An Investigation of the E ﬀ ect of Toughness and Brittleness Indexes on Ampere Consumption and Wear Rate of a Circular Diamond Saw

The circular diamond saw (CDS) is one of the major sawing machines in dimension stone processing plants. Predicting the performance of a circular diamond saw is very important to estimate the cost and the planning of the stone processing plants. The performance of a CDS depends on some important factors such as machine and tool characteristics, physical and mechanical characteristics of rock and tool wear rate. In this research, it is aimed to investigate the relationship between ampere consumption, brittleness indexes and toughness with the wear rate of a CDS. This aim was pursued by using a fully instrumented cutting rig to cut 14 types of hard rock at constant feed rates, cut depths and peripheral speeds. Wear rate, toughness and brittleness indexes were evaluated using simple and multiple curvilinear regression analysis and predicted models were developed. The results indicated that there is a signi ﬁ cant correlation between wear rate, ampere consumption and toughness. It is concluded that, the wear rate of a CDS can be reliably estimated using a multiple curvilinear model which includes ampere consumption and toughness.


Introduction
The prediction of dimension stone sawability is one of the most important factors in estimating the production cost and planning of the quarries and factories. Dimension stone sawability with a CDS dependent on parameters including; the rock and machine characteristics and saw operating characteristics. The investigation of these parameters in the stone industry is important for establishing the most suitable and economic usage of the sawing method in the future. Many studies have been done on the relations between a circular diamond saw (CDS)  The main goal of this research is to study the possibility of estimating the wear rate of a circular diamond saw from brittleness indexes and toughness in hard rock cutting process. First of all, the wear rate of a CDS was correlated with brittleness indexes and toughness using simple regression and then, multiple regression analysis was performed.

Toughness
Toughness refers to a material's ability to deform plastically and thus absorb the applied energy. To measure this property, a sample of the material must be subjected to a static test to acquire its characteristic stressstrain curve. The work done to fracture the sample can then be obtained by measuring the area under this curve. The volume-specifi c toughness is known as modulus of toughness and refers to the maximum work done (in inch-pounds) to rupture a unit volume (in cubic inch) of the material. For materials like cast iron and concrete, which have a parabolic shaped stress-strain curve, an accurate estimate of modulus of toughness, Mt, is given by Equation 1: (1) Where, σ a is the ultimate strength and ε f is the strain at failure (Deer and Miller 1966).
Tough materials have high strength and ductility while brittle materials have a generally low toughness due to their inability to absorb energy by undergoing plastic deformation (Jastrzebski 1959). The toughness of a material or rock is in fact the ability of its matrix to bind its constituent minerals and grains together but also depends on the strength of individual grains or minerals (Deer and Miller 1966). Thus, rocks with a strong matrix consisting of strong minerals are expected to be the toughest (Shepherd 1951). From a microscale point of view, it is the type and strength of binding forces between atoms, ions, or molecules that decide the hardness and toughness of a material. From a macroscale perspective, both hardness and toughness are closely correlated with the yield strength (Jastrzcbski 1959). One of the earliest toughness classifi cation systems was introduced in 1926 by Harley. In this system, toughness was expressed as the ft-lbs. of work that must be done to drill one cubic inch of rock, and was related to a grinding resistance that could be determined using a small grinding machine.

Brittleness indexes
Another important mechanical property of rocks is brittleness, which has been generally defi ned as the property of materials that rupture or fracture without any measurable plastic deformation (Glossary of Geology and Related Sciences 1960). However, alternative definitions of brittleness have also been introduced by researchers of different backgrounds for different applications (Yarali and Soyer 2011). The notable defi nitions of brittleness are the lack of ductility (Hetenyi 1966), the defi ciency in internal cohesion leading to an easy fracture (Ramsey 1967), and the property of materials such as cast iron and many rocks that fracture at or only slightly beyond the yield stress (Obert and Duvall 1967). As mentioned, brittleness can be described as a property of materials that rupture or fracture without undergoing measurable if any plastic deformation. The related literature contains a number of stress-strain curve based defi nitions for brittleness index (Baron 1962 (2) Where, B1, B2, B3 and B4 are brittleness, σ c is the uniaxial compressive strength (MPa), σ t is the Brazilian tensile strength (Mpa).

