Equilibrium stress/strain

Direct measurements of equilibrium stress-strain isotherms of SAH are complicated by the gel softness. Nevertheless, a number of experiments on compression and tension of the gels has been reported (see, for example, Refs. [18, 21, 42]). The method of dynamic light scattering is free from such inconveniences... 

These experimental results show conclusively that the deformation factor occurring in the theoretical equation of state offers only a crude approximation to the form of the actual equilibrium stress-strain curve. The reasons behind the observed deviation are not known. It does appear, however, from observations on other rubberlike systems that the type of deviation observed is general. Similar deviations are indicated in TutyP rubber (essentially a cross-linked polyisobutylene) and even in polyamides having network structures and exhibiting rubberlike behavior at high temperatures (see Sec. 4b). 

The equilibrium stress-strain isotherms in elongation, and the swelling ratios in benzene, were measured at 25°C for these networks. Network chain densities calculated from these measurements exceeded the values predicted from stoichiometry. 

Equilibrium stress-strain dependences were determined in extension using a stress relaxation arrangement described earlier (21). Dry non-extracted samples were measured at 150 C in nitrogen atmosphere and extracted samples swollen in dimethylformamide were measured at 25 C. The equilibrium value of stress 6 e was reached within 2-4 min except of a few dry samples with the lowest tig, for which the equilibrium stress was determined using an extrapolation procedure described earlier (21). 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

The cross-link density can be determined by equilibrium swelling or from equilibrium stress-strain measurements at low strain rate, elevated temperature, and sometimes in the swollen state3 °... 

When using equilibrium stress-strain measurements, the cross-link density is determined from the Mooney-Rivlin equation ... 

During this same period, the equilibrium stress-strain properties of well characterized cross-linked networks were being studied intensively. More complex responses than the neo-Hookean behavior predicted by kinetic theory were observed. Among other possibilities it was speculated that, in some unspecified way, chain entanglements might be a contributing factor. 

Smith,T.L., Frederick, . E. Ultimate tensile properties of elastomers. IV. Dependence of the failure envelope, maximum extensibility, and equilibrium stress-strain curve on network characteristics. J. Appl. Phys. 36,2996-3005 (1965). 

In the simplest study of this type, Al-ghamdi and Mark [138] studied reinforcement of PDMS by two zeolites of different pore sizes. The zeolites were a zeolite 3A (pore diameter 3 A) and a zeolite 13X (pore diameter 10 A), both with a cubic crystalline structure. They were simply blended into hydroxyl-terminated chains of PDMS which were subsequently end-linked with tetraethoxysilane to form an elastomeric network. These elastomers were studied by equilibrium stress-strain measurements in elongation at 25°C. Both zeolites increased the modulus and related mechanical properties of the elastomer, but the effect was larger for the zeolite with the larger pore size. 

Figure 2.46 Typical non-equilibrium stress-strain curve in elongation. After F. W. Billmeyer, Jr., Textbook of Polymer Science, 2nd Ed., Wiley-Interscience, New York, 1971. Reproduced by permission of John Wiley and Sons.

Weakly crosslinked epoxy-amine networks above their Tg exhibit rubbery behaviour like vulcanized rubbers and the theory of rubber elasticity can be applied to their mechanical behaviour. The equilibrium stress-strain data can be correlated with the concentration of elastically active network chains (EANC) and other statistical characteristics of the gel. This correlation is important not only for verification of the theory but also for application of crosslinked epoxies above their Tg. 

Fig. 9. Equilibrium stress-strain behavior of entangled networks in uniaxial extension and compression. The solid lines (p = 0.5, 0.75,1.00) were calculated for the primitive segment model (Eq. 62 and II-5). The short-dash line is the Doi-Edwards model (Eq. 40 and 11-11). The long-dash line is the affine Gaussian network model (Eq. 41 and n-12) adjusted to have the same initial modulus...

Incidentally, equation (9.4) fits Figure 9.7 much better than it does Figure 9.5. There are two reasons (a) Figure 9.7 represents a much closer approach to an equilibrium stress-strain curve, and (b) the much higher level of sulfur... 

Equilibrium stress-strain data on bulk samples were difficult to obtain because of the aforementioned problems. Therefore, the behavior of PBD networks in which the fluctuations of the junctions are highly constrained was explored with Network D-1, which is highly cross-linked. Although the junctions in highly cross-linked networks are subject to smaller relative constraint from entanglements, these junctions will remain highly constrained. Measurements on samples swollen with hexa-decane eliminate the problem of equilibrium and provide a measure of this limit the experimental data exhibit large departures from phantom behavior. But these departures are within the framework of the theory Network D-1 exhibits affine-like behavior. 

In the field of rubber elasticity both experimentalists and theoreticians have mainly concentrated on the equilibrium stress-strain relation of these materials, i e on the stress as a function of strain at infinite time after the imposition of the strain > This approach is obviously impossible for polymer melts Another complication which has thwarted the comparison of stress-strain relations for networks and melts is that cross-linked networks can be stretched uniaxially more easily, because of their high elasticity, than polymer melts On the other hand, polymer melts can be subjected to large shear strains and networks cannot because of slippage at the shearing surface at relatively low strains These seem to be the main reasons why up to some time ago no experimental results were available to compare the nonlinear viscoelastic behaviour of these two types of material Yet, in the last decade, apparatuses have been built to measure the simple extension properties of polymer melts >. It has thus become possible to compare the stress-strain relation at large uniaxial extension of cross-linked rubbers and polymer melts ... 

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 ... 

The equilibrium stress- strain curves of polymeric materials depart considerably from linearity in tension but not in shear. Although there can be considerable differences in quoted moduli depending on whether a tangential or secant modulus is quoted and to what elongation these refer, there is much less ambiguity in quoting the shear modulus. 

Figure 3.1. Typical Equilibrium Stress-Strain Behavior for a Highly Filled Pol3nner when the Direction of Strain is Reversed...

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