Analytical modeling of material flow in equal channel angular extrusion (ECAE)

We provide analytical forms for the plastic deformation and velocity gradients associated with a single pass of equal channel angular extrusion (ECAE). Three cases of plastic deformation are considered: ideal simple shear, a plastic deformation zone (PDZ) in the shape of a central fan of angle βm, and a two-part PDZ consisting of a central fan in the ‘upper’ region and a low intensity shear deformation in the ‘lower’ region. The analysis for simple shear considers a general die angle Φ, whereas the other two cases only consider Φ=90°. The tensors for deformation and velocity gradients completely describe the deformation, such as the directions and magnitudes of material stretching and rotations. From this analysis, one can calculate deformation and texture evolution. Texture evolution during flow through the central fan zone involves continuous rotation of the texture components causing the texture developed at the end of the extrusion to be rotated relative to the ideal simple shear case. The analysis of the two-part zone suggests inhomogeneity in texture evolution, in which features of the initial texture are retained and rotated in the lower region, while they are nearly erased in the upper region. These analytical flow patterns for a single pass can be repeatedly applied for any number of passes of any ECAE route.