Extension ratios

Figure C2.1.16. Tensile stress as a Hmction of the extension ratio registered for a sample of natural mbber (circles). The broken curve is calculated from equation (C2.1.20). (Data from [79].)...

The compositions were 60 40 and 50 50 rubber-plastic weight ratio. The stress at the break is the product of the ultimate extension ratio. 

The average force f(r) in the chain when the ends are held a distance r apart could then be obtained from Eq. (10) providing the appropriate configuration distribution function p(r) is known. In the limit of a small extension ratio, p(r) is approximately proportional to peq(r) ... 

Equation (13) is valid for r/Nlp < 0.25 (Fig. 3). At much higher extension ratios, the force must increase indefinitely since the molecule is almost straightened out. The thermodynamic approach to the problem of coil stretching for a freely-jointed chain was considered by Treloar [32], who obtained the following expression for the stress-strain relationship when the two chain ends are kept a distance r apart ... 

Derived from molecular arguments, Eq. (14) is correct for any extension ratio of the freely-jointed chain. In spite of its generality, the use of Eq. (14) is limited due to mathematical complexity. To account for the finite extensibility of the chain, the approximate finitely extensible nonlinear elastic (FENE) law proposed by Warner has gained popularity due to its ease of computation [33] ... 

From the standpoint of thermodynamics, the essential quantity which governs the course of a chemical reaction is the chemical potential of the system or, in the case of interest, the free energy storage within the molecular coil. This quantity, unfortunately, is difficult to evaluate in non-steady flow. At modest extension ratios (X < 4), the free energy storage of a freely-jointed bead-spring chain is... 

It is also comparatively straightforward to-calculate P200, P220, P420 and P o for a biaxially oriented aggregate of transversely isotropic units in terms of the principal extension ratios Xx, X2 and (with X,X2 3 = 1). 

Equation (32a) has been very successful in modelling the development of birefringence with extension ratio (or equivalently draw ratio) in a rubber, and this is of a different shape from the predictions of the pseudo-affine deformation scheme (Eq. (30a)). There are also very significant differences between the predictions of the two schemes for P400- In particular, the development of P400 with extension ratio is much slower for the network model than for the pseudo-affine scheme. 

A rubber network which is deformed to three independent extension ratios Xt, X2 and X3 takes the form of a biaxially oriented aggregate defined by coefficients P,mo. For example, we have... 

Me the molecular mass between cross-links M the primary molecular mass (28,492 Da) a the extension ratio... 

The state of macroscopic deformation may be characterized by considering the deformation of a rectangular prism, with extension ratios Xx, Xz, along the x, y,... 

The elastic free energy given by the elementary and the more advanced theories are symmetric functions of the three extension ratios Xx, Xy, and Xz. One may also express the dependence of the elastic free energy on strain in terms of three other variables, which are in turn functions of Xx, Xy, and Xz. In phenomenological theories of continuum mechanics, where only the observed behavior of the material is of concern rather than the associated molecular deformation mechanisms, these three functions are chosen as... 

X-ray diffraction pictures taken with a flat-film camera show that crosslinked SE-BR samples crystallize on stretching. Sharp reflections are observed at an extension ratio of 4 1 (Figure 4). With samples having different degrees of stereoregularity the order for increasing strain-induced crystallization is the same as the order for the rate of low temperature crystallization. 

Figure 10. X-ray diffraction pattern of BPR, gum vulcanizate. Extension ratio,...

Figure 19 shows the temperature dependence of the percent crystallinity for high trans SBR, prepared with a Ba-Li catalyst and containing 757. trans-1,4 content with 14 wt.7. styrene, at 3 extension ratios. The percent crystallinity that develops is temperature dependent, there being an increase in the amount of crystallinity with a decrease in temperature. However, the amount of crystallinity that develops is essentially independent of strain. The amount of crystallinity that develops at room temperature, regardless of the level of strain, is extremely small ( 9) ... 

Figure 20 shows the percent crystallinity as a function of temperature and extension ratio for high trans SBR (227. styrene, 877. trans) prepared with a Ba-Mg-Al catalyst. A... 

The main conclusions of the strain induced crystallization behavior of high trans polybutadiene based rubber and natural rubber are (1) the rate of crystallization is extremely rapid compared to that of NR (2) the amount of strain induced crystallization is small compared to that of NR, especially at room temperature and (3) for the high trans SBR s relative to NR, crystallization is more sensitive to temperature at low extension ratios, and crystallization is less sensitive to strain. 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

Figure 3. Modulus contributions from chemical cross-links (Cx, filled triangles) and from chain entangling (Gx, unfilled symbols) plotted against the extension ratio during cross-linking, A0, for 1,2-polybutadiene. Key O, GN, equibiaxial extension , G.v, pure shear A, Gx, simple extension Gx°, pseudo-equilibrium rubber plateau modulus for a polybutadiene with a similar microstructure. See Ref. 10.

The stress relaxation properties of a high molecular weight polybutadiene with a narrow molecular weight distribution are shown in Figure 1. The behavior is shown in terms of the apparent rubber elasticity stress relaxation modulus for three differrent extension ratios and the experiment is carried on until rupture in all three cases. A very wide rubber plateau extending over nearly 6 decades in time is observed for the smallest extension ratio. However, the plateau is observed to become narrower with increasing extension... 

Figure 1. Stress relaxation curves for three different extension ratios. Uncross-linked high-vinyl polybutadiene with a weight average molecular weight of 2 million and a reference temperature of 283 K. G is the apparent rubber elasticity modulus calculated from classical affine theory. (Solid line is data from Ref. 1).

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