Conical extrusion of a work-hardening material: an asymptotic analysis

We study steady-state conical extrusion of an isotropic, power-law hardening material with a Coulomb friction condition present at the die faces. An asymptotic theory is developed based on an axial velocity field which is nearly slug-like, i.e., a deformation field for which the transverse variations of the axial velocity are modest in size. However, although the velocity is slug-like, within the asymptotic limit considered the shear stresses are not negligible compared to the longitudinal deviatoric stresses. For this reason the theory accounts for the first manifestations of inhomogeneous deformation. In practical terms the validity of the asymptotic theory generally requires either the friction coefficient µ to be small or the die slope h/L to be small (where h is the radius reduction and L the die length). The primary result of the work is the set of equations (76)–(78). In addition, the present formulation enables for the first time the development of a model of inhomogeneous deformation in conical extrusion which is analogous to the very popular inhomogeneous deformation theory developed by Orowan for plane-strain sheet rolling. Results are presented for a number of examples illustrating the depature from a state of homogeneous compression which is typically found.