Nominal and true strain

In section 2.2.2, the strain e was defined as quotient of the change in length A/ and the initial length /q  

This is a sensible definition in the case of elastic deformations because the initial, undeformed state (length lo) is a reference state the material returns to upon unloading. 

During plastic deformation, atoms within the material rearrange, and the initial state of the material is not retained. Therefore, it is not helpful to relate all deformations to the initial state. Instead, strains should be calculated relative to the current state of the material. If, for example, a specimen is plastically lengthened and compressed to its original length, it seems to be in the original state macroscopicaUy, but, usually, not all of the atoms have returned to their initial positions. This shows that the current state of the material depends not only on the current strain, but also on the deformation history. 

In metals, for example, the states before and after the plastic deformation can usually be distinguished macroscopicaUy, for the stress needed for further plastic deformation (called the yield strength, see section 3.2) usually increases. To describe the deformation history, a plastic equivalent strain is defined, which always increases during plastic deformation. This equivalent strain is non-zero after the deformation. It wiU be discussed in section 3.3.5. 

A further reason not to relate plastic deformations to the initial length is that they are usually large. Especially in the case of several deformation steps, this would lead to incorrect results for the strains (see the example on page 65). 

---

Fig. 3.1. Comparison of nominal and true strain for a deformation of a tensile specimen in one or two steps, respectively. In total, the length is doubled during the deformation. The nominal strain differs (si + 2 / 12) while the true strain is identical (< i + < 2 = >12) for both deformation sequences...

If we expand the relation between nominal and true strain, equation (3.3), in a Taylor series and cut off after the second term, we find... 

This equation is given in terms of true stress and true strain. As we said in Chapter 8, tensile data are usually given in terms of nominal stress and strain. From Chapter 8 ... 

Chapter 8 Nominal and True Stress and Strain, Energy of Deformation... 

Figure 1.9 Comparison between nominal and true stress-strain.

Finally, Figure 14.5d shows a schematic of simple shear deformation. The specimen is clamped between steel blocks. The blocks must move parallel to each other in order to get a shear strain that is uniform along the waisted region. The shear stress is calculated as t = F/A, where F is the force applied to the plane of area A. In this test it is not necessary to distinguish between nominal and true stress because the shear strain does not affect A. The shear strain is defined as y = Ax/y, where Ax is the displacement of planes separated by a distance y. Ax being measured in the direction of the force applied, which is perpendicular to y. As in the case of the compression plane strain test, there is no change in the dimension of the sample along the z axis. 

Nominal and true stress/strain diagrams differ characteristically (Figure 11-15). 

Thus if we know a, the true stress as a function of X (i.e. the true stress-strain curve) can be computed. The nominal and true stress-strain curves are compared in Figure 11.2. 

To compare nominal strain and true strain, we investigate two different deformation processes of two rods The first rod is strained in two steps A/i and A/2, the second in a single step A/i 2 = A/i - - A/2-... 

The stress is nonoinal as the area A relates to the undeformed material. For small strains, the nominal and true stresses can be assumed equal. Then, when all six components of stress are acting, the energy is additive and, expressing the result in differential form, we have... 

Figure 6. Engineering stress (r0) vs. nominal strain (e0) and true stress (rn) vs. true strain (en) for ABS 1, 2, and 3 at 10,000 in./min...

Fig. 15 (a) Typical stress-strain curves of a surfactant-containing HM PDMA hydrogel formed using 2 mol% of C17.3M under compression, represented as the dependences of nominal true stresses (gray dashed curves) on the deformation ratio A SDS = 1 % (wA ). Results of 15 separate tests are shown. Red circles represent the points of failure of the gel samples. The inset shows rrtnie versus A curves of two successive tests conducted on the same gel sample for up to 99.99 % compression, (b) Stress-strain curve of the same HM PDMA hydrogel with 7 % (w/v) of SDS (solid blue curve) and without SDS (solid dark red curve), Co = 15 % (w/v), C17.3M = 2 mol%, NaCl = 0.5 M. From [41] with permission from Elsevier... 

