Application of equi-potential lines method for accurate definition of the deforming zone in the upper-bound analysis of forward extrusion problems

Abstract

Prediction of the optimal pressure is one of the objectives in the die design for the extrusion process. In this study, the concept of Equi-Potential Lines (EPLs) was applied for an accurate definition of the deforming zone in the upper-bound solution of the extrusion process. To implement the concept for the extrusion process, the initial and final shapes were considered and two different potentials were assigned to these. Then, the EPLs were drawn between the two shapes while minimizing the work path from the initial billet to the desired final shape. Bilinear, cubic Bezier, and polynomial curves were used for the definition of the deforming zone. It was shown that the coefficients of cubic Bezier curve could be replaced by the values obtained from the EPLs method. Moreover, accurate 3D streamlines representing material flow path in the deforming zone were obtained from the present method. Finally, forming pressures in the extrusion process were computed using the proposed geometry. To show the effectiveness of the proposed method, the estimated values were compared with experimental results. It was shown that the proposed method had a remarkable agreement with the experimental results.

Appendices

Appendix A

1.1 Velocity field

The general velocity vector for any arbitrary point in the deforming region (Fig. 2) can be defined as follow [19]:

$$ \overrightarrow{V}=\frac{V_xi+{V}_yj+{V}_zk}{{\left({V_x}^2+{V_y}^2+{V_z}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} $$

(11)

Using the procedure of [14], it can be shown that a kinematically admissible velocity field based on the deforming region (Eq. (5-d)) is given by:

$$ {V}_x=\frac{ ft}{h_t}{V}_z\kern0.5em ,\kern0.5em {V}_y=\frac{\mathit{\mathsf{g}}t}{h_t}{V}_z\kern0.5em ,\kern0.5em {V}_z=M\left(u,q,t\right) $$

(12)

Where M is a function obtained by considering compressibility conditions. Three-dimensional incompressibility relation is given by:

$$ \frac{\partial {V}_x}{\partial X}+\frac{\partial {V}_y}{\partial Y}+\frac{\partial {V}_z}{\partial Z}=0 $$

(13)

By using the introduced velocity field and after some manipulation (see Appendix B) and integration of the compressibility equation, the following equation is obtained for M [19]:

$$ M=\frac{C\left(u,q\right)}{\left({f}_u{\mathit{\mathsf{g}}}_q-{f}_q{\mathit{\mathsf{g}}}_u\right)+\frac{h_q}{h_t}\left({f}_t{\mathit{\mathsf{g}}}_u-{f}_u{\mathit{\mathsf{g}}}_t\right)+\frac{h_u}{h_t}\left({f}_q{\mathit{\mathsf{g}}}_t-{f}_t{\mathit{\mathsf{g}}}_q\right)} $$

(14)

Where C (u, q) is obtained from boundary condition at entry section (t = 0):

$$ C\left(u,q\right)={V}_0{\left[\left({f}_u{\mathit{\mathsf{g}}}_q-{f}_q{\mathit{\mathsf{g}}}_u\right)+\frac{h_q}{h_t}\left({f}_t{\mathit{\mathsf{g}}}_u-{f}_u{\mathit{\mathsf{g}}}_t\right)+\frac{h_u}{h_t}\left({f}_q{\mathit{\mathsf{g}}}_t-{f}_t{\mathit{\mathsf{g}}}_q\right)\right]}_{t=0} $$

(15)

By defining M(u, q, t), the velocity field is fully determined through the relation (12).

1.2 Upper-bound solution

The upper-bound value for extrusion power can be calculated as follow [19]:

$$ J={\dot{W}}_e+{\dot{W}}_x+{\dot{W}}_f+{\dot{W}}_i $$

(16)

Where ${\dot{W}}_e$ is the power due to the velocity discontinuity at entrance section:

$$ {\dot{W}}_e=\frac{Y}{\sqrt{3}}{\displaystyle \iint \varDelta {V}_e\varDelta {S}_e=}\frac{Y}{\sqrt{3}}{\displaystyle \iint {{\left[{V_x}^2+{V_y}^2+{\left({V}_z-{V}_0\right)}^2\right]}^{\frac{1}{2}}}_{t=0}d{S}_e} $$

