Yield strain stress-temperature curves

One way to obtain long-term information is through the use of the time-temperature-superposition principle detailed in Chapter 7. Indeed, J. Lohr, (1965) (the California wine maker) while at the NASA Ames Research Center conducted constant strain rate tests from 0.003 to 300 min and from 15° C above the glass transition temperature to 100° C below the glass transition temperature to produce yield stress master curves for poly(methyl methacrylate), polystyrene, polyvinyl chloride, and polyethylene terephthalate. It should not be surprising that time or rate dependent yield (rupture) stress master curves can be developed as yield (rupture) is a single point on a correctly determined isochronous stress-strain curve. Whether linear or nonlinear, the stress is related to the strain through a modulus function at the yield point (mpture) location. As a result, a time dependent master curve for yield, rupture, or other failure parameters should be possible in the same way that a master curve of modulus is possible as demonstrated in Chapter 7 and 10. 

The first point of zero slope on the curve (point C) is identified with material yielding and so its coordinates are called the yield strain and stress (strength) of the material. The yield strain and stress usually decrease as temperature increases or as strain rate decreases. The final point on the curve (point D) corresponds to specimen fracture. This represents the maximum elongation of the material specimen its coordinates are called the ultimate, or failure strain and stress. Ultimate elongation usually decreases as temperature decreases or as strain rate increases. 

Indeed, it has been observed that the onset of yielding of isotropic polymers is approximately constant, 0.02< [<0.025, which implies that 0.04shear yield strain, the plastic shear deformation of the domain satisfies a plastic shear law. For temperatures below the glass transition temperature, the continuous chain model enables the calculation of the tensile curve of a polymer fibre up to about 10% strain [6]. Figure 7 shows the observed stress-strain curves of PpPTA fibres with different moduli compared to the calculated curves. 

The material properties used in the simulations pertain to a new X70/X80 steel with an acicular ferrite microstructure and a uniaxial stress-strain curve described by er, =tr0(l + / )", where ep is the plastic strain, tr0 = 595 MPa is the yield stress, e0=ff0l E the yield strain, and n = 0.059 the work hardening coefficient. The Poisson s ratio is 0.3 and Young s modulus 201.88 OPa. The system s temperature is 0 = 300 K. We assume the hydrogen lattice diffusion coefficient at this temperature to be D = 1.271x10 m2/s. The partial molar volume of hydrogen in solid solution is... 

Depending on the material and deformation conditions (strain rate, temperature) other stress-strain curve shapes can be observed (Fig. 2b and c). In Fig. 2b, the plastic flow occurs at the same stress level as that required for the yielding so the strain softening does not exist. In the case shown in Fig. 2c, the strain hardening happens very close to yielding, suppressing both strain softening and plastic flow behaviour. 

The flow stresses of the intermetallic Ti5Si3 and TiSi2 compounds were determined in compression tests in air at the strain rate of =10"2 s"1 in the temperature range from 700 to 1500 °C. Figure 8 presents the true yield stress vs. temperature curves. 

Fig. 3.6 is an example of the ductile plastic tensile stress-strain curve. This curve identifies behavior so that as the strain increases, stress initially increases approximately proportionately (from point 0 to point A). Point A is called the proportional limit. From point 0 to point B, the behavior of the material is purely elastic/stretches but beyond point B, the material exhibits an increasing degree of permanent deformation/stretch. Point B is the elastic limit of the material. At point C the material is yielding and so its coordinates are called the yield strain and stress (strength) of the material. Point D relates to the S-S elongation at break/failure. Table 3.2 provides these type data at room temperature for different materials. 

