GATE 2022 ME (ME1) – Question 56
A 4 mm thick aluminum sheet of width $w = 100$ mm is rolled in a two-roll mill of roll diameter 200 mm each. The workpiece is lubricated with a mineral oil, which gives a coefficient of friction, $\mu = 0.1$. The flow stress ($\sigma_f$) of the material in MPa is $\sigma_f = 207 + 414\varepsilon$, where $\varepsilon$ is the true strain. Assuming rolling to be a plane strain deformation process, the roll separation force ($F$) for maximum permissible draft (thickness reduction) is _________ kN (round off to the nearest integer).
Use: $F = 1.15\bar{\sigma}_f\left(1 + \frac{\mu L}{2\bar{h}}\right)wL$, where $\bar{\sigma}_f$ is average flow stress, $L$ is roll-workpiece contact length, and $\bar{h}$ is the average sheet thickness.
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Correct answer: 350
Explanation
The maximum draft is $\Delta h_{max} = \mu^2R = 0.01 \times 100 = 1$ mm, so the sheet goes from 4 mm to 3 mm. The contact length is $L = \sqrt{R\Delta h} = \sqrt{100 \times 1} = 10$ mm and the average thickness is $\bar{h} = 3.5$ mm. The true strain is $\varepsilon = \ln\frac{4}{3} = 0.2877$, and the average flow stress is $207 + 414 \times \frac{0.2877}{2} = 266.6$ MPa. Then $F = 1.15 \times 266.6 \times \left(1 + \frac{0.1 \times 10}{7}\right) \times 100 \times 10 = 350.3 \times 10^3$ N $= 350$ kN.