(17)

${\dot{W}}_x$ is the power due to velocity discontinuity at exit section:

$$ {\dot{W}}_x=\frac{Y}{\sqrt{3}}{\displaystyle \iint \varDelta {V}_x\varDelta {S}_x=}\frac{Y}{\sqrt{3}}{\displaystyle \iint {{\left[\left({V_x}^2+{V_y}^2+{\left({V}_z-{V}_0\left(\frac{A_e}{A_x}\right)\right)}^2\right)\right]}^{\frac{1}{2}}}_{t=1}d{S}_x} $$

(18)

A _e and A _x are the areas of inlet and outlet sections, respectively. Moreover, ${\dot{W}}_f$, the power due to friction between working material and die surface is:

$$ {\dot{W}}_f=m\frac{Y}{\sqrt{3}}{\displaystyle \iint \varDelta {V}_f\varDelta {S}_f=m}\frac{Y}{\sqrt{3}}{\displaystyle \iint {{\left[\left({V_x}^2+{V_y}^2+{V_z}^2\right)\right]}^{\frac{1}{2}}}_{u=1}d{S}_f} $$

(19)

In ${\dot{W}}_f$ calculations, m is the friction factor. Finally, ${\dot{W}}_i$ the power due to internal deformation is calculated as follows:

$$ {\dot{W}}_i=\frac{2Y}{\sqrt{3}}{\displaystyle \iiint {\left[\left(\frac{{\varepsilon_{\dot{x}x}}^2+{\varepsilon_{\dot{y}y}}^2+{\varepsilon_{\dot{z}z}}^2}{2}\right)+{\varepsilon_{\dot{x}y}}^2+{\varepsilon_{\dot{y}z}}^2+{\varepsilon_{\dot{z}x}}^2\right]}^{\frac{1}{2}} dV} $$

(20)

The total extrusion pressure can be calculated as:

$$ \frac{P}{Y}=\frac{J}{ Y\pi {R}^2} $$

(21)

In relation (21), Y and R are the yield strength and radius of the billet material, respectively.

Appendix B

The incompressibility condition of the material in xyz coordinates system can be shown by dimensionless variables of u, q, and t as following:

$$ {\varepsilon}_{ii}=\frac{\partial {V}_x}{\partial x}+\frac{\partial {V}_y}{\partial y}+\frac{\partial {V}_z}{\partial z}=0 $$

(22)

$$ \frac{\partial {V}_i}{\partial {x}_k}={\displaystyle {\sum}_{j=1}\left(\frac{\partial {V}_i}{\partial {u}_j}.\frac{\partial {u}_j}{\partial {x}_k}\right)} $$

(23)

The above relation can be written in matrix form as follows:

$$ \frac{\partial {V}_i}{\partial {x}_k}=\left[\begin{array}{ccc}\hfill \frac{\partial {V}_x}{\partial x}\hfill & \hfill \frac{\partial {V}_x}{\partial y}\hfill & \hfill \frac{\partial {V}_x}{\partial z}\hfill \\ {}\hfill \frac{\partial {V}_y}{\partial x}\hfill & \hfill \frac{\partial {V}_y}{\partial y}\hfill & \hfill \frac{\partial {V}_y}{\partial z}\hfill \\ {}\hfill \frac{\partial {V}_z}{\partial x}\hfill & \hfill \frac{\partial {V}_z}{\partial y}\hfill & \hfill \frac{\partial {V}_z}{\partial z}\hfill \end{array}\right] $$

(24)

$$ \frac{\partial {V}_i}{\partial {u}_j}=\left[\begin{array}{ccc}\hfill \frac{\partial {V}_x}{\partial u}\hfill & \hfill \frac{\partial {V}_x}{\partial q}\hfill & \hfill \frac{\partial {V}_x}{\partial t}\hfill \\ {}\hfill \frac{\partial {V}_y}{\partial u}\hfill & \hfill \frac{\partial {V}_y}{\partial q}\hfill & \hfill \frac{\partial {V}_y}{\partial t}\hfill \\ {}\hfill \frac{\partial {V}_z}{\partial u}\hfill & \hfill \frac{\partial {V}_z}{\partial q}\hfill & \hfill \frac{\partial {V}_z}{\partial t}\hfill \end{array}\right] $$

