d3 1 P dei
When the tensions reach a maximum, Equation 5.22 becomes zero and the non-dimensional strain-hardening is, This is only valid for p gt -1. If p lt -1, the sheet will thicken and for a strain-hardening material, the tension will never reach a maximum. For a material that obeys the generalized stress-strain law it may be shown by substituting Equations 2.18 b and 2.19 c in Equation 5.24 that the relationship between principal stress and strain during proportional deformation in which a and p are...
112 Free expansion of a cylinder by internal pressure
Expansion of a round tube without change in length is analysed. The tube will deform in plane strain, i.e. the strain in the axial direction will be zero. Initially the tube will remain circular and the radius will increase. The expansion of a cylindrical element in this mode is illustrated in Figure 11.4. The strain and stress states, for an isotropic material, are s0 se 0 st - 1 se -se Figure 11.4 Element of a circular tube with internal pressure. If the material properties obey the...
622 Plane strain bending
If the flat sheet on either side of the bend in Figure 6.1 is not deforming it will constrain the material in the bend to deform in plane strain i.e. the strain parallel to the bend will be zero. In this work, plane strain conditions will be assumed, unless stated otherwise. The deformation process in bending an isotropic sheet is therefore Following Equations 2.18 b and 2.19 c , for, p 0, a 1 2, we obtain a i Of S and ei s 6.6 where S is the plane strain flow stress. Equation 6.6 assumes the...
913 Worked example the hydrostatic bulging test
Equipment designed to obtain an effective stress strain curve by bulging a circular diaphragm with hydrostatic pressure is shown, in part, in Figure 9.4. An extensometer measures the current diameter D of a small circle near the pole of original diameter D0 and a spherometer measures the height of the pole above this circle h. The current pressure is p and the original thickness of the sheet is t0. Assuming that within this circle a state of uniform biaxial tension exists and that the shape is...
542 Necking in biaxial tension
In the first quadrant of the strain diagram where both principal strains are positive or tensile, there is no direction of zero extension and, as discussed above, necking of the type illustrated in Figure 5.10 is not possible. Experimentally it is observed that necking still occurs under biaxial tension, but as shown in Figure 5.9 b , at a strain greater than the attainment of maximum tension and usually along a line perpendicular to the major tensile stress. To explain this, a different model...
111 The loadextension diagram
Figure 1.2 shows a typical load-extension diagram for a test on a sample of drawing quality steel. The elastic extension is so small that it cannot be seen. The diagram does not represent basic material behaviour as it describes the response of the material to a particular process, namely the extension of a tensile strip of given width and thickness. Nevertheless it does give important information. One feature is the initial yielding load, Py, at which plastic deformation commences. Initial...
114 Worked example tensile test properties
The initial gauge length, width and thickness of a tensile test-piece are, 50, 12.5 and 0.80 mm respectively. The initial yield load is 1.791 kN. At a point, A, the load is 2.059 kN and the extension is 1.22 mm. The maximum load is 2.94 kN and this occurs at an extension of 13.55 mm. The test-piece fails at an extension of 22.69 mm. Determine the following initial yield stress, 1A_5- 179 x 106 Pa 179 MPa initial cross-sectional area, 12.5 x 0.80 10 mm2 10 5m2 tensile strength, 2.94 x 103 10-5...
113 The true stressstrain 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. i 17 where A is the...
Common forming processes
Blanking and piercing. As sheet is usually delivered in large coils, the first operation is to cut the blanks that will be fed into the presses subsequently there may be further blanking to trim off excess material and pierce holes. The basic cutting process is shown in Figure I.1. When examined in detail, it is seen that blanking is a complicated process of plastic shearing and fracture and that the material at the edge is likely to become hardened locally. These effects may cause difficulty...