## 3 Results and Discussion
### Correlation between Material Properties and Breakage Rate Parameters
#### 3.5.1 Simple Regression Analysis
It is seen that weak correlations exist between \(x_{m}\) and \(V_{p}\) (R\({}^{2}\) = 0.08) as well as \(E_{t}\) (R\({}^{2}\) = 0.09) when inverse functions are used. Furthermore, similar results between \(x_{m}\) and \(R_{L}\) or \(UCS\) are also obtained, with R\({}^{2}\) 0.35 and 0.57, respectively. It is mentioned that very good to excellent correlations between optimum size \(x_{m}\) and other variables are obtained, if marble is excluded from regression analysis. Thus, it is deduced that the optimum size \(x_{m}\) increases with decreasing P-wave velocity, Schmidt rebound value, uniaxial compressive strength and tangent modulus of elasticity, for all materials tested except for marble. It is therefore obvious that as strength or stiffness of a material increases the optimum size at which the breakage rate obtains its maximum value becomes finer. Figure 12: Correlations between parameter \(\alpha\) and (**a**) P-wave velocity \(V_{p}\), (**b**) Schmidt rebound values \(R_{L}\), (**c**) uniaxial compressive strength \(UCS\) and (**d**) tangent modulus of elasticity \(E_{t}\); horizontal error bars show standard deviation of the measurements. Figure 14: Correlations between optimum size \(x_{m}\) and (**a**) P-wave velocity \(V_{p}\), (**b**) Schmidt rebound values \(R_{L}\), (**c**) uniaxial compressive strength \(UCS\) and (**d**) tangent modulus of elasticity \(E_{t}\); horizontal error bars show standard deviation of the measurements. Figure 13: Correlations between maximum breakage rate \(S_{m}\) and (**a**) P-wave velocity \(V_{p}\), (**b**) Schmidt rebound values \(R_{L}\), (**c**) uniaxial compressive strength \(UCS\) and (**d**) tangent modulus of elasticity \(E_{t}\); horizontal error bars show standard deviation of the measurements. The results for marble are indeed peculiar. It is known that the behavior of a material during breakage depends on several factors including mainly rock size, grain size and microstructure, crystallinity, porosity, mixtures of grinding media, presence of secondary mineral phases and microcracks [26; 60; 61]. In addition, breakage of mineral grains may be affected by the presence of inter-granular cementations [62]. Furthermore, the energy required for plastic deformation is different for each material and this affects breakage rate [63]. In a recent study, a spheroid model was used to identify the role of shape in particle size analysis [64]. One major difference between marble and the other three materials examined in our study is that the formation of marble is the result of metamorphism of sedimentary carbonate rocks, mainly limestones or dolomites. Twinning planes (planes along which two crystals meet) are also present in marbles. Furthermore, during grinding of marble ultra-fine particles are generated and this may affect breakage rate and final particle size distribution. In a recent study, it has been shown that the initial breakage rate of coarse particles increases with increasing proportion of fines in the mixture; the presence of a big proportion of fines indicates that they exhibit a much smaller collision cross-section compared to coarser particles [65]. All these factors definitely affect the behavior of marble during breakage and thus additional experimental studies are required to elucidate this aspect.