• Hydrostatic bulging test. In this test a circular disc is clamped around the edge and bulged to a domed shape by fluid pressure. From measurement of pressure, curvature
and membrane or thickness strain at the pole, a true stress–strain curve under equal biaxial tension can be obtained. The advantage of this test is that for materials that
have little strain-hardening, it is possible to obtain stress–strain data over a much larger strain range than is possible in the tensile test.
• Simulative tests. A number of tests have been devised in which sheet is deformed in a particular process using standard tooling. Examples include drawing a cup, stretching
over a punch and expanding a punched hole. The principles of these tests are covered in later chapters.
1.4 Exercises
Ex. 1.1
A tensile specimen is cut from a sheet of steel of 1 mm thickness. The initial width is 12.5 mm and the gauge length is 50 mm.
a The initial yield load is 2.89 kN and the extension at this point is 0.0563 mm.
Determine the initial yield stress and the elastic modulus. b
When the extension is 15, the width of the test-piece is 11.41. Determine the R
-value. [Ans: a: 231 MPa, 205 GPa; b 1.88]
Ex. 1.2 At 4 and 8 elongation, the loads on a tensile test-piece of half-hard aluminium
alloy are 1.59 kN and 1.66 kN respectively. The test-piece has an initial width of 10 mm, thickness of 1.4 mm and gauge length of 50 mm. Determine the K and n values.
[Ans: 174 MPa, 0.12]
Ex. 1.3 The K, n and m values for a stainless steel sheet are 1140 MPa, 0.35 and 0.01
respectively. A test-piece has initial width, thickness and gauge length of 12.5, 0.45 and 50 mm respectively. Determine the increase in load when the extension is 10 and the
extension rate of the gauge length is increased from 0.5 to 50 mmminute. [Ans: 0.27 kN]
Ex. 1.4 The following data pairs load kN; extension mm were obtained from the plastic
part of a load-extension file for a tensile test on an extra deep drawing quality steel sheet of 0.8 mm thickness. The initial test-piece width was 12.5 mm and the gauge length 50 mm.
1.57, 0.080; 1.90, 0.760; 2.24, 1.85; 2.57, 3.66;
2.78, 5.84; 2.90, 8.92
2.93,11.06; 2.94,13.49; 2.92, 16.59; 2.86, 19.48; 2.61, 21.82; 2.18, 22.69 Obtain engineering stress–strain, true stress, strain and log stress, log strain curves. From
these determine; initial yield stress, ultimate tensile strength, true strain at maximum load, total elongation and the strength coefficient, K, and strain-hardening index, n.
[Ans: 156 MPa, 294 MPa, 0.24, 45, 530 MPa, 0.24]
Material properties 13
2
Sheet deformation processes
2.1 Introduction
In Chapter 1, the appropriate definitions for stress and strain in tensile deformation were introduced. The purpose now is to indicate how the true stress–strain curve derived from
a tensile test can be applied to other deformation processes that may occur in typical sheet forming operations.
A common feature of many sheet forming processes is that the stress perpendicular to the surface of the sheet is small, compared with the stresses in the plane of the sheet the
membrane stresses. If we assume that this normal stress is zero, a major simplification is possible. Such a process is called plane stress deformation and the theory of yielding for
this process is described in this chapter. There are cases in which the through-thickness or normal stress cannot be neglected and the theory of yielding in a three-dimensional stress
state is described in an appendix.
The tensile test is of course a plane stress process, uniaxial tension, and this is now reviewed as an example of plane stress deformation.
2.2 Uniaxial tension
We consider an element in a tensile test-piece in uniaxial deformation and follow the process from an initial small change in shape. Up to the maximum load, the deformation
is uniform and the element chosen can be large and, in Figure 2.1, we consider the whole gauge section. During deformation, the faces of the element will remain perpendicular to
each other as it is, by inspection, a principal element, i.e. there is no shear strain associated with the principal directions, 1, 2 and 3, along the axis, across the width and through the
thickness, respectively.
dt t
w P
d w
3 d
l
1 2
l
Figure 2.1 The gauge element in a tensile test-piece showing the principal directions.
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