

CONTENTS 

Figure 1. A schematic of ECAE.  Figure 2. (a) The fan deformation pattern of ECAE and (b) a velocityvector 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  

the billet is extruded from the original position X_{0} to the completely deformed state X_{1}. In the second subprocess,
the billet at X_{1} is reoriented vertically to X_{2}. In the third subprocess,
the billet at X_{2} is rotated about its long axis through a certain angle (n90°) to a new position X_{3}, 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 f_{1} ( ), f_{2} ( ), and f_{3} ( ), the following relation exists
This means that if the forms of f_{1} ( ), f_{2} ( ), and f_{3} ( ) are determined, the onetoone mapping of f ( ) between the material particle in an original position (X_{0}) and its position after one ECAE cycle (represented by X_{3}) can be established. In other words, the movement of any material particle can be traced. The transformations f_{2} ( ) and f_{3} ( ) represent combinations of pure rigidbody translation and rotation and are easy to determined. The deformation process represented by f_{1} ( ) is associated with a twodimensional shear deformation. Unfortunately, the analytical form of f_{1} ( ) does not exist, and the relationship X_{0} X_{1} has to be numerically calculated. Suppose that X^{t} is an intermediate configuration of the billet between X_{0} and X_{1} at moment t, then the next configuration, X^{t}^{+t}, after time increment t can be determined by

As the velocityvector 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
in which
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) 8453645; fax (409) 8622418; email rgoforth@mengr.tamu.edu.
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