Fragment size

From this it can be concluded that the wide distribution of fragment sizes from milhng is inherent in the breakage process itself and that attempts to improve grinding efficiency by weakening the particles will result in coarser fragments which may reqiiire a further break to reach the desired size. 

On one hand, inherent flaws or perturbations in a fracturing body, which are the sites of internal fracture nucleation, have been recognized as important in determining characteristic fracture spacing and, consequently, the nominal fragment size in a fracture event. Theoretical work based on a physical description of these material imperfections has been actively pursued (Curran et al., 1977 Grady and Kipp, 1980). 

An acceptable reconciliation of inherent flaw and fracture energy concepts has not been achieved and provides an area of current study. The two theoretical concepts will be discussed, and several applications in fragment-size prediction will be described. We will make comparisons between the two fragmentation approaches and attempt to identify some conditions which determine when one or the other method applies. 

The fracture stress and fragment size data in Fig. 8.11 are observed to... 

Figure 8.11. Fragment size and fracture stress dependence on tensile loading strain rate for oil shale.

Assuming a complete transfer of the kinetic energy in (8.24) into energy dissipated during the spall fracture process in (8.25) provides an expression for the characteristic spall fragment size... 

Other methods for implementing the energy balance have been proposed and lead to moderate differences in predicted fragment size (Grady, 1982 Glenn and Chudnovsky, 1986). We contrast this earlier kinetic energy based theory with a more recent development in the following section. 

If it is assumed that the material carried into tension is predisposed to spall through a sufficient microscopic flaw structure when the energy condition is satisfied, then the expressions, (8.27) through (8.29), as equalities, can be solved for the spall properties. These are, for the fragment size... 

Note that this product of the spall stress and time is a constant provided F itself is constant. This relation is also useful for estimating fracture energy from dynamic spall data. Also, from (8.30) and (8.31) a relation for the fragment size... 

We note here that the fragment size predicted through the more recent energy-horizon theory ((8.30) or (8.34)) is between a factor of 2 to 3 smaller in nominal diameter than predicted through the earlier kinetic energy theory ((8.26)). This difference is more marked if a measure of fragment mass is... 

In (8.35) Y is the flow stress in simple tension (and may itself be a function of the temperature and strain rate) and is the critical volumetric strain at void coalescence (calculated within the model to equal 0.15 independent of material). Note that the ductile fragmentation energy depends directly on the fragment size s. With (8.35), (8.30) through (8.32) become, for ideal ductile spall fragmentation,... 

Further, if one considers the distance the tensile unloading wave can propagate over the time to fracture f from (8.46), a lower bound criterion for the fragment size can be established... 

A reasonable estimate of the number of flaws needed to sustain an equilibrium value of A can be made based on the concepts introduced earlier in this section. To achieve a nominal fragment size of d, the number of activated flaws per unit volume should be of order d From the equilibrium... 

Up to this point we have addressed primarily the flaw structure and energy concepts in stress-wave loaded solids governing the creation of new fracture surface area (or the mean fragment size) in catastrophic fragmentation events. It remains to consider a concept which is frequently the end concern in impulsive fracture applications, namely, the distribution in sizes of the particles produced in the dynamic fragmentation event. 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

Figure 8.22. Comparison of numerical cumulative fragment size data and aluminum expanding ring data.

At the conclusion of the calculation, a fragment size distribution as well as fragment number is provided. A cumulative number distribution is shown in Fig. 8.22 and compared with aluminum ring data acquired at = lO s (Grady and Benson, 1983). With the assumed fracture site nucleation law, the calculated distribution appears to agree reasonably well with the data. The calculation better predicts the tails of the distribution which have trends which deviate from strict exponential behavior as was noted in the previous section. 

The more common approach is the actual positioning of random lines on a surface to create a statistical distribution of fragment sizes. One example of this, suggested by Mott and Linfoot (1943), is a construction of randomly positioned and oriented infinite lines as illustrated in Fig. 8.23. If the random lines are restricted to horizontal or vertical orientation an analytic solution can be obtained for the cumulative fragment number (Mott and Linfoot,... 

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