Inhomogeneity Factors affecting the forming limit curve

e 1 e 2 e 2 e 1 High e f Low e f Figure 5.19 Diagram showing the effect of fracture strain ε f on the limit strains. e 1 e 2 Small imperfection Large imperfection Figure 5.20 Effect of the magnitude of the imperfections on the forming limit curves. reduction in thickness, but other forms of imperfection are possible, such as inclusions, local reductions in strength due to segregation of strengthening elements or texture vari- ation. Surface roughness may also be a factor. Whatever the form of the imperfection, it will also have a distribution both spatially and in size population; as the critically strained regions may only occupy a small area of the sheet, there is also a probabilistic aspect. The critical region may, or may not contain a large defect and therefore there is likely to be some scatter in measured limit strains and the forming limit curve is more properly a region of increasing probability of failure.

5.5.5 Anisotropy

The shape of the yield locus is shown to influence the forming limit in biaxial tension. This locus changes if the material becomes anisotropic. If a quadratic yield function is used, as in Equation 2.11, anisotropy in the sheet, characterized by an R-value 1, will cause the locus to be extended along the biaxial stress axis as shown in Figure 5.21a. The effect of this in a numerical analysis of the forming limit would be to reduce the biaxial strain limit. This is not observed experimentally and it appears that a different yield function employing higher exponents, 6–8, is more realistic for certain materials. The shape of a yield locus for a high exponent law is shown in Figure 5.21b and for such a model it Load instability and tearing 77 e 1 e 1 e 2 e 2 R 1 R = 1 R 1 R = 1 a b Figure 5.21 Plane stress yield loci. The dotted ellipses are for a quadratic function for an isotropic material; the bold lines are for a high R-value material with a a quadratic yield function and b a high exponent function. n e 1 e 2 e = n Forming limit Figure 5.22 Diagram showing the strain envelope in which material data can be obtained from the tensile test. has been shown that changes in the R-value do not have a significant influence on the forming limit curve.

5.5.6 Other considerations

In the tensile test, stress strain data can only be obtained up to the onset of diffuse necking, i.e. up to an effective strain ε ≤ n. The envelope of this strain is shown in Figure 5.22 78 Mechanics of Sheet Metal Forming