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> metal punch pressure p b dead metal zone
Indentation using a Constrained Punch
A hodograph can be constructed for the example of a constrained punch. This is similar to hardness testing. The situation is not easy to analyse, with five blocks of metal all moving and eight planes on which there is energy dissipation through shear.
Symmetryaxis metal pressure p punch dead metal zone b b/2 Symmetryaxis
Indentation using a Constrained Punch
Rule 1:
Look for symmetry and reduce the geometry accordingly.
A B C D E Q Q' R S θ θ O
Indentation using a Constrained Punch
Rule 2:
Label regions of the model which move relative to each other.
A particle will travel a path as shown. It is sheared when it meets the line BC to move by shear parallel to CD and then sheared along DB to move by shear parallel to DE
O
Indentation using a Constrained Punch
Rule 3:
Define an origin of the hodograph, O, corresponding to a stationary component of the system.
O q
Indentation using a Constrained Punch
Rule 4:
Draw the velocity vector of the unknown force Q, or unit length (Oq), on the hodograph.
O q r s q' O q
Indentation using a Constrained Punch
Rule 5:
Draw vectors in the known directions of the moving components (Or and Os), relative to the origin O and to each other (q'r and rs) on the hodograph.
Indentation using a Constrained Punch
Rule 6:
Each vertex where these vectors intersect represents one (or more) of the labelled regions of the model.
Oq and Oq' define motion of particles in region Q and Q'.

θ θ θ Completed hodograph
Indentation using a Constrained Punch
Rule 7:
Velocity vectors must be oriented parallel to slip planes due to conservation of matter.
b/2 θ θ θ
Indentation using a Constrained Punch
1. Look for symmetry and reduce the geometry accordingly. 2. Label regions of the model which move relative to each other. 3. Define an origin of the hodograph, corresponding to a stationary component of the system. 4. Draw the velocity vector of the unknown force, or unit length, on the hodograph. 5. Draw vectors in the known directions of the moving components, relative to the origin and to each other on the hodograph. 6. Each vertex where these vectors intersect represents one (or more) of the labelled regions of the model. 7. Velocity vectors must be oriented parallel to slip planes (conservation of matter).