Strains extension ratios

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 ... 

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 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) ... 

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. 

PP bead foams were subjected to oblique impacts (167), in which the material was compressed and sheared. This strain combination could occur when a cycle helmet hit a road surface. The results were compared with simple shear tests at low strain rates and to uniaxial compressive tests at impact strain rates. The observed shear hardening was greatest when there was no imposed density increase and practically zero when the angle of impact was less than 15 degrees. The shear hardening appeared to be a unique function of the main tensile extension ratio and was a polymer contribution, whereas the volumetric hardening was due to the isothermal compression of the cell gas. Eoam material models for FEA needed to be reformulated to consider the physics of the hardening mechanisms, so their... 

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). 

One of the simplest criteria specific to the internal port cracking failure mode is based on the uniaxial strain capability in simple tension. Since the material properties are known to be strain rate- and temperature-dependent, tests are conducted under various conditions, and a failure strain boundary is generated. Strain at rupture is plotted against a variable such as reduced time, and any strain requirement which falls outside of the boundary will lead to rupture, and any condition inside will be considered safe. Ad hoc criteria have been proposed, such as that of Landel (55) in which the failure strain eL is defined as the ratio of the maximum true stress to the initial modulus, where the true stress is defined as the product of the extension ratio and the engineering stress —i.e., breaks down at low strain rates and higher temperatures. Milloway and Wiegand (68) suggested that motor strain should be less than half of the uniaxial tensile strain at failure at 0.74 min.-1. This criterion was based on 41 small motor tests. 

Other interesting features of elastomeric networks can be revealed using the plots of the reduced stress, crred = /( — -2) against inverse extension ratio 1. This can be analyzed from the stress-strain behavior described by a phenomenological expression suggested by Mooney [78] and Rivlin and Saunders [79] ... 

It should be considered that in the case of plotting 1 = a/ X — X 2) against inverse extension ratio (X 1), the nominal stress a is defined as the force divided by the undeformed cross-sectional area of the sample and X is the extension ratio, defined as the ratio of deformed to the undeformed length of the sample stretched in the uniaxial direction (as shown in Fig. 3). For PTFE powder, the intrinsic strain is deduced from (2) by defining X - 1 ... 

In their study of the NMR T2 and T2 of crosslinked cis-polyisoprene sheets under extension, von Meerwall and Ferguson 65) found that T2 of the rubber had much smaller anisotropy ( magic angle effect) than that of trace penetrants at the same extension ratio X < 3. However, the penetrant diffusion (referred to the strained dimensions) was within experimental error isotropic these findings are equally valid for C6F6 and n-hexadecane as penetrant. The authors concluded that segment orien-... 

Valanis,K.C., Landel,R.F. The strain-energy function of a hyperelastic material in terms of extension ratios. J. Appl. Phys. 38,2997-3002 (1967). 

Equation (55) differs from Eq. (39) by the term (—yaT) and hence (AU/W)V T should be independent of extension ratio X, since (—yaT) is not a function of deformation. Equation (54) leads also to the following expression for the strain-induced volume dilation... 

It is very important to stress that the decrease of the internal energy contribution with increasing extension ratio is due to a decrease of the intermolecular interaction with deformation, since the intramolecular contribution is independent of the deformation in full accord with the statistical theory. At very high strains, the /.-dependent part of fu/f approaches a limiting value of —0.68 for the Mooney-Rivlin and —0.07 for the Valanis-Landel materials. 

Statistical network theory leads to the expression of the strain energy density (energy stored in unit volume of the rubber) in terms of the extension ratios ... 

It should be noted that the stresses usually used are engineering stresses calculated from the ratio of force and original cross section area whereas true stress is the ratio of the force and the actual cross sectional area at that deformation. Clearly, the relationship between stress and strain depends on the definition of stress used and taking the case of tensile strain, for example, the true stress is equal to the engineering stress multiplied by the extension ratio. 

W can be obtained by extending a test piece without a nick and plotting a stress/strain curve, W being derived from the area under the stress/strain curve up to the extension ratio. 

Meier (9) has modeled the spherical domain morphology by a simple cubic lattice in which domains are arranged on the lattice sites. The tie molecules run between nearest-neighbor domains and are assumed to be confined by pairs of infinite, parallel walls. The extension ratio for the interdomain region is set equal to the macroscopic extension ratio divided by the volume fraction of the interdomain material. The ratio of the initial interdomain dimension to the domain dimension is set equal to the ratio of the volume fractions of the interdomain and domain material. Using this three-chain model, Meier calculates the stress-strain relation by differentiating his entropy expression with respect to the interdomain extension ratio. The Meier calculation has some difficulties the interdomain deformation fails to vanish in the absence of an applied macroscopic deformation the relation between the ratio of the domain dimension to the initial interdomain dimension and the ratio of volume fractions is incorrect and the differentiation should be carried out with respect to the macroscopic extension ratio. 

In rubber elasticity, it is usual to use extension ratios rather than strain. If a block of robber is stretched parallel to a single axis, such that its length changes from lQ to / (Figure 13-54), the extension ratio is defined to be (Equation 13-41) ... 

Consider a cube of cross-linked elastomer with unit dimensions. This specimen is sub jected to a tensile force /. The ratio of the increase in length to the unstretched length is the nominal strain e (epsilon), but the deformation is sometimes also expressed as the extension ratio A (lambda) ... 

PS, and in the range 142-155 °C for PC. The following mechanical test has been adopted the specimens are stretched at constant elongational strain rate (0.05 s l for PS and 0.025 s l for PC) up to a final extension ratio L/Lq close to 2.5 for all of the samples. The stress is then allowed to relax at constant deformation. 

To this point the craze fibril volume fraction Vf and fibril extension ratio X have discussed as if they were true constants of the craze. While this view is approximately correct, one would expect the draw ratio of the polymer fibrils to depend somewhat on the stress at which they are drawn, since the polymer in the fibrils should have a finite strain hardening rate. Experimental evidence for just such stress effects on X, is discussed below. 

Correlation function Elongational strain Strain rate Craze strain Elastic strain Shear strain Interfacial length Extension ratio... 

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