The engineering stress–strain curve

50 100 150 200 250 300 5 10 15 20 25 30 35 40 45 50 Engineering strain, Engineering stress, MPa E Tot. E u s f TS a Engineering strain Engineering stress e y s f E e eng. s eng. b Engineering strain e eng. 0.2 s proof s eng. Engineering stress c Figure 1.3 a Engineering stress–strain curve for the test of drawing quality sheet steel shown in Figure 1.2. b Initial part of the above diagram with the strain scale magnified to show the elastic behaviour. c Construction used to determine the proof stress in a material with a gradual elastic, plastic transition. 4 Mechanics of Sheet Metal Forming In some materials, the transition from elastic to plastic deformation is not sharp and it is difficult to establish a precise yield stress. If this is the case, a proof stress may be quoted. This is the stress to produce a specified small plastic strain – often 0.2, i.e. about twice the elastic strain at yield. Proof stress is determined by drawing a line parallel to the elastic loading line which is offset by the specified amount, as shown in Figure 1.3c. Certain steels are susceptible to strain ageing and will display the yield phenomena illustrated in Figure 1.4. This may be seen in some hot-dipped galvanized steels and in bake-hardenable steels used in autobody panels. Ageing has the effect of increasing the initial yielding stress to the upper yield stress σ U ; beyond this, yielding occurs in a discontinuous form. In the tensile test-piece, discrete bands of deformation called L¨uder’s lines will traverse the strip under a constant stress that is lower than the upper yield stress; this is known as the lower yield stress σ L . At the end of this discontinuous flow, uniform deformation associated with strain-hardening takes place. The amount of discontinuous strain is called the yield point elongation YPE. Steels that have significant yield point elongation, more than about 1, are usually unsuitable for forming as they do not deform smoothly and visible markings, called stretcher strains can appear on the part. Engineering strain e eng. s L YPE s u Engineering stress s eng. Figure 1.4 Yielding phenomena in a sample of strain aged steel.

1.1.3 The true stress–strain curve

There are several reasons why the engineering stress–strain curve is unsuitable for use in the analysis of forming processes. The ‘stress’ is based on the initial cross-sectional area of the test-piece, rather than the current value. Also engineering strain is not a satisfactory measure of strain because it is based on the original gauge length. To overcome these disadvantages, the study of forming processes is based on true stress and true strain; these are defined below. True stress is defined as σ = P A 1.7 where A is the current cross-sectional area. True stress can be determined from the load–extension diagram during the rising part of the curve, between initial yielding and Material properties 5 the maximum load, using the fact that plastic deformation in metals and alloys takes place without any appreciable change in volume. The volume of the gauge section is constant, i.e. A l = Al 1.8 and the true stress is calculated as σ = P A l l 1.9 If, during deformation of the test-piece, the gauge length increases by a small amount, dl, a suitable definition of strain is that the strain increment is the extension per unit current length, i.e. dε = dl l 1.10 For very small strains, where l ≈ l , the strain increment is very similar to the engineering strain, but for larger strains there is a significant difference. If the straining process con- tinues uniformly in the one direction, as it does in the tensile test, the strain increment can be integrated to give the true strain, i.e. ε = dε = l l dl l = ln l l 1.11 The true stress–strain curve calculated from the load–extension diagram above is shown in Figure 1.5. This could also be calculated from the engineering stress–strain diagram