(25)

Then:

$$ \frac{\partial {u}_j}{\partial {x}_k}={J}^{-1}=\frac{1}{ \det J}\left[\begin{array}{ccc}\hfill {I}_{11}\hfill & \hfill {I}_{12}\hfill & \hfill {I}_{13}\hfill \\ {}\hfill {I}_{21}\hfill & \hfill {I}_{22}\hfill & \hfill {I}_{23}\hfill \\ {}\hfill {I}_{31}\hfill & \hfill {I}_{32}\hfill & \hfill {I}_{33}\hfill \end{array}\right] $$

(26)

Where J is the Jacobian for conversion of x, y, and z coordinates to u, q, and t. The components of the matrix in relation (26) can be defined as:

$$ \begin{array}{l}{I}_{11}={g}_q{h}_t-{g}_t{h}_q\\ {}{I}_{12}={f}_q{h}_t+{f}_t{h}_q\\ {}{I}_{13}={f}_q{g}_t-{f}_t{g}_q\\ {}{I}_{21}=-{g}_u{h}_t+{g}_t{h}_u\\ {}{I}_{22}={f}_u{h}_t-{f}_t{h}_u\\ {}{I}_{23}={f}_u{g}_t+{f}_t{g}_u\\ {}{I}_{31}={g}_u{h}_q-{g}_q{h}_u\\ {}{I}_{32}=-{f}_u{h}_q+{f}_q{h}_u\\ {}{I}_{33}={f}_u{g}_q-{f}_q{g}_u\end{array} $$

(27)

$$ J=\left[\begin{array}{ccc}\hfill {f}_u\hfill & \hfill {f}_q\hfill & \hfill {f}_t\hfill \\ {}\hfill {g}_u\hfill & \hfill {g}_q\hfill & \hfill {g}_t\hfill \\ {}\hfill {h}_u\hfill & \hfill {h}_q\hfill & \hfill {h}_t\hfill \end{array}\right] $$

(28)

$$ \det J=\frac{\partial \left(x,y,z\right)}{\partial \left(u,q,t\right)}=\left[\begin{array}{ccc}\hfill \frac{\partial x}{\partial u}\hfill & \hfill \frac{\partial x}{\partial q}\hfill & \hfill \frac{\partial x}{\partial t}\hfill \\ {}\hfill \frac{\partial y}{\partial u}\hfill & \hfill \frac{\partial y}{\partial q}\hfill & \hfill \frac{\partial y}{\partial t}\hfill \\ {}\hfill \frac{\partial z}{\partial u}\hfill & \hfill \frac{\partial z}{\partial q}\hfill & \hfill \frac{\partial z}{\partial t}\hfill \end{array}\right] $$

(29)

By replacing the (24), (25), and (26) in (23):

$$ \begin{array}{l}\left[\begin{array}{ccc}\hfill \frac{\partial {V}_x}{\partial x}\hfill & \hfill \frac{\partial {V}_x}{\partial y}\hfill & \hfill \frac{\partial {V}_x}{\partial z}\hfill \\ {}\hfill \frac{\partial {V}_y}{\partial x}\hfill & \hfill \frac{\partial {V}_y}{\partial y}\hfill & \hfill \frac{\partial {V}_y}{\partial z}\hfill \\ {}\hfill \frac{\partial {V}_z}{\partial x}\hfill & \hfill \frac{\partial {V}_z}{\partial y}\hfill & \hfill \frac{\partial {V}_z}{\partial z}\hfill \end{array}\right]=\frac{1}{ \det J}\left[\begin{array}{ccc}\hfill \frac{\partial {V}_x}{\partial u}\hfill & \hfill \frac{\partial {V}_x}{\partial q}\hfill & \hfill \frac{\partial {V}_x}{\partial t}\hfill \\ {}\hfill \frac{\partial {V}_y}{\partial u}\hfill & \hfill \frac{\partial {V}_y}{\partial q}\hfill & \hfill \frac{\partial {V}_y}{\partial t}\hfill \\ {}\hfill \frac{\partial {V}_z}{\partial u}\hfill & \hfill \frac{\partial {V}_z}{\partial q}\hfill & \hfill \frac{\partial {V}_z}{\partial t}\hfill \end{array}\right]\times \left[\begin{array}{ccc}\hfill {I}_{11}\hfill & \hfill {I}_{12}\hfill & \hfill {I}_{13}\hfill \\ {}\hfill {I}_{21}\hfill & \hfill {I}_{22}\hfill & \hfill {I}_{23}\hfill \\ {}\hfill {I}_{31}\hfill & \hfill {I}_{32}\hfill & \hfill {I}_{33}\hfill \end{array}\right]\\ {}\end{array} $$

