Statistical theory of elasticity

It is actually possible within the framework of the statistical theory of elasticity to deduce an expression similar to Eq, (3.33) that considers the experimentally observed decrease in modulus. This is done by using a model different from the affine deformation model, known as the phantom network model. In the phantom network the nodes fluctuate around mean... 

In the statistical theory of elasticity for spatially structured polymers, the modulus of elasticity is a function of the number of effective cross-links ... 

The principal elastic constants for a nematic liquid crystal have already been defined in Sec. 5.1 as splay (A , j), twist(/ 22) and bend(fc33). In this section we shall outline the statistical theory of elastic constants, and show how they depend on molecular properties. The approach follows that of the generalised van der Waals theory developed by Gelbart and Ben-Shaul [40], which itself embraces a number of earlier models for the elasticity of nematic liquid crystals. Corresponding theories for smectic, columnar and biaxial phases have yet to be developed. 

The statistical theories of elasticity have shown that the principal elastic constants depend on the single particle distribution functions and the intermolecular forces. The former can be accounted for in terms of order parameters, but intermolecular parameters are more diffieult to interpret in terms of molecular properties. Results for hard particle potentials relate the elastic constants to particle dimensions, but the depen-... 

It is also possible to estimate the cross-link density from the stress-strain data, using the statistical theory of rubber-like elasticity [47,58]. For a swollen rubber the relationship is... 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

The statistical theory of crosslinking used in the last section also gives the theoretical concentration of elastically-active chains, N, which in turn determines the rubbery modulus E = 3NRT (R is the gas constant and T is the absolute temperature). At 70% reaction one calculates E - 2 x 10 dyn/cm1 2 3 4 5 6 7 8 9 10, in agreement with the apparent level in Figure 1. 

Simultaneous IPN. According to the statistical theory of rubber elasticity, the elasticity modulus (Eg), a measure of the material rigidity, is proportional to the concentration of elastically active segments (Vg) in the network [3,4]. For negligible perturbation of the strand length at rest due to crosslinking (a reasonable assumption for the case of a simultaneous IPN), the modulus is given by ... 

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

In the current statistical theory of rubber elasticity, it is suggested that the front-factor molecular forces. They have proposed a semiempirical equation of state taking into account the dependence... 

In other statistical theories of rubber elasticity (see e.g. reviews 29,34)) the Gaussian statistics is not valid even at small deformations and the intramolecular energy component is dependent on deformation. 

The statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

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. 

Kvasnikov, I. A. The application of Ising s model in the statistical theory of high elasticity. Vysokomolekul. Soedin. 3, 1617 (1961). 

Landau LD, Lifshitz EM (1969) Statistical physics, 2nd edn. Oxford, Pergamon Landau LD, Lifshitz EM (1987a) Fluid mechanics, 2nd edn. Oxford, Pergamon Landau LD, Lifshitz EM (1987b) Theory of Elasticity, 3rd edn. Oxford, Pergamon Landau LD, Lifshitz EM, Pitaevskii LP (1987) Electrodynamics of continuous media, 2nd edn. Oxford, Pergamon... 

The average length (or molecular weight) of network chains in a crosslinked polymer can be experimentally determined from the equilibrium rubbery modulus. This relationship is a direct result of the statistical theory of rubber-like elasticity . In the last decade or so, modem theories of rubber-like elasticity 2127) further refined this relationship but have not altered its basic foundation. In essence, it is... 

Stress-strain properties for unfilled and filled silicon rubbers are studied in the temperature range 150-473 K. In this range, the increase of the modulus with temperature is significantly lower than predicted by the simple statistical theory of rubber elasticity. A moderate increase of the modulus with increasing temperature can be explained by the decrease of the number of adsorption junctions in the elastomer matrix as well as by the decrease of the ability of filler particles to share deformation caused by a weakening of PDMS-Aerosil interactions at higher temperatures. 

The presence of filler in the rubber as well as the increase of the surface ability of the Aerosil surface causes an increase in the modulus. The temperature dependence of the modulus is often used to analyze the network density in cured elastomers. According to the simple statistical theory of rubber elasticity, the modulus should increase twice for the double increase of the absolute temperature [35]. This behavior is observed for a cured xmfilled sample as shown in Fig. 15. However, for rubber filled with hydrophilic and hydrophobic Aerosil, the modulus increases by a factor of 1.3 and 1.6, respectively, as a function of temperature in the range of 225-450 K. It appears that less mobile chain units in the adsorption layer do not contribute directly to the rubber modulus, since the fraction of this layer is only a few percent [7, 8, 12, 21]. Since the influence of the secondary structure of fillers and filler-filler interaction is of importance only at moderate strain [43, 47], it is assumed that the change of the modulus with temperature is mainly caused by the properties of the elastomer matrix and the adsorption layer which cause the filler particles to share deformation. Therefore, the moderate decrease of the rubber modulus with increasing temperature, as compared to the value expected from the statistical theory, can be explained by the following reasons a decrease of the density of adsorption junctions as well as their strength, and a decrease of the ability of filler particles to share deformation due to a decrease of elastomer-filler interactions. 

In contrast to the filled samples, the deformation energy for the unfilled ones increases proportionally to the increase in the absolute temperature according to the prediction of the simple statistical theory of rubber elasticity. Thus, it appears that the change of the modulus and the deformation energy with increasing temperature reveals a decrease of the density of adsorption junctions in the elastomer matrix, as well as a decrease of the ability of filler particles to share deformation, resulting from a weakening of elastomer-filler interactions. 

This, model contradicts with the statistical theory of rubber elasticity, and the artificial assumption oia = 3k TJb has been used. 

Statistical Mechanics of Elasticity by J. H. Weiner, John Wiley Sons, New York New York, 1983. Weiner s book has a number of interesting and useful insights into the meeting point between continuum mechanics and microscopic theories. 

Filled Rubbers and the Statistical Theory of Rubber Elasticity. . . 186... 

There exists as yet no rigorous extension of the statistical theory of rubber elasticity to a filled elastomer. Nevertheless, many attempts have been made to apply the theory to data on filled rubbers, usually with the objective of obtaining at least an approximate estimate of the number of filler-rubber attachments. These attempts have been discussed in earlier reviews (17, 126) and will not be considered here in full detail. We only restate briefly some of the experimental and theoretical difficulties inherent in this approach. 

In the formulation of the statistical theory of rubber elasticity,5 11 the following simplifying assumptions are made ... 

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