# Why is Maximum Roll Pressure Not Always at the Plastic-Elastic Boundary?

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.

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.

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.