Classical theory of rubber elasticity

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

For the free energy of elastic deformation of network chains, Fel, the following expression of the classical theory of rubber elasticity can be used [21] ... 

During the last decade, the classical theory of rubber elasticity has been reconsidered significantly. It has been demonstrated (see, e.g. Ref.53>) that, for the phantom noninteracting network whose chains move freely one through the other, the equations of state of Eqs. (28) and (29) for simple deformation as well as for W, Q and AIJ [Eqs. (30)-(32) and (35)—(37)] are proportional not to v but to q, which is the cycle rank of the network, i.e. the number of independent circuits it contains. For a perfect phantom network of uniform functionality cp( > 2)... 

Usually the deformation behaviour of rubbery crosslinked polymers with long chains between crosslinks obeys the classical theory of rubber elasticity. However, the situation for densely crosslinked polymers is not so simple. 

These theories are based on the classical theories of rubber elasticity of macromolecular solids, wherein permanent chemical crosslinks connect segments of molecules, forcing them to move together. This central idea can be applied to polymeric liquids. However in this case, the interactions between molecules are assumed to be localized at junctions and are supposed to be temporary. Whatever their nature, physical or topological, these crosslinks are continually created and destroyed but, at any time, they ensure sufficient connectivity between the molecules to give rise to a certain level of cooperative motion. 

In this work we used polystyrene-based ionomers.-Since there is no crystallinity in this type of ionomer, only the effect of ionic interactions has been observed. Eisenberg et al. reported that for styrene-methacrylic acid ionomers, the position of the high inflection point in the stress relaxation master curve could be approximately predicted from the classical theory of rubber elasticity, assuming that each ion pah-acts as a crosslink up to ca. 6 mol %. Above 6 mol %, the deviation of data points from the calculated curve is very large. For sulfonated polystyrene ionomers, the inflection point in stress relaxation master curves and the rubbery plateau region in dynamic mechanical data seemed to follow the classical rubber theory at low ion content. Therefore, it is generally concluded that polystyrene-based ionomers with low ion content show a crosslinking effect due to multiplet formation. More... 

For the free energy Fel of elastic deformation of the polymer gel the expression of classical theory of rubber elasticity modified by Birshtein is... 

The deformation ability of networks strongly swollen with benzene and those slightly swollen in cyclohexane was unexpectedly found to be the same. What is surprising here is the absence of any correlation between the volume increase of model networks on swelling and their deformation under compression or elongation [130], as it would have to foUow from the classic theory of rubber elasticity. This theory does not predict any difference between the extensional modulus and the shear modulus that controls the swelling. Nevertheless, the experimental ratio of Ce(CH)/ Ce(BZ) = 6 is twice as large as the ratio of E(CH)/E(BZ) = 3 (irrespective ofp) [123]. 

Samulski [90, 91] described these interactions in the mean field approximation by an additional intermolecular potential from the classical theory of rubber elasticity. A similar expression is proposed for the elastic free energy. 

Affine deformation This model assumes that the deformation of each configuration of the chains is affine in the macroscopic deformation. It is not compatible with known classical theories of rubber elasticity. 

The presence of positive cooperativity, explained by the interconversion between different conformational states, also demonstrates that these entropic elastic model proteins are not properly described as random chain networks, as the adherents of the relevance of the classical theory of rubber elasticity to protein elasticity are compelled to argue. 

The classical theories of rubber elasticity rest on two basic assumptions [4] ... 

SBR elastomer with known crosslinking densities was studied in dynamic shear and tensile creep and data collected from -30 to 70 °C used to construct TTS master curves. In addition to a temperature shift factor a vertical shift factor was required from 10 to 30 °C to account for changes in density. Linear viscoelastic properties were observed in accordance with the classical theory of rubber elasticity. Standard vertical shift factors were required in a comparative TTS test with uncrosslinked polybutadiene and poly(ethylene-cu-propylene-co-diene monomer) (EPDM). ... 

The molecular theory of rubber elasticity on the basis of affine deformation assumption is the affine network theory, or the classical theory of rubber elasticity. 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

In the Doi-Edwards theory the environment of a chain is modeled as a tube with a diameter which is constant over the tube length. Each subchain, which is the part of the polymer chain between localized entanglements, resides in a tube segment. The subchains react to an instantaneous strain affine deformation, as in the classical theory of rubber elasticity. Therefore, immediately after the imposition of a simple elongational strain at t 0, the stress is given by Eq. (6), with G 0 Se(X) is given by Eq. (8), and, in analogy with Eq. (9), the relaxational part of the modulus at t = 0 equals ... 

An ideal elastomer is a network of amorphous segments crosslinked at specific sites along the polymer chain. This network obeys the classic theory of rubber elasticity as described by Flory (28) and may be represented as follows. 

According to the classical theory of rubber elasticity [41, p. 234], the equilibrium shear modulus for infinitesimal deformations is ... 

The classical theory of rubber elasticity as used for calculating the mechanical properties of swollen and non-swollen, cross-linked, non-crystalline polymeric materials, is essentially based on a model of a random collection of macromolecular chains which are macroscopically homogenously cross-linked. Although this model has been rather successful, it has become quite evident in recent years, that such systems are not entirely disordered. 

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