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, μ = 0.1. The flow stress (σ) of the material in MPa is σ = 207 + 414 𝜀, where 𝜀 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:

 $F = 1.15 \barσ \left( 1 + \frac{\mu L}{2 \bar h} \right)$ wL, where $\bar \sigma$ is average flow stress, L is roll-workpiece contact length, and $\bar h$ is the average sheet thickness

A $$10$$ $$mm$$ deep cylindrical cup with diameter of $$15$$ $$mm$$ is drawn from a circular blank. Neglecting the variation in the sheet thickness, the diameter (upto $$2$$ decimal points accuracy) of the blank is _________ $$mm.$$

In a sheet metal of $$2$$ $$mm$$ thickness a hole of $$10$$ $$mm$$ diameter needs to be punched. The yield strength in tension of the sheet material is $$100$$ $$MPa$$ and its ultimate shear strength is $$80$$ $$MPa.$$ The force required to punch the hole (in $$kN$$) is ______________

A rectangular hole of size $$100\,\,mm \times 50\,\,mm$$ is to be made on a $$5$$ $$mm$$ thick sheet of steel having ultimate tensile strength and shear strength of $$500$$ $$MPa$$ and $$300$$ $$MPa,$$ respectively. The hole is made by punching process. Neglecting the effect of clearance, the punching force (in $$kN$$) is