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N. Zabaras, Spring 2000

In general the prediction of external forces needed to cause metal ow is needed. Such prediction is difcult due to uncertainties introduced from frictional effects and non-homogeneous deformation as well as from not knowing the true manner of strain hardening. Each solution method is based on several assumptions. The easiest method is the ideal work method. The work required for deforming the workpiece is equated to the external work. The process is considered ideal in the sense that the external work is completely utilized to cause deformation only. Friction and non-homogeneous deformation are neglected. AXISYMMETRIC EXTRUSION AND DRAWING1

Figure 1: Illustration of direct or forward extrusion assuming ideal deformation. Let us consider axisymmetric extrusion (Fig. 1) where the diametral area is reduced from . The ideal work is



where and is the percent area reduction, i.e. . Note that the nal axial strain is usually called the homogeneous strain and denoted as , i.e. .

For axisymmetric problems, we will assume that


we nally can write:

Note that if there is no hardening (




The external work (actual work) applied is or per unit volume (



where is the applied extrusion pressure. For an ideal process,

, i.e.

lower bound In reality:



Similar results can be obtained for rod or wire drawing (Fig. 2). The external work/volume in drawing is and so in general we have:

Figure 2: Illustration of rod or wire drawing. 2



is the applied drawing stress.

Friction, Redundant Work and Efciency The actual work


Redundant (non-homogeneous deformation) as follows: (8)

and are usually combined. We dene the mechanical efciency

The efciency

is a function of the die, lubrication, reduction rate, etc. Usually

Figure 3: Comparison of ideal and actual deformation to illustrate the meaning of redundant deformation. Generalizing the formulas given above for the extrusion pressure and drawing stress, we can write the following:




Figure 4: The stress-strain behavior is depicted in (c), the metal obeying , and is to be considered as the true stress needed to reduce to ( is the corresponding true strain). by pulling through Example: A round rod of initial diameter, can be reduced to diameter a conical die with a necessary load, , as shown in sketch 4(a). A similar result can occur by applying a uniaxial tensile load, as shown in sketch 4(b). Using the ideal-work method for both the drawing and tensile operations, compare the load with the load (or the drawing stress with the tensile stress ) needed to produce equivalent reductions. For drawing we showed that

(11) (12) (13)

For tension From the two equations above But,

(strain at ultimate load max strain to avoid necking). So nally,




Maximum drawing reduction in axisymmetric drawing

Figure 5: The tensile stress-strain curve and the drawing stress-strain behavior for two levels of deformation efciency. The intersection points, , are the limit strains in drawing. increases. Its value cant be higher than the yield With greater reduction the drawing stress 2 stress of the material at the exit. From the previous analysis


The maximum possible value of is , where we denote as axial strain corresponding to maximum reduction. From the above equations

the nal



and maximum reduction per pass


(perfect drawing) the maximum reduction is given as and for For (perfectly plastic material no hardening) we have that: .

The yield condition for axisymmetric problems has the form: Y.S., where the yield stress of the material at any location inside the deformation zone and the die pressure. Note that , which together with the yield condition at the exit implies that (at the exit) Y.S. at the exit.

Figure 6: Inuence of semi-die angle on the actual work , during drawing where the individual contributions of ideal, , frictional, , and redundant work, , are shown.

Figure 7: Effect of semi-die angle on drawing efciency for various reductions; note the change in the optimal die angle, .


Figure 8: Plane strain drawing. The calculations and previous denitions are applicable to plane strain problems with only minor modications. The differences arise from the new form of the yield condition and the new expression for the equivalent strain. They have as follows: Yield condition: the deformation zone. Equivalent strain:

Y.S., where Y.S. is the yield stress of the material at any location in

The above changes will modify the nal results as follows: Plane strain extrusion: Extrusion Pressure:


where For For

, with the homogeneous strain


(rigid plastic material):

(power law hardening):

Plane strain drawing: Drawing Stress:

(20) where

where For For

, with the homogeneous strain (-strain)

(rigid plastic material):

(power law hardening):

For max reduction:


from which we nally conclude that:


Note that the max reduction is the same for both plane strain and axially symmetric problems.