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Noncommercial-Share Alike 2.0 Licence UK: England & Wales

https://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Extrusion with Dead Metal Zone at 45°

A hodograph can be constructed for plane strain extrusion
when there is friction on die faces. Here, we assume a dead metal zone
develops in the container adjacent to the die.

Extrusion with Dead Metal Zone at 45°

Rule 1:

Look for symmetry and reduce the geometry accordingly.

Look for symmetry and reduce the geometry accordingly.

Extrusion with Dead Metal Zone at 45°

Rule 2:

Label regions of the model which move relative to each other.

Label regions of the model which move relative to each other.

A particle will travel a path as shown. It is sheared
when meeting line AB to move parallel to AC, then sheared at BC to move
parallel to CD.

Extrusion with Dead Metal Zone at 45°

Rule 3:

Define an origin of the hodograph, O, corresponding to a stationary component of the system.

Define an origin of the hodograph, O, corresponding to a stationary component of the system.

Extrusion with Dead Metal Zone at 45°

Rule 4:

Draw the velocity vector of the unknown force, i.e. unit length (Op), on the hodograph.

Draw the velocity vector of the unknown force, i.e. unit length (Op), on the hodograph.

Extrusion with Dead Metal Zone at 45°

Rule 5:

Draw vectors in the known directions of the moving components (dq and Or), relative to the origin O and to each other (pq and qr ) on the hodograph.

Draw vectors in the known directions of the moving components (dq and Or), relative to the origin O and to each other (pq and qr ) on the hodograph.

Extrusion with Dead Metal Zone at 45°

Rule 6:

Each vertex where these vectors intersect represents one (or more) of the labelled regions of the model.

Each vertex where these vectors intersect represents one (or more) of the labelled regions of the model.

Oq
defines motion of particles in region Q .

Extrusion with Dead Metal Zone at 45°

Rule 7:

Velocity vectors must be oriented parallel to slip planes due to conservation of matter.

Velocity vectors must be oriented parallel to slip planes due to conservation of matter.