Rubber-like elasticity theory

Rubber-like Elasticity Theory and Highly Crosslinked Epoxies.120... 

Thus, the level of sophistication which one may consider for the application of rubber-like elasticity theory to epoxy networks may depend on the application. For highly crosslinked systems (M < 1,000), a quantitative dependence of the rubbery modulus on network chain length has recently been demonstrated , but the relevance of higher order refinements in elasticity theory is questionable. Less densely crosslinked epoxies, however, are potentially suitable for testing modern elasticity theories because they form via near quantitative stepwise reactions. Detailed investigations of such networks have been reported by Dusek and coworkers in recent studies ... 

Reactivity, independent 21 Relative rigidity 90, 94 P-Relaxation 139 Rubber-like elasticity theory 118 —modified epoxy resins 99 Rubbery cohesive fracture 125 — fracture energies 143, 146... 

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 gelation process that leads to the network structures required for rubber-like elasticity have been extensively studied, by experiments, theory, and simulations.245-249 In some case, the gelation can be made to be reversible.250... 

Priss LS (1981) Molecular origin of constants in the theory of rubber-like elasticity considering network chains steric interactions. J Pure Appl Chem 53 1581-1596 Priss LS, Gamlitski YuA (1983) Mechanism of conformation transitions in polymer chains. Polym Sci USSR 25 117-123... 

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... 

The mean-squared, end-to-end distance in Equation [9] is the simplest average property of interest for a polymer chain. Among other physical properties, this quantity appears in the equations of statistical mechanical theories of rubber-like elasticity. In Equation [9], the angle brackets denote the ensemble (or time) average over all possible conformations. The subscript 0 indicates that the average pertains to an unperturbed chain (theta conditions no excluded volume effects are present). (See Figure... 

C. Menduina, C. McBride, and C. Vega (2001) Correctly averaged Non-Gaussian theory of rubber-like elasticity - application to the description of the behavior of poly(dimethylsiloxane) bimodal networks. Phys. Chem,. Chem. Phys. 3, p. 1289... 

According to the theory of rubber-like elasticity, the relationship between the stress, f, and the elongation is... 

The calculation of the elastic term is based on the available theories of rubber-like elasticity, which calculate the change of elastic free energy AFg/ due to dilation X. The elastic contribution to the chemical potential is then given by... 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

M. Shen, W. F. Hall, and R. E. De Wames, Molecular theories of rubber-like elasticity and... 

Despite of the approximations, the statistical theory is of fundamental significance for understanding of the molecular mechanisms causing rubber-like elasticity. It serves as a starting point for generalizations that agree more precisely with experiments. One generalization is the Mooney-Rivlin equation. After Equation (35), we have ... 

There are several molecular theories of rubber-like elasticity The simplest one is based on a Gaussian distribution function for the end to end separation of the network chains [23] (the dimensions of the free chains as unperturbed by excluded volume effect are represented by (r )o)... 

Applying the kinetic theory of rubber-like elasticity (62,109) to the entanglement network one can determine the molecular weight between entanglements Me from the plateau compliance Jn or plateau modulus Gn (62) ... 

The Theory of Kuhn and Grun. The theory of birefringence of deformed elastomeric networks was developed by Kuhn and Griin and by Treloar on the basis of the same procedure as that used for the development of the classical theories of rubber-like elasticity (48,49). The pioneering theory of Kuhn and Griin is based on the affine network model that is, upon the application of a macroscopic deformation the components of the end-to-end vector for each network chain are assumed to change in the same ratio as that of the corresponding dimensions of the macroscopic sample. 

Birefringence of Phantom Networks. This theory is the basis for all theories that deal with birefringence of elastomeric polymer networks. It is based on the phantom network model of rubber-like elasticity. This model considers the network to consist of phantom (ie, non-interacting) chains. Consider the instantaneous end-to-end distance r for the ith network chain at equilibrium and at fixed strain. For a perfect (ie, no-defects) phantom network the birefringence induced... 

Di-tert-alkyl and di-tert-aryl peroxides (e.g., di-tert-butyl peroxide and dicumyl peroxide) find some use in the production of heat-stable vulcanizates of natural rubber. However, the particular significance of these reagents is that their mode of reaction is rather straightforward and it has been possible to establish a direct quantitative relationship between the number of cross-links in a vulcanizate and its elastic modulus. This correlation is important to the theory of rubber-like elasticity. 

There are several important postulates that have been used in the development of the molecular theories of rubber-like elasticity [9]. 

The simplest molecular theories of rubber-like elasticity are based on the Gaussian distribution function... 

In summary, the anomalous upturn in modulus observed for crystallizable polymers such as natural rubber and cw-1,4-polybutadiene is largely, if not entirely, due to strain-induced crystallization. In the case of the noncrystaUizable PDMS model networks it is clearly due to the limited chain extensibility, and thus the results on this system will be extremely useM for reliable evaluation of the various non-Gaussian theories of rubber-like elasticity. 

Experimental results on networks of natural rubber in shear deformation [134] are not well accounted for by the simple molecular theory of rubber-like elasticity. The constrained-junction theory, however, was found to give excellent agreement with experiment. Shear measurements have also been reported for some unimodal and... 

In addition to timescale shifts with temperature, the magnitude of the compliance or modulus can change. The kinetic theory of rubber-like elasticity suggests that the entropically based contribution of the modulus to the viscoelastic response should increase in direct proportion to the absolute temperature. Correspondingly, the reciprocal of the steady-state recoverable compliance should be directly proportional to the absolute temperature. This is true at temperatures that are greater than 2Tg, but, between 1.2Tg and 2Tg, the steady-state recoverable compliance Js is essentially independent of temperature. At still lower temperatures a strong decrease of Js, is seen [51]. 

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