Collet chuck

Collet chucks form a tapered collar to close on the tube when an outer pipe slides over the taper.

The image at the right shows a standard Eaton Leonard style collet chuck with inserts.

From Wikipedia "A collet...is a holding device—specifically, a subtype of chuck—that forms a collar around the object to be held and exerts a strong clamping force on the object when it is tightened via a tapered outer collar. It may be used to hold a workpiece or a tool."