Mooney-Rivlin relationship

Figure 2. The results of Figure 1 represented as suggested by the semi-empirical Mooney-Rivlin relationship (8,28-30) which usually gives a modulus Lf D decreasing linearly with decreasing a (28).

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. 

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>... 

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. 

Mooney-Rivlin correction. In the 1940 s, Mooney and Rivlin showed that, generally, the basic force-elongation relationship must be corrected by a term proportional to the reciprocal extension ratio ... 

An analytical elastic membrane model was developed by Feng and Yang (1973) to model the compression of an inflated, non-linear elastic, spherical membrane between two parallel surfaces where the internal contents of the cell were taken to be a gas. This model was extended by Lardner and Pujara (1980) to represent the interior of the cell as an incompressible liquid. This latter assumption obviously makes the model more representative of biological cells. Importantly, this model also does not assume that the cell wall tensions are isotropic. The model is based on a choice of cell wall material constitutive relationships (e.g., linear-elastic, Mooney-Rivlin) and governing equations, which link the constitutive equations to the geometry of the cell during compression. 

Overall cross-linking density data (Pt) of sulfur vulcanized (V)-EPTM and ENB-EPDM were obtained from stress-strain measurements performed under equilibrium conditions and u g the Mooney-Rivlin and Guth-Einstein uations. Chemical cross-linking density data (Pg) were obtained through the emiarical relationship found by... 

F. De Candia, A. Taglialatela, and V. Vittoria, Structure-Property Relationships in Crosslinked Networks from cis-1, 4-Polybutadiene and Methacrylic Acid. Swelling Behavior, J. Appl. Polym. Sci. 20, 831 (1976). Simultaneous crosslinking of polybutadiene and polymerization of methacrylic acid. Swelling studied via Mooney-Rivlin equation. 

The theory of rubber elasticity explains the relationships between stress and deformation in terms of numbers of active network chains and temperature but cannot correctly predict the behavior on extension. The Mooney-Rivlin equation is able to do the latter but not the former. While neither theory covers all aspects of rubber deformation, the theory of rubber elasticity is more satisfying because of its basis in molecular structure. 

Typically, for many polymers, comparison of the Mooney—Rivlin parameters, O and T, reveal the relationship 0 < R < (0.2) . 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

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