Mooney—Rivlin theory

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. 

Regarding the topological constraint, Grosberg et al. [57] modified the classical theory by incorporating the crumpled globule state and the Mooney-Rivlin type energy contribution. 

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. 

Theories based on these concepts all have to take into account the phenomenology of the stress-strain behaviour of networks. In unilateral extension as well as compression one observes, even at moderate extension (1.1 deviations from the Gaussian behaviour, which can be empirically described by the so-called Mooney-Rivlin equation ... 

There is at the moment no quantitative theory to account for the behaviour shown in Fig. 34. As long as such a theory is absent, it is not possible to obtain vei from a Mooney-Rivlin type stress-strain relation. 

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 ... 

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. 

At high extensions, departure from the Gaussian chain approximation becomes significant and has led to the development of a more general but semiempirical theory based on experimental observations. This is expressed in the Mooney, Rivlin, and Saimders (MRS) equation... 

Due to the dual filler and crosslinking nature of the hard domains in TPEs, the molecular deformation process is entirely different than the Gaussian network theories used in the description of conventional rubbers. Chain entanglements, which serve as effective crosslinks, play an important role in governing TPE behavior. The stress-strain results of most TPEs have been described by the empirical Mooney-Rivlin equation ... 

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... 

Vapour-sorption experiments on different polymer plus solvent systems have shown that the elastic component of the solvent chemical potential exhibits a maximum, contrary to the phantom network theories or the Mooney-Rivlin equation. Furthermore, evidence has been found that the localisation and height of the maximum is dependent upon the nature of the diluent. 

It should be emphasized that the Mooney-Rivlin equation is empirical and is based on phenomenological principles it does not involve molecular concepts and is not an explanation for the deviations from molecular theories. 

Despite of the approximations, the statistical theory is of fundamental significance for understanding of the molecular mechanisms causing rubber-like elasticity. It serves as a starting point for generalizations that agree more precisely with experiments. One generalization is the Mooney-Rivlin equation. After Equation (35), we have ... 

Solution cross-linked elastomers also exhibit stress-strain isotherms in elongation that are closer in form to those expected from the simplest molecular theories of rubberlike elasticity. Specifically, there are large decreases in the Mooney-Rivlin 2C correction constant described in... 

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. ... 

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 ... 

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