mechanics_of_sheet_metal_forming_1_6798

P Perfect strip
eU

Load

ETotal. Real test-piece

E
0 Elongation

Figure 5.6 Load extension diagram for a perfect strip and one containing an imperfection.

functions in Equation 5.12, we obtain

(n − εU) ≈ −n dAo (5.13)
A0

As dA0 is a negative quantity, the root of Equation 5.13 is real and the difference between
the maximum strain in the uniform region and the strain-hardening index n is dependent

on the strain-hardening index and the magnitude of the imperfection.

5.3.1 (Worked example) maximum uniform strain

The length, width and thickness of the parallel reduced section of a tensile test-piece are
100, 12.5 and 0.8 mm respectively. The material has a stress, strain curve fitted by the
relation σ1 = 700ε10.22 MPa. Over a small length, the width is 0.05 mm less than elsewhere.
Estimate the strain in the uniform region of the test-piece after the strip has been tested
to failure.

Solution. The initial cross-sectional area is 0.8 × 12.5 = 10 mm2. The difference in area
at the imperfection is 0.8 × 0.05 = 0.04 mm2. The imperfection ratio is dA0/A0 = −0.004.

From Equation 5.13,

0.22 − εU = −0.22 (−0.004) = 0.03 or εU = 0.19 (5.14)

If the imperfection did not exist, the uniform strain would be 0.22. We thus see that an
imperfection of only 0.4%, reduces the maximum uniform strain by (0.03/0.22) × 100 =
13.5%. This demonstrates a phenomenon often observed in sheet metal forming: very small
changes in initial conditions can give large changes in the final result. It is also found
that if repeated tests are performed on apparently uniform material, there is considerable
scatter in the observed maximum uniform strain. This is probably due to differences in
the magnitude of imperfections that exist in individual test-pieces.

5.3.2 The effect of rate sensitivity

The analysis above assumes that the material strain-hardens, but is insensitive to changes
in strain rate. In sheet metal at room temperature, rate sensitivity is small, but it can affect
necking as strain rates in the neck can become quite high when uniform straining ceases

66 Mechanics of Sheet Metal Forming