In this article we will discuss about how to analyze compression of circular disc in forging operations of metals.
Equilibrium Equation with Slab Method:
In this method it is assumed that plane sections remain plane during compression and any non-uniformity in deformation is neglected. Also it is assumed that the material is rigid perfectly plastic and effects of strain hardening and strain rate are neglected. Now consider a small circular element of work piece (slab) bounded by two radial lines with included angle and ‘dθ’ two circular arcs with radii ‘r’ and ‘r + dr’, (Fig. 6.25).
As the disc is compressed the material moves out radially. The different stresses acting on the slab (Fig. 6.25) are (i) die pressure pr, (ii) the frictional stress τr directed towards center, (iii) radial stresses σr and σr + dσr on inner and outer curved faces respectively and (iv) circumferential stress σθ on side faces. Consideration of equilibrium of these forces in the radial direction gives the following equation.
Since dθ is very small, we may take sin (dθ/2) ≈ dθ/2. Also by neglecting the terms which have two or more differentials, the above equation may be simplified to the following.
Here it is taken that σr, σθ and pr are the principal stresses, which is an approximation.
Compression of Circular Disc with Slipping Friction:
The value of τr depends on the friction conditions at the interface between die and the material. In case of compression in cold state with polished and well lubricated dies we may put τr = μ pr. This is called slipping friction or Coulomb friction condition. By substituting this in Eqn. (6.30) we obtain,
The average die pressure pm may be obtained by dividing PT by area of disc. Thus
Compression of Circular Disc with Sticking Friction:
When the coefficient of friction is high, which is the case in hot forging, it is quite possible that interfacial friction over the entire disc may be sticking friction. In such cases frictional stress τr is constant on the entire surface and is equal to yield strength of the material in shear (K). Substituting this in Eqn. (6.30) we get,
The mean die pressure (pm) may be obtained by dividing the above equation by the area of disc, which is πR2. The maximum pressure occurs at the center of the disc while the minimum pressure occurs on the outer edge of the disc. Some authors believe that friction on either side of center changes from positive to negative over a small region. The pressure curve for such a case has rounded top in stead of a sharp corner.
Compression of Circular Disc with Slipping and Sticking Friction:
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In most of the hot forging operations both slipping friction and sticking friction are present. In the slipping region which is towards the outer boundary the die pressure pr is given by Eqn. (6.34).
As we move from outer boundary (i.e. r = R) towards the center of disc the value of pr increases exponentially and so does the frictional stress μ pr. The increase continues up to a point where the sticking starts, i.e. where μ pr becomes equal to K = yield strength of material in shear. This is illustrated in Fig. 6.28.
The pressure distribution on the dies is similar to that illustrated in Fig. 6.24, for strip forging except that at outer edge the pressure is equal to σ0. The diagram consists of two zones, i.e. slipping zone and sticking zone.
The sticking zone extends from center to the radius Rs. The die pressure in this zone is given by Eqn. (6.47). The zone of slipping friction extends from radius Rs to the edge of disc, i.e. from Rs to radius R. The total die load is obtained by integrating pr over both the zones. Thus,
The forging load calculated from the above expressions gives load at the start of forging. If it is desired to be determined the forging load after compression, first we have to determine the height and diameter of forged disc. If height of forged component is given the diameter may be determined by equating the volumes of forged disc and un-forged disc. If it is cold forging then the yield strength of material after the deformation (strain hardening) should also be determined and used in the above expressions.