(30)

By differentiating (30) $\left({V}_x=\frac{f_t}{h_t}{V}_z\kern0.5em ,\kern0.5em {V}_y=\frac{g_t}{h_t}{V}_z\kern0.5em ,\kern0.5em {V}_z=M\left(u,q,t\right)\right)$ with respect to u, q, t it can be written as:

$$ \begin{array}{l}\frac{\partial {V}_x}{\partial u}=\frac{\left({f}_{tu}.M+{f}_t.{M}_u\right).{h}_t-{f}_t.M.{h}_{tu}}{h_t^2}\hfill \\ {}\frac{\partial {V}_x}{\partial q}=\frac{\left({f}_{tq}.M+{f}_t.{M}_q\right).{h}_t-{f}_t.M.{h}_{tq}}{h_t^2}\hfill \\ {}\frac{\partial {V}_x}{\partial t}=\frac{\left({f}_u.M+{f}_t.{M}_t\right).{h}_t-{f}_t.M.{h}_{tt}}{h_t^2}\hfill \\ {}\frac{\partial {V}_y}{\partial u}=\frac{\left({g}_{tu}.M+{g}_t.{M}_u\right).{h}_t-{g}_t.M.{h}_{tu}}{h_t^2}\hfill \\ {}\frac{\partial {V}_y}{\partial q}=\frac{\left({g}_{tq}.M+{g}_t.{M}_q\right).{h}_t-{g}_t.M.{h}_{tq}}{h_t^2}\hfill \\ {}\frac{\partial {V}_y}{\partial t}=\frac{\left({g}_{tt}.M+{g}_t.{M}_t\right).{h}_t-{g}_t.M.{h}_{tt}}{h_t^2}\hfill \\ {}\frac{\partial {V}_z}{\partial u}={M}_u\hfill \\ {}\frac{\partial {V}_z}{\partial q}={M}_q\hfill \\ {}\frac{\partial {V}_z}{\partial t}={M}_t\hfill \end{array} $$

(31)

Expanding the relation (30) gives:

$$ \begin{array}{l}\frac{\partial {V}_x}{\partial x}=\frac{1}{ \det J}\left({I}_{11}\frac{\partial {V}_x}{\partial u}+{I}_{21}\frac{\partial {V}_x}{\partial q}+{I}_{31}\frac{\partial {V}_x}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_x}{\partial y}=\frac{1}{ \det J}\left({I}_{12}\frac{\partial {V}_x}{\partial u}+{I}_{22}\frac{\partial {V}_x}{\partial q}+{I}_{32}\frac{\partial {V}_x}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_x}{\partial z}=\frac{1}{ \det J}\left({I}_{13}\frac{\partial {V}_x}{\partial u}+{I}_{23}\frac{\partial {V}_x}{\partial q}+{I}_{33}\frac{\partial {V}_x}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_y}{\partial x}=\frac{1}{ \det J}\left({I}_{11}\frac{\partial {V}_y}{\partial u}+{I}_{21}\frac{\partial {V}_y}{\partial q}+{I}_{31}\frac{\partial {V}_y}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_y}{\partial y}=\frac{1}{ \det J}\left({I}_{12}\frac{\partial {V}_y}{\partial u}+{I}_{22}\frac{\partial {V}_y}{\partial q}+{I}_{32}\frac{\partial {V}_y}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_y}{\partial z}=\frac{1}{ \det J}\left({I}_{13}\frac{\partial {V}_y}{\partial u}+{I}_{23}\frac{\partial {V}_y}{\partial q}+{I}_{33}\frac{\partial {V}_y}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_z}{\partial x}=\frac{1}{ \det J}\left({I}_{11}\frac{\partial {V}_z}{\partial u}+{I}_{21}\frac{\partial {V}_z}{\partial q}+{I}_{31}\frac{\partial {V}_z}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_z}{\partial y}=\frac{1}{ \det J}\left({I}_{12}\frac{\partial {V}_z}{\partial u}+{I}_{22}\frac{\partial {V}_z}{\partial q}+{I}_{32}\frac{\partial {V}_z}{\partial t}\right)\hfill \\ {}\frac{\partial {V}_z}{\partial z}=\frac{1}{ \det J}\left({I}_{13}\frac{\partial {V}_z}{\partial u}+{I}_{23}\frac{\partial {V}_z}{\partial q}+{I}_{33}\frac{\partial {V}_z}{\partial t}\right)\hfill \end{array} $$

