Mooney-plot

This conclusion is based on the assumption that 2Cj is the actual network modulus at infinite deformation. Flory has pointed out that 2Q is the extrapolation of a straight line which is only a small part (in a narrow range of deformations) of a more general sigmoidal curve. Due to the use of 7. as independent variable, the Mooney plot gives more importance to small deformations and generally this extrapolation... 

In practice, it is found that the results for rubbers are typically similar to those shown in Figure A2.1 where the Mooney equation is adequate for 1/A > 0.45. In spite of this limitation, Mooney plots are a very useful way of dealing with data for the deformation of rubber networks. Such plots are often used as a first step to fitting results to more complicated relationships such as those proposed by Edwards and Vilgis [4], which provide a molecular understanding of rubber elasticity. 

Figure A2.1 Mooney plots for rubbers covering a wide degree of vulcanization (Gumbrell, Mullins, and Rivlin 1953). Reproduced with permission from Treloar, the Physics of Rubber Elasticity, 3rd edn.. Clarendon Press, Oxford, 1975...

Fig. 7.12. Mooney-plot of compression and extension data obtained for natural rubber. Results from Rivlin and Saunders [75]...

The suggestion to expand the treatment in this way is due to Mooney. It was readily accepted and applied with success for the description of tensile stress-strain curves. Figure 7.12 presents an example which also reveals, however, a major deficiency. Data are shown in the form of a Mooney-plot , which is based on Eq. (7.105), written as... 

Let us discuss the experimental consequences of this result briefly. It is convenient to use for experimental comparisons the so-called Mooney plot, based on the expression... 

The basic information that one can get out of the stress-strain relation is firstly the slippage which dominates at small strain (given by the softening in the Mooney plot — see Figure 14) and secondly the tube diameter, via the ratio a/b. This is very useful information, because this quantity, widely... 

FIGURE 12.IS Plot of relative Mooney viscosity at 120°C versus volume fraction of the fiber for ethylene-propylene-diene monomer (EPDM) mbber-melamine fiber compounds. (From Rajeev, R.S., Bhowmick, A.K., De, S.K., Kao, G.J.P., and Bandyopadhyay, S., Polym. Compos., 23, 574, 2002. With permission.)... 

The change in Mooney viscosity is plotted as a function of turns of the mixer (i.e., rpm X cummulative minutes of mixing). The results are presented in Figures 16.2 and 16.3. Under the LTM conditions httle discrimination is observed between triplicate batches of the control (no additive) and the batches containing disulfidic peptizer. However, even under the LTM condition. 

In order to check this prediction, stress-strain measurements were made up to moderate strains at room temperature. The obtained data are plotted in the usual manner as a versus 1/X in Figure 8. Table V gives the Mooney-Rivlin constants 2C and 2C calculated from these plots and also the ratio C./Cj. 

Figure 8. Mooney-Rivlin plots for strain dependent measurements at 298 K. Key A, PDMS-BI , PDMS-B2 X, PDMS-B3 O, PDMS-B5 , PDMS-B6 , PDMS-B7 V, PDMS-B8 A, PDMS-B9 PDMS-B10.

Number-average molecular weights are Mn = 660 and 18,500 g/ mol, respectively (15,). Measurements were carried out on the unswollen networks, in elongation at 25°C. Data plotted as suggested by Mooney-Rivlin representation of reduced stress or modulus (Eq. 2). Short extensions of the linear portions of the isotherms locate the values of a at which upturn in [/ ] first becomes discernible. Linear portions of the isotherms were located by least-squares analysis. Each curve is labelled with mol percent of short chains in network structure. Vertical dotted lines indicate rupture points. Key O, results obtained using a series of increasing values of elongation 0, results obtained out of sequence to test for reversibility. 

Figure 11 shows plots according to equation(lO) of stress-strain data for triol-based polyester networks formed from the same reactants at three initial dilutions (0% solvent(bulk), 30% solvent and 65% solvent). Only the network from the most dilute reactions system has a strictly Gaussian stress-strain plot (C2 = 0), and the deviations from Gaussian behaviour shown by the other networks are not of the Mooney-Rivlin type. As indicated previously, they are more sensibly interpreted in terms of departures of the distribution of end-to-end vectors from Gaussian form. 

Figure 11. Mooney-Rivlin plot of stress-strain data (32) for three triol-based polyester networks prepared from sebacoyl chloride and LHT240 at various initial dilutions in diglyme as solvent. Conditions P100 is 0% solvent P130 is 30% solvent PI 65 is 65% solvent.