In this equation, is the original gauge length of the specimen and Lf is the gauge length at the stress-strain point of interest. True, strain is always less than nominal strain see strain), but the difference is small unless the strain is large. For example, a nominal strain of 0.5 (50%) corresponds to a true strain of 0.41 (41%). 

A variety of specimen shapes and sizes can be used but the most common is a smooth-bar tensile coupon, as described in ASTM E 8, Test Method for Tension Testing of Metallic Materials." In smooth-bar tests, the changes are measured in terms of time to failure, ductility (elongation or reduction-in-area), maximum load achieved, and area bounded by a nominal stress-elongation curve or a true stress/ true strain curve, which are often supported by fractogra-phic examination. The specimen is exposed to the environment while it is stressed under a constant displacement rate. 

If we compare the calculated strains of both specimens, which have the same length initially and eventually, the true strains are identical (v = 0.693), but the nominal strains are not (0.833 1.000). Only by using the true strains can identical strain values be calculated independent of the deformation history. The distinction between true and nominal strain is also discussed in exercise 7. 

It was pointed out in Section 10.4.3 that wall slip can cause a large error in the determination of the strain in step-strain experiments, and the true strain maybe much less than the nominal strain inferred from the displacement of a rheometer surface. The observation that N /a is independent of time does not, by itself, imply that there is no slip unless this ratio is also equal to the nominal strain applied. And when the Lodge-Meissner rule is not obeyed, it is often taken as evidence that slip is occurring, and the stress ratio Nj/cris used in place of the nominal strain as the independent variable in reporting shear stress and normal stress difference data [40]. 

The experimental mechanical techniques most commonly used for network characterization are uniaxial extension and compression,and also biaxial strain.A sketch of a rubber sample under extension is shown in Figure 10(a). The nominal stress a is defined as the ratio of the force/to the cross-sectional area Aq of the undeformed specimen, and the strain e as the ratio of the length change AL to the original length Lq. These definitions are given in equations (67) and (68). The deformation is also often expressed in terms of the extension ratio X defined in equation (69). The cross-sectional area of the specimen varies with deformation. A true stress, defined as the ratio of the force to the real deformed area, is also frequently used. 

Assuming that a diamond-pyramid hardness test creates a further nominal strain, on average, of 0.08, and that the hardness value is 3.0 times the true stress with this extra strain, construct the curve of nominal stress against nominal strain, and find ... 

Figure 1-2 records some typical stress-strain curves for different polymer types. Some polymers exhibit a yield maximum in the nominal stress, as shown in part (c) of this figure. At stresses lower than the yield value, the sample deforms homogeneously. It begins to neck down at the yield stress, however, as sketched in Fig. 11-20. The necked region in some polymers stabilizes at a particular reduced diameter, and deformation continues at a more or less constant nominal stress until the neck has propagated across the whole gauge length. The cross-section of the necking portion of the specimen decreases with increasing extension, so the true stress may be increasing while the total force and the nominal stress...

When a material is subjected to small deformations, the cross-sectional area of the unstrained sample, Aq, coincides with the cross-sectional area of the strained sample, A. However, in the case of elastomers, in which the deformations can be extremely high, account has to be taken of the change in the cross section of the sample. Consequently, the value of the stress a, calculated by using Eq. (3,33) and called nominal stress, does not coincide with the true tensile stress (A (Fig. 3.10). 

Equation (14.7) corresponds to the slope of the tangent to the curve cr, vs. e drawn from the point e = -1 or X, = 0. Figure 14.7 shows the true stress versus nominal strain curves for polymer samples A and B. Curves B] and B2 are compatible with curve B of Figure 14.6. The so-called Considere construction, Eq. (14.7), is satisfied with the tangent to the curves drawn from E = — 1. The tangential point corresponds to the maximum observed in the curve vs. and therefore with the maximum load that the specimen can support. In practice, the Considere construction is used as a criterion to decide when a polymer will form an unstable neck or form a neck accompanied by cold drawing. 

---