Modeling of mechanical alloying: Part II. Development of computational modeling programs
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I.
INTRODUCTION a (r) = R v
IN the first article of this series, tl] we described the basic physics of deformation, coalescence, and fragmentation taking place during a ball-powder-ball collision of the type that occurs in mechanical alloying (MA) or mechanical milling (MM). This led to specification of conditions requisite for each of these events and to equations predicting how these events would affect the dimensions, morphology, and properties of the powder. The conditions given are for a single impact in the sense that they specify physical requirements to be met during the course of a given impact. But with repetitive impacts, powder particles change their shape, hardness, and size. For each impact, then, it is necessary to determine new powder particle dimensions and powder properties and to test whether or not conditions have been met for a fragmentation* or coalescence event. *Equation [16] of Part I, because it is written explicitly in terms of a critical deformation, obscures that the critical deformation is a cumulative one; Le, the left side of said equation is in reality a product of successive collisions. To illustrate this, we note that Eq. [1] (of this article and of part I) specifies the deformation (a) in a specific collision. In contrast, the a of Eq. [16] in part I is different. For example, suppose that odho = 0.2 ' for each and every collision according to Eq. [1] of part I. Thus, for the first collision, a/ho (Eq. [16]) also is equal to 0.2. However, for the second collision, the left-hand side of Eq. [16] is equal to 0.2 + 0.2 (1 - 0.2) = 0.36 and the left-hand side of Eq. [16] after the third collision is equal to 0.36 + 0.2 (1 - .36) = 0.488, etc. We regret this confusion in the original article and apologize for any consequent misunderstandings.
In addition to conditions changing with successive impacts, they vary with position within the contact area during a single impact. The most basic "event" during milling is plastic deformation. The deformation of the powder trapped between two balls is given by[1]
D. MAURICE, formerly Graduate Student, Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903, is NRC Research Associate, Albany Research Center, U. S. Bureau of Mines, Albany, OR 97321. T.H. COURTNEY, formerly Professor, Department of Materials Science and Engineering, University of Virginia, is Professor and Chair, Department of Metallurgical and Materials Engineering, Michigan Technological University, Houghton, MI 49931. Manuscript submitted June 17, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS A
( ~ v v ) 1/2
r2 R
[1]
where v is the relative collision velocity, r the distance from the center of contact, R the radius of the balls, PBthe density of the balls, and Hv the powder hardness. It is apparent that any computational scheme must monitor the deformation, and hence any events dependent on deformation, as a function of particle position during the impact. In this article, we expand our "single impact" treatments of part I and
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