How to Analyze Drawing of Circular Wires?

For design of wire drawing machinery it is necessary to know the force and power required to draw the wire. The process is illustrated in Fig. 9.8. The total pull is due to different deforming processes taking place in the die.

These are as follows:

(i) Before entry into die, the surface particles of wire are moving parallel to wire axis. At entry into the deformation zone these particles suffer shear deformation and their velocity becomes parallel to conical die surface. The other particles in the interior layers of wire also suffer shear deformation but to a lesser degree. The shear deformation decreases towards centre line of wire where particles suffer no shear deformation. The shear deformation contributes to the pull required to draw the wire.

(ii) In the conical portion of die the wire cross-section is reduced under the action of pull and pressure of die surface. A major portion of pulling force is due to deformation in the conical portion of die. Also the wire rubs against the die surface which leads to frictional dissipation hence it also contributes to pull required for the process.

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(iii) At the exit from the conical portion of die the particles again suffer shear deformation and start moving parallel to die axis in the land portion of die.

(iv) In the land portion the material rubs against the die surface and hence the frictional force increases the pulling force.

All these factors are analyzed here.

1. Stress Due to Shear Deformation at the Entry:

Let us consider the material in a small slab width dx before the entry into the die (Fig. 9.8). Here we take that the zone of plastic deformation is contained between the contacting die surfaces and two planes BF and CG perpendicular to the die- axis at entry and exit from the conical portion of die respectively.

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After the slab of width dx enters the conical deformation zone its shape changes from rectangular cross section to trapezoidal cross section. Different layers of material suffer different shear strain (Fig. 9.9).

The material contacting the die surface shears through an angle α- the semi die angle, while the material at the center suffers zero shear. At any radius r the material suffers shear strain γ ≈ tan θ = r/A. This is on the assumption that material particles are flowing towards the apex point of conical surface of die. The dimension A is the axial distance from entry section to the apex point 0.

where σ₀₂ is the value of yield strength at the exit from die. The above expressions were first derived by Korber & Eichinger by taking τ₀ = σ₀/2 instead of τ₀ = σ₀/√3.

2. Drawing Stress Due to Conical Portion of Die:

In the following, we use the slab method of analysis for which the following assumptions are made:

(i) Plane sections before the deformation remain plane during the plastic deformation.

(ii) The material of wire is rigid perfectly plastic material. The work hardening may be taken into account by taking the average of the values of yield strength before and after the drawing process.

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Figure 9.10 shows a circular wire being drawn through a conical die having semi-cone angle ‘α’. Consider a slab of width dx, shown hatched in the figure. The slab is contained by a portion of contacting die surface and two planes normal to wire axis and which are located at a distance x and x + dx from the exit end of the conical portion of die.

The forces acting on this slab are due to the following stresses:

(i) Axial stress σ_x acting on smaller flat face and σ_x+ dσ_x on the bigger flat face of the slab. These stresses are parallel to wire axis.

(ii) The pressure p_x which acts normal to the contacting die surface.

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(iii) The frictional stress τ_v which acts tangentially on the surface contacting the die and in the direction opposite to metal flow.

Let us, now, consider the equilibrium of these forces in the axial direction of the wire.

For this the following additional assumptions are made:

(i) The acceleration forces in the process are very small compared to other stresses and may be neglected.

(ii) The stresses σ_x and σ_x + dσ_x do not vary on the areas on which they are acting.

With the above assumptions, the balance of forces along the axis gives the following equation:

The value of above expression at the end of conical portion of die may be obtained by substituting r = R₂ in Eqn (9.11). Now the drawing stress at the exit of conical portion of die also includes the increase in stress due to shear deformation at the exit plane which is given by Eqn (9.5). Thus the expression for drawing stress (σ_d) at the end of conical portion of the die becomes,

The work hardening effect may be accounted for by taking a mean value of yield strengths at entry and at exit from the die.

3. Effect of Land Portion of Die on Drawing Stress:

After passing through the conical portion, the wire passes through a parallel portion of die called land. The cross section of land portion is same as that at the exit of the conical portion.

The material inside the land portion remains in plastic state because elastic recovery occurs only after exit from the land. Therefore, the wire material in the land portion of die is all the time pressing against the die-surface. The drawing stress increases because of the friction between the material and die surface over the length of the land.

For analyzing the effect of land on the drawing stress let us consider a slab of width dx at a distance ‘x’ from the end of conical portion as shown in Fig. 9.11. The radius of wire in the land portion is R₂.

The stresses acting on the slab are (i) axial stresses σ_x and σ_x + dσ_x on the two flat faces of the slab, (ii) die pressure p_x which acts on the circular surface and (iii) frictional stress τ_x= µp_x which also acts on the circular surface. Consideration of equilibrium of forces in the axial direction results in the following equation.

4. Optimum Die Angle:

The experimental investigation carried out by Wistriech has shown that as we increase the die angle from a very low value, the drawing stress at first decreases and afterwards starts increasing. For given conditions of friction, reduction and yield strength of the metal being drawn, the drawing stress is minimum at a particular die angle which is called the optimum die angle.

Figure 9.12 shows the effect of die angle on drawing stress. Analytically the optimum die angle may be solved from the equation obtained by differentiating the expression for σ_d(Eqn.9.12) with respect to α and equating it to zero.

Since the explicit expression for optimum die angle (2α_opt) is difficult to obtain from the above equation, we resort to numerical calculations and find the angle at which the drawing stress is minimum for any particular case. For numerical calculations the expression of drawing stress (σ_d) at the exit from the die with shear deformation is repeated for convenience of reader.