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In the rolling of steel, a sheet of steel is compressed plastically, so that it is thinner upon exit than at entry. This necessitates a change in velocity to maintain mass flow. As a result, certain portions of the roll can be considered pushing or pulling the strip from friction due to mismatch between the velocities. At a certain point in the roll, these velocities are considered equal, and is thus termed the neutral point.

The roll pressure is not uniform throughout the roll bite, and has a triangular distribution like the one shown below, known as the friction hill, since friction is proportional to the force applied. This relation is deemed appropriate as the peak of the friction hill happens to be the neutral point. As such, the roll can be considered pushing the strip forward preceding the neutral point and pulling the strip back proceeding the neutral point.

Friction Hill

This qualitative description makes sense to me so far, but I have trouble understanding it in respect to material deformation. With a stress strain curve, like the one shown below, deformation increases monotonically until its ultimate strength. Prior to that point, any decrease in stress will relax the material so that it decompresses elastically to its new size.

Stress-Strain Curve

When passing through a roll stand, the friction hill dictates that at some point, pressure applied to the strip decreases. A decrease in pressure (and therefore stress) should necessitate that either a transition from plastic to elastic behavior, or exceeding the ultimate strength of the material. Given that a sheet of steel can be rolled through multiple stands in quick succession, it stands to reason that the latter is not the case, else the sheet should fail in compression. However, since the sheet is still compressed, and furthermore, plastically compressed past the neutral point, it would seem it should experience an increase, rather than decrease, in roll pressure.

Why does maximum roll pressure not occur at the plastic-elastic boundary? This appears to be a natural consequence of the assumption that maximum pressure occurs at the neutral point. I can accept that the neutral point may occur well before the plastic-elastic boundary, but I don't know why maximum pressure must occur at the neutral point.

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1) During the pass, the elastic limit is reached just in the beggining (it correspond about 0.2% of deformation, and a typical pass has easily 20%). The material is all the time in the plastic region. So forget about elastic-plastic boundary.

2) The stress strain curve is valid under uniaxial stress. If a test sample had also transverse stresses for example, the curve would be different. In order to plastic deformation happens, shear stresses must reach a minimum value. And they result from the difference of main stresses. It is easy in the uniaxial case because the side stresses are zero.

The material inside the contact arc are under vertical compression due to the rolls. But it is also highly constrained to move sideways due to friction with the rolls. So, most of the material flow is longitudinal.

The roll RPM is constant, but the speed of the bar increases during the pass. So, in the first part, the bar is slower than the rolls and the friction force in the surface is in same direction of rolling. The opposite after the neutral point. The result is building up a longitudinal compressive force in the bar that is maximum at the neutral point.

Now we have a triaxial compressive stress state. If all stresses were equal, there would be no plastic deformation. But as the friction forces have a limit, the vertical stress increases until the difference to the horizontal stress is enough to deform the material (or break the roll...). That is the reason for the friction hill.

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