Stress-extension/strain curves

Show that from measurements of the stress versus strain curve and the force required to keep a rubber at constant extension as temperature is varied (and pressure is held constant) (dH/dl)T p, (dS/dl)T p, and (dG/dl)T p can he obtained. (Hint the F function is useftd for this problem.)... 

Fig. 2. Rubbery tensile stress versus strain curves for DGEBA/DDS epoxy networks at T = T, + 45 K. Ordinate (true stress) normalized by 3 cRT. O Extension data Recovery data. After LeMay >)...

Figure 7 A typical load/extension (stress versus strain) curve for a yam from the Victory sail. Typically, a test length of 2.5cm was set...

In a plot of (T, against A (Figure 53) yield will occur according to eqn 5.7 at point M that is to say the engineering stress-strain curve will show a maximum only if a tangent can be drawn from A => 0 to touch the true stress-extension ratio curve at a point such as M. 

Fig. 4. Strain-energy density is given by the area under a stress-extension ratio curve...

Fig. 3.21 Stress versus strain curves a for tj-SiC samples with varying chemical disrader at a strain rate of 10 s . b Stress versus strain curves for 3C-SiC, nc-SiC and melt-quenched a-SiC with an extension rate of 100 m/s [18, 29]. (dpa is displacement per atom)...

Fig. 2. Nominal stress versus strain curves for S-B-S triblock copolymer (Kraton 101) showing mechanical hysteresis (upper curve) and extension to break (lower curve) at 220°C and constant strain rate e = 0.444 min . ...

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

Fig. 6. The effect of rate of extension on the stress—strain curves of rayon fibers at 65% rh and 20°C. The numbers on the curves give the constant rates of...

Using both condensation-cured and addition-cured model systems, it has been shown that the modulus depends on the molecular weight of the polymer and that the modulus at mpture increases with increased junction functionahty (259). However, if a bimodal distribution of chain lengths is employed, an anomalously high modulus at high extensions is observed. Finite extensibihty of the short chains has been proposed as the origin of this upturn in the stress—strain curve. 

Stress—Strain Curve. Other than the necessity for adequate tensile strength to allow processibiUty and adequate finished fabric strength, the performance characteristics of many textile items are governed by properties of fibers measured at relatively low strains (up to 5% extension) and by the change ia these properties as a function of varyiag environmental conditions (48). Thus, the whole stress—strain behavior of fibers from 2ero to ultimate extension should be studied, and various parameters should be selected to identify characteristics that can be related to performance. 

Figure 10.6. Effect of temperature on the tensile stress-strain curve for polyethylene. (Low-density polymer -0.92g/cm . MFI = 2.) Rate of extension 190% per minute ...

The phenomena of brittle and tough fracture give rise to fairly characteristic stress-strain curves. Brittle fracture in materials leads to the kind of behaviour illustrated in Figure 7.1 fairly uniform extension is observed with increasing stress, there is minimal yield, and then fracture occurs close to the maximum on this graph. 

Density is also found to increase in this region, thus providing additional evidence of crystallisation. Certain synthetic elastomers do not undergo this strain-induced crystallisation. Styrene-butadiene, for example, is a random copolymer and hence lacks the molecular regularity necessary to form crystallites on extension. For this material, the stress-strain curve has a different appearance, as seen in Figure 7.12. 

Figure 18.1 is the typical stress-strain curves of the filled rubber (SBR filled with fine carbon black, HAF),

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

Nevertheless, it is obviously shown in Figure 18.1 that the stress-strain curve of the filled mbber gives the clear stress upturn, thus its tensile strength becomes 30 MPa. Therefore, the fundamental question is what happens or what stmcture is produced in the carbon black-filled mbber under large extension, which newly generates the stress upturn. In the case of the fine carbon black-fiUed system, when carbon blacks are dispersed ideally, the carbon gel makes the continuous phase at the... 

It has often been pointed out for a long time that the hysteresis energy given from the hysteresis loop under large extension is too big compared with the viscoelastic dissipation energy. For example, the hysteresis loop given from the stress relaxation is only 20%-30% of that from the stress-strain curve, when both measurements are performed at the same relaxation time and the... 

FIGURE 21.3 Concept of nanofishing, (a) Schematic drawing and (b) force-extension (stress-strain) curve. 

Figure 1. Stress-time data from stress-strain curves measured in simple tension at 30°C on the LHT-240 polyurethane elastomer at seven extension rates, A from 9.4 X t° 9.4 min 1. Key 0,9, stress as a function of time ( — 1)/X, at the indicated values of strain, ( — 1).

Figure 4 shows stress-strain curves measured at an extension rate of 94% per minute on the TIPA elastomer at 30°, —30°, and —40°C. With a decrease in temperature from 30° to -40°C, the ultimate elongation increases from 170% to 600%. The modulus Ecr(l), evaluated from a one-minute stress-strain isochrone, obtained from plots like shown in Figure 1, increases from 1.29 MPa at 30°C to only 1.95 MPa at —40°C. This small increase in the modulus and the large increase in the engineering stress and elongation at fracture results from viscoelastic processes. 

Figure 4. Stress-strain curves for the TIP A polyurethane elastomer measured at the indicated temperatures at an extension rate of 0.94 min 1. Arrows indicate...

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

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