Uniaxial tension, β = −12 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 The effective strain ε for any deformation process such as the one illustrated in Figure 3.1 can be calculated from the principal strains and the strain ratio using Equation 2.19c. As shown in Section 2.8, if the material is isotropic, the effective stress–strain curve is coincident with the uniaxial true stress–strain curve and a variety of mathematical relations may be fitted to the true stress–strain data. Some of the more common empirical relations are shown in Figure 3.4 and in these diagrams elastic strains are neglected. In the diagrams shown, the experimental curve is represented by a light line, and the fitted curve by a bold line. e s e e e e s s s s = K e + e n s = K e n s = Y + P e s = Y a b c d Figure 3.4 Empirical effective stress–strain laws fitted to an experimental curve.

3.5.1 Power law

A simple power law σ = Kε n 3.6 will fit data well for some annealed sheet, except near the initial yield; this is shown in Figure 3.4a. The exponent, n, is the strain-hardening index as described in Section 1.1.3. The constants, K and n, are obtained by linear regression as explained in the section referred to. The only disadvantage of this law is that at zero strain, it predicts zero stress and an infinite slope to the curve. It does not indicate the actual initial yield stress.

3.5.2 Use of a pre-strain constant

Although it requires the determination of three constants, a law of the type σ = K ε + ε n 3.7 is useful and will fit a material with a definite yield stress as shown in Figure 3.4b. The constant ε has been termed a pre-strain or offset strain constant. If the material Deformation of sheet in plane stress 37