Curved-neck specimen

Fig. 3.1. Single fiber compressive tests with (a) parallel-sided and (b) curved-neck specimen.

If we load a material in compression, the force-displacement curve is simply the reverse of that for tension at small strains, but it becomes different at larger strains. As the specimen squashes down, becoming shorter and fatter to conserve volume, the load needed to keep it flowing rises (Fig. 8.6). No instability such as necking appears, and the specimen can be squashed almost indefinitely, this process only being limited eventually by severe cracking in the specimen or the plastic flow of the compression plates. 

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. 

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 the development of the neck starts, the applied load on the specimen ceases to increase as the strain increases, reaching a maximum in the curve of ci vs. s . Mathematically the condition for obtaining a maximum is met when dF/dz = 0 or dGn/ds = 0. If this condition is applied to Eq. 

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. 

Fig. 7. Typical load-displacement curve of a DENT specimen with indication of the total work partition into yielding related (Wy) and necking/tearing (W J related components.

At the peak force (position A in Fig. 8.5a), there are two possibilities for the next strain state Elastic unloading along path AU, and further plastic straining along the path AN. A non-uniform strain state develops, as parts of the specimen elastically unload, and the plastic strain in one region increases to form a neck. The plastic deformation of the neck is partially driven by elastic energy release from the rest of the specimen. The condition that A is at the maximum in the force-extension or force-strain curve can be written... 

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