**3. Wear mechanism that is formed during the use of circular saw, grinding mills, grinding stones**

Circular saws is the cutting tool that is used the most in cutting and sizing of natural stones that are segments containing diamonds' welded around circular metal body. Grinding mills are used for process such as cutting, graving, shaping… etc. In the abrading operation mentioned until here, the process of wear results from simple geometrical situations with a linear movement between abrasive grain and material. When grinding mills are analyzed, if abrading operation occurs on the edge of grinding mill, the situation is simple. But if abrading is on the disc, the situation is complex. In the literature, abrading mechanism that occurs here is mentioned with the word grinding. The wear that occurs here is defined as a micro-scaled grinding.

Salmon (1992) tried to make a mathematical modeling for the abrading operation on the surface of grinding mill.

Situation of grinding mill during the operation is geometrically shown in Figure 7.

Theories on Rock Cutting, Grinding and Polishing Mechanisms 193

According to the geometrical structure in the figure;

Calculated as such:

grains is calculated as:

be:

*K Db C* 

Figure 9.

0.5 ( / ) (4 / ) *w s t d D V DV Crt*

As d, is at a small value according to D,

If abrasive grain is represented as such (Figure 8);

**Figure 8.** Frontal view of the presented grain (Salmon, 1992)

. . '. equation turns to *K DrtC*

is reached. . . *V D <sup>s</sup>*

0.5 2 2 *AB D D d* ( / 2) ( / 2 )

0.5 *AB d D d* ( )

Cutting arc is accepted to be a straight line and the mill -along a peripheral line- that contain K amount of abrasive grain, proceeds as much as f with 1/K turn. Cutting depth of abrasive

0.5 *t f Sin f AB D f d D d D* ( ) ( ) / ( / 2) 2 ( ) /

0.5 2 ( / ) /( ) *<sup>w</sup> t f d D f V Kw*

Number of abrasive grains along a peripheral line ise determined with the equation below.

*K Db C* . . '.

t/2 is used as average grain cutting depth. So, the ratio of grain width to cutting depth will

*r bt* 2 '/

 so;

Cutting geometry when abrasive grain is considered, cutting geometry is presented in

. /2 . If this is put in Equation 7;

<sup>2</sup> (4 / ) *w s t V d V Crl* (8)

0.5 ( 2 )( ) // *<sup>w</sup> t V Kw d D can be wri n tte* (7)

**Figure 7.** Geometric presentation of wear produced during grinding mill application (Salmon, 1992)

Symbols that are shown here and that will be used hereafter is shown below:


Vw: Feedrate of material

K: Number of abrasive grains along a peripheral line


Salmon (1992) made these approaches;

Length of cutting trajectory can be determined as below:

$$\begin{aligned} \mathbf{d}^2 &= \mathbf{D}^2 \,/\, 4 - \left(\mathbf{D} \,/\, 2 - d\right)^2 + d^2 \\\\ &= \mathbf{D} \,. \mathbf{d} \\\\ \mathbf{l} &= \left(\mathbf{D} \,. \mathbf{d}\right)^{0.5} \end{aligned} \tag{6}$$

According to the geometrical structure in the figure;

$$AB = \left( (D/2)^2 - (D/2 - d)^2 \right)^{0.5}$$

Calculated as such:

192 Tribology in Engineering

Situation of grinding mill during the operation is geometrically shown in Figure 7.

**Figure 7.** Geometric presentation of wear produced during grinding mill application (Salmon, 1992)

2 2 2 2 *lD D d d* /4 ( /2 )

*D d*.

0.5 *l Dd* ( .) (6)

Symbols that are shown here and that will be used hereafter is shown below:

D: Diameter of grinding mill (mm)

Vw: Feedrate of material

l: Length of cutting trajectory

Vs: Peripheral speed of grinding mill (m/s) w: Angular speed of grinding mill (rad/s)

C: Number of active abrasive grains per unit t: Cutting depth of abrasive grains (mm)

d: Cutting depth of grinding mill (mm) b': Theoretical width of each grain (mm) b: Cutting width of abrasive grain (mm) Salmon (1992) made these approaches;

K: Number of abrasive grains along a peripheral line

Length of cutting trajectory can be determined as below:

$$AB = \left(d(D-d)\right)^{0.5}$$

Cutting arc is accepted to be a straight line and the mill -along a peripheral line- that contain K amount of abrasive grain, proceeds as much as f with 1/K turn. Cutting depth of abrasive grains is calculated as:

$$t = f\left(\operatorname{Sim}\theta\right) = f\left(AB\right) / \left(D \mid \mathcal{D}\right) = \mathcal{D}f\left(d(D-d)\right)^{0.5} / D^2$$

As d, is at a small value according to D,

$$t = 2 \, f \, (d \, / \, D)^{0.5} \quad f = V\_w \, / \, \text{(Kw)}$$

$$t = (2V\_w \, / \, \text{Kw}) (d \, / \, D)^{0.5} \quad \text{can be } written \tag{7}$$

Number of abrasive grains along a peripheral line ise determined with the equation below.

$$K = \pi .D \, b^\circ \, C$$

If abrasive grain is represented as such (Figure 8);

**Figure 8.** Frontal view of the presented grain (Salmon, 1992)

t/2 is used as average grain cutting depth. So, the ratio of grain width to cutting depth will be:

$$r = 2b^\prime/t$$

*K Db C* . . '. equation turns to *K DrtC* . /2 . If this is put in Equation 7;

0.5 ( / ) (4 / ) *w s t d D V DV Crt* is reached. . . *V D <sup>s</sup>* so;

$$dt^2 = \left(4V\_w d / V\_s \text{Crl}\right) \tag{8}$$

Cutting geometry when abrasive grain is considered, cutting geometry is presented in Figure 9.

