Worked example maximum uniform strain

and deformation becomes concentrated in necks. To illustrate the effect, we consider here a non-strain-hardening material having a tensile stress–strain rate curve σ 1 = B ˙ε m 1 5.15 where ˙ε 1 is strain rate, i.e. ˙ε 1 = dε 1 dt = dl l dt = v l 5.16 In Equation 5.16, t is time, v the velocity or cross-head speed of the testing machine and l is the length of the parallel reduced section of the test-piece. If we consider an imperfect strip as illustrated in Figure 5.4, then the force transmitted through both regions is P = σ 1 A = σ 1 + dσ 1 A + dA 5.17 from which we obtain dσ 1 σ 1 = − dA A 5.18 Thus the difference in stress for the two regions is proportional to the magnitude of the imperfection. Differentiating the material law, Equation 5.15, we obtain d ˙ε 1 ˙ε 1 = 1 m dσ 1 σ 1 = − 1 m dA A 5.19 This indicates that for a given imperfection, the difference in the strain rate between the imperfection and the uniform region is inversely proportional to the strain-rate sensitivity index m. If m is small, as it is in most sheet metal at room temperature, the difference in strain rate will be large and the imperfection will grow rapidly. In a special class of alloys, termed superplastic, m is unusually high, about 0.3; strips of these materials can be extended several hundred per cent as the growth of imperfections is very gradual. In some non-metals, such as molten glass, m ∼ 1, and these can be drawn out almost indefinitely. In materials with low values of m, rate sensitivity will not greatly influence the max- imum uniform strain, because, as shown in Figure 5.5, the strain and hence strain rate are approximately similar in both regions up to maximum load. Beyond this point, the necking process will be affected by rate sensitivity and it is found that the post-uniform elongation is higher in materials with greater rate sensitivity.

5.4 Tensile instability in stretching continuous sheet

It is shown in the previous section that in a tensile strip, provided some imperfections exist, diffuse necking will start when the load reaches a maximum. On the other hand, in sheet that is stretched over a punch, diffuse necking is not observed. The tension in the sheet may reach a maximum, but the punch will exert a geometric constraint on the strain distribution that can develop. This is illustrated in Figure 5.7. If a diffuse neck did develop, the increased strain would lead to the sheet moving away from the punch and this is implausible. It is possible that the strain may accelerate in some region compared with Load instability and tearing 67