Figure:(a) The strain diagram showing the different deformation modes corresponding to different strain ratios. (b) Equibiaxial stretching at the pole of a stretched dome. (c) Deformation in plane strain in the side-wall of a long part. (d) Uniaxial extension of the edge of an extruded hole. (e) Drawing or pure shear in the flange of a deep-drawn cup, showing a grid circle expanding in one direction and contracting in the other. (f) Uniaxial compression at the edge of a deep-drawn cup. (g) The different proportional strain paths shown in Figure plotted in an engineering strain diagram.
which β = 1. As stated above, the principal tension and principal stress in the direction, 1, will always be tensile or positive, i.e. σ1 ≥ 0. For the extreme case in which σ1 = 0 we find from Equations 2.6 and 2.14, that α = −∞ and β = −2. Therefore all possible straining paths in sheet forming processes will lie between 0A and 0E in Figure(a) and the strain ratio will be in the range −2 ≤ β ≤ 1.
Equal biaxial stretching, β = 1
The path 0A indicates equal biaxial stretching. Sheet stretched over a hemispherical punch will deform in this way at the centre of the process shown in Figure (b). The membrane strains are equal in all directions and a grid circle expands, but remains circular. As β = 1, the thickness strain is ε3 = −2ε1, so that the thickness decreases more rapidly with respect to ε1 than in any other process. Also from Equation 2.19(c), the effective strain is ε = 2ε1 and the sheet work-hardens rapidly with respect to ε1.
Plane strain, β = 0
In this process illustrated by path, 0B, in Figure (a), the sheet extends only in one direction and a circle becomes an ellipse in which the minor axis is unchanged. In long, trough-like parts, plane strain is observed in the sides as shown in Figure (c). It will be shown later that in plane strain, sheet is particularly liable to failure by splitting.
Uniaxial tension, β = −1/2
The point C in Figure (a) is the process in a tensile test and occurs in sheet when the minor stress is zero, i.e. when σ2 = 0. The sheet stretches in one direction and contracts in the other. This process will occur whenever a free edge is stretched as in the case of hole extrusion in Figure (d).
Constant thickness or drawing, β = −1
In this process, point D, membrane stresses and strains are equal and opposite and the sheet deforms without change in thickness. It is called drawing as it is observed when sheet is drawn into a converging region. The process is also called pure shear and occurs in the flange of a deep-drawn cup as shown in Figure (e). From Equation 3.1(b), the thickness strain is zero and from Equation 2.19(c) the effective strain is ε = 2/√3ε1 = 1.155ε1 and work-hardening is gradual. Splitting is unlikely and in practical forming operations, large strains are often encountered in this mode.
Uniaxial compression, β = −2
This process, indicated by the point E, is an extreme case and occurs when the major stress σ1 is zero, as in the edge of a deep-drawn cup, Figure (f). The minor stress is compressive, i.e. σ2 = −σf and the effective strain and stress are ε = −ε2 and σ = −σ2 respectively. In this process, the sheet thickens and wrinkling is likely.
Thinning and thickening
Plotting strains in this kind of diagram, Figure (a), is very useful in assessing sheet forming processes. Failure limits can be drawn also in such a space and this is described in a subsequent chapter. The position of a point in this diagram will also indicate how thickness is changing; if the point is to the right of the drawing line, i.e. if β > −1, the sheet will thin. For a point below the drawing line, i.e. β < −1, the sheet becomes thicker. The engineering strain diagram
In the sheet metal industry, the information in Figure (a) is often plotted in terms of the engineering strain. In Figure (g), the strain paths for constant true strain ratio paths have been plotted in terms of engineering strain. It is seen that many of these proportional processes do not plot as straight lines. This is a consequence of the unsuitable nature of engineering strain as a measure of deformation and in this work, true strains will be used in most instances. Engineering strain diagrams are still widely used and it is advisable to be familiar with both forms. In this work, true strain diagrams will be used unless specifically stated.
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