9.5 HODOGRAPHS FOR SLIP-LINE FIELDS

Construction of hodographs for slip-line fields is necessary to

1. Assure the field is kinematically admissible;

2. Determine where in the field most of the energy is expended; and

3. Predict distortion of material as it passes through the field.

In constructing hodographs it may be noted that

1. The velocity is constant within a constant pressure zone;

2. In leaving a field of changing ? ₂ there may or may not be a sudden change of velocity;

3. The magnitude of the velocity everywhere along a given slip line is constant though the direction may change;

4. In a field of curved lines, both the magnitude and direction of the velocity change;

5. There is always shear on the boundary between the deforming material inside the field and the rigid material outside of it; and

6. The vector representing a velocity discontinuity must be parallel to the discontinuity itself.

Figure 9.10 shows half of the field and the corresponding hodograph. Region OAD moves downward with the velocity, V ₀ , of the punch. There is a discontinuity along OA such that the absolute velocity is parallel to the arc AE at A, and the velocity just to the right of OA differs from that in triangle OAD by a vector parallel to OA. The discontinuity between the material in the field at A and outside the field is equal to the absolute velocity inside the field at A. This discontinuity...