Small-strain modulus

Concluding, we can state that the absolute values of the small-strain moduli, which are greater for networks having comblike crosslinks, than for those with tetrafunctional junctions, are understandable, if we assume that the fluctuations of junctions are restricted by the very short chains. The strain dependent measurements do not agree quantitatively with the recent theory, although the trends are in accordance. An exact correspondence... 

It is shown that model, end-linked networks cannot be perfect networks. Simply from the mechanism of formation, post-gel intramolecular reaction must occur and some of this leads to the formation of inelastic loops. Data on the small-strain, shear moduli of trifunctional and tetrafunctional polyurethane networks from polyols of various molar masses, and the extents of reaction at gelation occurring during their formation are considered in more detail than hitherto. The networks, prepared in bulk and at various dilutions in solvent, show extents of reaction at gelation which indicate pre-gel intramolecular reaction and small-strain moduli which are lower than those expected for perfect network structures. From the systematic variations of moduli and gel points with dilution of preparation, it is deduced that the networks follow affine behaviour at small strains and that even in the limit of no pre-gel intramolecular reaction, the occurrence of post-gel intramolecular reaction means that network defects still occur. In addition, from the variation of defects with polyol molar mass it is demonstrated that defects will still persist in the limit of infinite molar mass. In this limit, theoretical arguments are used to define the minimal significant structures which must be considered for the definition of the properties and structures of real networks. 

Sharaf, M. A., The Effects of Network Imperfections on the Small-Strain Moduli of Polydimethylsiloxane Elastomers Having High Functionality Cross-Links. Int. J. Polym. Mater. 1992,18(3-4), 237-252. 

FIGURE 22.6 Payne effect of butyl composites with various amounts of N330, as indicated (left) [28]. Scaling behavior of the small-strain modulus of the same composites right). The obtained exponent 3.5 confirms the cluster-cluster aggregation model. (From Kliippel, M. and Heinrich, G., Kautschuk, Gummi, Kunststoffe, 58, 217, 2005. With permission.)... 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

Since the stiffness of the bonds transfers to the stiffness of the whole filler network, the small strain elastic modulus of highly filled composites is expected to reflect the specific properties of the filler-filler bonds. In particular, the small strain modulus increases with decreasing gap size during heat treatment as observed in Fig. 32a. Furthermore, it exhibits the same temperature dependence as that of the bonds, i.e., the characteristic Arrhenius behavior typical for glassy polymers. Note however that the stiffness of the filler network is also strongly affected by its global structure on mesoscopic length scales. This will be considered in more detail in the next section. 

In the framework of the approximation given by the rigidity condition, a simple power law relation can be derived for the dependency of the small strain modulus G 0 of the composite on filler concentration . It is obtained,... 

FIGURE 36.4. Flocculation behavior of the small strain modulus at 160°C of uncross-linked S-SBR-composites of various molar mass with 50 phr N234, as indicated (left) and strain dependence of the annealed samples after 60 min (right). Reproduced from M. Kluppel and G. Heinrich, Kautschuk, Gummi, Kunstsoffe 58, 217-224 (2005) with permission from Huthig. 

To account for the intermediate behavior between the affine and phantom modulus of a real network, Graessley has proposed an expression for the small strain modulus G as a function of an empirical paran ter having a value between 0 (afSne) and 1 (jdiantom). Specifically,... 

Monaco P, Marchetti S, Totani G, Marchetti D (2009) Interrelationship between small strain modulus Go and operative modulus. In International conference on performance based design in earthquake geotechnical engineering, Toyko... 

Its small strain modulus reaches a tiue equilibrium value at room temperature. 

Another interesting aspect of the Gibbs-DiMarzio " " theory is that it predicts an effect of deformation on the glass transition temperature due to changes in the entropy of the system upon straining the network. For uniaxial extension, the result in terms of the small-strain modulus G, measured at a temperature Tq, is ... 

At small strains the cell walls at first bend, like little beams of modulus E, built in at both ends. Figure 25.10 shows how a hexagonal array of cells is distorted by this bending. The deflection can be calculated from simple beam theory. From this we obtain the stiffness of a unit cell, and thus the modulus E of the foam, in terms of the length I and thickness t of the cell walls. But these are directly related to the relative density p/ps= t/lY for open-cell foams, the commonest kind. Using this gives the foam modulus as... 

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