Upper bound analysis of wire flat rolling with experimental and FEM verifications

Abstract

In this paper, an upper bound analysis is developed to calculate the rolling torque in the wire flat rolling process. Streamlines are defined parametrically in the deforming region and the formulation of the admissible velocity field are developed. The internal and frictional power terms are obtained from the upper bound solution. Through the analysis, the rolling torque is obtained. Furthermore, the effects of the roll speed, wire diameter, and reduction in height on the rolling torque are studied. In addition, the experimental and numerical investigations are performed to verify the proposed theoretical model. A very good agreement is obtained between the analytical results with those from the simulation and the experiment.

Appendix

The general velocity vector equation for any arbitrary point in the deformation zone is expressed as:

$$\vec{V} = \frac{1}{{\left( {V_{x}^{2} + V_{y}^{2} + V_{z}^{2} } \right)^{1/2} }} \times \left( {V_{x} i + V_{y} j + V_{z} k} \right).$$

(36)

The tangent vector for each point in the flow line is as follows:

$$\vec{T} = \frac{1}{{\left( {T_{1}^{2} + T_{2}^{2} + T_{3}^{2} } \right)^{1/2} }} \times \left( {T_{1} i + T_{2} j + T_{3} k} \right).$$

(37)

where

$$\begin{aligned} T_{1} & = \frac{\partial X}{\partial t} \\ T_{2} & = \frac{\partial Y}{\partial t} \\ T_{3} & = \frac{\partial Z}{\partial t} \\ \end{aligned}$$

(38)

Assuming that each particle that flows along a flowline in the deforming region and that the velocities of all these particles are tangent to their respective flowlines at all times, the following admissible velocity field may be derived with the volume constancy condition:

$$\frac{{V_{x} }}{{T_{1} }} = \frac{{V_{y} }}{{T_{2} }} = \frac{{V_{z} }}{{T_{3} }}$$

(39)

The Eq. (39) can be rewritten as follows:

$$\begin{aligned} V_{y} & = \frac{{T_{2} }}{{T_{3} }}V_{z} \\ V_{x} & = \frac{{T_{1} }}{{T_{3} }}V_{z} \\ \end{aligned}$$

(40)

Therefore, the velocity field for each arbitrary particle in the deformation zone is expressed as:

$$V_{x} = \frac{{f_{t} }}{{h_{t} }}V_{z} \quad V_{y} = \frac{{g_{t} }}{{h_{t} }}V_{z} \quad V_{z} = M\left( {u,q,t} \right)$$

(41)

where M(u, q, t) is an unknown function that satisfies the incompressibility conditions expressed as follows:

$$\frac{{\partial V_{x} }}{\partial X} + \frac{{\partial V_{y} }}{\partial Y} + \frac{{\partial V_{z} }}{\partial Z} = 0$$

(42)

Introducing partial differentiation in three dimensions:

$$\left( {\partial V_{i} /\partial X_{k} ) = \mathop \sum \limits_{j = 1} (\left( {\partial V_{i} /\partial U_{j} } \right) \cdot \left( {\partial U_{j} /\partial X_{k} } \right)} \right)$$

(43)

or in matrix format:

$$\frac{{\partial V_{i} }}{{\partial X_{k} }} = \left[ {\begin{array}{*{20}c} {\frac{{\partial V_{x} }}{\partial x}} & {\frac{{\partial V_{x} }}{\partial y}} & {\frac{{\partial V_{x} }}{\partial z}} \\ {\frac{{\partial V_{y} }}{\partial x}} & {\frac{{\partial V_{y} }}{\partial y}} & {\frac{{\partial V_{y} }}{\partial z}} \\ {\frac{{\partial V_{z} }}{\partial x}} & {\frac{{\partial V_{z} }}{\partial y}} & {\frac{{\partial V_{z} }}{\partial z}} \\ \end{array} } \right]$$

(44)

$$\frac{{\partial V_{i} }}{{\partial u_{j} }} = \left[ {\begin{array}{*{20}c} {\frac{{\partial V_{x} }}{\partial u}} & {\frac{{\partial V_{x} }}{\partial q}} & {\frac{{\partial V_{x} }}{\partial t}} \\ {\frac{{\partial V_{y} }}{\partial u}} & {\frac{{\partial V_{y} }}{\partial q}} & {\frac{{\partial V_{y} }}{\partial t}} \\ {\frac{{\partial V_{z} }}{\partial u}} & {\frac{{\partial V_{z} }}{\partial q}} & {\frac{{\partial V_{z} }}{\partial t}} \\ \end{array} } \right]$$

