Modulus strain curves

FIGURE 2.41 Stress-strain and modulus-strain curves for steam-treated drawn nylon-6 filaments. (Solid curve.) Untreated filaments (a) constant filament length (b) free shrinkage. (From Schultz-Gebhart, F., Faserf. Textiltechn., 1911, 28, 467. With permission.)... 

Mod-2 second maximum of the modulus-strain curve (mN/tex) El-mod-2 elongation at the second maximum of the modulus-strain curve (%)... 

The first five mechanical properties of the set mentioned before are related to the stress-strain behavior of the yarns. In Fig. 2b, the first derivative of the stress-strain curve (i.e., the modulus-strain curve) was given. Clearly, two maxima are seen that can be attributed to two different mechanisms. According to unpublished work [9], the first maximum can be related to straining of an entanglement network in the sense of Ball et al. [10] (see Fig. 16). After breaking up the entanglements, the tie molecules proper, connecting... 

In the second part of the modulus-strain curve, the tie molecules are directly addressed. The strain at which the maximum occurs can vary considerably, ranging in this series from 3% to 14%. For a maximum, positioned at a high strain level, the force is built up very gradually and the increase of force per unit of increase of strain, being the modulus, is low. This is illustrated in Fig. 17. Therefore, a low orientation, which is the main reason for a high... 

Elongation at the Second Maximum of the Modulus-Strain Curve El-Mod-2)... 

Of the elongation at which the second peak in the modulus-strain curve is found, only the two major contributions are well understood. As described earlier for Mod-2, we assume that the maximum is related to fracture of the shortest tie molecules. The combination of a low level of orientation and a narrow distribution provides the situation where the shortest tie molecules have, relatively, the greatest contour length, as illustrated by the asterisk in Fig. 19. Therefore, the first fracture of tie molecules can be expected to occur at relatively high strain values for structures with low orientations combined with narrow tie chain-length distributions. 

Figure 2.2 shows the tensile stress-strain curves of the NBR [2], where the measurements were made with three (high) speeds and at the three temperatures. The same data are replotted in Figure 4.1 as the modulus-strain curves. This conversion is made because the modulus is the material property and the stress is only the manifestation of the property. The... 

Figure 4.1 Tensile modulus-strain curves of NBR sample B.

Modulus-strain curves of another sample of NBR are shown in Figure 4.3 [2]. The shapes of the curves are different from that in Figure 4.1. In Figure 4.3, the modulus is shown to increase with the increase of strain, i.e., strain-hardening. 

The ratio of stress to strain in the initial linear portion of the stress—strain curve indicates the abiUty of a material to resist deformation and return to its original form. This modulus of elasticity, or Young s modulus, is related to many of the mechanical performance characteristics of textile products. The modulus of elasticity can be affected by drawing, ie, elongating the fiber environment, ie, wet or dry, temperature or other procedures. Values for commercial acetate and triacetate fibers are generally in the 2.2—4.0 N/tex (25—45 gf/den) range. 

Fig. 3. Stress—strain curve of typical polyesterether elastomer showing the three main regions (I, II, and III) (181), where A is the slope (Young s modulus)...

Fig. 41. Typical stress—strain curve. Points is the yield point of the material the sample breaks at point B. Mechanical properties are identified as follows a = Aa/Ae, modulus b = tensile strength c = yield strength d = elongation at break. The toughness or work to break is the area under the curve.

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. 

For a fiber immersed in water, the ratio of the slopes of the stress—strain curve in these three regions is about 100 1 10. Whereas the apparent modulus of the fiber in the preyield region is both time- and water-dependent, the equiUbrium modulus (1.4 GPa) is independent of water content and corresponds to the modulus of the crystalline phase (32). The time-, temperature-, and water-dependence can be attributed to the viscoelastic properties of the matrix phase. 

Proportion of Hard Segments. As expected, the modulus of styrenic block copolymers increases with the proportion of the hard polystyrene segments. The tensile behavior of otherwise similar block copolymers with a wide range of polystyrene contents shows a family of stress—strain curves (4,7,8). As the styrene content is increased, the products change from very weak, soft, mbbedike materials to strong elastomers, then to leathery materials, and finally to hard glassy thermoplastics. The latter have been commercialized as clear, high impact polystyrenes under the trade name K-Resin (39) (Phillips Petroleum Co.). Other types of thermoplastic elastomers show similar behavior that is, as the ratio of the hard to soft phase is increased, the product in turn becomes harder. 

ISOCHRONOUS STRESS - STRAIN CURVE CREEP MODULUS - TIME CURVE... 

The stiffness of a plastic is expressed in terms of a modulus of elasticity. Most values of elastic modulus quoted in technical literature represent the slope of a tangent to the stress-strain curve at the origin (see Fig. 1.6). This is often referred to as Youngs modulus, E, but it should be remembered that for a plastic this will not be a constant and, as mentioned earlier, is only useful for quality... 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

The mechanical properties can be studied by stretching a polymer specimen at constant rate and monitoring the stress produced. The Young (elastic) modulus is determined from the initial linear portion of the stress-strain curve, and other mechanical parameters of interest include the yield and break stresses and the corresponding strain (draw ratio) values. Some of these parameters will be reported in the following paragraphs, referred to as results on thermotropic polybibenzoates with different spacers. The stress-strain plots were obtained at various drawing temperatures and rates. 

The mechanical properties were obtained using a tensile machine at room temperature and for a strain rate of 1000%/h. Each reported value of the modulus was an average of five tests. The tensile modulus Et was taken as the slope of the initial straight line portion of the stress-strain curve. 

For the past century one successful approach is to plot a secant modulus that is at 1% strain or 0.85% of the initial tangent modulus and noting where they intersect the stress-strain curve (Fig. 2-2). However for many plastics, particularly the crystalline thermoplastics, this method is too restrictive. So in most practical applications the limiting strain is decided based on experience and/or in consultation between the designer and the plastic material manufacturer. Once the limiting strain is known, design methods based on its creep curves become rather straightforward (additional information to follow). 

Fig. 2-2 (a) Example of the modulus of elasticity determined on the initial straight portion of the stress-strain curve and secant modulus and (b) secant modulus for two different plastics that are 85% of the initial tangent modulus. 

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