Theories on Rock Cutting, Grinding and Polishing Mechanisms 195

According to Salmon (1992), it is possible to solve wear problems in grinding mill

In terms of energy need, in order to remove material from the surface, the most efficient phase is this cutting phase. Minimum specific energy is used in this way. Here, specific energy is the energy that is needed for removing unit material form the surface and unit is

According to Salmon (1992), energy used during abrading in which chip is shaped can occur

Another approach for grinding mills is developed by Chen and Rowe (1996). According to Chen and Rowe (1996), when a abrasive grain on the surface of a moving chip is thought; firstly, abrasive grain combines on the material with a narrow curve. In this way, more material is removed. Secondly, productive contact point of the surface of abrading changes

So, as can be seen in Figure 10, while abrasive grain made "ploughing" at the beginning of

**Figure 10.** Phases of chip formation on the edge to grinding mill (Chen and Rowe, 1996)

applications and present alternative solutions.

joule/mm3 or Btu/in3.

 Heating on working material, Heat occurs at grinding unit, Heat that occurs at chips, Kinetic energy at chips, Radiation diffused around,

Energy spent on producing new surface,

as long as it moves on this contact arc.

this arc, the rest of is can make cutting.

Residual stress in the surface and chip's lattice structure.

in these ways:

**Figure 9.** Cutting geometry of a abrasive grain. (Salmon, 1992)

Here;

b: Cutting width (mm)

t: Thickness of chip that is not deformed (mm)

u: Specific energy (Jmm-3);

For cutting operation at one point;

$$u = F\_c V\_c / b + V\_c$$

$$\text{Cutting Force} = F\_c - \text{ubt}$$

Cutting force in abrading is used as tangential force. So, equation is arranges as s u u' F V /V bd .

Cutting Force = Fcu'bdVu/Vs

Work done in unit of time: u'Vubd

Number of abrasive grain working at unit of time: VsCb

Work done by one single grain = u'Vud / VsC

Average force affecting one single grain F" is calculated by dividing the work done by one grain into the length of cutting arc.

$$F'' = \text{\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\mathbin{g}} \text{\tiny\mathbin{g}} \text{\tiny\mathbin{g}} \text{\tiny\mathbin{g}} \text{\tiny\mathbin{g}} \text{\tiny\phantom{\Lambda}} \text{\tiny\phantom{\Lambda}} \text{\tiny\phantom{\Lambda}} \text{\tiny\phantom{\Lambda}} \text{\tiny\phantom{\Lambda}} \text{\tiny\phantom{\Lambda}} \text{\small} \text{\small} \tag{9}$$

From Equation 8:

<sup>2</sup> 2 4 / 4 / *s s t Vud V Crl l Vud V Cr t*

When l is put into the place in Equation 9,

$$F'' = \mathfrak{u}'\mathfrak{t}^2 r \quad \text{/} \text{ 4} \tag{10}$$

is obtained.

According to Salmon (1992), it is possible to solve wear problems in grinding mill applications and present alternative solutions.

In terms of energy need, in order to remove material from the surface, the most efficient phase is this cutting phase. Minimum specific energy is used in this way. Here, specific energy is the energy that is needed for removing unit material form the surface and unit is joule/mm3 or Btu/in3.

According to Salmon (1992), energy used during abrading in which chip is shaped can occur in these ways:


194 Tribology in Engineering

Here;

b: Cutting width (mm)

s u u' F V /V bd .

From Equation 8:

is obtained.

Cutting Force = Fcu'bdVu/Vs

Work done in unit of time: u'Vubd

grain into the length of cutting arc.

u: Specific energy (Jmm-3);

For cutting operation at one point;

**Figure 9.** Cutting geometry of a abrasive grain. (Salmon, 1992)

Number of abrasive grain working at unit of time: VsCb

Work done by one single grain = u'Vud / VsC

When l is put into the place in Equation 9,

/ *cc c u FV b V*

Cutting Force *<sup>c</sup> F ubt*

Cutting force in abrading is used as tangential force. So, equation is arranges as

Average force affecting one single grain F" is calculated by dividing the work done by one

<sup>2</sup> 2 4 / 4 / *s s t Vud V Crl l Vud V Cr t*

" ' / *<sup>s</sup> F u Vud V Cl* (9)

<sup>2</sup> *F utr* " ' / 4 (10)

t: Thickness of chip that is not deformed (mm)


Another approach for grinding mills is developed by Chen and Rowe (1996). According to Chen and Rowe (1996), when a abrasive grain on the surface of a moving chip is thought; firstly, abrasive grain combines on the material with a narrow curve. In this way, more material is removed. Secondly, productive contact point of the surface of abrading changes as long as it moves on this contact arc.

