The Mooney-Rivlin model

As an alternative to the molecular approach of the three models described above, a phenomenological model of elasticity may be used. In such a 

The model developed by Mooney and Rivlin starts from three strain invariants  

The free energy density of the network Fj V is written as a power series in the difference of these invariants from their values in the undeformed network (A c = A., = A = 1)  

The second term in this series is analogous to the free energy of the classical models [Eg. (7.23)]  

The true stress in the Mooney-Rivlin model can be obtained from the free energy density  

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In uniaxial extension, the Mooney-Rivlin model predicts Eq. (2) to hold, where Q and C2 are two material constants. 

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.

Fig. 22.14 Experimental data (O) and best fit (full line) with the Mooney-Rivlin model in uniaxial extension.

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. 

Comparison with values of and of the NAST model described earlier supports this hypothesis [36]. However, unlike the NAST model, the Mooney-Rivlin model fails to predict compression data as the linearity shown in Figure 9.21 continues for a > 1, whereas the experimental data show a maximum and a decrease in the function plotted in Figure 9.21. There have been many other constitutive relations for rubbers based on different representation of the strain energy function... 

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). 

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. 

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 ... 

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

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 deformation dependence of the confining potential [Eq. (7.62)] results in a non-classical stress strain dependence of the non-affine tube model. The prediction of this model for the stress-elongation relation in tension is qualitatively similar to the Mooney-Rivlin equation [Eg. (7.59)]... 

The preceding equations provided a reasonable foundation for predicting DE behavior. Indeed the assumption that DEs behave electronically as variable parallel plate capacitors still holds however, the assumptions of small strains and linear elasticity limit the accuracy of this simple model. More advanced non-linear models have since been developed employing hyperelasticity models such as the Ogden model [144—147], Yeoh model [147, 148], Mooney-Rivlin model [145-146, 149, 150] and others (Fig. 1.11) [147, 151, 152]. Models taking into account the time-dependent viscoelastic nature of the elastomer films [148, 150, 151], the leakage current through the film [151], as well as mechanical hysteresis [153] have also been developed. 

It has already been mentioned in Sect. 2 that the simplest assumption, affine deformation of the tubes d = dgk, yields the Mooney-Rivlin equation (1). The value V = 1/2 was obtained by a microscopic model which is briefly discussed in Sect. 2. It is suitable for the description of moderately but almost completely cross-linked networks (e.g. sulphur-, peroxide-, or radiation-crosslinked NR, PB and PDMS chains of very high degree of polymerization). 

A point worth noting here is that several of the molecular models that will be described in the subsequent sections are Neo-Hookean in form. Normally, dry rubbers do not exhibit Neo-Hookean behavior. As for the Mooney-Rivlin form of strain energy density function, rubbers may follow such behavior in extension, yet they do not behave as Mooney-Rivlin materials in compression. In Fig. 29.2, we depict typical experimental data for a polydimethylsiloxane network [39] and compare the response to Mooney-Rivlin and Neo-Hookean behaviors. The horizontal lines represent the affine and the phantom limits (see Network Models in Section 29.2.2). The straight line in the range A <1 shows the fit of the Mooney-Rivlin equation to the experimental data points. 

An alternative model which also describes stress-strain data for larger deformation is presented by the Mooney-Rivlin equation [40, 41], The equation describes the rubber elasticity of a polymer network on the basis that the elastomeric sample is incompressible and isotropic in its unstrained state and that the sample behaves as Hookean solid in simple shear. In a Mooney-Rivlin plot of a uniaxial deformation, the experimental measured stress cr, divided by a factor derived from classical models, is plotted as function of the reciprocal deformation 1/A ... 

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 ... 

For both the phantom and the affine networks, the reduced stress is calculated to be independent of deformation. However, stress-strain measurements carried out in uniaxial extension of dry and swollen networks have revealed departures from these predictions of simple models 5. These observations then gave rise to phenomenological equations like the Mooney-Rivlin expression, i.e. 

It is interesting to compare eq. (3.54) with the expressions obtained from the statistical theories (Fig. 3.20). According to both the affine network model and the phantom network model of James and Guth, the reduced stress remains constant and independent of strain, which is not the case for the Mooney-Rivlin equation. 

Plot of tensile and compressive data for a silicone rubber sample (replotted from Gottlieb, Macosko, and Lepsch, 1981) compared with predictions of the neo-Hookean (C s 101 kPa) and Mooney-Rivlin models (Ci = 17.5 kPa.C2 = 16.5kPa). 

Example 1.6.1 shows that both the neo-Hookean and Mooney-Rivlin models are of the Valanis-Landel form. 

Mooney—Rivlin model would describe the behaviors exhibited (89—91). The reader is referred to reviews of various proposed strain energy functions (89,92,93) for further information. 

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

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