Indentation pressure of a smooth circular punch

A smooth, flat, rigid punch under increasing normal load presses against a half space of a perfectly plastic material which obeys Tresca’s yield criterion. An admissible velocity field is constructed for an arbitrary smooth punch hence, for any particular case, a limit design theorem of Drucker, Prager and Greenberg may be used to compute an upper bound for the punch indentation pressure. A lower bound for any convex area of indentation has been given by Shield and Drucker. The results of the present paper are used to compute an upper bound for a punch with circular cross section. It is conjectured that this is an upper bound for an arbitrary punch.