Fig. 5. 23 shows an idealized stress-strain curve for a ductile polymer sample. In this case the nominal stress (r is plotted against the strain e. The change in the cross-section of a parallel-sided specimen is also sketched schematically at different stages of the deformation. Initially the stress is proportional to the strain and Hooke s law is obeyed. The tensile modulus can be obtained from the slope. As the strain is increased the curve decreases in slope until it reaches a maximum. This is conventionally known as the yield point and the yield stress and yield strain, Oy and Cy, are indicated on the curve. The yield point for a polymer is rather difficult to define. It should correspond to the point at which permanent plastic deformation takes place, but for polymers a permanent set can be found in specimens loaded to a stress, below the maximum, where the curve becomes non-linear. The situation is further complicated by the observation that even for specimens loaded well beyond the yield strain the plastic deformation can sometimes be completely recovered by annealing the specimen at elevated temperature. In practice, the exact position of the yield point is not of any great importance and the maximum point on the curve suffices as a definition of yield. The value of the yield strain for polymers is typically of the order of 5-10% which is very much higher than that of metals and ceramics. Yield in metals normally occurs at strains below 0.1%. 

As a pipeline is heated, strains of such a magnitude are iaduced iato it as to accommodate the thermal expansion of the pipe caused by temperature. In the elastic range, these strains are proportional to the stresses. Above the yield stress, the internal strains stiU absorb the thermal expansions, but the stress, g computed from strain 2 by elastic theory, is a fictitious stress. The actual stress is and it depends on the shape of the stress-strain curve. Failure, however, does not occur until is reached which corresponds to a fictitious stress of many times the yield stress. 

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. 

Consequently, changing the temperature or the strain rate of a TP may have a considerable effect on its observed stress-strain behavior. At lower temperatures or higher strain rates, the stress-strain curve of a TP may exhibit a steeper initial slope and a higher yield stress. In the extreme, the stress-strain curve may show the minor deviation from initial linearity and the lower failure strain characteristic of a brittle material. 

At higher temperatures or lower strain rates, the stress-strain curve of the same material may exhibit a more gradual initial slope and a lower yield stress, as well as the drastic deviation from initial linearity and the higher failure stain characteristic of a ductile material. 

Brittleness Brittle materials exhibit tensile stress-strain behavior different from that illustrated in Fig. 2-13. Specimens of such materials fracture without appreciable material yielding. Thus, the tensile stress-strain curves of brittle materials often show relatively little deviation from the initial linearity, relatively low strain at failure, and no point of zero slope. Different materials may exhibit significantly different tensile stress-strain behavior when exposed to different factors such as the same temperature and strain rate or at different temperatures. Tensile stress-strain data obtained per ASTM for several plastics at room temperature are shown in Table 2-3. 

Fig. 6.2. Yield strengths from tensile tests at 23 °C are plotted against the glass transition temperatures (T,max) of the five polymers [] result of extrapolated stress-strain-curve...

As shown in Sect. 2, the fracture envelope of polymer fibres can be explained not only by assuming a critical shear stress as a failure criterion, but also by a critical shear strain. In this section, a simple model for the creep failure is presented that is based on the logarithmic creep curve and on a critical shear strain as the failure criterion. In order to investigate the temperature dependence of the strength, a kinetic model for the formation and rupture of secondary bonds during the extension of the fibre is proposed. This so-called Eyring reduced time (ERT) model yields a relationship between the strength and the load rate as well as an improved lifetime equation. 

Atomic force microscopy and attenuated total reflection infrared spectroscopy were used to study the changes occurring in the micromorphology of a single strut of flexible polyurethane foam. A mathematical model of the deformation and orientation in the rubbery phase, but which takes account of the harder domains, is presented which may be successfully used to predict the shapes of the stress-strain curves for solid polyurethane elastomers with different hard phase contents. It may also be used for low density polyethylene at different temperatures. Yield and rubber crosslink density are given as explanations of departure from ideal elastic behaviour. 17 refs. 

Williams, Landel, and Ferry equation (WLF) Used for predicting viscoelastic properties at temperatures above Tg when these properties are known for one specific temperature, yield point Point on a stress-strain curve below which there is reversible recovery. 

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