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

Determination of the velocity field

The scheme of upsetting between parallel plates is shown in Fig. 1.

  

Figure 1: Work piece before and after upsetting

The points of the deformity zone can be given in a cylindrical system of coordinates by the points $(r, z)$, where $z\in (0, h)$, $r\in (0, f(z))$ (see Fig. 1.). We denote the velocity field by $w(r,z)=[w_r(r,z), w_z(z)]$. The components of $w$ can be determined by using the following model assumptions:

1. The material is incompressible.

2. The deformation is axisymmetric, that is

$$w_r(0, z) = 0, \qquad \mbox{for all z}$$ (1)

3.The $z$ component of velocity at the contact of the piece and the plates is as follows:

$$w_z(0)=0, \qquad w_z(h)=-v_0$$ (2)

4. At $z=h/2$, $w_z$ has a point of inflexion, that is, the deformation velocity $\dot{\varepsilon}$ has an extremum at this point. 5. At $z=h/2$ the radial velocity component $w_r$ has a maximum when the work piece has a bilge form. 6. The upset material is homogenous and isotropic. 7. The $z$-component of velocity can be written in the form of

$$w_z(z)=a\cdot z^3 +b\cdot z^2 +c\cdot z +d$$ (3)

because this function is the simplest assimetric one to describe the vwlocity field of the deformation of the work piece. Here $a$, $b$, $c$ and $d$ are provisionally unknowns to be determined. From Assumption 3. we get that

$$d=0$$ (4)

$$a\cdot h^3 + b\cdot h^2 + c\cdot h= -v_0$$ (5)

From Assumption 4.:

 

$$\frac{\partial^2 w_z}{\partial z^2}=3ah+2b=0$$ (6)

From equations (4, 5, 6), the values of $a$ and $b$ can be determined, whence:

$$w_z(z)=2\frac{(c h + v_0) z^3}{h^3} - 3 \frac{(c h +v_0)z^2}{h^2} +c z$$ (7)

Assumption 1. im;ies that the velocity field is divergence free:

$$\dot{\varepsilon}_{ii}= \frac{\partial w_r}{\partial r} +\frac{w_r}{r} +\frac{\partial w_z}{\partial z} =0$$ (8)

By substituting the expression of $w_z(z)$ into his equation, for the component of $w_z(r,z)$ we obtain first-order differential equation, that contains $z$ as a parameter. Condition 2. can be regarded as an initial condition for this equation. So the problem given by formulas (8) and (1) can be solved uniquely for every $z$. It is easy to see that the solution is as follows:

$$w_r(r,z)=-3\,\frac{r z^2 c}{h^2} - 3\,\frac{r z^2 v_0}{h^3} + 3\,\frac{r z c}{h} +3 \frac{r z v_0}{h^2} -\frac12\,r c$$ (9)

where $c$ is a provisionally arbitrary constant to be determined. Introducing the nondimensional parameter $k$ by

 

$$k=-c\frac{h}{v_0}$$ (10)

Equations (7,9) can be rewritten as:

$$w_z(z)= \frac{z(-2 z^2 k v_0 + 2 z^2 v_0 +3 z k v_0 h -3 z v_0 h -k v_0 h^2)}{h^3}$$ (11)

$$w_r(z,r)=-\,\frac12\,\frac{r(-6 z^2 k v_0 + 6 z^2 v_0 +6 z k v_0 h -6 z v_0 h -k v_0 h^2)}{h^3}$$ (12)

  

Figure 2: Radial component of the velocity at different values of $k$ ($v_0 = 1$, $h = 5.3$, $r = 8$)

In the above example, at $k = 1$, the work piece remains cylindrical, while at $k = 0.6$ gets bilge and at $k = 1.4$ the mantle surface of the piece becomes concave. In case of convex shape, at $z=h/2$, the deformation velocity $\dot{\varepsilon}_z$ has a maximum, that is, $w_z(z)$ is steepest in the point of inflexion. (Fig.3).

  

Figure 3: Axial component of velocity at different values of $k$ ($v_0 =1$, $h = 5.3$, $r = 8$)

In case $k = 1$ equations (11 and 12) gives the typical forms of homogenous deformations, which is well-known, see e.g. [2,3]:

$$w_z(z)=-\,\frac{z v_0}{h}$$ (13)

$$w_r(r)=\frac12\,\frac{r v_0}{h}$$ (14)

The exact value of $k$ can be determined by minimization the power requirement of the forming.

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