(45)

$$\frac{{\partial X_{k} }}{{\partial u_{j} }} = \left[ {\begin{array}{*{20}c} {\frac{\partial x}{\partial u}} & {\frac{\partial x}{\partial q}} & {\frac{\partial x}{\partial t}} \\ {\frac{\partial y}{\partial u}} & {\frac{\partial y}{\partial q}} & {\frac{\partial y}{\partial t}} \\ {\frac{\partial z}{\partial u}} & {\frac{\partial z}{\partial q}} & {\frac{\partial z}{\partial t}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {f_{u} } & {f_{q} } & {f_{t} } \\ {g_{u} } & {g_{q} } & {g_{t} } \\ {h_{u} } & {h_{q} } & {h_{t} } \\ \end{array} } \right] = J$$

(46)

where J is the Jacobian for conversion of coordinates from x, y, z to u, q, t. Therefore:

$$\frac{{\partial u_{j} }}{{\partial X_{k} }} = J^{ - 1} = \frac{1}{\left| J \right|}\left[ {\begin{array}{*{20}c} {I_{11} } & {I_{12} } & {I_{13} } \\ {I_{21} } & {I_{22} } & {I_{23} } \\ {I_{31} } & {I_{32} } & {I_{33} } \\ \end{array} } \right]$$

(47)

where

$$\left| J \right| = \frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {u,q,t} \right)}} = \left| {\begin{array}{*{20}c} {\frac{\partial x}{\partial u}} & {\frac{\partial x}{\partial q}} & {\frac{\partial x}{\partial t}} \\ {\frac{\partial y}{\partial u}} & {\frac{\partial y}{\partial q}} & {\frac{\partial y}{\partial t}} \\ {\frac{\partial z}{\partial u}} & {\frac{\partial z}{\partial q}} & {\frac{\partial z}{\partial t}} \\ \end{array} } \right|$$

(48)

Replacing (44), (45) and (47) in (43):

$$\left[ {\begin{array}{*{20}c} {\frac{{\partial V_{x} }}{\partial x}} & {\frac{{\partial V_{x} }}{\partial y}} & {\frac{{\partial V_{x} }}{\partial z}} \\ {\frac{{\partial V_{y} }}{\partial x}} & {\frac{{\partial V_{y} }}{\partial y}} & {\frac{{\partial V_{y} }}{\partial z}} \\ {\frac{{\partial V_{z} }}{\partial x}} & {\frac{{\partial V_{z} }}{\partial y}} & {\frac{{\partial V_{z} }}{\partial z}} \\ \end{array} } \right] = \frac{1}{\left| J \right|}\left[ {\begin{array}{*{20}c} {\frac{{\partial V_{x} }}{\partial u}} & {\frac{{\partial V_{x} }}{\partial q}} & {\frac{{\partial V_{x} }}{\partial t}} \\ {\frac{{\partial V_{y} }}{\partial u}} & {\frac{{\partial V_{y} }}{\partial q}} & {\frac{{\partial V_{y} }}{\partial t}} \\ {\frac{{\partial V_{z} }}{\partial u}} & {\frac{{\partial V_{z} }}{\partial q}} & {\frac{{\partial V_{z} }}{\partial t}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {I_{11} } & {I_{12} } & {I_{13} } \\ {I_{21} } & {I_{22} } & {I_{23} } \\ {I_{31} } & {I_{32} } & {I_{33} } \\ \end{array} } \right]$$

(49)

The elements of the J⁻¹ matrix can be calculated as follows:

$$\begin{aligned} I_{11} & = g_{q} h_{t} - g_{t} h_{q} \\ I_{12} & = - f_{q} h_{t} + f_{t} h_{q} \\ I_{13} & = f_{q} g_{t} - f_{t} g_{q} \\ I_{21} & = - g_{u} h_{t} + g_{t} h_{u} \\ I_{22} & = f_{u} h_{t} - f_{t} h_{u} \\ I_{23} & = - f_{u} g_{t} + f_{t} g_{u} \\ I_{31} & = g_{u} h_{q} - g_{q} h_{u} \\ I_{32} & = - f_{u} h_{q} + f_{q} h_{u} \\ I_{33} & = f_{u} g_{q} - f_{q} g_{u} \\ \end{aligned}$$

