Deformation uniaxially deformed model network

Model networks, synthesized by endlinking processes, contain few structural defects and are close to ideality. Spring-suspended bead models seem to fit adequately with the structural data obtained on labelled model networks and with the swelling and uniaxial deformation behavior of these networks. (67 refs.)... 

Neutron scattering experiments were also carried out on some model networks with labelled crosslinks, in the unswollen state13 26 To achieve such measurements the dry polystyrene networks are heated above their glass transition temperature, submitted to uniaxial deformation, quenched under stress and studied in the neutron scattering apparatus. 

Contrary to the extent of sweUing, the deformation of swollen model networks under uniaxial compression (or elongation) strongly depends on the quantity of danghng chains (Fig. 1.11). The elastic modules Ep of swollen samples decrease Hnearly with increasing portion of danghng chains up to about p = 0.45 and then decrease even more rapidly [129]. Indeed, the elastic modulus is proportional to the number of elastically active chains... 

The model network employed is described in detail in Gao and Weiner [2] and [3], Briefly put, the model chains are freely jointed, and the covalent bonds are represented by a linear, stiff spring of equilibrium length a the noncovalent interaction is the repulsive portion of a Lennard-Jones potential which approximates a hard-sphere interaction of diameter a. The network corresponds to the familiar three-chain model of rubber elasticity (see Treloar [10]). In the reference state, three chains, one in each coordinate direction, have their end atoms fixed in the center of the faces of a cube of side L periodic boundary conditions are employed to remove surface effects as is customary in molecular dynamics simulations. The system is siibjected to a uniaxial deformation at constant volume so that the cube side in the x direction has length XL while the other two sides have lengtn... 

Whereas k = 1.3 is derived from the above-presented NSE data, k = 2.75 is expected for a four-functional PDMS network of Ms = 5500 g/mol on the basis of Eq. (67). Similar discrepancies were observed for a PDMS network under uniaxial deformation [88]. Elowever, in reality this discrepancy may be smaller, since Eq. (67) provides the upper limit for k, calculated under the assumption that the network is not swollen during the cross-linking process due to unreacted, extractable material. Regardless of this uncertainty, the NSE data indicate that the experimentally observed fluctuation range of the cross-links is underestimated by the junction constraint and overestimated by the phantom network model [89],... 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Some applications are listed to illustrate these potentials. They may be classified in various ways. The most direct approach consists in working with labelled (deuterated), model systems. Real materials (designed for industrial applications and processed at a large scale) are often multicomponent, complex systems, which may be relatively ill defined at the molecular scale. Thus, working with chemically well defined, labelled materials (which, however, often have relatively poor mechanical properties by themselves) is a way to isolate and study the various parameters which play a role in rubber properties. Studies are done both in the relaxed state and in constrained (uniaxially deformed) states. This approach is illustrated in Section 15.3. Examples of studies performed in model, single component networks are presented. However, even in this case, the sensitivity of the method is such that it may detect the presence of a few percent of molecular defects. 

Figure 12. Intrinsic atomic stresses <7n, Vjj. <733 as determined from molecular dynamics simulation of tetrafunctional network model in uniaxial volume deformation and for corresponding melt. (After Ref. [19].)...

By fitting experimental data for different deformation modes to these functions, the three network parameters of unfilled polymer networks Gc, Ge, and ne/Te can be determined. The validity of the concept can be tested if the estimated fitting parameters for the different deformation modes are compared. A plausibility criterion for the proposed model is formulated by demanding that all deformation modes can be described by a single set of network parameters. The result of this plausibility test is depicted in Fig. 44, where stress-strain data of an unfilled NR-vulcanizate are shown for the three different deformation modes considered above. Obviously, the material parameters found from the fit to the uniaxial data provide a rather good prediction for the two other modes. The observed deviations are within the range of experimental errors. 

The success of the developed model in predicting uniaxial and equi-biaxi-al stress strain curves correctly emphasizes the role of filler networking in deriving a constitutive material law of reinforced rubbers that covers the deformation behavior up to large strains. Since different deformation modes can be described with a single set of material parameters, the model appears well suited for being implemented into a finite element (FE) code for simulations of three-dimensional, complex deformations of elastomer materials in the quasi-static Emit. 

