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

The neccessary power of forming

The power requirement of forming is composed as a sum of two components:

$$P(k)=P_{id}(k)+P_s (k)$$ (15)

where $P_{id}$ is the pure power requirement of forming, while $P_s$ is the friction power which arises between the contact surfaces of the piece and the pressure plates. Calculation of components see [3] for details:

• $P_{id}$ is the power of pure deformation:
  
  $$P_{id}=\int_{V} k_f \dot{\varepsilon}_{e}\,dV$$
  
  
  where $k_f$ is the forming strength of the material and $\dot{\varepsilon}_{e}$ is the comparative deformation velocity.
• $P_s$ is the friction power:
  
  $$P_s=\int_{A} \mu k_f \vert v_{rel}\vert\, dA$$
  
  
  assuming the $\tau=\mu\sigma_n\approx \mu k_f$ which is the Coulomb friction. The $\vert v_{rel}\vert$ is the relative displacement of the piece and the die at the contacting surfaces.

In details, with considering deformation strength at the mean value:

$$P_{id}=2\pi\overline{k_f} \int_{z=0}^{h}\int_{r=0}^{f(z)} \dot{\varepsilon}_{e} r dr dz$$ (16)

where $\dot{\varepsilon}_{e}$ comparative deformation velocity can be computed from deformation velocity components $\dot{\varepsilon}_{ij}$. In case of axisymmetric piece we getget:

$$\dot{\varepsilon}_{e}=\sqrt{\frac23( \dot{\varepsilon}_... ...t{\varepsilon}_{\Theta}^2 +2\dot{\varepsilon}_{rz}^2 \right)}$$ (17)

The values of $\dot{\varepsilon}_{ij}$ deformation velocity components are as follows:

$$\dot{\varepsilon}_{ij}=\frac12\left( w_{i,j} + w_{j,i}\right)$$ (18)

Taking into account that the piece is in contact with the die in two sides, and, $\vert v_{rel}\vert=w_r(r,h)=w_r(r,0)$, the friction power can be expressed as follows:

$$P_s=2 P_{s1}= 4 \pi \mu \overline{k_f} \int_0^R r w_r dr$$ (19)

For the sake of simplicity we perform the calculations for solid cylindrical pieces at $k = 1$. A similar example solution is available in [8]. Equation (18) takes the following simple scalar form:

$$\dot{\varepsilon}_z= \frac{\partial w_z}{\partial z} = -\,\frac{v_0}{h}$$ (20)

$$\dot{\varepsilon}_r= \frac{\partial w_r}{\partial r} = \frac12\,\frac{v_0}{h}$$ (21)

$$\dot{\varepsilon}_{\Theta}= \frac{w_r}{r} = \frac12\,\frac{v_0}{h}$$ (22)

$$\dot{\varepsilon}_{r z}+\frac12\left( \frac{\partial w_r}{\partial z} + \frac{\partial w_z}{\partial r} \right)$$ (23)

$$\dot{\varepsilon}_{r \Theta}=0, \qquad \dot{\varepsilon}_{\Theta z}=0$$

The value of $\dot{\varepsilon}_{e}$ comparative deformaition velocity can be expressed from (20, 21, 22, 23) as:

$$\dot{\varepsilon}_{e}= \sqrt{\frac23\left(\dot{\varepsi... ...Theta}^2 +2\dot{\varepsilon}_{rz}^2 \right)} = \frac{v_0}{h}$$ (24)

From equations (5, 6, 7), we obtain: $\dot{\varepsilon}_z+\dot{\varepsilon}_r+\dot{\varepsilon}_{\Theta}=0$, so the introduced velocity field satisfies the condition of incompressibility. The comparative deformation velocity come the components of power, can be expressed in the following form:

$$P_{id}=2 \pi \overline{k_f}\int_{z=0}^h \int_{r=0}^R \frac{v_0}{h}\,r\,dr\,dz= R^2 \pi \overline{k_f} v_0$$ (25)

$$P_s=4 \pi \mu \overline{k_f} \int_0^R r \,\frac12\,\frac{... ...}{h}\,dr = \frac23\,\frac{\pi \mu \overline{k_f} R^3 v_0}{h}$$ (26)

The total power $P$ can be determined from the velocity $v_0$ and the mean force $\overline{F}$, acting on contact surface.

$$P= \overline{F} v_0$$ (27)

So the power requirement of forming is:

$$\overline{F}=\frac{P_{id}+P_s}{v_0}= \overline{k_f} R^2 \pi \left(1+\frac{2 \mu R}{3 h} \right)$$ (28)

Equation (28) can be derived also by the average stress method, and is known as Siebel-formula (see [2,3]). If $k\neq1$, the velocity field changes according to the actual value of $k$. Best value of $k$ bz the upper bound method minimiyes the following function.

$$P(k)=P_{id}(k)+P_s(k)$$ (29)

In the case of $k\neq1$ the exprressions for the power are more complicated, because initial conditions (11, 12) are also more complicated, and

$$\dot{\varepsilon}_{r z}=\frac12\left(\frac{\partial w_r}{\partial z} + \frac{\partial w_z}{\partial r} \right)\neq 0.$$

Calculations were performed by using mathematical software MapleV (see [9]), see also [1] for a similar industrial applications.

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