Mooney-Rivlin coefficients

Figure 15.13 Slopes P=A/( k2- k i) versus the elastic modulus 2(Cj+C2) (Cj and C2 are the Mooney-Rivlin coefficients), in bimodal PDMS networks composed of a mixture of long (Mn = 25000 g.mol"1) and short chains (Mn = 3000 g.mol"1). Either long (capital letters) or short (small letters) chains are deuterated. The long chain volume fractions are 15% (d, D), 50% (m, M) and 75% (s, S)...

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... 

The predictions of the classical models of rubber elasticity correspond to horizontal lines in a Mooney-Rivlin plot (C2 = 0). Plots for experimental data show a positive slope (C2 > 0). This indicates a stress softening with increasing deformation (as the reciprocal deformation 1 /X decreases). A comparison of (4) and (6) shows that, for the classical models, the Mooney-Rivlin coefficient 2Ci, corresponds to the shear modulus G given by (5), and the Mooney-Rivlin plot presents another method to determine crosslink density Vc or the average molecular mass of a network chain Me ... 

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),... 

CKF Theory Coefficients and C j. which are Mooney-Rivlin in character, whereas the crosslinK network appears to be simply neo-Hookean described by the CKF Theory Coefficient C. only. It was noted in early experiments " "... 

As described previously, t ne N network of strands is assigned the CKF Theory Coefficients and Cpf, the Y network of Vy strands is taken as neo-Hookean with the CKF Co-efficient C-y only, since it is in compression, following similar reasoning as used previously for the Vy network for conditions when = 0. Reservations concerning use of the Mooney-Rivlin formulation and its justification for convenience have been described elsewhere. Then assuming front factors of unity, the strand densities are related to the CKF Theory Coefficients as follows ... 

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. 

Elastomers as a class require special comment. Because of their low T s airi hence low they are normally evaluated well above T/ . Any structure present below has been destroyed. We covered Mooney-Rivlin C and C2 constants as well as stress optical coefficients (SOC) in an earlier discussion of order.26 Any evidence of order reported then had to be of the stress-strain induced type and hence not immediately relevant to our present purposes. In many instances properties were determined for elastomer-diluent systems, with a further lowering of Tu by the diluent. 

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