Stress-strain plot

This ambiguity in the stress space loading criterion may be illustrated by considering a stress-strain plot corresponding to simple tension, as shown schematically in Fig. 5.3. With each point on the stress-strain curve past the initial elastic limit point A, there is associated a point on the elastic limit surface in stress space and a point on the elastic limit surface in strain space. On the hardening portion of the stress strain curve AB, both the stress and the strain are increasing, and the respective elastic limit surfaces are moving... 

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 influence of the annealing process at a higher temperature was also studied, and important changes in the properties of a sample annealed at 55°C during 5 days (PTEB-CR sample) were observed [33]. The PTEB-CR specimens were stretched at 1 and 10 cm/min, but the behavior was similar in both cases. Typical stress-strain plots are shown in Fig. 14 for samples stretched at 23°C... 

Figure 14 Stress-strain plots of several PTEB specimens, stretched at I cm/min and two drawing temperatures. The inset shows the two CR specimens at low deformations.

Tough fracture, which is alternatively known as ductile fracture, by contrast, gives the type of behaviour illustrated in Figure 7.2. After the maximum in the stress-strain plot has been reached, there is a substantial amount of yielding, before the sample eventually breaks. 

Figure 7.1 Stress-strain plot for brittle fracture...

Higher extent of silica generation with high TEOS concentration improves the mechanical properties severalfolds as illustrated by the tensile stress-strain plots on ACM-sdica hybrid nanocomposites on increasing TEOS concentrations in Figure 3.6. 

FIGURE 3.6 Tensile stress-strain plots of acrylic mbber (ACM)-silica hybrid nanocomposites using different tetraethoxysilane (TEOS) concentrations. The number in the legends indicates wt% TEOS concentrations. (From Bandyopadhyay, A., Bhowmick, A.K., and De Sarkar, M., J. Appl. Polym. Sci., 93, 2579, 2004. Courtesy of Wiley InterScience.)... 

FIGURE 3.13 Tensile stress-strain plots for acrylic rubber (ACM)-siUca and epoxidized natural rubber (ENR)-sibca hybrid composites synthesized from various solvents (a) ACM-siUca and (b) ENR-siUca. The letters after the rubbers in the legend indicate solvents used T = THF, M = methyl ethyl ketone (MEK), D = DME, E = EAc, CH = CHCl3, CC CCLj. (From Bandyopadhyay, A., De Sarkar, M., and Bhowmick, A.K., J. Appl. Polym. Sci., 95, 1418, 2005 and Bandyopadhyay, A., De Sarkar, M., and Bhowmick, A.K., J. Mater. Sci., 40, 53, 2005. Courtesy of Wiley InterScience and Springer, respectively.)... 

FIGURE 4.8 Comparative tensile stress-strain plot of polychloroprene-ordinary zinc oxide (ZnO) and poly-chloroprene-nano-ZnO system. (From Sahoo, S., Kar, S., Ganguly, A., Maiti, M., and Bhowmick, A.K., Polym. Polym. Compos., 2007 (in press). Courtesy of Smithers Rapra Technology Ltd.)... 

FIGURE 9.8 Typical stress-strain plots for a strip of recombinant resilin tested in phosphate-buffered saline (PBS). Sample cycled to 225%, showing resilience of 97% (solid curve) and later tested to failure showing extension at break of 313% (dotted curve). (FromElvin, C.M., Carr, A.G., Huson, M.G., Maxwell, J.M., Pearson, R.D., Vuocolo 1, T., Liyon, N.E., Wong, D.C.C., Merritt, D.J., and Dixon, N.E., Nature, 437, 999, 2005.)... 

The result is a hysteresis loop on a stress-strain plot (Fig. 8). The energy loss per cycle is the area of the hysteresis loop and is given by ... 

Most engineering materials, particularly metals, follow Hooke s law by which it is meant that they exhibit a linear relationship between elastic stress and strain. This linear relationship can be expressed as o = E where E is known as the modulus of elasticity. The value of E, which is given by the slope of the stress-strain plot, is a characteristic of the material being considered and changes from material to material. 

This expression indicates that U, the strain energy per unit volume is given by the area under the stress-strain plot over the strain range e = 0 to e = e. If Hooke s law holds, i.e., if a = E e, it follows that... 

The shaded area in the stress-strain plot shown in Figure 1.4 is numerically equal to the modulus of resilience. It is to be noted that for a given value of E, Ur directly proportional to cPL while for a given value of cPL, Ur is inversely proportional to E (stiffness). 

Figure 11 shows plots according to equation(lO) of stress-strain data for triol-based polyester networks formed from the same reactants at three initial dilutions (0% solvent(bulk), 30% solvent and 65% solvent). Only the network from the most dilute reactions system has a strictly Gaussian stress-strain plot (C2 = 0), and the deviations from Gaussian behaviour shown by the other networks are not of the Mooney-Rivlin type. As indicated previously, they are more sensibly interpreted in terms of departures of the distribution of end-to-end vectors from Gaussian form. 

FIGURE 14.1 Stress-strain plots for a Hookean spring (a) where E (Equation 14.1) is the slope, and a Newtonian dashpot (b) where is a constant (Equation 14.3). 

FIGURE 14.2 Stress-strain plot for stress relaxation for the Maxwell model (a) and Voigt-Kelvin model (b). 

Several stress-strain plots are shown in Fig. 1-10. Four important quantities characterize the stress-strain behavior of a polymer ... 

Fig. 1-10 Stress-strain plots for a typical elastomer, flexible plastic, rigid plastic, and fiber.

Figure 3.3 shows representative stress-strain curves for a variety of polymeric materials. At normal use temperatures, such as room temperature, rigid polymers such as polystyrene (PS) exhibit a rapid increase in stress with increasing strain until sample failure. This behavior is typical of brittle polymers with weak interchain secondary bonding. As shown in the top curve in Figure 3.3, the initial stress-strain relation in such polymers is approximately linear and can be described in terms of Hooke s law, i.e., S = Ee, where E is Young s modulus, typically defined as the slope of the stress-strain plot. At higher stresses, the plot becomes nonlinear. The point at which this occurs is called the proportional limit. 

Figure 3.3 Stress-strain plots representative of brittle, ductile, and elastomeric polymeric materials. Failure is denoted by . (After J. Fried, Plastics Engineering, July 1982, with permission.)...

Fig. 8 Stress-strain plots of layered-silicate-filled NBR cured by sulfur and peroxide (a), sulfur in the presence of different zinc organic salts (b), and peroxide in the presence of different vulcanizing ingredients (c)...

Fig. 39 Improvement of physical properties of various rubbers through incorporation of clay or swollen clay 50% modulus (a), tensile strength (b), elongation at break (c), and the stress-strain plots for natural rubber (d). The strain was measured on the basis of clamp distance measurements...

For both EPDM-LDH and XNBR-LDH nanocomposites, the various tensile properties are summarized in Table 13 and their typical stress-strain plots are shown in Fig. 58 [104]. In Fig. 58a, the gum vulcanizates of both rubber systems showed typical NR-like stress-strain behavior with a sharp upturn in the stress-strain plot after an apparent plateau region, indicating strain-induced crystallization. With the addition of LDH-C10 in the XNBR matrix, the stress value at all strains increased significantly, indicating that the matrix undergoes further curing (Fig. 58b). 

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