Mooney-Rivlin curve

In unfilled rubbers, which are not capable of strain-induced crystallization, the upturns on Mooney-Rivlin curves have shown to be absent 92 95). They disappear also in crystallizable rubbers at elevated temperatures and in the presence of solvents. On the other hand, the upturns do not appear for butadiene, nitrile and polyurethane rubbers if the limited chain extensibility function is introduced in the Mooney-Rivlin expression 97). Mark 92) has concluded that in the absence of selfreinforcement due to strain-induced crystallization or domains the rupture of the networks occurs long before the limited chain extensibility can be reached. 

Fig. 22, The same data as in Fig. 21 plotted as force versus deformation ratio. If one identifies the compression modulus with the Gaussian constant 3 (A vk T/Lt) (( >o) the experimental curve in extension lies below the Gaussian curve. The C part of the Mooney-Rivlin curve in extension lies again below the experimental curve...

The transition of bulk network from Gaussian to Langevin statistics accompanied with transfer of the part of a load to inner hard polymer layer at the filler surface (in the range of Mooney-Rivlin curve minimum). 

Eq. 11 reproduces experimental results for simple extension (a> 1) fairly well. The behavior pf virtually all types of rubbers is remarkably similar in this respect. The observed force plotted against elongation typically displays greater curvature than the theory predicts. The two theoretical curves shown are in fact the same curve plotted on different ordinates, as is Indicated. The one is scaled to match the initial slope of the Mooney-Rivlin curve, and the other converges to the ultimate slope of that cvirve as a increases indefinitely. 

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 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... 

Even when the above complications are negligible or properly accounted for and when strain-induced crystallization is absent, the stress-strain curves for networks seldom conform to Eq. (7.3). The ratio //(a — 1/a2) generally decreases with elongation. An empirical extension of Eq. (7. IX the Mooney-Rivlin equation, has been used extensively to correlate experimental results ... 

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...

For gum rubbers and lightly filled compounds, the Mooney-Rivlin equation often models the tensile stress-strain curve well up to extensions of 150% or more. However, for more highly filled compounds (and almost always for commercially important compounds) this simple function only works well up to about 50% strain. A much better fit over an extended strain range can be obtained by taking the next logical term in the infinite series of the general expression. Using ... 

The stress-strain curve for unfilled NR exhibits a large increase in stress at higher deformations. NR displays, due to its uniform microstructure, a very unique important characteristic, that is, the ability to crystallise under strain, a phenomenon known as strain-induced crystallization. This phenomenon is responsible for the large and abrupt increase in the reduced stress observed at higher deformation corresponding, in fact, to a self-toughening of the elastomer because the crystallites act as additional cross-links in the network. This process can be better visualized by using a Mooney-Rivlin representation, based on the so-called Mooney-Rivlin equation ... 

Stress-strain curves for the various models are plotted in the Mooney-Rivlin fashion in Fig. 9. 

Figure 6-10. Mooney-Rivlin plots of natural rubber filled with MT carbon black Top set actual data without using the strain amplification factor. Bottom curves after reduction using the strain amplification factor, equation (6-95). [After L. Mullins and N. R. Tobin, J. Appl. Polym. Sci., 9, 2993 (1965), by 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. 

Fig. 22.13 Reduced stress representation of the stress-strain curves for the four adhesives. The broken line is an illustrative fit of the data with the Mooney-Rivlin model.

The parameters obtained from fitting Eq. (4) to the experimental data are shown in Fig. 22.16. The fits are not as good as the Mooney-Rivlin fits [45, 48] but nevertheless capture reasonably weU the stress-strain curves in approximately the same ranges of extension ratios. The results can be interpreted as follows the parameter Gg, which is directly related to the volume density of fixed crosslink points, varies significantly between the pure triblock adhesive and the high diblock content adhesives, where the fit gives a value close to zero. On the other hand the parameter Gg is much higher and nearly independent of the diblock content. This shows that the low-strain modulus is essentially con-... 

The nonlinear elastic properties can be described by both the Mooney-Rivlin model and the molecularly based slip-tube model. Both of these models stress the fact that the low-strain modulus of the adhesives is controlled by the entanglement structure of the isoprene -i- resin phase, while the high-strain modulus is controlled by the physical crossHnk structure. The incorporation of diblocks in the adhesive dramatically reduces the density of crossHrrks and causes a more pronounced softening in the high-strain part of the stress-strain curve. 

FIGURE 57.4 Pressure-radius curves for a Mooney-Rivlin tube with the approximate dimensions of the carotid. 

FIGURE 29.5. Mooney-Rivlin reduced stress plot showing comparison of experimental data with modified constrained chain model (MCC) predictions for dry (o) and swollen ( ) natural rubber networks [112,117]. Swelling agent n-Decane. continuous lines are theoretical curves calculated with paremeters /cT/l/o = 0.17MPa and kq =2.0. 

Schematic stress-strain isotherms in elongation for a unimodal elastomer in the Mooney-Rivlin representation of modulus against reciprocal elongation. The isotherms are represented as the dependence of the reduced stress ([f ] = f /(a - on reciprocal elongation. (f = f/A, f = elastic force, A = undeformed area, a = elongation). The top three are for a crystallizable network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent. Isotherm D is for an unswollen unimodal network that is inherently noncrystallizable.

Schematic Mooney-Rivlin isotherms for a noncrystaUizable bimodal network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent.

Fig. 4.8 Mooney-Rivlin plot of a cross-linked natural rubber. The curves A-G have different degrees of cross-linking with sulfur content covering from 3% to 4%. (Reprinted with permission from Gumbrell, S. M. Mullins, L. Rivlin, R. S., Trans. Faraday Soc. 49, 1495 (1953).)...

The form of the stress-strain curve for rubbers subjected to unidirectional extension shows significant departures from theory. This is illustrated in Fig. 1 where curves calculated from eg. 8, shown dashed, are compared with the solid curve calculated according to the Mooney-Rivlin empirical equation which can be written... 

Fig. 1.41. Mooney-Rivlin isotherms for PDMS elastomers filled with in situ-generated silica, with each curve labeled with the amount of filler precipitated into it [173]. Filled symbols are for results obtained out of sequence in order to establish the amount of elastic irreversibility, a common occurrence with reinforcing fillers. The vertical lines locate the rupture points.

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