Methodology of study
In this study, the relationship of wear rate of a CDS with ampere consumption, brittleness indexes and toughness is investigated. This goal is pursued by using a fully instrumented cutting rig with a maximum spindle motor power of 4 kW and spindle speed of 3000 rpm to cut 14 types of hard rock at constant feed rates, cut depths and peripheral speeds, and measuring the uniaxial compressive strength (UCS), Brazilian tensile strength (BTS), and Young's modulus (YM) as suggested by (ISRM 1981). To determine the UCS, the fi ve standard NX core samples (length to diameter ratio of 2.5:1) were taken using a diamond rotating drill from a block sample. The mechanical tests were carried out by a servo controlled testing machine designed for rock test. The standard uniaxial compressive strength test of core samples was carried out under a loading rate of 1 Mpa/s. The tangent Young's modulus at a stress level equal to 50% of the ultimate UCS is used in this study.
The circular diamond saw used in this study was 250 mm in diameter and had a 50 mm thick steel core with a standard narrow radial slot and 18 pieces of 35 mm×2.5 mm×6.0 mm diamond impregnated segments fi xed around its periphery, giving a grit size of about 50/60 US mesh at 35 percent concentration. In the course of this study, fourteen rock specimens were cut by the described saw. In the cutting process for each sample, the length of cutting was 120 cm in three steps and the cutting time was 60 seconds at each cutting step.
During the cutting test, ampere consumption was monitored by ampere meter. A digital ampere meter installed on the cutting machine was used to measure ampere consumption. For each sample, the average electrical current consumption was calculated in terms of ampere. The wear rate of the circular diamond saw was determined based on one segment. The measurement was taken for a selected segment, at the beginning and end of each cutting test. Changes of segment height, width and length of the circular diamond saw were measured for determination of the wear rate by a digital micrometer (resolution of 0.001mm). During the sawing trials, the cutting operating parameters such as periph-eral speed (3000 rpm), feed rate (0.67 cm/s), and depth of cut (0.5 cm) were considered constant. The results of laboratory study are given in Table 1.
Wear rate, ampere consumption, toughness and brittleness indexes were evaluated using simple and multi-

Simple regression
In this statistical analysis, the wear rate of the circular diamond saw, toughness and brittleness indexes were correlated by the least squares regression method. We tested the linear, logarithmic, exponential and power regressions and determined, for each regression, the approximation equation that yields the highest correlation coefficient. The wear rate versus type of brittleness are shown in Figures 2 to 5. Figure 6 shows the relationship between the wear rate and toughness. Equations 6 to 9 express the relationship between the wear rate and type of brittleness, and Equation 10 expresses dependence of wear rate on toughness. These equations are presented by Equations 6 to 10: Where Wr is the wear rate of the diamond saw, mm3, B1, B2, B3 and B4 are brittleness indexes.

Multiple curvilinear regression analysis
In this section, the wear rate of the diamond saw at different brittleness indexes, toughness and ampere consumption were analysed using multiple curvilinear regression. Regression analysis was performed by the computing software "Statistical Package for the Social Sciences (SPSS)". The regression models including two independent variables are presented in Equations 11 to 15: Where I is ampere consumption, A.
Equations 11 to 15 were derived from B1, B2, B3, B4, toughness and I. the wear rate of the circular diamond saw increases with an increase in the ampere consumption. The ampere consumption is easy to detect during the sawing process, so it can be used in the stone processing plants to predict the wear rate in different sawing conditions.

Validation of models
Validation of the developed models was performed while considering the F test, the t test, correlation coefficient (R 2 ), and the plots of predicted values versus the actual values. The statistical results of the fi ve models are given in Table 2.
The R 2 of Equations 11 to 15 are 0.56, 0.56, 0.57, 0.57, and 0.58, respectively. These values are fair, however, they show better values in comparison with Equations 6 to 10. The t test was used to determine the sig-nifi cance of R 2 . For the use of this test, the tabulated t value (t*) according to the confi dence level is compared with the computed t value to accept or reject the null hypothesis. A computed t value that is greater than the tabulated t value disproves the null hypothesis; in this case, R² is important. Otherwise, the null hypothesis is not rejected, and R 2 is not signifi cant. With a 90% confidence level, the tabulated t value ±1.363 for equations

Conclusion
The circular diamond saw is one of the most important machines used in stone processing plants. The performance prediction of these saws is important in the cost estimation and the planning of the plants. An accurate estimation of wear rate helps to make the planning of the rock sawing programs more effectual. In this study, the relationship between wear rate of a circular diamond saw, toughness, and brittleness indexes of hard rocks were investigated using simple and multiple curvilinear regression analysis. According to the results of simple regression analysis, B3, B4, and toughness indexes show a better correlation with the wear rate. For practical considerations, ampere consumption was measured during the sawing process. Some models were obtained using multiple curvilinear regression with respect to ampere consumption, toughness, and brittleness indexes. Finally, validation of the developed models were analysed while considering the F test, the t test, correlation coefficient, and the plots of predicted values versus actual values and the best model was selected. The results showed that the wear rate of a CDS can reliably be predicted from ampere consumption and toughness. It is important to emphasize, the developed model can be applied only on granite rocks. More research must be done to check the validity of the obtained model for other types of rocks and the effect of the type of saw.