Worked example empirical laws

c By substitution, at ε , σ = 500ε 0.26 = 100 MPa∴ε = 0.20 1 0.26 = 0.002. At ε = 0.40, σ = Kε + ε n gives σ = 395 MPa; i.e. the difference in the laws is neg- ligible at large strains.

3.6 The stress diagram

It has already been mentioned that a diagram in which the strains are plotted, e.g. Figure 3.2c, is valuable in the study of a process. In the same way, a diagram in which the stress state associated with each strain point is shown is very useful in understanding the forces involved in a process. Such a diagram is shown in Figure 3.5. Like Figure 3.3a, this is not a diagram for a particular process, but is used to illustrate the link between the strain and stress diagrams. Also, contours of equal effective stress are shown, which are of course yield loci for particular values of flow stress. During deformation, plastic flow will start from the initial yield locus shown as a continuous line, i.e. when σ = σ f and the loading path will be along a radial line of slope 1α. B A C D E a = − ∞ a = 12 a = − 1 a = 1 a = s 1 s 2 s f 1 s f a Figure 3.5 The processes shown in the strain space, Figure 3.2, illustrated here in the stress space. The current yield ellipse is shown as a broken line.. To plot a point in this diagram, the stress ratio is calculated from the strain ratio, Equation 2.14. The effective strain is determined from Equation 2.19c, and from the known material law, the effective stress determined and the principal stresses calculated from Equation 2.18b. The current state of stress is shown as a point on the ellipse given as a broken line. This yield locus intercepts the axes at ±σ f . The principal stresses are σ 1 ; σ 2 = α.σ 1 and σ 3 = 0 3.11 and each path in the strain diagram, Figure 3.3a has a corresponding path in the stress diagram as detailed below. Deformation of sheet in plane stress 39

3.6.1 Equal biaxial stretching, α = β = 1

At A, the sheet is stretching in equal biaxial tension and σ 1 = σ 2 = σ 3.12 In an isotropic material, each stress is equal to that in a simple tension test.

3.6.2 Plane strain, α = 12, β = 0

For plane strain, i.e., zero strain in the 2 direction, the stress state is indicated by the point B and σ 1 = 2 √ 3 σ = 1.15σ and σ 2 = 1 2 σ 1 3.13 For a material of given flow stress, the magnitude of the major stress, σ 1 , is greater in this process than in any other.

3.6.3 Uniaxial tension, α = 0, β = −12

This point is illustrated by C in Figure 3.5; the major stress is equal to the flow stress σ f and the minor stress is zero. The process occurs in the tensile test, and as mentioned, at a free edge.

3.6.4 Drawing, shear or constant thickness forming, α = −1, β = −1

Along the left-hand diagonal at D, the membrane stresses and strains are equal and opposite and there is no change in thickness. The stresses are σ 1 = 1 √ 3 σ f = 0.58σ f = 0.58σ and σ 2 = − 1 √ 3 σ f = −0.58σ f = −0.58σ 3.14 It will be noted that the magnitudes of stresses to cause deformation are at a minimum in this process, i.e. in magnitude, they are only 58 of the stress required to yield a similar element in simple tension. This can be considered an ideal mode of sheet deformation as the stresses are low, there is no thickness change, and, as will be shown later, failure by splitting is unlikely.

3.6.5 Uniaxial compression, α = −∞, β = −2

This mode mostly occurs at a free edge in drawing a sheet as the stress on the edge of the sheet is zero. The minor stress is equal to the compressive flow stress, i.e. σ 1 = 0 and σ 2 = −σ f = −σ 3.15 As indicated, high compressive stresses are often associated with wrinkling of the sheet. 40 Mechanics of Sheet Metal Forming