The Mooney-Rivlin Equation

The non-linear response of elastomers to stress can also be handled by abandoning molecular theories and using continuum mechanics. In this approach, the restrictions imposed by Hooke s law are eliminated and the derivation proceeds through the strain energy using something called strain invariants (you don t want to know ). The result, called the Mooney-Rivlin equation, can be written (for uniaxial extension)—Equation 13-60  

You should notice that the first term has the same form as that given by simple theories of rubber elasticity. The equation fits extension data for deformations up to about 300% very well, but cannot fit compression data using the same values of the constants C, and C2. Attempts to obtain the second term (in this form) using a molecular theory have not, as yet, been very successful, so we ll say no more about it. 

The most important fact that you should grasp from this discussion is the entro-pic nature of rubber elasticity. Although the agreement between the simple model described here and experiment is not that great you have to keep in mind that there are both theoretical assumptions (e.g., affine deformation) and mathematical approximations (Gaussian chain statistics) that have 

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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. 

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... 

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 ... 

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 well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

Shen 391 has considered the thermoelastic behaviour of the materials described by the Mooney-Rivlin equation and has shown that the energetic component is given... 

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... 

Orientational entropy of crosslinks and the Mooney-Rivlin equation. 

A very common simplification of the equation is generally referred to as the Mooney-Rivlin equation and consists of the first two terms and written in the form ... 

The use of the Mooney Rivlin equation and alternative relations will be considered in Section 3 on data for finite element analysis. 

The main interest in finite element analysis from a testing point of view is that it requires the input of test data. The rise in the use of finite element techniques in recent years is the reason for the greatly increased demand for stress strain data presented in terms of relationships such as the Mooney-Rivlin equation given in Section 1 above. 

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 ... 

The Mooney-Rivlin equation is readily available from the Constitutive Equation for isotropic elastic materials ... 

Table 13.7 shows the values of Ci and C2 for different families of elastomers. Obviously the Mooney-Rivlin equation may also be written as... 

TABLE 13.7 Constants of the Mooney-Rivlin equation (numerical values derived from Blokland (1968))... 

Equation (4-46) is valid for small extensions only. The actual behavior of real cross-linked elastomers in uniaxial extension is described by the Mooney-Rivlin equation which is similar in form to Eq. (4-46) ... 

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 Mooney-Rivlin equation is frequently used to interpret experimental results related to stress softening ... 

Creep, Swelling, and Extraction Studies. Additional indications of component interaction were found in the results of creep experiments with the xl series shown in Tables VI and VII. Creep in the xl series is fully recoverable, i. e. there is no permanent set, consistent with a PUx continuous phase. The increase in recoverable compliance, however, indicates a reduction in apparent crosslink density with increasing delay time before irradiation. This result is reinforced by the data in Table VII Ci and Cg are the material constants in the Mooney-Rivlin equation. The rubbery plateau modulus and the crosslink density of PUx prepared in BHA, which mimics xl formation, is less than that of PU prepared neat. 

When the direction of the compressive force is parallel and perpendicular to the pearl chain structure, a deviation has been found from the ideal mechanical behavior. The nominal stress does not obey a linear dependence with the quantity D, as demonstrated in Fig. 14. This kind of mechanical behavior can be described by the Mooney-Rivlin equation with C2 < 0. 

The Ci term of the Mooney-Rivlin equation is often identified with the shear modulus of the statistical equation we see that both Cx s depend on Vr in the same manner [compare equation (6-92) with (6-87) or (6-93) with (6-88)]. 

Computer) Examine Figure 6-5 carefully and note that the data at higher strains tend to fall below the line representing the statistical theory for an ideal elastomer. However, the data might be accurately represented by the Mooney-Rivlin equation. 

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... 

It may be recalled that Eq. (2.77) was derived for an ideal network. The actual behavior of real cross-linked elastomers, however, shows much better accord with the Mooney-Rivlin equation (Mooney, 1948 Rivlin, 1948) ... 

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 ... 

Cj term of the Mooney-Rivlin equation has been interpreted as meaning that these systems are under-entangled with respect to . This approach may be fruitful as the starting point of a theory for networks consisting of cycles trapped by linear chains which passed through them prior to being end-linked or of networks... 

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