It is demonstrated that the quasi-static stress-strain cycles of carbon black as well as silica filled rubbers can be well described in the scope of the theoretic model of stress softening and filler-induced hysteresis up to large strain. The obtained microscopic material parameter appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler-filler bonds, and the polymer network chain density. In particular it is shown that the model fulfils a plausibility criterion important for FE applications. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. 

In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio X of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n, of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n 5 por polymer SI). 

By writing these equations in terms of the shear modulus, the form of the stress-elongation relation becomes quite general. Many other network elasticity models also predict stress elongation relations of this form, with different predictions for the shear modulus. For this reason, we refer to Eqs (7.32) and (7.33) as the classical stress-elongation forms. As demonstrated in Fig. 7.3, this classical form describes the small deformation uniaxial data on... 

For classical models, the Mooney-Rivlin coefficients are 2Ci=G and 2 = 0. However, experimental data plotted in Fig. 7.13, in the form suggested by Eq. (7.59), show that C2>0. In this Mooney-Rivlin plot, the stress divided by the prediction of the classical models is plotted as a function of the reciprocal deformation 1/A. The predictions of the affine, phantom, and Edwards tube network models correspond to horizontal lines on the Mooney-Rivlin plot (C2 = 0). Experimental data on uniaxial... 

Fig. 7. The Doi-Edwards model applied to the strands of a network. The end-to-end vector of each strand changes according to the macroscopic deformation, in this case a uniaxial extension. The path length in-...

Small-angle neutron scattering has also been applied to the analysis of networks that were relaxing after a suddenly applied constant uniaxial deformation (Boue et al., 1991). Results of dynamic neutron scattering measurements of Allen et al. (1972, 1973, 1971) indicate that segments of network chains diffuse around in a network, and the activation energies of these motions are smaller than those obtained for the center of mass motion of the whole chains. Measurements by Ewen and Richter (1987) and Oeser et al. (1988) on PDMS networks with labeled junctions show that the fluctuations of junctions are substantial and equate approximately to those of a phantom network model. Their results also indicated that the motions of the junctions are diffusive and... 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

The eKperimental situation can be summarised as follows Crosslinked and swollen polystyrene gels exhibit a chain deformation which is less than that given by all phantom network models Early experiments by Clough et al, on radiation crosslinked polystyrene networks had been interpreted to be consistent with the free-fluctuation phantom scattering law. Most of the scattering experiments which were performed on uniaxially stretched end-linked siloxane networks showed much smaller molecular deformations than would be predicted even from Eq. (78) To... 

Both the affine and the phantom network models predict that the reduced stress, [/ ], measured in uniaxial deformation is independent of the deformation ratio. However, it... 

Here X = L/Lo is the extension ratio of the sample. Note that the corresponding strain is = (L -Lo) /To = A - 1. The proportional factor G is the shear modulus of the sample. Equation (4) describes small deformation uniaxial data on polymer networks quite well. With a fit of experimental stress-strain data for low extensions it is possible to predict crosslink properties because the classical models show that the shear modulus G is proportional to both temperature and crosslink density Vc ... 

An alternative model which also describes stress-strain data for larger deformation is presented by the Mooney-Rivlin equation [40, 41], The equation describes the rubber elasticity of a polymer network on the basis that the elastomeric sample is incompressible and isotropic in its unstrained state and that the sample behaves as Hookean solid in simple shear. In a Mooney-Rivlin plot of a uniaxial deformation, the experimental measured stress cr, divided by a factor derived from classical models, is plotted as function of the reciprocal deformation 1/A ... 

Stress-strain measurements in uniaxial extension have revealed that real networks have a behavior closest to the affine limit at small deformations and approach the phantom limit at large deformations. The recent molecular theory developed by Flory and Erman accounts for this transition. In this model, the restrictions on junction... 

The molecular model of an elastomeric network with local intermolecular correlations, given by Flory, is used to calculate the components of the molecular deformation tensor and molecular orientation. Effects of molecTilar parameters such as severity of entanglements, network inhomogeneities and conditions during cross-linking are discussed. Components of molecular deformation and orientation are calculated for a network under uniaxial stress. 

For both the phantom and the affine networks, the reduced stress is calculated to be independent of deformation. However, stress-strain measurements carried out in uniaxial extension of dry and swollen networks have revealed departures from these predictions of simple models 5. These observations then gave rise to phenomenological equations like the Mooney-Rivlin expression, i.e. 

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