(32)

Moreover, the strain and velocity relations can be defined as:

$$ \begin{array}{l}{\varepsilon}_{xx}=\frac{\partial {V}_x}{\partial x}\hfill \\ {}{\varepsilon}_{xy}=\frac{1}{2}\left(\frac{\partial {V}_x}{\partial y}+\frac{\partial {V}_y}{\partial x}\right)\hfill \\ {}{\varepsilon}_{xz}=\frac{1}{2}\left(\frac{\partial {V}_x}{\partial z}+\frac{\partial {V}_z}{\partial x}\right)\hfill \\ {}{\varepsilon}_{yz}=\frac{1}{2}\left(\frac{\partial {V}_y}{\partial z}+\frac{\partial {V}_z}{\partial y}\right)\hfill \\ {}{\varepsilon}_{yy}=\frac{\partial {V}_y}{\partial y}\hfill \\ {}{\varepsilon}_{zz}=\frac{\partial {V}_z}{\partial z}\hfill \end{array} $$

(33)

Using (31), (32), and (33) the incompressibility condition can be re-written as:

$$ \begin{array}{l}{\varepsilon}_{ii}=\frac{\partial {V}_x}{\partial x}+\frac{\partial {V}_y}{\partial y}+\frac{\partial {V}_z}{\partial z}=\frac{1}{ \det J}\left({I}_{11}\frac{\partial {V}_x}{\partial u}+{I}_{21}\frac{\partial {V}_x}{\partial q}+{I}_{31}\frac{\partial {V}_x}{\partial t}+{I}_{12}\frac{\partial {V}_y}{\partial u}+{I}_{22}\frac{\partial {V}_y}{\partial q}+{I}_{32}\frac{\partial {V}_y}{\partial t}\right.\hfill \\ {}\left.+{I}_{13}\frac{\partial {V}_z}{\partial u}+{I}_{23}\frac{\partial {V}_z}{\partial q}+{I}_{33}\frac{\partial {V}_z}{\partial t}\right)=0\hfill \end{array} $$

(34)

$$ \begin{array}{l}{\varepsilon}_{ii}=\frac{1}{ \det J}\left[\left({g}_q{h}_t-{g}_t{h}_q\right)\frac{\left({f}_{tu}.M+{f}_t.{M}_u\right).{h}_t-{f}_t.M.{h}_{tu}}{h_t^2}\right.\hfill \\ {}\left(-{f}_qh{}_t+{f}_t{h}_q\right)\frac{\left({f}_{tq}.M+{f}_t{M}_q\right).{h}_t-{f}_t.M.{h}_{tq}}{h_t^2}\hfill \\ {}+\left({g}_u{h}_q-{g}_q{h}_u\right)\frac{\left({f}_{tt}.M+{f}_t.{M}_1\right).{h}_t-{f}_t.M.{h}_{tt}}{h_t^2}\hfill \\ {}+\left(-{f}_q{h}_t+{f}_t{h}_q\right)\frac{\left({g}_{tu}.M+{g}_t.{M}_u\right).{h}_t-{g}_t.M.{h}_{tu}}{h_t^2}\hfill \\ {}+\left({f}_u{h}_t-{f}_t{h}_u\right)\frac{\left({g}_{tq}.M+{g}_t.{M}_q\right).{h}_t-{g}_t.M.{h}_{tq}}{h_t^2}\hfill \\ {}+\left(-{f}_u{h}_q+{f}_q{h}_u\right)\frac{\left({g}_{tt}.M+{g}_t.{M}_t\right).{h}_t-{g}_t.M.{h}_{tt}}{h_t^2}\hfill \\ {}+\left({f}_q{g}_q-{f}_t{g}_q\right){M}_u+\left(-{f}_u{g}_t+{f}_t{g}_u\right){M}_q\hfill \\ {}+\left.\left({f}_u{g}_q-{f}_q{g}_u\right){M}_t\right]=0\hfill \end{array} $$

