Many researchers have made use of Upper-bound theorem for solution of forging, extrusion, wire and tube drawing etc. In forging alone there are a large number of papers. It is not possible to describe all of them, it is rather difficult even to give references of all of them.
Here we mention a few fundamental ones only. Avitzur has given upper bound solutions for compression of circular disc and infinite strip taking into account the non-uniformity of metal flow along the thickness of work piece, which results in barreling of work piece. Juneja has solved the compression of non-circular discs, i.e. regular polygonal, any polygonal disc with an in-circle and rectangular discs taking into account the non-uniformity of flow in the lateral direction, which results in bulging of work piece.
The work was extended by Juneja and Sagar for compression of any four sided disc and for closed die compression of polygonal discs. In actual compression of all non-circular discs both bulging and barreling occur, however, it is a bit difficult to deal the both together.
Compression of Polygonal Discs between Two Flat Dies:
In the following analysis we consider compression of a regular polygonal disc. However, the analysis is equally applicable (with slight modifications) to irregular polygon discs provided all the sides touch an in-circle, i.e. the bisectors of corner angles meet at a point. The following analysis is due to Juneja.
Consider a ‘n’ sided regular polygonal disc (Fig. 6.29). It can be divided into V symmetrical triangles, each bounded by one side of polygon and two radial lines from the end- points of the side to the in-center of the polygon. The metal flow in each of these triangles is similar, therefore, let us consider only one of these triangles say, FOG.
Here FG is the side of polygon, OF and OG are the radial lines, point O is the in-center of the polygon. Let us draw ON perpendicular to the side FG. The line ON divides the triangle FOG into two identical triangles. The metal flow patterns in FON and GON are mirror images of each other. In the following we consider the triangle FON.
For describing the flow, we take cylindrical co-ordinates r, θ and z with origin at O. The metal flow at any point S(r, θ, z) is given as follows.
The second factor in the square bracket is very small compared to unity because λ is quite small. For exact calculations we may carry out numerical integration, it is also worthwhile to simplify the expression by neglecting it. Also λ is not a function of r, θ and z. Therefore, we may simplify by taking the expression under square-root sign as unity.
Since it is an upper-bound solution it is not proper to neglect contribution of Uθ, and at the same time there is an over whelming simplification of expressions and we shall be avoiding numerical integration if we take (6.59) as following. The value of following expression would only be slightly more than if we take square root of sum of squares of the two velocities.
The value of λ in the above expression represents the degree of non-uniformity of metal flow during compression. For extreme values of friction factor m and A/h, λ, approaches asymptotically to 0.5 which is the maximum value of λ. For λ = 0.5, the extension in the radial lines (like OF and OP) becomes zero, Bulging is maximum at λ = 0.5
For values A/h = 1.0 and 8.0 and for different values of m, Fig. 6.30 shows the values of λ, for different polygonal discs. Figure 6.31 compares the values of pm/σ0, for different polygonal discs for values of m from 0 to 1. Figure 6.32 shows the values of pm/σ0 for a square disc, for different values of m and A/h.
The graph shows that with increasing of number of sides the value of λ goes down. Thus its value is maximum for triangular discs. With increase in number of sides the metal flow becomes more uniform around the circumference and hence λ goes down. For a circular disc its value is zero.
Shapes of Deformed Discs:
Shapes of the disc after compression may be determined by following calculations. In small time ΔT let the compression be δh by top platen moving at the velocity Ů.
Thus the final position of particle at point C can be located. We may, similarly, determine the final positions of all points on the boundary after the compression.
For sake of illustration (Fig. 6.34) and (Fig. 6.35) show the shapes of square and hexagonal discs after compression.