So, as can be seen in Figure 10, while abrasive grain made "ploughing" at the beginning of this arc, the rest of is can make cutting.

**Figure 10.** Phases of chip formation on the edge to grinding mill (Chen and Rowe, 1996)

In Figure 10, production of chip by grain on grinding mill during the movement of chip is seen. Cutting arc length of grain is shown with lk, thickest chip thickness that hasn't changed shape is shown with hm, tangential turning speed of mill is represented with Vs, progress speed of processed material is represented with Vw, cutting depth is represented with a.

**Figure 11.** Behavior of circular grain in abrading (Chen and Rowe, 1996)

Movement of a circular grain on the edge of mill during abrading is shown in Figure 11. Cutting depth of grain in here is represented with t, amount of pressure with R, its horizontal component with Ft, vertical component with Fn, cross sectional area of chip that hasn't gone under any change with A, diameter of circular area which is the section of this on the surface with b, angle between power of pressure and vertical with . Pressure force affecting grain is represented with

$$R = \pi b \,\, H(\text{C}^\prime/\text{\textdegree C}) \tag{11}$$

Theories on Rock Cutting, Grinding and Polishing Mechanisms 197

(15)

(16)

When friction force is taken as µRCos, specific energy in friction is obtained as;

4 3 *<sup>g</sup>*

*t* 

total specific energy for grain is formulized as below.

literature about grinding mills.

**saw** 

3 ' 4 3 *<sup>f</sup> b C e H Cos t*

*b C e H Sin Cos*

In parallel with common use of grinding mills in the industry, there are many studies in the

**4. Wear mechanism formed in the process of cutting with diamond frame** 

Generally rock cutting mechanism is explained by the formation of indentation with plastic deformation and breaking mechanism of rock. When cutting depth of diamond is deep enough to produce visible cracks on a rock, breakages occur and chips are formed as a result of this. As can be seen schematically in Figure 12, there is a plastic deformation under the channel that is produced by the tangential movement of abrasive grain along the surface and there are two main crack systems named radial and lateral that are produced from this zone. Radial cracks are formed with wedge wear type when high normal force is used and when this force is removed, these cracks can continue to spread because of permanent tensile stress at the edge of crack. Lateral cracks start to be formed when he force is removed

and can continue to spread with the effect of permanent tension (Konstanty, 2002).

**Figure 12.** Schematic view of plastic deformation zone formed during cutting (Konstanty, 2002)

3 '

 

Here C is the strain factor defined as the rate of average pressure affecting contact area to normal stress. Necessary specific energy is defined as;

$$e\_c = F\_t / A \tag{12}$$

When <sup>4</sup> 3 *A bt* , if necessary specific energy is used for cutting,

$$e\_{cc} = \frac{3RS\sin\theta}{4bt} \tag{13}$$

When is put in the equation, specific energy is calculated as below:

$$e\_{cc} = \frac{3\pi}{4} \frac{b}{t} H\left(\frac{C}{3}\right) \text{Sim}\theta\tag{14}$$

When friction force is taken as µRCos, specific energy in friction is obtained as;

$$e\_f = \frac{3\pi}{4} \mu \frac{b}{t} H\left(\frac{C}{3}\right) \text{Cov}\theta \tag{15}$$

total specific energy for grain is formulized as below.

196 Tribology in Engineering

In Figure 10, production of chip by grain on grinding mill during the movement of chip is seen. Cutting arc length of grain is shown with lk, thickest chip thickness that hasn't changed shape is shown with hm, tangential turning speed of mill is represented with Vs, progress speed of processed material is represented with Vw, cutting depth is represented with a.

Movement of a circular grain on the edge of mill during abrading is shown in Figure 11. Cutting depth of grain in here is represented with t, amount of pressure with R, its horizontal component with Ft, vertical component with Fn, cross sectional area of chip that hasn't gone under any change with A, diameter of circular area which is the section of this on the surface with b, angle between power of pressure and vertical with . Pressure force

'

( / 3) (11)

/ *c t e FA* (12)

(13)

(14)

 *R bHC* 

Here C is the strain factor defined as the rate of average pressure affecting contact area to

3 4 *cc*

3 ' 4 3 *cc b C e H Sin t* 

*e*

*RSin*

*bt* 

**Figure 11.** Behavior of circular grain in abrading (Chen and Rowe, 1996)

normal stress. Necessary specific energy is defined as;

*A bt* , if necessary specific energy is used for cutting,

When is put in the equation, specific energy is calculated as below:

affecting grain is represented with

When <sup>4</sup>

3

$$e\_g = \frac{3\pi}{4} \frac{b}{t} H\left(\frac{C}{3}\right) \left(\text{Sin}\theta + \mu \text{Cos}\theta\right) \tag{16}$$

In parallel with common use of grinding mills in the industry, there are many studies in the literature about grinding mills.