(50)

Differentiating (41) with respect to u, q, t gives:

$$\begin{aligned} \frac{{\partial V_{x} }}{\partial u} & = \frac{{\left( {f_{tu} .M + f_{t} .M_{u} } \right)h_{t} - f_{t} .M.h_{tu} }}{{h_{t}^{2} }} \\ \frac{{\partial V_{x} }}{\partial q} & = \frac{{\left( {f_{tq} .M + f_{t} .M_{q} } \right)h_{t} - f_{t} .M.h_{tq} }}{{h_{t}^{2} }} \\ \frac{{\partial V_{x} }}{\partial t} & = \frac{{\left( {f_{tt} .M + f_{t} .M_{t} } \right)h_{t} - f_{t} .M.h_{tt} }}{{h_{t}^{2} }} \\ \frac{{\partial V_{y} }}{\partial u} & = \frac{{\left( {g_{tu} .M + g_{t} .M_{u} } \right)h_{t} - g_{t} .M.h_{tu} }}{{h_{t}^{2} }} \\ \frac{{\partial V_{y} }}{\partial q} & = \frac{{\left( {g_{tq} .M + g_{t} .M_{q} } \right)h_{t} - g_{t} .M.h_{tq} }}{{h_{t}^{2} }} \\ \frac{{\partial V_{y} }}{\partial t} & = \frac{{\left( {g_{tt} .M + g_{t} .M_{t} } \right)h_{t} - g_{t} .M.h_{tt} }}{{h_{t}^{2} }} \\ \frac{{\partial V_{z} }}{\partial u} & = M_{u} , \quad \frac{{\partial V_{z} }}{\partial q} = M_{q} ,\quad \frac{{\partial V_{z} }}{\partial t} = M_{t} \\ \end{aligned}$$

(51)

Expanding (49):

$$\begin{aligned} \frac{{\partial V_{x} }}{\partial x} = \frac{1}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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}{\left| J \right|}\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{aligned}$$

(52)

Replacing from (50), (51) and (52) into (42):

$$\begin{aligned} & \frac{1}{\left| J \right|}\left[ {\left( {g_{q} h_{t} - g_{t} h_{q} } \right)\frac{{\left( {f_{tu} \cdot M + f_{t} \cdot M_{u} } \right)h_{t} - f_{t} \cdot M \cdot h_{tu} }}{{h_{t}^{2} }}} \right. \\ & \quad + \left( { - g_{u} h_{t} + g_{t} h_{u} } \right)\frac{{\left( {f_{tq} \cdot M + f_{t} \cdot M_{q} } \right)h_{t} - f_{t} \cdot M \cdot h_{tq} }}{{h_{t}^{2} }} \\ & \quad + \left( {g_{u} h_{q} - g_{q} h_{u} } \right)\frac{{\left( {f_{tt} \cdot M + f_{t} \cdot M_{t} } \right)h_{t} - f_{t} \cdot M \cdot h_{tt} }}{{h_{t}^{2} }} \\ & \quad + \left( { - f_{q} h_{t} + f_{t} h_{q} } \right)\frac{{\left( {g_{tu} \cdot M + g_{t} \cdot M_{u} } \right)h_{t} - g_{t} \cdot M \cdot h_{tu} }}{{h_{t}^{2} }} \\ & \quad + \left( {f_{u} h_{t} - f_{t} h_{u} } \right)\frac{{\left( {g_{tq} \cdot M + g_{t} \cdot M_{q} } \right)h_{t} - g_{t} \cdot M \cdot h_{tq} }}{{h_{t}^{2} }} \\ & \quad + \left( { - f_{u} h_{q} + f_{q} h_{u} } \right)\frac{{\left( {g_{tt} \cdot M + g_{t} \cdot M_{t} } \right)h_{t} - g_{t} \cdot M \cdot h_{tt} }}{{h_{t}^{2} }} \\ & \quad \left. { + \left( {f_{q} g_{t} - f_{t} g_{q} } \right)M_{u} + \left( { - f_{u} g_{t} + f_{t} g_{u} } \right)M_{q} + \left( {f_{u} g_{q} - f_{q} g_{u} } \right)M_{t} } \right] = 0 \\ \end{aligned}$$

