Curves, true stress-strain

True stress-strain curves for plastic flow... 

Corrosion effect of forming Elongation X gauge length Standard hydropress specimen test True stress-strain curve Uniformity of characteristics... 

Conventional stress-strain curves are necessarily similar to the load-deformation curves from which they are derived. True stress-strain curves can also be derived in which the stress is based on the actual or installtaueous area of the cross-section. Such curves do not have a maximum corresponding to C, but increase continuously to the breaking load. 

Typical true stress-strain curves for amorphous polymers are shown in Fig. 2. 

Figure 7.11. True stress-strain curve for aorta. Total, elastic, and viscous stress-strain curves for aorta. The total stress-strain curves (open boxes, top) were obtained by collecting all the initial, instantaneous, force measurements at increasing time intervals and then dividing by the initial cross-sectional areas and multiplying by 1.0 + the strain. The elastic stress-strain curves (closed diamonds, middle) were obtained by collecting all the force measurements at equilibrium and then dividing by the initial cross-sectional areas and multiplying by 1.0 + the strain. The viscous component curves (closed squares, bottom) were obtained as the difference between the total and the elastic stresses. Error bars represent one standard deviation of the mean.

The true stress-strain curves for DGER-DAP (P = 1) cured up to alop at different thermal prehistory and temperatures 12,15) are summarized in Fig. 17 (a, b). These curves are typical, although extension at break eb for the networks cured with wPhDA or DADPhS is usually lower (eb = 4.5 5,5% at e = 10 2 min 1 and T = 25 °C). The small eb values are not determined by the network structure as was shown in Sect. 5.1.2 and 5.1.3, but they result from the high rate of the fracture processes, which is typical for any glassy substance. 

Fig. 17a and b. True stress-strain curves (extension rate, s = 10 2 min-1) for glassy DGER-DAP (P = 1) network at different test temperatures, annealed (cooling at the rate T = 10 K/h from 160 °C to 70 °C) and quenched (jump-cooling) samples... 

A number of materials, however, show stress-strain curves of the shape sketched in Fig. 24.10. After the normal convex first part, the stress-strain curve shows an inversion point, after which the stress increases rapidly with strain. This phenomenon is sometimes called "strain hardening". In this case, a straight line through the origin can intersect the stress-strain curve at two points A and B. This means that only the intersection points A and B are possible conditions. The intermediate intersection point C is unstable. So in this case two parts of the specimen, e.g. a fibre, with different draw ratios and, hence, different cross sections can coexist. If the fibre is stretched, part of the material with a cross section of point A is converted into material with a cross section of point B. In contrast, in Chap. 13 the Considere plot is defined as the tangent line on the true stress-strain curve. 

Figure 5. True stress-strain curves of CP Ti after HPT and annealing.

Figure 3 shows the true stress strain curves for the consolidated microcrystalline powder. Both tensile and compression results show that sample 2B gives the best UTS and ductility. In tension, the 2B sample shows nearly ideal elasto-plastic response without necking until about 17 percent elongation. The... 

The tensile and compression true stress strain curves for the nano particle Cu consolidates are shown in Figure 4. The 2B material gives a tensile UTS of over 700 MPa, but failed at only 1.2 percent strain. The response in... 

Practically, tensile samples were allowed to swell in Millipore water for 24h in order to reach the equilibrium prior mechanical testing. In such conditions, only strain due to mechanical deformation, and not to swelling process, was measured. Figure 11 presents typical true stress-strain curves. It comes out that the introduction of PCL segments enliances the overall mechanical properties in comparison to the PDMAEMA hydrogel. 

Figure 11 True stress-strain curves recorded by tensile testing for swollen PDMAEMA (S = 95.4%, grey line) and PDMAEMA-l-PCL (S = 35.4 %, black... 

Figure 13 True stress-strain curves recorded by tensile testing for the swollen PDMAEMA-l-PCL hydrogels atr.t. and containing PCLDMA cross-linker characterized by aMn of1700 ( ), 4300 (o) and 5800 g.mof (d).

Young modulus as determined from the slope of the true stress-strain curve (Figure 13) in the region of 10% strain... 

