The Stress-Strain Curve

The Stress-Strain Curve.—Stress-strain results obtained by Treloar on natural rubber vulcan- 

In turning to the last-mentioned results, it is to be noted that the theory given above should, in principle, be equally applicable to a deformation for which a l, provided that expansions are allowed to occur in each of the transverse directions to the extent The 

Anthony, Caston, and Guth obtained considerably better agreement between the experimental stress-strain curve for natural rubber similarly vulcanized and the theoretical equation over the range a = 1 to 4. KinelP found that the retractive force for vulcanized poly-chloroprene increased linearly with a — l/a up to a = 3.5. 

Throughout all of the experiments cited above on isothermal stretch- 

For the purpose of testing the validity of what we have called the deformation factor, it is convenient to plot rl a — l/a ) vs. a if rubbers swollen permanently (not merely for purposes of equilibration) are included, the quantity roV2 /(a — l/a ) is to be preferred, where To is the tension referred to the cross section when unswollen and unstretched, and V2 is the volume fraction of polymer in the swollen mixture (see Appendix B, Eq. B-5). According to theory, both of these quantities should be independent of the deformation, and the latter should be independent of the degree of swelling as well. On the basis 

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The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

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...

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. 

The abihty of a fiber to absorb energy during straining is measured by the area under the stress—strain curve. Within the proportional limit, ie, the linear region, this property is defined as toughness or work of mpture. For acetate and triacetate the work of mpture is essentially the same at 0.022 N/tex (0.25 gf/den). This is higher than for cotton (0.010 N/tex = 0.113 gf/den), similar to rayon and wool, but less than for nylon (0.076 N/tex = 0.86 gf/den) and silk (0.072 N/tex = 0.81 gf/den) (3). 

Most extmded latex fibers are double covered with hard yams in order to overcome deficiencies of the bare threads such as abrasiveness, color, low power, and lack of dyeabiUty. During covering, the elastic thread is wrapped under stretch which prevents its return to original length when the stretch force is removed thus the fiber operates farther on the stress—strain curve to take advantage of its higher elastic power. Covered mbber fibers are commonly found in narrow fabrics, braids, surgical hosiery, and strip lace. 

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. 

Typical stress—strain curves are shown in Figure 3 (181). The stress— strain curve has three regions. At low strains, below about 10%, these materials are considered to be essentially elastic. At strains up to 300%, orientation occurs which degrades the crystalline regions causing substantial permanent set. 

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. 

Fig. 4. The stress—strain curves of a wool fiber at different relative humidities.

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. 

For stainless steel, the stress-strain curve (see Fig. 26-37) has no sharp yield point at the upper stress limit of elastic deformation. Yield strength is generally defined as the stress at 2 percent elongation. 

Proportional limit the point on the stress-strain curve at which will commence the deviation in the stress-strain relationship from a straight line to a parabolic curve (Figure 30.1). 

Yield point a point on the stress-strain curve that defines the mechanical strength of a material under different stress conditions at which a sudden increase in strain occurs without a corresponding increase in the stress (Figure 30.1). 

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... 

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

When metals are rolled or forged, or drawn to wire, or when polymers are injection-moulded or pressed or drawn, energy is absorbed. The work done on a material to change its shape permanently is called the plastic work- its value, per unit volume, is the area of the cross-hatched region shown in Fig. 8.9 it may easily be found (if the stress-strain curve is known) for any amount of permanent plastic deformation, e. Plastic work is important in metal- and polymer-forming operations because it determines the forces that the rolls, or press, or moulding machine must exert on the material. 

The energy expended in deforming a material per unit volume is given by the area under the stress-strain curve. For example,... 

The nominal stress at yielding. In many materials this is difficult to spot on the stress-strain curve and in such cases it is better to use a proof stress. 

We now turn to the other end of the stress-strain curve and explain why, in tensile straining, materials eventually start to neck, a name for plastic instability. It means that flow becomes localised across one section of the specimen or component, as shown in Fig. 11.5, and (if straining continues) the material fractures there. Plasticine necks readily chewing gum is very resistant to necking. 

Finally, mild steel can sometimes show an instability like that of polythene. If the steel is annealed, the stress/strain curve looks like that in Fig. 11.10. A stable neck, called a Luders Band, forms and propagates (as it did in polythene) without causing fracture because the strong work-hardening of the later part of the stress/strain curve prevents this. Luders Bands are a problem when sheet steel is pressed because they give lower precision and disfigure the pressing. 

In compression, a single large flaw is not fatal (as it is tension). As explained in Chapter 17, cracks at an angle to the compression axis propagate in a stable way (requiring a progressive increase in load to make them propagate further). And they bend so that they run parallel to the compression axis (Fig. 20.7). The stress-strain curve therefore rises (Fig. 20.8), and finally reaches a maximum when the density of... 

Fig. 20.8. The stress-strain curve for cement or concrete in compression. Cracking starts at about half the ultimate strength.

At temperatures 50°C or so below T, thermoplastics become plastic (hence the name). The stress-strain curve typical of polyethylene or nylon, for example, is shown in Fig. 23.10. It shows three regions. 

Fig. 25.2. The stress-strain curve of o continuous fibre composite (heavy line), showing how it relates to those of the fibres and the matrix (thin lines). At the peak the fibres are on the point of failing.

Many fibrous composites are made of strong, brittle fibres in a more ductile polymeric matrix. Then the stress-strain curve looks like the heavy line in Fig. 25.2. The figure largely explains itself. The stress-strain curve is linear, with slope E (eqn. 25.1) until the matrix yields. From there on, most of the extra load is carried by the fibres which continue to stretch elastically until they fracture. When they do, the stress drops to the yield strength of the matrix (though not as sharply as the figure shows because the fibres do not all break at once). When the matrix fractures, the composite fails completely. 

When a foam is compressed, the stress-strain curve shows three regions (Fig. 25.9). At small strains the foam deforms in a linear-elastic way there is then a plateau of deformation at almost constant stress and finally there is a region of densification as the cell walls crush together. 

Fig. 25.10. Cell wall bending gives the linear-elastic portion of the stress-strain curve.

Cellular materials can collapse by another mechanism. If the cell-wall material is plastic (as many polymers are) then the foam as a whole shows plastic behaviour. The stress-strain curve still looks like Fig. 25.9, but now the plateau is caused by plastic collapse. Plastic collapse occurs when the moment exerted on the cell walls exceeds its fully plastic moment, creating plastic hinges as shown in Fig. 25.12. Then the collapse stress (7 1 of the foam is related to the yield strength Gy of the wall by... 

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