In this article we will discuss about how to analyze plain strain compression in forging process of metals.
Compression with Coulomb Friction:
In slab method of analysis it is assumed that plane sections before the deformation remain plane during deformation. The non-uniformity in metal flow as described above is not taken into account. Nevertheless, the method is highly useful because it gives simple closed form solutions which are reasonably reliable.
In Coulomb friction it is taken that frictional force between two surfaces is proportional to normal pressure. The constant of proportionality is called coefficient of friction which is generally assumed to be constant. This condition may exist when the forming tools are well lubricated and compression is done at ambient temperatures. This is also called slipping friction.
Figure 6.20 shows a cross-section of the strip being compressed between two flat dies. The region shown hatched is the slab of width ‘dx’ at a distance ‘x’ from the centerline. The stresses acting on the faces of the slab are, (i) px-the die pressure, (ii) τx-the frictional stress at the die-material interface, (iii) σx and σx + dσx are the lateral stresses acting on the vertical faces of the slab. Let L denote the half the width of strip cross section and let b denote the dimension of strip perpendicular to the paper. No generality is lost even if we take b = 1. Let h be the thickness of strip.
Since the strip is symmetric about the center line, we may only consider the slab in the right hand half of the strip. The equilibrium of forces acting on the slab in the x-direction gives the following equation.
With Coulomb friction, τx is equal to μ.px. Substitution of this in the Eqn. (6.2) gives the following:
By substituting the value of C in Eqn. (6.9) and after simplifying it, we get the following expression for specific die pressure px.
A similar expression may be obtained for the left half of the strip by reversing the positive direction of x. A plot of die pressure for different values of coefficient of friction µ is shown in Figure 6.21. The total die load (PT) may be obtained by integrating px over the interface area between the die and the disc. Thus
The average die pressure pm is obtained by dividing PT by the area of contact which is equal to 2Lb. Thus the following equation is obtained for average die pressure.
From Eqn. (6.12) it is clear that (2μL/h) is the main factor that affects the mean die pressure. Moreover, the effect of coefficient of friction is similar to that of the ratio (2L/h). For very large values of h or very small values of μ, the value of pm will be nearer to σ0‘, while it would be much larger than σ0‘ when L is very large compared to h and or μ is large. In case of frictionless compression, μ = 0 and hence pm equals σ0‘. The maximum value of die-pressure ‘Pmax‘ occurs at the center line of the specimen being compressed, i.e. at x = 0
Compression with Sticking Friction:
In hot forging of steel and other alloys the coefficient of friction is high and also little or no lubricant is used. In such a condition, a layer of metal contacting the die-surface may stick onto the die and flow may take place just under the surface layer. This condition of friction is called sticking friction.
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In many cases, a part of contacting surface may be in slipping friction and another part in sticking friction. In case of sticking friction, the frictional stress on the interface is equal to K = the yield strength of the metal in shear. We again refer to Fig. 6.20, in which, the frictional stress τx is now equal to K. The Eqn. (6.2) becomes,
Compression with Slipping and Sticking Friction:
With intermediate values of μ and suitable ratios of L/h both the slipping and sticking friction regions may be present on the interface between die and the metal. The interface pressure increases as we go from edge of the strip to its centerline. Therefore, the frictional stress which is equal to μ.px also increases. With high value of μ and high L/h ratio or both, px may increase to a very high value.
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And the frictional stress μ.px may reach K. This is illustrated in Fig. 6.23 which shows the frictional stress diagram. The frictional stress on the right half is shown negative because the direction of frictional stress is opposite to that of x while frictional stress on left half is shown positive because on this side the direction of friction is same as the + ve direction of ‘x’. Sticking starts from a point where magnitude of μpx reaches K. Let the sticking start at a distance Xs from center, then
Example 1:
A 200 mm wide, 500 mm long and 10 mm thick strip is compressed between two flat dies in plane strain such that the dimension 500 remains constant. The coefficient of friction between dies and the strip is 0.1 and yield strength of material in compression is σ0 = 200 N/mm2. Determine the mean die pressure and the maximum die pressure. Determine change in mean and maximum pressures when µ is changed to half.
