Mooney-Rivlin isotherm

Figure 4 Mooney-Rivlin isotherms for the two chains described in Figure 3, showing the resulting increases in modulus due to the chains being stretched by the presence of the filler particles.

Schematic Mooney-Rivlin isotherms for a noncrystaUizable bimodal network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent.

Fig. 1.41. Mooney-Rivlin isotherms for PDMS elastomers filled with in situ-generated silica, with each curve labeled with the amount of filler precipitated into it [173]. Filled symbols are for results obtained out of sequence in order to establish the amount of elastic irreversibility, a common occurrence with reinforcing fillers. The vertical lines locate the rupture points.

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. 

Number-average molecular weights are Mn = 660 and 18,500 g/ mol, respectively (15,). Measurements were carried out on the unswollen networks, in elongation at 25°C. Data plotted as suggested by Mooney-Rivlin representation of reduced stress or modulus (Eq. 2). Short extensions of the linear portions of the isotherms locate the values of a at which upturn in [/ ] first becomes discernible. Linear portions of the isotherms were located by least-squares analysis. Each curve is labelled with mol percent of short chains in network structure. Vertical dotted lines indicate rupture points. Key O, results obtained using a series of increasing values of elongation 0, results obtained out of sequence to test for reversibility. 

For this reason, stress-strain isotherms are frequently represented by the semi-empirical Mooney-Rivlin relationship81,85... 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

Figyre 3.16 Mooney-Rivlin plots [Eq. (3.38)] showing the effect of the temperature on stress-strain isotherms for model PDMS networks (15,16). The filled circles represent the reversibility of the elastic measurements, and the vertical lines locate the fracture points. (From Ref. 15.)... 

TABLE 29.2 Mooney-Rivlin parameters of the stress-strain isotherms for different networks systems [114],... 

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... 

Schematic stress-strain isotherms in elongation for a unimodal elastomer in the Mooney-Rivlin representation of modulus against reciprocal elongation. The isotherms are represented as the dependence of the reduced stress ([f ] = f /(a - on reciprocal elongation. (f = f/A, f = elastic force, A = undeformed area, a = elongation). The top three are for a crystallizable network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent. Isotherm D is for an unswollen unimodal network that is inherently noncrystallizable.

Fig. 1.13. The modulus shown as a function of the reciprocal elongation as suggested by the semi-empirical Mooney-Rivlin equation [/ ] = 2Ci -h lC20i [2, 66]. The elastomer is natural rubber, both unswollen and swollen with n-decane [66]. Each isotherm is labeled with the volume fraction of polymer in the network.

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