Uniaxial compression, α = −∞, β = −2 The stress diagram

3.7 Principal tensions or tractions

In Section 3.1.1, the use of ‘tensions’ in the analysis of sheet metal forming was introduced. The principal tensions on a sheet element are illustrated in Figure 3.1c. Tension is the force per unit length transmitted in the sheet and has the units of [force][length]; a typical unit used is kNm. These tensions govern deformation in the sheet and the forces acting on the tooling. It is found more convenient to model processes in terms of tension rather than stress and for this reason, the determination of tensions for the different processes illustrated in Figures 3.2a and 3.5 is described. We shall be concerned here only with tensions in the principal directions. As thickness will change, we must calculate both the current principal stresses and thickness in order to determine the principal tensions. For any region, as in Figure 3.1, the principal stresses and current thickness can be determined using the relations given above. The tensions will be in proportion to the stresses, i.e. T 1 = σ 1 t ; T 2 = σ 2 t = αT 1 3.16 For a region of a sheet in which the thickness is uniform, i.e. t = constant, the principal tensions will satisfy a tension yield locus that is geometrically similar to the yield stress locus as given in Figures 2.6 and 2.7. If the material obeys a von Mises yield condition, the principal tensions in the sheet at yield will satisfy a generalized or effective yielding tension relation given by T = σ t = T 2 1 − T 1 T 2 + T 2 2 = 1 − α + α 2 T 1 3.17 This is illustrated in Figure 3.6. T 1 T 2 T O Figure 3.6 Relation between the principal tensions for an element deforming in a proportional process with a current effective tension of T = σ t. Deformation of sheet in plane stress 41 T 1,2 e 1 e 1 ∗ = n 1 + b T 1 T 2 Figure 3.7 Principal tensions versus the major strain for a proportional process. For any particular stress ratio and major strain, the effective stress and the thickness can be obtained using the relations given above. For a material in which the stresses and strains obey the power law, Equation 3.6, the major tension can be determined as T 1 = σ 1 t = Kε n t exp [ − 1 + β ε 1 ] √ 1 − α + α 2 3.18 This may be derived using equations 2.12d, 2.5 and 2.6. From Equations 3.16 and 3.18, the principal tensions can be found and are illustrated in Figure 3.7; in this case the strain ratio is positive. As discussed earlier, the major tension T 1 will always be equal to or greater than zero. Depending on the value of the stress or strain ratios, the minor tension T 2 will be positive or negative. For a given material and process, Equation 3.18 can be reduced to the form T 1 = const ε n 1 exp [ − 1 + β ε 1 ] 3.19 Differentiating this expression, we find that tensions reach a maximum only for processes in which the sheet thins, i.e. when β −1. When this is the case, the strain at maximum tension, denoted by ∗ , will be ε ∗ 1 = n 1 + β 3.20 For uniaxial tension, β = −12, the maximum in tension is at ε ∗ 1 = 2n, and for plane strain, β = 0, maximum tension is when ε ∗ 1 = n. The relation between maximum tension and necking is discussed in Chapter 5.

3.7.1 Worked example tensions

Compare the tensions and the thickness at the point of maximum tension in a sheet initially of 0.8 mm thickness with a stress strain characteristic of, σ = 530ε 0.246 MPa when deformed in equal biaxial tension and plane strain. Solution. For equal biaxial tension, α = 1, β = 1. At maximum tension, ε ∗ 1 = n 1 + β = n 2 = 0.123.The effective strain is ε = 2ε 1 = 0.246. The thickness at maximum tension is, t = t exp − 1 + β ε ∗ 1 = 0.8 exp −0.246 = 0.626 mm 42 Mechanics of Sheet Metal Forming