(35)

Relation (35) in a simple way can be written as:

$$ A\left(u,q,t\right)M+B\left(u,q,t\right).\frac{\partial M}{\partial t}=0 $$

(36)

Where:

$$ \begin{array}{l}\begin{array}{l}A={h}_t\left({g}_q{f}_{tu}{h}_t-{g}_q{h}_{tu}{f}_t-{g}_t{h}_q{f}_{tu}-{g}_u{h}_t{f}_{tq}+{g}_u{h}_{tq}{f}_t\right.\hfill \\ {}+{g}_t{h}_u{f}_{tq}+{g}_u{h}_q{f}_{tt}-{g}_q{h}_u{f}_{tt}-{f}_q{g}_{tu}{h}_t+{f}_q{h}_{tu}{g}_t\hfill \\ {}\left.+{f}_t{h}_q{g}_{tu}-{f}_u{g}_{tq}{h}_t-{f}_u{h}_{tu}{g}_t-{f}_t{h}_u{g}_{tq}-{f}_u{h}_q{g}_{tt}+{f}_q{h}_u{g}_{tt}\right)\hfill \\ {}+{g}_t{h}_q{h}_{tu}{f}_t-{g}_t{h}_u{h}_{tq}{f}_t-{g}_q{h}_u{h}_{tt}{f}_t-{g}_u{h}_q{f}_t{h}_{tt}\hfill \\ {}-{f}_t{h}_q{h}_{tu}{g}_t+{f}_t{h}_u{h}_{tq}{g}_t+{g}_q{h}_u{h}_{tt}{f}_t-{f}_u{h}_u{h}_{tt}{g}_t\hfill \end{array}\hfill \\ {}B={h}_t\left({h}_t\left({f}_u{g}_q-{f}_q{g}_u\right)+{h}_q\left({f}_t{g}_u-{f}_u{g}_t\right)+{h}_u\left({f}_q{g}_t-{f}_t{g}_q\right)\right)\hfill \end{array} $$

(37)

By integrating relation (36) we have:

$$ \int \frac{\partial M}{M}=-\int \frac{A}{B}\partial t $$

(38)

$$ A=\frac{\partial B}{\partial t}-{A}_1\kern0.5em ,\kern0.5em {A}_1=2{h}_{tt}\left({h}_t\left({f}_u{g}_q-{f}_q{g}_u\right)+{h}_q\left({f}_t{g}_u-{f}_u{g}_t\right)+{h}_u\left({f}_q{g}_t-{f}_t{g}_q\right)\right) $$

(39)

$$ \frac{A}{B}=\frac{\left(\partial B/\partial t\right)-{A}_1}{B} $$

(40)

$$ \frac{A}{B}=\frac{\left(\partial B/\partial t\right)}{B}-\frac{2{h}_{tt}}{h_t} $$

(41)

$$ \int \frac{\partial M}{M}=-\int \left(\frac{\left(\partial B/\partial t\right)}{B}-\frac{2{h}_{tt}}{h_t}\right)\partial t $$

(42)

$$ Ln(M)=- Ln(B)+ Ln\left({h}_t\right)+ Ln(C) $$

(43)

In relation (43) C is a function of u and q (C = C(u,q)) that yields:

$$ M=\frac{h_t^2C}{B} $$

(44)

Finally, replacing B from relation (37) in (44) gives:

$$ M=\frac{C\left(u,q\right)}{\left({f}_u{g}_q-{f}_q{g}_u\right)+{h}_q/{h}_t\left({f}_t{g}_u-{f}_u{g}_t\right)+{h}_u/{h}_t\left({f}_q{g}_t-{f}_t{g}_q\right)} $$

(45)