Most of us are quite conversant with the simple tension test which is the most common and is conducted to determine the yield strength, ultimate tensile strength, percent elongation and fracture strength of metals. Figures 1.6 and 1.7 show typical stress-strain curves obtained in tension tests on two different alloys. Figure 1.6 is obtained for a low carbon steel or mild steel test specimen and Fig. 1.7 is obtained for an aluminium alloy test specimen.
In these two diagrams the stress and strain are defined as given below:
Stress = Force/Original area of cross-section of specimen.
Strain = Change in length/Original length of specimen.
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In case of mild steel, for loading up to point A (Fig. 1.6), there is a straight line or linear relationship between stress and strain. That is why the point A is also called proportional limit. Thereafter, the relationship between stress and strain is nonlinear up to a point B which is very near to A. Also up to the point B the deformations are largely elastic and on unloading the specimen regains the original dimensions.
But beyond the point B the metal yields, it suffers plastic deformation. This is indicated by a sudden bend in the curve. Most of the strain after this point is plastic strain which is not recovered on unloading. The value of stress at B is called upper yield strength and point B is called upper yield point.
With further increase in strain (beyond B) the stress may fall a bit to a lower level at C (Fig. 1.6). This is due to formation of Luder bands. With increase in tension, localized plastic flow takes place in a narrow band with boundary planes inclined at a certain angle with the axis of test specimen. Consequently the load falls a bit. However, due to strain hardening of the material in the band, the load again increases till another Luder band appears in the test piece.
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This goes on till the whole specimen is full of Luder bands. Thus between the points C and D the stress is oscillating between two narrow limits. This occurs in alloys having interstitial solid solution structure like low carbon steels. The stress at the point C is called lower yield strength.
However with further increase in strain beyond the point D, when the test piece is full of Luder bands, the load or stress again starts increasing. The distinction between the two yields may disappear with strain hardening and only a small kink may remain on the stress strain curve. Some authors prefer to take stress value at C as the flow stress at the yield point, however, the data given in material standards generally refers to upper yield point.
After the point D the stress-strain curve moves upwards, however, with further deformation, its slope gradually decreases to zero at the point E which is the highest point on the curve. After E the curve goes down. Before the point E, increase in strain increases the load on the specimen due to strain hardening.
Even after the point E, the strain hardening is still there but at some point the area of cross section of the test piece starts decreasing much faster and a neck formation starts, with the result, the force that the test piece can bear decreases continuously with further deformation resulting in an unstable condition.
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After some elongation in the neck, the specimen fractures at the point F. Since we have defined stress as force divided by original area of cross section the stress value thus calculated also decreases after E, however, if we take true stress, it will be much higher.
The stress at the point E is known as ultimate tensile strength. At point E the actual area of cross section is smaller than the original area of cross-section. In order to obtain realistic values of material strength, we must plot true stress-true strain curve.
Most of the nonferrous metals and alloys do not show the sudden bend or kink in their stress strain curve like the one we get for low carbon steel. Instead there is a gradual transition from elastic to plastic state (Fig. 1.7). In such cases, the yield strength may be taken at the point of intersection of tangents to elastic and plastic lines.
However the change from elastic to plastic state is very gradual, the point of intersection is rather ambiguous. Different values may be obtained if tangents are drawn at different points. Therefore, an offset yield point is obtained at a strain of 0.002 (0.2%). A straight line is drawn parallel to initial portion of stress-strain curve at the strain value of 0.002 and the point where it intersects the stress-strain curve is taken as yield point.
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The curves of Figs. 1.6 and 1.7 are drawn with the stress defined as load divided by original area of cross-section and the longitudinal strain is defined as δl/l0, where l0 is the original length of test specimen. The curves would look very much different if we use true stress on ordinate and true strain on abscissa.
The true stress and true strain are defined below:
For accurate calculations, the true stress-true strain curve for the metal should be drawn to determine the yield strength. Figure 1.6 redrawn on true stress-true strain axes would look like the one shown in Fig. 1.8. Also there are standard specifications for the shape and dimensions of test specimen, which should be adapted in order to obtain meaningful results.
Besides, in all above type of tests the following factors should also be noted:
(i) Temperature at which the test is conducted.
(ii) The strain rate during the test.
(iii) Accuracy of load measuring instrument.
(iv) Accuracy of instrument which measures elongation.
A mild steel rectangular specimen of length 100 mm is extended to 120 mm. neglecting the elastic deformation and taking that material is isotropic, determine the true strains in length, width and thickness directions.
Determine the true strains in length, width and thickness directions of a sheet metal test specimen if it is elongated to 130% of its original length. The anisotropy ratio ԑw/ԑt = 1.5
Also determine the per cent decrease in area of cross-section.
