Plane strain, β = 0 Modes of deformation

3.4.4 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 3.3e. From Equation 3.1b, the thickness strain is zero and from Equation 2.19c the effective strain is ε = 2 √ 3 ε 1 = 1.155ε 1 and work-hardening is gradual. Splitting is unlikely and in practical forming operations, large strains are often encountered in this mode.

3.4.5 Uniaxial compression, β = −2

This process, indicated by the point E, is an extreme case and occurs when the major stress σ 1 is zero, as in the edge of a deep-drawn cup, Figure 3.3f. The minor stress is compressive, i.e. σ 2 = −σ f and the effective strain and stress are ε = −ε 2 and σ = −σ 2 respectively. In this process, the sheet thickens and wrinkling is likely.

3.4.6 Thinning and thickening

Plotting strains in this kind of diagram, Figure 3.3a, 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.

3.4.7 The engineering strain diagram

In the sheet metal industry, the information in Figure 3.3a is often plotted in terms of the engineering strain. In Figure 3.3g, 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.

3.5 Effective stress–strain laws

In the study of a process, the first step is usually to obtain some indication of the strain distribution, as in Figure 3.2c. As mentioned, this may be done by measuring grids or from some geometric analysis. The next step is to determine the stress state associated with strain at each point. To do this, one must have stress–strain properties for the material and Chapter 2 indicates how the tensile test data can be generalized to apply to any simple process using the effective stress–strain relations. In numerical models, the actual stress–strain curve can be used as input, but in a mechanics model it is preferable to use a simple empirical law that approximates the data. Here we consider some of these laws. 36 Mechanics of Sheet Metal Forming