Nominal Stress-Strain Curve vs True Stress-Strain Curve. 29... 

Rg. 8.4 The Considere construction. Possible forms of true-stress-strain curves (upper curves), the tangents drawn to them from the point e = -1 on the strain axis and the corresponding nominal-stress-strain curves. The curves are schematic for clarity in showing the construction for real polymers the initial part of the curve would generally be much steeper and the maximum, when it occurs, is generally at a much lower value of strain than that shown. See example 8.1 for a more realistic version of the curves in (c). 

There remains the question of whether the drop in load observed at yielding arises from the purely geometrical strain softening associated with a true-stress-strain curve of the form shown in fig. 8.4(c), where there is no drop in the true stress but merely a reduction in slope of the stress train curve, or whether there is actually a maximum in the true-stress strain curve as shown in fig. 8.4(d). Experiments on polystyrene and PMMA in compression, under which the geometrical effect cannot take place, show that a drop in load is still observed. Results from extensive studies of PET under a variety of loading conditions also support the idea that a maximum in the true-stress train curve may occur in a number of polymers. 

Differentiation of the first expression for fTn(e) shows that the stress increases monotonically until e = 356.6/(2 x 998) = 0.179 and then begins to fall monotonically. The minimum must therefore lie in the region described by the second expression. Differentiation shows that it lies at e = d/A = 3.09. The true-stress-strain curve is obtained by multiplying On(e) by 1 -f e and any minimum in it must lie in the region described by the second expression for cr ie). Differentiation of (1 + e)cTn(e) leads to the condition (a — h) + (4 — 2h)e + 6e ... 

Figure 5.109 illustrates a calculation of the true draw ratio by following the changes of the cross-section at various positions along the fiber, starting at the point of initial necking. In Fig. 5.110 the true stress-strain curves are plotted as calculated... 

Van Ruiten and co-workers [45] studied the drawability and attainable mechanical properties of PA 4,6 yarn using true stress-strain curves. The concept of the molecular network was applied to an analysis of these fibres and yarns and the mechanical properties of yarns with different draw ratios was evaluated in terms of the network draw ratio, which was determined by matching true stress-strain curves. The validity of the molecular network concept for these yarns and its suitability for predicting fibre drawability was assessed. A method for predicting the maximum attainable tenacity of drawn yarns under given drawing conditions from precursor mechanical properties is proposed. 

Experimental observations indicate that, in general, the true stress-strain curves of ductile materials coincide. Brittle materials, among them brittle ceramics, do not show a similar behavior. Usually, experimental observations indicate that brittle materials are stronger under compression than under tension. 

True stress-strain curves of tension and compression have been experimentally observed to coincide. Yet, this observation is not tme for brittle materials (that behave like glass), in which fracture and yield stress coincide. Thus, in brittle materials, e.g., ceramics, the strain seldom exceeds 1 %, depending to a large degree on the type of deformation involved. Classical Hookean behavior is observed in such brittle materials. The presence of imperfections of various kinds, including porosities, has a profound effect on the mechanical behavior of ceramics. 

The output signal, which is proportional to the average diameter of the specimen, is fed into an existing true stress-strain computer and can be read visually on a graduated dial or, when used in conjunction with load data from the testing machine, can provide an autographic curve of true stress-strain on a Baldwin MD-2 recorder. 

Fig. 3. True stress-strain curves at various temperatures for iodide titanium.

Of special interest is the different character of the true stress-strain curves at -452°F compared to those obtained at higher temperatures. Noteworthy here is the high rate of linear strain hardening at high strains, the high values of strain at maximum load compared to what would be expected based on extrapolation from higher temperature data and the observation that in some cases fracture occurs at or just beyond maximum load. 

If the deformation process were continuous at any one location, a maximum load would occur at the normal strain, but this strain is exceeded because of the elastic loading and interrupted nature of the deformation. The true stress-strain curve therefore does not represent the strain hardening properties of one particular region of the bar in the same sense as at higher temperatures, but rather represents the upper envelope of flow stress for various austenite-martensite structures. 

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