Submitted to HEJ Manuscript no.: MET-990617-A

Modeling of the deformation process

For the sake of simplicity in the modeling of the deformation process we assumed that the coefficient of friction remains constant during the upsetting. Under such a condition the value of $k$ depends only on the current height of the work piece. If the value of $v_0 dt$ is small enough, the value of $k$ can be considered to be constant during upsetting. For investigation of the forming process, before upsetting at the cross section of the piece we adopted a set of ponts, and, by using relations (11, 12), we determined the new position of the adopted point for displacement $v_0 dt$. Then we considered the new geometry, and so the value of $k$ was determined. The procedure can be repeated while we remain in the validity range of the relation (11, 32). At the calculations the value of $v_0 dt$ was $0.1 \mbox{mm}$.

  

Figure 4: The variation of the specific power requirement of forming depending on the value of $k$ at different degrees of upsetting. ($\mu = 0.12$, $H_0 = 5.3 \mbox{mm}$, $R_0 = 8 \mbox{mm}$)

The way of calculation can be facilitated by expressing the value of $k$ as a function of the coefficient of friction $\nu$ and the height $h$ of the piece to be upset. For a given initial geometry, in a certain range of $\mu$ friction coefficients and height $h$, shown on thw Fig. 6., $k$ can be approximated with acceptable accuracy by regression calculation using a function of the type:

$$k=c_1\mu +c_2 h +c_3$$ (32)

type of functions.

  

Figure 5: The variation of the specific power requirement of forming depending on the value of $k$ at different values of friction. The upset height of the work piece is $h = 3.3 \mbox{mm}$. ( $H_0 = 5.3 \mbox{mm}$, $R_0 = 8 \mbox{mm}$)

  

Figure 6: Values of $k$ as a function of friction coefficient $mu$ and the instantaneous $h$ height of the piece ( $H_0 = 5.3 \mbox{mm}$, $R_0 = 8 \mbox{mm}$, $k=1.23602 - 0.977926 \mu -0.0865165 h$)

Applying the above described method we used an AutoLISP program. The program upsets the piece for the desired degree (to $h$ height) with the given coefficient of friction and in its final stadium draws the picture of the deformed web of dots and the field of velocity (Fig. 7 refers). By means of the deformed set of dots the local deformations can also be studied.

  

Figure 7: Deformation of the set of points and the instantaneous field of velocity

The above modeling makes it possible to investigate the effects of friction coefficient as well.

  

Figure: Effects of friction coefficients ( $H_0=5.3 \mbox{mm}$, $R_0=8 \mbox{mm}$, $h=2.6 \mbox{mm}$)

Here I present 3 animations with different coefficients of friction:

• $\eta=0.06$, click here
• $\eta=0.12$, click here
• $\eta=0.24$, click here

Submitted to HEJ Manuscript no.: MET-990617-A