Mooney—Rivlin plot

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.

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.

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... 

Fig. 7.2 Mooney-Rivlin plot for the Slip-Link model...

Fig. 21. Mooney-Rivlin plots (Eq. IV-1) for a chemically crosslinked polyurethane rubber (from polyoxypropyleneglycol, toluene diisocyanate and trimethylol propane). Note the differences in Cx and C2 values in extension and compression...

Fig. 23. Mooney-Rivlin plots of extended natural rubber vulcanizates. The upswing at small A 1 values, i.e. large strains, is due to finite extensibility [Mullins (72S)]. At high crosslinking densities (upper curves) the upswing occurs at a smaller strain...

Fig. 24. Mooney-Rivlin plots of extended and swollen rubbers. The upswing is again due to finite extensibility, which comes into play at an earlier stage the higher the swelling (lower curves) [Mullins 128)1...

The elastic moduli of the component polymers were obtained by extrapolating Mooney-Rivlin plots (8, 9) of the stress to zero strain ... 

Figure 12.10. Mooney-Rivlin plots for pureNR and 1 phr MWNTs/NR composite.

Figure 7.19. Mooney-Rivlin plot for 7S and IIS soy fractions at 30% moisture content (Copyright 1998 from Understanding phase transitions and ehemieal complexing reactions in 7S and 1 IS soy protein fractions by Morales-Diaz and Kokini (In Phase/State Transitions in Foods, Rao and Hartel (Eds.)). Reproduced by permission of Routledge/Taylor Francis Group, LLC.)...

Figure 3.12 Mooney-Rivlin plots for various rubbers in uniaxial extension (8). (From Ref. 2.)...

Figyre 3.16 Mooney-Rivlin plots [Eq. (3.38)] showing the effect of the temperature on stress-strain isotherms for model PDMS networks (15,16). The filled circles represent the reversibility of the elastic measurements, and the vertical lines locate the fracture points. (From Ref. 15.)... 

For classical models, the Mooney-Rivlin coefficients are 2Ci=G and 2 = 0. However, experimental data plotted in Fig. 7.13, in the form suggested by Eq. (7.59), show that C2>0. In this Mooney-Rivlin plot, the stress divided by the prediction of the classical models is plotted as a function of the reciprocal deformation 1/A. The predictions of the affine, phantom, and Edwards tube network models correspond to horizontal lines on the Mooney-Rivlin plot (C2 = 0). Experimental data on uniaxial... 

Mooney-Rivlin plots for uniaxial tension data on three networks prepared from radiation-crosslinking a linear polybutadiene melt with------------------... 

Figure 6-6. Simulated Mooney-Rivlin plot, equation (6-82), for C2/C = 2. Solid line is the undistorted response of the material "data" are the results with a 1% (standard deviation) random error incorporated into both the force and length "measurements."...

Figure 6-8. Effect of swelling on the Mooney-Rivlin plot of natural rubber where Vr is the volume fraction of elastomer in the swollen sample, (swelling liquid, n-decane). [After Mullins, J. Appl. Polym. Sci., 2, 257 (1959), by the permission of John Wiley Sons, Inc.]...

That crystallization increases the elastic stress has already been demonstrated in Figure 6-8, in which the Mooney-Rivlin plot shows a rise at high extension ratios. However, it should be remembered that part of this increase is due to finite extensibility of network chains. In Figure 6-13 we show the stress-strain curves of natural rubber at two temperatures. At 0 °C there is considerable strain-induced crystallization, and we observe a dramatic rise in the elastic stress above X = 3.0. Wide-angle X-ray measurements show the appearance of crystallinity above this strain. At 60 °C there is little or no crystallization, and the stress-strain curve shows a much smaller upturn at high strains. The latter is presumably due only to the finite extensibility of the polymer chains in the network. 

The data for this figure are provided above and in the CD in file NR-Mod.TXT. Using a spreadsheet, generate a Mooney-Rivlin plot and calculate C, and C2. In addition, fit the data to the equivalent multivariant linear form... 

Computer) The Mooney-Rivlin plots often show poor linearity at low strains. Considering that a likely systematic error is an inaccurate value of L0> the unstretched length of the sample, the observed A should be corrected by a multiplicative constant close to 1.0 to get the true stretch ratio a A. Using the form of the Mooney-Rivlin equation... 

Figure 2.31 A Mooney-Rivlin plot with the data of Problem 2.42. (After Sperling, 1986.)...

Make a Mooney-Rivlin plot and determine the constants C and 2- Obtain an estimate of the degree of crosslinking of the mbber band. (Density of mbber band = 1.09 g/cm. ) Answer ... 

Figure 5. Mooney-Rivlin plots of stress-strain data(lO) for two tetrol-based polyurethane networks from system 5 of Figures 2 and 4, prepared at various dilutions in nitrobenzene as solvent. Networks swollen in nitrobenzene at 4QOC...

Equation (29.7) makes obvious the reasons for the representation of experimental data in the so-called Mooney-Rivlin plot. If the material has a Mooney-Rivlin strain energy density function then a plot of (cn — straight line with the slope and intercept at A = 1 determining 2C2 and (2Ci -I- 2C2), respectively. 

Hence, in the Mooney-Rivlin plot, the stress-strain data are reduced to a line of slope zero. 

FIGURE 29.2. Comparison of typical stress-strain data for PDMS rubber [39] in a Mooney-Rivlin plot with Neo-Hookean and Mooney-Rivlin strain energy function descriptions. (See text for discussion). 

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