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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.

Indentation using a Constrained Punch

Rule 1:

Look for symmetry and reduce the geometry accordingly.

Look for symmetry and reduce the geometry accordingly.

Indentation using a Constrained Punch

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 it meets the line BC to move by shear parallel to CD and then sheared
along DB to move by shear parallel to DE

Indentation using a Constrained Punch

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.

Indentation using a Constrained Punch

Rule 4:

Draw the velocity vector of the unknown force Q, or unit length (Oq), on the hodograph.

Draw the velocity vector of the unknown force Q, or unit length (Oq), on the hodograph.

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.

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.

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'.

Indentation using a Constrained Punch

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.

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).