The following article is a component of the August 1998 (vol. 50, no. 8) JOM and is presented as JOM-e. Such articles appear exclusively on the web and do not have print equivalents.

The Three-Dimensional Simulation of Flow Pattern in Equal-Channel Angular Extrusion

CONTENTS
INTRODUCTION THE MATHEMATICS OF ECAE THE SELF-CONSISTENT VELOCITY FIELD FLOW PATTERN CONCLUSION References

Equal-channel angular extrusion is an innovative forming process to extrude material without substantial change in its geometric shape. Multiple extrusions from equal-channel angular extrusion allows heavy plastic strain to the bulk material. More importantly, by changing the orientation of the billet between successive extrusions, sophisticated microstructures and textures can be developed in the material. According to the orientation chosen after each pass, four fundamental equal-channel angular extrusion routes (A, B, C, and C´) are defined and used for different purposes. It is important to understand the flow pattern after each route because it significantly impacts the formation and orientation of the microstructures. this article, we establish a three-dimensional computational model to predict the flow pattern. To this end, a unified mathematical description for all equal-channel angular extrusion routes will first be given. After that, we construct the self-consistent velocity field for equal-channel angular extrusion. Based on the results from numerical calculation, the flow pattern can be graphically simulated for any route and pass number.

INTRODUCTION

Figure 1. A schematic of ECAE.	Figure 2. (a) The fan deformation pattern of ECAE and (b) a velocity-vector diagram.

ECAE has many noticeable technological benefits.² Since the cross section of the billet remains nearly unchanged after each extrusion, this process is repeatable, and, hence, heavy plastic strain can be obtained by multiple extrusions. More importantly, if the cross sections of the channels are designed as square, the orientation of the shear plane can be easily changed by rotating the billet (± n 90°, n = 1 or 2) for subsequent extrusions. This simple fact makes it possible to use different routes (A, B, C, and C´), which are defined based on the different combinations of the shear planes, to develop different microstructures and textures within the bulk material.

It is expected that material properties can be estimated after extrusion. An efficient way to do this is to study the formation and orientation of the microstructures, which can be characterized by the flow patterns.

THE MATHEMATICS OF ECAE

Figure A. A typical cycle of ECAE. Figure A shows a typical cycle of the ECAE deformation process. X0, X1, X2, and X3 represent billet positions in a global coordinate system at different stages during the forming process r. The cycle can be decomposed into three sub-processes. In the first subprocess, f1: X0 X1 the billet is extruded from the original position X0 to the completely deformed state X1. In the second subprocess, f2: X1 X2 the billet at X1 is reoriented vertically to X2. In the third subprocess, f3: X2 X3 the billet at X2 is rotated about its long axis through a certain angle (n90°) to a new position X3, which is ready for the next extrusion cycle. Based on the rotated angle chosen in the third subprocess, the four different routes can be defined as follows: A route, n = 0; B route, n = +1 at even pass, n = -1 at odd pass; C route, n = 2; C' route, n = 1. Following the mathematical meanings of f1 ( ), f2 ( ), and f3 ( ), the following relation exists X3 = f3 (X2) = f3 (f2 (X2) ) = f3 (f2 (f1 (X0))) = f (X0) (A) This means that if the forms of f1 ( ), f2 ( ), and f3 ( ) are determined, the one-to-one mapping of f ( ) between the material particle in an original position (X0) and its position after one ECAE cycle (represented by X3) can be established. In other words, the movement of any material particle can be traced. The transformations f2 ( ) and f3 ( ) represent combinations of pure rigid-body translation and rotation and are easy to determined. The deformation process represented by f1 ( ) is associated with a two-dimensional shear deformation. Unfortunately, the analytical form of f1 ( ) does not exist, and the relationship X0 X1 has to be numerically calculated. Suppose that Xt is an intermediate configuration of the billet between X0 and X1 at moment t, then the next configuration, Xt+t, after time increment t can be determined by Xt+t = Xt + U = Xt + V(Xt)t (B) The velocity of the material particle [V(Xt)] is derived from the ECAE self-consistent velocity field.

THE SELF-CONSISTENT VELOCITY FIELD

As the velocity-vector diagram in Figure 2 indicates, the velocity of the material particle within the fan area will vary its direction continuously and keep the same magnitude (i.e., |u| =|w|). The relationships between the magnitude of x and u, x, and y can be derived from the constant volume constraint

|x| sin |AC| = |u| |AC|=|y| sin |BD|

in which

FLOW PATTERN

Figure 3. A cross section of the material billet.	Figure 4a. The flow pattern after four passes following route A	Figure 4b. The flow pattern after four passes following route B
Figure 5. Illustration of the surface effect after the (a) first, (b) second, and (c) fourth passes following route B.

CONCLUSION

References

ABOUT THE AUTHORS

H.J. Cui, R.E. Goforth, and K.T. Hartwig are currently at the Department of Mechanical Engineering at Texas A&M University, College Station, Texas.

For more information, contact R.G. Goforth, Texas A&M University, Materials Science and Engineering, College Station, Texas 77843; (409) 845-3645; fax (409) 862-2418; e-mail rgoforth@mengr.tamu.edu.