Mooney-Rivlin ratio

Dependence of Mooney-Rivlin ratio, 2C2/2C1, on the molecular weight between cross links. The factor 2C measures the departure from affineness as the elongation increases, and 2Cj approximates the high-deformation modulus. The ratio decreases with decrease in network chain molecular weight, and with increase in junction functionality, as predicted by theory. ... 

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. 

The two network precursors and solvent (if present) were combined with 20 ppm catalyst and reacted under argon at 75°C to produce the desired networks. The sol fractions, ws, and equilibrium swelling ratio In benzene, V2m, of these networks were determined according to established procedures ( 1, 4. Equilibrium tensile stress-strain Isotherms were obtained at 25 C on dumbbell shaped specimens according to procedures described elsewhere (1, 4). The data were well correlated by linear regression to the empirical Mooney-Rivlin (6 ) relationship. The tensile behavior of the networks formed In solution was measured both on networks with the solvent present and on networks from which the oligomeric PEMS had been extracted. 

The observed deviations from Gaussian stress-strain behaviour in compression were in the same sense as those predicted by the Mooney-Rivlin equation, with modulus increasing as deformation ratio(A) decreases. The Mooney-Rivlin equation is usually applied to tensile data but can also be applied compression data(33). According to the Mooney-Rivlin equation... 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

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. 

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 ... 

It is very important to stress that the decrease of the internal energy contribution with increasing extension ratio is due to a decrease of the intermolecular interaction with deformation, since the intramolecular contribution is independent of the deformation in full accord with the statistical theory. At very high strains, the /.-dependent part of fu/f approaches a limiting value of —0.68 for the Mooney-Rivlin and —0.07 for the Valanis-Landel materials. 

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...

Some further remarks concerning the Mooney-Rivlin equation are in place (14, 112). In dry rubber networks Ca in extension is often of the same order of magnitude as Cx, so that we are by no means confronted with a minor correction. In unilateral compression C2 is almost zero, and perhaps slightly negative. The constant Cx increases with the crosslinking density and with the temperature the ratio C2/C( in extension seems... 

Simple linear FEA programmes, as used for stress analysis of metals, take Young s modulus and Poisson s ratio as input but this is not satisfactory for rubbers because the strains involved cannot be considered as small and the Poisson s ratio is very close to 0.5. Non-linear FEA programmes for use with rubbers take data from a model such as the Mooney-Rivlin equation. More sophisticated programmes will allow a number of models to be used and may also allow direct input of the stress strain data. 

Mooney-Rivlin correction. In the 1940 s, Mooney and Rivlin showed that, generally, the basic force-elongation relationship must be corrected by a term proportional to the reciprocal extension ratio ... 

For the analysis of experimental force-deformation data, it is necessary to use a suitable constitutive equation for the material under test. The constitutive equation relates the stresses and strains that are generated in the wall during compression, and therefore relates the tensions and stretch ratios. For example, Liu et al. (1996) used a Mooney-Rivlin constitutive equation to investigate the compression of polyurethane microcapsules and the functions f, /2 and fa are produced in... 

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. 

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... 

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-... 

Figure 11-7. Ratio Cil C of the Mooney-Rivlin constants of different elastomers as a function of the chain cross section for 2Ci = 0.2 MPa. (After R. F. Boyer and R. L. Miller.)...

The phantom network can account qualitatively for many properties of crosslinked elastomers, but the quantitative explanation of basic properties is wrong. For example, stress-strain properties, especially in simple extension, show departures from the phantom network results even at extension ratios covered by the Gaussian chain model. The explanation of these departures, phenomenologically described by the famous Mooney-Rivlin Eq. (1)... 

Related problems which have received more interest recently are the swelling dependence of the C2 term in the Mooney-Rivlin equation, and the swelling dependence of the product of the dilation ratio and of the elastic contribution to the chemical potential of the solute in a swollen network. Classical network theories predicted either constant or monotonically increasing values, whereas the experiments give a sharp and pronounced maximum ... 

In Chapter 3 it was shown that when a sample of rubber is stressed in tension the stress / may be related to the extension ratio A by the semi-empirical Mooney-Rivlin equation ... 

Because of the limited usefulness of the Mooney-Rivlin equation, it is probably not worthwhile to seek a molecular interpretation of the coefficients C and C2 the deviations from neo-Hookean behavior should be examined in some other framework. However, if the deviations are expressed in terms of the ratio = C2/(C] + C2), this quantity can be correlated rather successfully with the relative numbers of trapped entanglements and cross-links in the network. It may be inferred from this and other studies that both trapped entanglements and cross-links contribute to C, but that C2 is associated with trapped entanglements only. 

Figure 1.11 The dependence of the molecular weight of the chain part between clusters on the Mooney-Rivlin equation constants ratio ICJIC for (1) PC (2) poly(arylate sulfone) (PAS) and (3) HOPE [56]...

Figure 4.12 The dependences of the reduced stress f on the reciprocal value of the drawing ratio A, corresponding to the Mooney-Rivlin Equation 1.22 for samples of (1) PCP-2 and (2) PCP-4 [59]...

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