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Materials processing and joining

A plate 20 mm thick is cold rolled in several passes to 5.0 mm, assuming the width does not change. Which value is closest to the magnitude of the true strain in the thickness direction, ln(h₀/h)?

  1. A0.75
  2. B1.39
  3. C3.0
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Answer: B. 1.39

The true strain is ln(h₀/h) = ln(20/5.0) = ln 4 ≈ 1.39. 0.75 is the fractional reduction in thickness (20 − 5)/20, and 3.0 is the nominal strain (4 − 1)/1 when the length becomes 4 times at constant volume. True strain has the advantage that intermediate steps can be added together. For example, rolling in two passes, 20 mm → 10 mm → 5 mm, gives a true strain of ln 2 ≈ 0.69 for each pass, and the sum equals the overall 1.39, whereas adding reductions of 50% for each pass does not give the overall 75%. Also, since the volume is constant, the true strain in the length direction is ln 4 ≈ 1.39 as well, equal in magnitude to the thickness strain and opposite in sign.

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