Mooney-Rivlin

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

In TPE, the hard domains can act both as filler and intermolecular tie points thus, the toughness results from the inhibition of catastrophic failure from slow crack growth. Hard domains are effective fillers above a volume fraction of 0.2 and a size <100 nm [200]. The fracture energy of TPE is characteristic of the materials and independent of the test methods as observed for rubbers. It is, however, not a single-valued property and depends on the rate of tearing and test temperature [201]. The stress-strain properties of most TPEs have been described by the empirical Mooney-Rivlin equation... 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

The elastic free energy of the so-called Mooney-Rivlin solid is obtained from Equation (46) as... 

Figure 4 Mooney-Rivlin isotherms for the two chains described in Figure 3, showing the resulting increases in modulus due to the chains being stretched by the presence of the filler particles.

The reduced stress is defined as the force per cross-sectional area of the undeformed sample, divided by the term X-X- with X being the relative elongation L/L0. With undiluted rubber, this is not found experimentally. In most cases, however, the elastic behaviour in a moderate elongation range is satisfactorily described fcy the empirical Mooney-Rivlin equation, which predicts a linear dependence of on reciprocal elongation X- (32-34)... 

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.

Mooney-Rivlin constants obtained from strain dependent measurements at 298 K... 

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. 

Figure 9. Dependence of the Mooney-Rivlin constant on extent of cross-linking. Mn is 21,600 PDMS. Key , 2C, X, 2CS (3).

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. 

Networks were prepared in all cases using the amount of endlinking agent necessary to give a minimum Mc. Values of Mc were calculated from the Mooney-Rivlin elasticity coefficient Cj, determined from tensile stress-strain measurements (10),... 

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... 

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. 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

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. 

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 using equilibrium stress-strain measurements, the cross-link density is determined from the Mooney-Rivlin equation ... 

In the case of filled systems, the two latter effects provide a substantial contribution to C2 compared with the influence of trapped entanglements [80]. For filled systems, the estimated or apparent crosslinking density can be analyzed with the help of the Mooney-Rivlin equation using the assumption that the hard filler particles do not undergo deformation. This means that the macroscopic strain is lower than the intrinsic strain (local elongation of the polymer matrix). Thus, in the presence of hard particles, the macroscopic strain is usually replaced by a true intrinsic strain ... 

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... 

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