Relationships between Particle Size and Fracture Energy for Single Particle Crushing t

An experimental study of single-particle crushing at slow compression rate was carried out for two kinds of glassy and five kinds of natural materials. The specimens were almost spherical particles of0.5 to 3.0 em in diameter. The relationships between particle size and fracture energy (strain energy) were calculated by using the results of the size effect ranging from about 10 em to 0. 0030 em of single particle crushing, as shown in the previous papers. The experimental results are summarized as follows:


Introduction
The importance of fracture characteristics in determining optimum design or operating condition in comminution process has been extensively recognized in various industries. The energy required for grinding materials or the grinding resistance is generally expressed as a function of particle size through size reduction. As well accepted, however, prolonged comminution has the tendency to show the finite limit of size reduction peculiar to the equipment used 6 ). This is not only due to the absorption of the energy into powder bed and the reduction of energy distribution for each fractured fragment, but also due to the increase in fracture strength resulted from size reduct This report was originally printed in Kagaku Kogaku Ron bun-shu, 10, 108-112 (1984) in Japanese, before being translated into English with the permission of the editorial committee of the Soc. Chemical Engineers, Japan. tion, that is, the effect of size on strength.
In the present paper, the relationship between compressive strength of spherical specimens of 0.5 to 3 em in nominal diameter and fracture energy was experimentally investigated. Furthermore, the energy required for single particle crushing and the specific fracture energy was calculated by use of the experimentally obtained relationship between compressive strength of spheres with an approximate diameter range of 30 f.J.m to I 0 em presented in the literatures 10 • 13 ). The result produced was that specific fracture energy in-creased with increasing particle size, showing a remarkable rise especially within the finer range of the particle size.
The experimental or calculated results obtained in this work were based on the condition where spherical or near-spherical specimens were fractured under static compressive load 9 ). However, the actual fracture is accompanied by the phenomenon that irregularly shaped particles are fractured under dynamic or impact load. Therefore the results of this paper will be available in practice as far as the difference between the ideal and actual phenomenon is taken into consideration to reasonable extent.

Sample and experimental method
The specimens used and their mechanical properties are listed in Table 1. The relationship between compressive strength of spheres and fracture energy is studied by experiment and calculation by use of the nominal diameter and number of specimenst:l prepared as indi- Table 1 Properties of samples  " ' Yamaguchi 8 ) reported that 20 specimens must be required for one point to obtain the confidence limit up to 95% by use of granite. Our study is for the time being is based on his results which might depend on the material kind.
KONA No. 3 (1985) cated in Table 2. Fracture experiments were carried out with Simadzu universal testing machine, Model REH-30, to obtain load-displacement variation which was presented on an X-Y recorder through differential transducer. The manufacturing procedure and its accuracy of the specimens, and the experimental method were fully described in the literatures 3 • 9 ).

Calculation of fracture energy
When the sphere of x in diameter is compressed between parallel plates as shown in Fig. 1, the elastic theory provides the displacement 6 as a function of the load Pas follows: where v is Poisson's ratio and Y is Young's modulus. For an overall or net energy for fracture, various definitions have been pro-posed4·5), though the final concept has been never obtained yet. In this paper, the term energy required for fracture is defined as elastic strain energy accumulated in a particle until the fracture will take place. That is: Accordingly, the fracture energy per unit mass E/M is given by Fig. 1 Crushing of sphere where p is the particle density. The compressive strength of spheres-Q-was presented by Hiramatsu el al. 2 ) as Substituting Eq. (4) into Eq. (3), E/M is determined from the compressive strength as follows: The strength of material, which is well known to be sensitive to its stracture 14 ) and thus dependent upon its size, is generally expressed as 1 • 7 ) (6) where S 0 is the strength of a specimen having a unit volume V 0 and m is Weibull's coefficient of uniformity with the value greater than 1. As shown in Eq. (6), the effect of specimen volume on strength decreases with increasing m, and the material becomes perfectly uniform in the limiting case as m = oo -Q--Q-By use of Eqs. (5) and (6), the fracture energy and the specific fracture energy at size x are rewritten respectively as follows: The ratio E/M increases over the finer region of x with a decrease in the coefficient m or the lack of uniformity of the material in question.

1 Experimental results of compressive strength of spheres and fracture energy
The typical values of E/M of borosilicate glass, quartz, feldspar, and marble plotted -tt The value S can be calculated under compressive load which is applied on points even if the shape of particles are irregular.  It is clear from Figs. 2 and 3 that the agreement between the experiment and the calculation seems to be satisfactory and thus these materials can be considered as semi-elastic solids. The similar tendency was also given for quartz glass. For marble, on the other hand, its experimental values are greater than the calculated line as indicated in Fig. 3. This reason was stated in the previous paper 12 ) as follows; the authors numerically integrated the load-displacement curve to obtain the whole fracture energy and determined the ratio of the energy required for plastic deformation to the whole energy by use of extrapolation. The calculated ratio in the case of marble was about 0.63, and this implies that the fracture energy measured would consist of the large amount of plastic deformation. From this reason, the experimental values might be greater than the calculated solutions.
For limestone and gypsum, the ratios were 0.55 and 0.78 respectively and both of their experimental results gave greater values than Summarizing the above discussion is that when the material has plastic characteristics, the fracture energy obtained experimentally is greater than the calculated solution. Therefore Eq. (5) will be available for estimating the fracture energy, only when the portion of plastic deformation energy and elastic strain energy can be determined in advance as stated in the previous paper 12 ).

2 Relationship between particle size and fracture energy
By use of Eqs. (7) and (8), Weibull's coefficient of uniformity m yields the fracture energy of a single particle E and the specific fracture energy E/M. A typical relationship between the specimen vdlume and the strength presented in the previous paper 13 ) is indicated in Fig. 4, which shows a set of line segments obtained by the least squares method. Such a relation was found to be expressed as a single straight line with a constant slope for glass materials and as a set of line segments with plural slopes for natural materials -c:r. In any case, however, the results imply that the strength will increase with KONA No. 3 ( 1985) decreasing particle volume or particle size. The calculated solutions of E and E/M by use of Eqs. (7) and (8) Fig. 7 Relationships between x and E/M orE with a particle size less than 500 J.lm, and this implies the large amount of energy required for producing fine particles. The results obtained in this work are based on the ideal fracture experiment under static, compressive load. It should be noticed that a practical fine comminution is accompanied by a fracture of irregular shaped particles under dynamic or impact load and the values v, Y and S in Eqs. (6), (7) and (8) are dependent upon the loading speed u). It should be also remarked that these results will be available for practical use only if the condition used is reasonably restricted.

Conclusion
The relationship between compressive strength of spheres and fracture energy was experimentally investigated under static compressive load by using 7 kinds of specimens of 0.5 to 3 em in diameter. Furthermore, the values of fracture energy per a single particle and unit mass of the particles with a diameter range of 0.0030 to 10 em were calculated by use of the relationship between particle volume (particle diameter) and strength which had been previously reported. Summarizing the results of this study: 1) The fracture energies were larger than the values calculated from the theoretical equations for plastic-like materials such as limestone, marble and gypsum, while these values were in reasonable agreement for quartz glass, borosilicate glass, quartz, and feldspar. 2) For natural materials, the specific fracture energies rapidly increased with decreasing particle size within the range of particle size smaller than about 500 J,Lm, and this implies the large amount of energy required for producing fine particles. Although the results obtained in this work are based on the ideal fracture behavior and seem to differ from the actual one, the resultant tendency may be reasonably available for practical use.