Network chains finite extensibility

According to our definition the end-to-end distances of the network chains have an a priori probability distribution which is Gaussian. The effect of finite extensibility of the chains will be postponed to Chapter IV, because it is a special aspect of non-Gaussian behaviour. 

To the authors knowledge there are no dry rubberlike networks which obey the stress-strain relation derived in the preceding section (Eq. III-l 1) over an appreciable range of deformation ratios (e.g. from Ax — 1 to 2), whereas this relation should hold for an ideal network at least over such a range before the finite extensibility of the chains impairs the derivation. 

Fig. 26. The ratio of the real force / to the Gaussian force fB according to the non-Gaussian cubical network chain model with energy effects of Krigbaum and Kanbko (171). A negative value of K (= Aej lhT) indicates a preference for compact conformations and vice versa. Swelling induces the upturn, due to finite extensibility, to occur at an earlier stage...

A number of workers have treated non-Gaussian networks theoretically in terms of this finite extensibility problem. The surprising conclusion is that the effect on simple statistical theory is not as severe as might be expected. Even for chains as short as 5 statistical random links at strains of up to 0.25, the equilibrium rubbery modulus is increased by no more than 20-30 percent (typical epoxy elastomers rupture at much lower strains). Indeed, hterature reports of highly crosslinked epoxy M, calculated from equilibrium rubbery moduh are consistently reasonable, apparently confirming this mild finite extmsibiUty effect. 

That crystallization increases the elastic stress has already been demonstrated in Figure 6-8, in which the Mooney-Rivlin plot shows a rise at high extension ratios. However, it should be remembered that part of this increase is due to finite extensibility of network chains. In Figure 6-13 we show the stress-strain curves of natural rubber at two temperatures. At 0 °C there is considerable strain-induced crystallization, and we observe a dramatic rise in the elastic stress above X = 3.0. Wide-angle X-ray measurements show the appearance of crystallinity above this strain. At 60 °C there is little or no crystallization, and the stress-strain curve shows a much smaller upturn at high strains. The latter is presumably due only to the finite extensibility of the polymer chains in the network. 

At very high deformations, the main contribution to reinforcement comes from the change of network chain statistics due to finite chain extensibility, but supported here with filler particle as... 

Edwards and Vilgis [11] subsequently extended this theory to include the limitation of finite extensibility for the network that they regarded as arising when the network chains are ftdly extended, i.e. ex where n is the number of links in the polymer chain between junction points (see Equation (3.17) above). This introduces another constant a = [(Aj + -I-Aj) / ]max into Equations... 

The use of the inverse Langevin function, as for example to derive eqn (3.20) overcomes the objection that the Gaussian analysis does not take into account the finite extensibility of the network chains. However in order to make the Langevin statistics reasonably mathematically tractable assumptions have to be introduced which strictly make the statistics invalid under non-Gaussian conditions. Their justification must lie in the fact that the resultant expressions give far better fits to the experimental data than the Gaussian statistics. 

The Phan-Thien/Tanner (PTT) model is one of many generalizations that have been introduced to deal with the deficiencies of the basic Maxwell model constant viscosity, quadratic first normal stress difference, zero second normal stress difference, and infinite tensile stress at a finite extension rate. Many of these models, including the PTT, are derived from microstructural models that attempt to account for aspects of chain morphology and interactions. PTT is a network model, in which the chains are assumed to interact at entanglement points. There are kinetic expressions... 

Edwards and Vilgis [16] subsequently extended this theory to include the limitation of finite extensibility for the network that they regarded as arising when the network chains... 

The mesoscale model consists of a set of crosslink nodes (i.e., junctions) connected via single finite-extensible nonlinear elastic (FENE) bonds (that can be potentially cross-linked and/or scissioned), which represent the chain segments between crosslinks. In addition, there is a repulsive Lennard-Jones interaction between all crosslink positions to simulate volume exclusion effects. The Eennard-Jones and FENE interaction parameters were adjusted and the degree of polymerization (p) for a given length of a FENE bond calibrated until the MWD computed from our network matched the experimental MWD of the virgin material [112]. 

Finite chain extensibility is the major reason for strain hardening at high elongations (Fig. 7.8). Another source of hardening in some networks is stress-induced crystallization. For example, vulcanized natural rubber (cw-polyisoprene) does not crystallize in the unstretched state at room temperature, but crystallizes rapidly when stretched by a factor of 3 or more. The extent of crystallization increases as the network is stretched more. The amorphous state is fully recovered when the stress is removed. Since the crystals invariably have larger modulus than the surrounding... 

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

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