Mooney-Rivlin relation

The measured results of a stretching experiment are normally treated with the empirical Mooney-Rivlin relation (Mooney 1940 Rivlin 1949), as given by... 

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

For the analysis of experimental force-deformation data, it is necessary to use a suitable constitutive equation for the material under test. The constitutive equation relates the stresses and strains that are generated in the wall during compression, and therefore relates the tensions and stretch ratios. For example, Liu et al. (1996) used a Mooney-Rivlin constitutive equation to investigate the compression of polyurethane microcapsules and the functions f, /2 and fa are produced in... 

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 ... 

We can also derive an expression on the basis of the Mooney-Rivlin strain energy function for swollen elastomers. A dry elastomer sample will undergo two types of deformation one due to swelling and the other due to extension. The strain energy function per unit volume of swollen elastomer is related to that of the dry sample by... 

The parameters obtained from fitting Eq. (4) to the experimental data are shown in Fig. 22.16. The fits are not as good as the Mooney-Rivlin fits [45, 48] but nevertheless capture reasonably weU the stress-strain curves in approximately the same ranges of extension ratios. The results can be interpreted as follows the parameter Gg, which is directly related to the volume density of fixed crosslink points, varies significantly between the pure triblock adhesive and the high diblock content adhesives, where the fit gives a value close to zero. On the other hand the parameter Gg is much higher and nearly independent of the diblock content. This shows that the low-strain modulus is essentially con-... 

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 ... 

Plastic stress-strain relations (NeoHooke, Mooney-Rivlin, Kilian)... 

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 ... 

The strain measures for dry (unswollen) vulcanizates of a large number of natural rubbers, butadiene-styrene and butadiene-acrylonitrile copolymers, polydimethylsiloxanes, polymethylmethacrylates, polyethylacrylates and polybutadienes with different degrees of crosslinking and measured at various temperatures re confined within the shaded area in Fig. 1. These measures were determined from the stress as a function of extension at (or near) equilibrium, i.e. by applying Eq. (7). Therefore they only reproduce the equilibrium stress-strain relation for the crossllnked rubbers. In all cases the strain dependence of the tensile force (and hence of the tensile stress) was expressed in terms of the well-known Mooney-Rivlin equation, equating the equilibrium tensile stress to ... 

In Chapter 3 it was shown that when a sample of rubber is stressed in tension the stress / may be related to the extension ratio A by the semi-empirical Mooney-Rivlin equation ... 

In the same manner by means of Eq. (56) the tmidimensional stress-strain relation can be obtained for Mooney-Rivlin and Yeoh model. These results are shown in Table 4. 

We add to these set of equations the constitutive equations that relate stresses to strains. One form of constitutive equations for isotropic highly deformable materials is of the generalized Mooney-Rivlin type [2] in which the strain energy density W is expressed in terms of the strain invariants ... 

When (f) is plotted versus l/X] (as suggested by the Mooney-Rivlin empirical relation), as shown in Figure 10, the effect of large strain on reducing the elasticity of die netwoik can be clearly observed. An analogous procedure used for the uniaxial extension of ar-PS gave a similar result. The value of... 

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... 

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