Rubbers ideal

Id. The Ideal Rubber.—The data available at present as summarized above show convincingly that for natural rubber (dE/dL)T,v is equal to zero within experimental error up to extensions where crystalhzation sets in (see Sec. le). The experiments of Meyer and van der Wyk on rubber in shear indicate that this coefficient does not exceed a few percent of the stress even at very small deformations. This implies not only that the energy of intermolecular interaction (van der Waals interaction) is affected negligibly by deformation at constant volume—which is hardly surprising inasmuch as the average intermolecular distance must remain unchanged—but also that con-... 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

Equations (44), (45), and (45 ) are alternative expressions for the theoretical equation of state for an ideal rubber. Equations of state for swollen networks, derived in Appendix B, have the same form. Only the magnitude predicted for r at a given elongation is affected by swelling. 

The theoretical equation of state for an ideal rubber in tension, Eq. (44) or (45), equates the tension r to the product of three factors RT, a structure factor (or re/Eo, the volume of the rubber being assumed constant), and a deformation factor a—l/a ) analogous to the bulk compression factor Eo/E for the gas. The equation of state for an ideal gas, which for the purpose of emphasizing the analogy may be written P = RT v/Vq) Vq/V), consists of three corresponding factors. Proportionality between r and T follows necessarily from the condition dE/dL)Ty=0 for an ideal rubber. Results already cited for real rubbers indicate this condition usually is fulfilled almost within experimental error. Hence the propriety of the temperature factor... 

In this equation e is the longitudinal strain and er is the strain in the width (transverse) direction or the direction perpendicular to the applied force It can be shown that when Poisson s ratio is 0.50, the volume of the specimen remains constant while being stretched. This condition of constant volume holds for liquids and ideal rubbers. In general, there is an increase in volume, which is given by... 

Show that the corresponding equation for an ideal rubber band is... 

The first term (dE/d/) in Equation 9.43 is important in the initial low modulus stretching process, and the second term [T (dG/d/)] predominates in the second high modulus stretching process. For an ideal rubber, only the second term is involved. 

Poly-isobutylene (PIB) is a very useful rubber because of its very low gas permeability. Co-polymerised with small amounts of isoprene (to enable vulcanisation with sulphur) to butyl rubber (HR), it is the ideal rubber for inner tubes. If PIB would crystallise, it could not be used as a technical rubber The same holds for the rubbers BR and IR. 

For an ideal network T is in the numerator of the formula for E, so that E is proportional to the absolute temperature. The log E - T curve thus shows a positive slope (not a straight line because of the log-scale but slightly curved upward). In reality this simple picture is often disturbed by deviations from ideal rubber-elastic behaviour. 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

Ideally, rubber toughening should be accomplished without substantial sacrifices in modulus. For each modified resin, flexural and Young moduli and plane-strain fracture toughness were determined. Examination of various fracture surfaces by scanning electron microscopy showed the effects of modifier composition on the morphology of these multi-phase materials as well as the prominent features of the fracture process. 

In a similar manner, for an ideal rubber band with... 

The above relationships result from an approach based on at least eight hypotheses that define an ideal rubber. Thermosets are generally far from the ideal case for the following reasons. 

Despite the preceding remarks and the fact that thermosets are far from being ideal rubbers, the basic rubber elasticity theory works surprisingly well in most practical cases, as illustrated by the data of Table 10.9. 

An ideal rubber is defined as one for which I —-1 = 0. What does this definition... 

Derive a formula for the entropy change for stretching an ideal rubber for which... 

The force resisting extension for an ideal rubber is completely entropically derived. Because it certainly requires a positive force to stretch a rubber,... 

Show that the first law of thermodynamics requires that heat be evolved when an ideal rubber is stretched. 

Show that a rubber for which the force required to achieve a given elongation (at a particular pressure) is proportional to the temperature is an ideal rubber. 

The last equality follows for an ideal rubber, in which, by definition, the internal energy does not depend on the distortion. 

Fully reversible negative expansion is shown by a strained ideal rubber, as a result of the entropy-elasticity, discussed in 5.1. 

Stress relaxation is the time-dependent change in stress after an instantaneous and constant deformation and constant temperature. As the shape of the specimen does not change during stress relaxation, this is a pure relaxation phenomenon in the sense defined at the beginning of this section. It is common use to call the time dependent ratio of tensile stress to strain the relaxation modulus, E, and to present the results of the experiments in the form of E as a function of time. This quantity should be distinguished, however, from the tensile modulus E as determined in elastic deformations, because stress relaxation does not occur upon deformation of an ideal rubber. 

The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation ... 

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