In this contribution, we report equilibrium modulus and sol fraction measurements on diepoxidet-monoepoxide-diamine networks and polyoxypropylene triol-diisocyanate networks and a comparison with calculated values. A practically zero (epoxides) or low (polyurethanes) Mooney-Rivlin constant C and a low and accounted for wastage of bonds in elastically inactive cycles are the advantages of the systems. Plots of reduced modulus against the gel fraction have been used, because they have been found to minimize the effect of EIC, incompleteness of the reaction, or possible errors in analytical characteristics (16-20). A full account of the work on epoxy and polyurethane networks including the statistical derivation of various structural parameters will be published separately elsewhere. 

In Figure 3, o/(X—X-z) is plotted against 1/X to obtain the constants 2Cj and 2C2 in the Mooney-Rivlin equation. The intercepts at 1/X = 0 and the slopes of the lines give the values of 2Cj and 2C2, respectively, listed in Table I. If these plots actually represent data accurately as X approaches unity, then 2(Cj + C2) would equal the shear modulus G which in turn equals E/3 where E is the Young s (tensile) modulus. An inspection of the data in Table I shows that 2(Cj + C2)/(E/3) is somewhat greater than one. This observation is in accord with the established fact that lines like those in Figure 3 overestimate the stress at small deformations, e.g., see ref. 15. 

When trying to determine the flow behaviour of a material suspected of exhibiting wall slip, the procedure is first to establish whether slip occurs and how significant it is. The magnitude of slip is then determined and by subtracting the flow due to slip from the measured flow rate, the genuine flow rate can be determined. The standard Rabinowitsch-Mooney equation can then be used with the corrected flow rates to determine the tw-jw curve. Alternatively, the results can be presented as a plot of tw against the corrected flow characteristic, where the latter is calculated from the corrected value of the flow rate. 

Plot of apparent fluidity against 1/di to determine the slip velocity (Mooney s method)... 

Other interesting features of elastomeric networks can be revealed using the plots of the reduced stress, crred = /( — -2) against inverse extension ratio 1. This can be analyzed from the stress-strain behavior described by a phenomenological expression suggested by Mooney [78] and Rivlin and Saunders [79] ... 

Fig. 49 Mooney-Rivlin plots of reduced stress ((7red) against deformation ( ) for (a) CR gum and PTFEokGy—CR at 10, 20, and 30 phr loading and (b) comparison of PTFE500kGy-CR with PTFEokGy CR and PTEE20kGy-CR. C is the contribution to ared arising from chemical crosslinking...

If material is neo-Hookean, its Mooney-Rivlin plot ought to give a horizontal line and hence yield C2 = 0. Thus one is tempted to consider that nonzero C2 must be associated in one way or another with the deviation of a given material from the idealized network model, and it is understandable why so many rubber scientists have concerned themselves with evaluating the C2 term from the Mooney-Rivlin plot of uniaxial extension data. However, the point is that a linear Mooney-Rivlin plot, if found experimentally, does not always warrant that its intercept and slope may be equated to 2(9879/,) and 2(91V/9/2), respectively. This fact is illustrated below with actual data on natural rubber (NR) and styrene-butadiene copolymer rubber (SBR). 

Fig. 27. Mooney-Rivlin plot of uniaxial extension data for NR (A) compared with the sum of dWIdli and 3W/3/2, where bW/blf were extrapolated for uniaxial extension from biaxial data. The contributions of bW/bli and l dW/bI2 to their sum are also shown...

There is no reason to anticipate that, in general, linear Mooney-Rivlin plots are obtained at least over a certain range of relatively small stretch ratios. Though not illustrated here, our data on the carbon-filled SBR gave the Mooney-Rivlin plots of markedly upward curvature, and again this curvature was found to be due mainly to the dependence of BW/bli on Xj. 

Figure 28 41 depicts the isochronal Mooney-Rivlin plots for SBR-1, where the extrapolated values of bW/bli and X lbW/bI2 are represented by solid lines and the sum of them by broken lines. As above, these sums are equivalent to the Mooney-Rivlin plot of uniaxial data. We again find that the slope of the sum curves depends mainly on the Xj dependence of bW/dli and therefore the slope is not equal to... 

These values were estimated from biaxial data in same manner as in Fig. 27. Sum curves are equivalent to Mooney-Rivlin plot, and C1 and C2 may be determined. Note that C is apparently independent of time t, while actual values of bWjbly are not... 

First, the Mooney-Rivlin plot was prepared with the paired values of ax and taken on the uniaxial line (X2 - (Xx ) 1/2) in Fig. 29A. The values of Cx and C2 determined from it were taken equal to bW/dIl and dW/dI2, respectively, and then... 

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