The thickness of a plate is reduced from $$30$$ $$mm$$ to $$10$$ $$mm$$ by successive cold rolling passes using identical rolls of diameter $$600$$ $$mm.$$ Assume that there is no change in width. If the coefficient of friction between the rolls and the work piece is $$0.1,$$ the minimum number of passes required is

In a rolling process, the roll separating force can be decreased by

During open die forging process using two flat and parallel dies, a solid steel disc of initial radius $$({R_{IN}})\,\,200mm$$ and Initial height $$({H_{IN}})\,\,50mm$$ attains a height $$({H_{FN}})$$ of $$30mm$$ and radius of $${R_{FN}}$$. Along the die-disc interfaces

(i) The coefficient of friction $$\left( \mu \right)$$ is :
$$\,\,\,\,\,\,\,\mu = 0.35\left[ {1 + {e^{ - {R_{IN}}/{R_{FN}}}}} \right]$$

(ii) In the region $${R_{SS}}\,\, \le \,\,r\,\, \le \,\,{R_{FN}},$$ sliding friction prevails and
$$\,\,\,\,\,\,P = \sqrt 3 .K.{e^{2\mu \left( {{R_{IN}} - r} \right)/{H_{FN}}}}$$ and $$\tau = \mu \,p$$

Where $$p$$ and $$\tau $$ are the normal and the shear stress respectively; $$K$$ is the shear yield strength of steel and $$r$$ is the radial distance of any point
(i) In the region $$0\, \le \,r\, \le \,{R_{IN}}.$$ sticking condition prevails

The value of $$RSS$$ (in $$mm$$ ), where sticking condition changes to sliding friction is

Using direct extrusion process, a round billet of $$100$$ $$mm$$ length and $$50mm$$ diameter is extruded. Considering an ideal deformation process (no friction and no redundant work), extrusion ratio $$4$$ and average flow stress of material $$300$$ $$MPa,$$ the pressure $$(MPa)$$ on the $$ram$$ will be