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Theoretical Considerations

Mechanics of Extrusion

The ram force F, or the input force exerted on the ram, during a forward extrusion process can be calculated. The value of the calculated force depends on the model used.

The input power is the power supplied by the ram force as it moves with a velocity uo:

This total input power is transformed into:

1. Ideal power consumed by the plastic deformation.
2. Frictional power dissipated due to friction along the die angle.
3. Redundant work due to inhomogeneous deformation.

In general, the total ram force depends on the above three components. There is a die angle, see figure below, for which the ram force is minimum. However, unless each component of the powers is know as a function of the die angle, it is very hard to determine the optimum angle.

Ideal Deformation

Let Ao be the cross-sectional area of the billet (material before being extruded), and let Af be the cross-sectional area of the extruded piece. An extrusion ratio is defined as:

Then, the absolute value of the true stain, , is given by:

= ln(R)

If Y denotes the yield stress of the perfectly plastic material, the energy dissipated in plastic deformation per unit volume is:

E = Y

The power due to plastic work of deformation is:

Pplastic = uo Ao E = uo Ao Y

In the ideal case, we assume that the total power input is equal to the power due to plastic work of deformation. Recall that the input power is:

Where p is the extrusion pressure at the ram. Equating Pplastic and Pinput, we find that:

Note that the value of extrusion pressure is equal to the area under the true stress / true strain curve for the material.

Ideal Deformation and Friction

When friction at the die-billet interface is accounted for, the power input is equal to the sum of the plastic deformation and the frictional power.

Because the billet is forced through a die with a substantial reduction in its cross-section, a dead zone in the metal flow pattern develops at the die exit region.

We assume that the material flow in that region takes place at a 45 degrees, this is an "effective die angle", and that the friction stress is equal to the shear yield stress k = Y/2 of the material. The power dissipated due to friction along the die angle is:

Pfriction = (uo/cos(45)) Ao (Y/2)

Equating the power input to the sum of the power of plastic deformation and the power of friction force:

p Ao uo = uo Ao Y + (uo/cos(45)) Ao (Y/2)

It follows immediately that the extrusion pressure is:

In this analysis, the force required to overcome friction at the billet-container interface was neglected. It can be easily calculated if we assume again that the frictional stress is equal to the shear yield stress of the material, k, and we let AL denotes the lateral surface of the billet remaining in the die, then an additional ram pressure, pf, due to wall friction is given by:

Thus, the total extrusion pressure becomes:

Actual Forces

The derivation of analytical expressions, including friction, die angle, and redundant work due to inhomogeneous deformation of the material, can be difficult. Consequently, a convenient empirical formula has been developed:

where a and b are constants determined experimentally. Approximate values for a and b are 0.8 and 1.2 to 1.5, respectively.

For strain hardening materials, Y in the above expressions should be replaced by the average flow stress.

Source: Kalpakjian, Manufacturing Processes for Engineering Materials.