinkedPIB after imposition of a shear strain = 2.5. Linear fitting yields 3.07 GPa - for the stress optical coefficient. (Top) Ratio of refractive index components (1 is the flow direction and 2 the direction of the gradient), which equals the value of the shear strain, in accord with the Lodge-Meissner relation (Eq. (6.66)) (Balasubramanian et al., 2005). 

The Lodge-Meissner relation holds namely, Gs t) = as indi-... 

Before this condition is fully realized, G i(f, A) should show a higher noise level than Gs t,X) as indeed observed in the simulations. As the ideal is never fully realized in the practice, one may regard the Monte Carlo simulations as showing that the Lodge-Meissner relation holds only within some fluctuating noise. 

Simulations of the entanglement-free Praenkel-chain model and experiments of an entangled system studied by Osaki et al. have been compared with respect to the damping factor, the ratio -N2 t, X)/Ni t, A) and the Lodge-Meissner relation over the entropic region. As revealed, the sameness or close similarity in magnitudes and behaviors between the two... 

Appendix 18.A — Proof of The Lodge-Meissner Relation as Applied to the Praenkel-Chain Model... 

While the denominator of Eq. (18. A.9) contains only even terms, the numerator contains just an odd term. Thus, A(A) = 0 for all A leading to the conclusion that the Lodge-Meissner relation Gs t,X) = G i(t, A) holds. 

This proportionality relation is called the Lodge-Meissner relation [45,46]. 

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. 

Hiis relation, first found by Lodge and Meissner using a phenomenological argument, has been well confirmed. The ratio between the second normal stress difference N2(t, y) and the first normal stress difference Ni(t, y) is shown in Fig. 7.17. The experimental values are again in reasonable agreement with the theory. 

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