(53)

or more simply:

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

(54)

where

$$\begin{aligned} A & = h_{t} (g_{q} f_{tu} h_{t} - g_{q} h_{tu} f_{t} - g_{t} h_{q} f_{tu} - g_{u} h_{t} f_{tq} + g_{u} h_{tq} f_{t} + g_{t} h_{u} f_{tq} \\ & \quad + g_{u} h_{q} f_{tt} - g_{q} h_{u} f_{tt} - f_{q} g_{tu} h_{t} + f_{q} h_{tu} g_{t} + f_{t} h_{q} g_{tu} + f_{u} g_{tq} h_{t} \\ & \quad - f_{u} h_{tq} g_{t} - f_{t} h_{u} g_{tq} - f_{u} h_{q} g_{tt} + f_{q} h_{u} g_{tt} ) + g_{t} h_{q} h_{tu} f_{t} \\ & \quad - g_{t} h_{u} h_{tq} f_{t} - g_{u} h_{q} f_{t} h_{tt} + g_{q} h_{u} f_{t} h_{tt} - f_{t} h_{q} g_{t} h_{tu} \\ & \quad + f_{t} h_{u} g_{t} h_{tq} + f_{u} h_{q} g_{t} h_{tt} - f_{q} h_{u} g_{t} h_{tt} \\ \end{aligned}$$

(55)

$$B = h_{t} \left[ {h_{t} \left( {f_{u} g_{q} - f_{q} g_{u} } \right) + h_{q} \left( {f_{t} g_{u} - f_{u} g_{t} } \right) + h_{u} \left( {f_{q} g_{t} - f_{t} g_{q} } \right)} \right]$$

(56)

Rearranging and integrating (54) yields:

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

(57)

It is evident from (55) and (56) that:

$$A = \frac{\partial B}{\partial t} - A_{1}$$

(58)

where

$$A_{1} = 2h_{tt} \left[ {h_{t} \left( {f_{u} g_{q} - f_{q} g_{u} } \right) + h_{q} \left( {f_{t} g_{u} - f_{u} g_{t} } \right) + h_{u} \left( {f_{q} g_{t} - f_{t} g_{q} } \right)} \right]$$

(59)

Therefore

$$\frac{A}{B} = \frac{{\frac{\partial B}{\partial t} - A_{1} }}{B}$$

(60)

or

$$\frac{A}{B} = \frac{\partial B/\partial t}{B} - \frac{{2h_{tt} }}{{h_{t} }}$$

(61)

Replacing (61) in (62):

$$\int {\frac{\partial M}{M}} = - \int {\left( {\frac{\partial B/\partial t}{B} - \frac{{2h_{tt} }}{{h_{t} }}} \right)} \cdot \partial t$$

(62)

It follows that

$$Ln\left( M \right) = - Ln\left( B \right) + 2 \times Ln\left( {h_{t} } \right) + Ln\left( C \right)$$

(63)

in which C is the constant of integration and a function of the other two parameters u and q;

$$C = C\left( {u,q} \right)$$

(64)

Rewritten (63):

$$M = \frac{{h_{t}^{2} \cdot C}}{B}$$

(65)

From (56):

$$M = \frac{{C \cdot h_{t}^{2} }}{{h_{t} \left[ {h_{t} \left( {f_{u} g_{q} - f_{q} g_{u} } \right) + h_{q} \left( {f_{t} g_{u} - f_{u} g_{t} } \right) + h_{u} \left( {f_{q} g_{t} - f_{t} g_{q} } \right)} \right]}}$$

(66)

or

$$M = \frac{{C \cdot h_{t} }}{{h_{t} \left( {f_{u} g_{q} - f_{q} g_{u} } \right) + h_{q} \left( {f_{t} g_{u} - f_{u} g_{t} } \right) + h_{u} \left( {f_{q} g_{t} - f_{t} g_{q} } \right)}}$$

(67)