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Theoretical Considerations

Analyzing the stresses and loads involved in a forging process are important for two major reasons. It allows to determine what materials are possible candidates based on whether or not they can sustain the stresses involved in the process. Also, it gives a clear idea of the forces, and thus the energy input and the equipment needed to perform the operation.

Various methods of analysis are available to determine those stresses and loads. We present one analysis method, the slab method, for the case of simple compression with friction of a rectangular workpiece in plane strain. other methods are the slip-line analysis, the upper-bound technique, and finite element methods. For more information on those methods we refer the reader to Kalpakjian in his book " Manufacturing Processes for Engineering Materials."

Stress Analysis of a rectangular workpiece forged in plane strain

The total volume of the workpiece does not change as it is being compressed between two flat dies. Thus, as its thickness reduces, the workpiece expands laterally. Frictional forces are generated at the die-workpiece interface ot oppose the relative movement of the relative movement of the workpiece. To simplify the analysis, a plane strain deformation is assumed.

The stresses acting on an element in plane-strain compression between the flat dies are shown in the figure below:

Where the lateral stress distribution is assumed to be uniform along the height h.

As the element is in static equilibrium, the balance of horizontal forces assuming unit width gives:

A second equation is found by applying the distortion-energy yield criteria for plane strain, assuming that and are principal stresses:

-= (2/ 3 .5)Y = Y'

Thus,

The assumption that is a principal stress is not strictly true since a shear stress is also acting on the same plane due to friction. However, it is acceptable for low values of the coefficient of friciton .

Combining the two equations, one gets:

Hence,

= C e -(2 x/h)

The boundary conditions are:

Therefore, solving for C one gets:

C = Y' e(2 a/h)

Consequently,

and

It is clear from the expression for the pressure p that it increases exponentially toward the center of the part forming what is known as the friction hill, also it increases as the ratio a/h and the friction increases.

The compression force per unit width of the workpiece is given by the area under the pressure curve. However, usually an approximate expression for the average pressure pav is used:

Hence,

It is important to note that h is an instantaneous height measure. Thus the force at any h during a continuous compression operation must be calculated individually.

Source: Kalpakjian, Manufacturing Processes for Engineering Materials