Gaussian stress-strain

The deviations from Gaussian stress-strain behaviour introduce uncertainties into the values of Mc/M discussed previously in this paper. However, such uncertainties have been shown to be of secondary importance compared with the ranges of Mc/Mc values found for networks from different reaction systems(25,32). 

The observed deviations from Gaussian stress-strain behaviour in compression were in the same sense as those predicted by the Mooney-Rivlin equation, with modulus increasing as deformation ratio(A) decreases. The Mooney-Rivlin equation is usually applied to tensile data but can also be applied compression data(33). According to the Mooney-Rivlin equation... 

Figure 11 shows plots according to equation(lO) of stress-strain data for triol-based polyester networks formed from the same reactants at three initial dilutions (0% solvent(bulk), 30% solvent and 65% solvent). Only the network from the most dilute reactions system has a strictly Gaussian stress-strain plot (C2 = 0), and the deviations from Gaussian behaviour shown by the other networks are not of the Mooney-Rivlin type. As indicated previously, they are more sensibly interpreted in terms of departures of the distribution of end-to-end vectors from Gaussian form. 

The deviations from Gaussian stress-strain behaviour for the tetrafunctional polyurethane networks of Figure 9 are qualitatively similar to these found for the trifunctional polyester networks (Z5), and the error bars on the data points for systems 4 and 5 in Figure 9 indicate the resulting uncertainties in Mc/Mc. It is clear that such uncetainties do not mask the increases in Mc/Mc with amount of pre-gel intramolecular reaction. 

Interesting deviations from Gaussian stress-strain behaviour in compression have been observed which related to the Me of the networks formed, rather than their degrees of swelling during compression measurements. 

Experimental results on reactions forming tri- and tetrafunctional polyurethane and trifunctional polyester networks are discussed with particular consideration of intramolecular reaction and its effect on shear modulus of the networks formed at complete reaction. The amount of pre-gel intramolecular reaction is shown to be significant for non-linear polymerisations, even for reactions in bulk. Gel-points are delayed by an amount which depends on the dilution of a reaction system and the functionalities and chain structures of the reactants. Shear moduli are generally markedly lower than those expected for the perfect networks corresponding to the various reaction systems, and are shown empirically to be closely related to amounts of pre-gel intramolecular reaction. Deviations from Gaussian stress-strain behaviour are reported which relate to the low molar-mass of chains between junction points. 

The values of Mq used in Figure 4 for network 5-17 and other networks derived from systems 4 and 5 were obtained from moduli evaluated from linear least-squares lines through the origins of the respective Gaussian stress-strain plots (a versus A- A ). 

Eqn(7) is the Gaussian stress-strain function, with a the force per unit area of the undeformed network, G the shear modulus and A the deformation ratio. In eqn(8), A is the so-called front factor, p the density of the... 

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

Stress upturn in the stress-strain relation of carbon black-fiUed rubbers can be reasonably revealed in terms of the non-Gaussian treatment, by regarding the distance between adjacent carbon particles as the distance between cross-links in the theory. 

These Monte Carlo distributions can be used in the standard three-chain model for rubber-like elasticity to generate, for example, stress-strain isotherms [5]. Non-Gaussian effects can cause large increases in modulus at high... 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

Theories based on these concepts all have to take into account the phenomenology of the stress-strain behaviour of networks. In unilateral extension as well as compression one observes, even at moderate extension (1.1 deviations from the Gaussian behaviour, which can be empirically described by the so-called Mooney-Rivlin equation ... 

Quested et al. [16] have conducted an extensive experimental program on the stress-strain behavior of the elastomer solithane while subjected to an ambient at high pressure. Some of their experimental results are reproduced in Fig. 13. (Note that the reported stress is the deviatoric, not the total, stress as observed from the fact that the reported stress is zero for X = 1 for the various imposed ambient pressures). For the classic ideal affine network model (all stress caused by ideal Nc Gaussian chains in a volume v with no nonbonded interactions)... 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

Figure 3.15 Isothermal stress-strain curve. Continuous line, non-Gaussian curve [Eq. (3.44) with G = NkeT = 0.273 MPa and n = 75]. (From Ref. 2.)...

Fig. 9. Eqiulibrium stress-strain behavior of entangled networks in uniaxial extensicm and compression. The solid lines (p = 0.5, 0.75,1.00) were calculated for the primitive segment model (. 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...

This model characterizes surface contacts, deals with agglomerates, and explains rubber swelling. It was further developed to characterize reinforcement as a non-Gaussian phenomenon. The model deals with intra-cluster forces and the stress-strain cycle. It is used in the experimental part on uniaxial compression to... 

The theoretical approach for determining the deformation behaviour of a network due to swelling or due to a mechanical force (stress-strain measurements, compression experiment) is based on a hypothetical phantom network. A phantom network is, by definition, a network with the fictitious property that chains and junctions can move freely through one another without destroying the cormectivity of the network. Usually, the network chains behave as Gaussian chains. Within the phantom network model, three network types can be distinguished ... 

The statistical approach, using a Gaussian distribution, thus appears to predict the stress-strain response except at moderately high elongations. 

Due to the dual filler and crosslinking nature of the hard domains in TPEs, the molecular deformation process is entirely different than the Gaussian network theories used in the description of conventional rubbers. Chain entanglements, which serve as effective crosslinks, play an important role in governing TPE behavior. The stress-strain results of most TPEs have been described by the empirical Mooney-Rivlin equation ... 

The phantom network can account qualitatively for many properties of crosslinked elastomers, but the quantitative explanation of basic properties is wrong. For example, stress-strain properties, especially in simple extension, show departures from the phantom network results even at extension ratios covered by the Gaussian chain model. The explanation of these departures, phenomenologically described by the famous Mooney-Rivlin Eq. (1)... 

Fig. 8.20 The elastomeric stress-strain curve of PET at 353 K, re-plotted against the Gaussian strain function g X), showing near-ideal rubbery behavior. The slope suggests an entanglement molecular weight of Me = 2342 g/mole.

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