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

The calculation of velocity field for ring shaped pieses

The axial component of velocity described by equation (11) can also be applied to ring shaped pieces. The radial component of velocity we can determine from the following initial condition for the differential equation (8) is

$$w_r(R_s,z)=0, \mbox{for all z}$$

, where $R_s$ is the radius of the cylindrical surface separating material flowing inwards and outwards [10]. When we already know the actual $R_s$ value, the displacement of any point of the workpiece can be determined with the help of equation (3). The value of $R_s$ can be expressed by the friction factor and the actual height of the workpiece (see [11,12]).

$$R_s=a \mu^b h^c$$ (33)

The values $a$, $b$, $c$ in (33) at certain initial geometry and a friction factor $nu$ can be found in the literature (see [12] for details). In the case of ring shaped pieces the radial velocity component can be obtained by solving the following differential equation:

 

$$w_r(r,z) = -\,\frac{1}{2 r h^3} ( -6r^2 z^2 k v_0 +6 r^2 z^2 v_0 + 6 r^2 z k v_0 h -6 r^2 z v_0 h - r^2 k v_0 h^2$$ (34)

$$+ 6 R^2_s z^2 k v_0 -6 R_s^2 z^2 v_0 -6 R_s^2 z k v_0 h + 6 R_s^2 z v_0 h - R_s^2 k v_0 h^2 )$$

Fig. 9 shows some rings with different friction coefficients.

  

Figure: Modeling material flow for ring shape upsetting ( $H_0 = 5.3 \mbox{mm}$, $r_0 = 4 \mbox{mm}$, $R_0 = 8 \mbox{mm}$, $h = 2.5 \mbox{mm}$)

Here I present 3 animations with different coefficients of friction:

• $\eta=0.06$, click here
• $\eta=0.12$, click here
• $\eta=0.20$, click here

As far as we know, investigation of deformation during upsetting between parallel pressure plates has not been carried out followed the process by animation.

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