On the Question of Rock Cutting

V. I. Zvarygin, S. S. Sulakshin

(Presented by the scientific seminar of the Department of Prospecting Techniques)

Existing theories of rock cutting do not provide a sufficiently clear explanation of the difference between cutting along a free surface and cutting under face conditions. In some theories the horizontal force acting on the cutter is not considered at all. This often leads to inaccuracies in examining the problem.

The distribution of stresses in the rock when cutting on the surface of specimens and when cutting under face conditions is not of the same character. The most exact and convincing method for investigating this question, in our opinion, is the method of the theory of elasticity.

In the present article an attempt is made to explain the mechanism of rock fracture and to reveal the mathematical dependence between certain cutting parameters.

In the process of cutting rocks, the cutter shears off in front of itself certain volumes of rock. Moreover, during some interval of time of the cutter’s action these volumes have different magnitudes: at first the cutter shears off small elements, which increase toward the end of the cycle to a maximum.

The horizontal forces, depending on the spalling element, likewise have different magnitudes: from a minimum to a maximum. In connection with this, difficulties arise in calculating the cutting forces. For simplicity we shall calculate the forces only according to the maximum value of the horizontal force. This is justified by the fact that, when calculating cutters for strength, one should be guided by the upper limit.

Let us first consider the process of rock cutting under face conditions.

In the process of cutting rock, axial forces \(P\) act, tending to drive the cutter into the rock, and horizontal forces \(F\), which shear off the rock element. The axial and horizontal forces constitute the total resultant. The process of rock cutting may be considered as the penetration of the cutter as a result of the action upon it of the resultant \(R\); therefore, in the present case, for calculating the value of \(R\), the formula derived by us for penetration of the cutter into rock may be used:

\[ R = \frac{2\tau\sigma h(b + l)}{3\cos\alpha}, \tag{1} \]

here \(\sigma\) is the critical stress of the given rock under shear;

\[ h \]

is the depth of penetration of the cutter;

\[ b \]

is the width of the cutter;

\[ l \]

is a constant quantity;

\[ \alpha \]

is the cleavage angle of the rock element.

If the cutting process takes place along the free surface of the specimen, then the horizontal force may be regarded as a force acting on a rectangular wedge in a direction asymmetric (with respect to the axis of symmetry of the wedge) (Fig. 1, \(a\)—according to Frocht).

Fig. 1.

According to the theory of elasticity, in the case of an asymmetric direction of the applied forces acting on a wedge, in the latter the centers of the lines of equal stresses are located not along the direction of the applied force, but on a straight line displaced from the direction of the forces by a certain angle.

M. M. Frocht [1] calculated that, for the case of application of a load to a rectangular wedge with the direction of the force parallel to one of the faces of the wedge, the angle between the axis of symmetry of the wedge and the neutral axis (a straight line perpendicular to the direction of action of the force) is \(12^\circ 30'\) (Fig. 1, \(a\)). The straight line of the lines of equal stresses is displaced from the direction of the force by \(45^\circ - 12^\circ 30' = 32^\circ 30'\).

Thus, as a result of the action of the horizontal force, the stresses are distributed only over part of the sphere (Fig. 1, \(b\)), while as a result of the action of the axial and horizontal forces a complex picture is obtained

of equal stresses (Figs. 1, 2). Directly at the cutter, in the lower part, the lines of equal stresses are distributed analogously to the lines of equal stresses during penetration of the cutter. On the surface of the specimen, and at some distance, the isochromes are distributed according to the law of stress distribution under a load applied asymmetrically to a rectangular wedge. The total stress at point \(A\) (the point of chipping on the surface of the specimen) can be calculated on the basis of Boussinesq’s formula [2].

\[ \sigma=\frac{3P}{2\pi d^{2}}=\frac{3P}{2\pi d'd''}, \tag{2} \]

where \(d\) is the diameter of the sphere of stress distribution;

\(d'\) is the minor axis of the ellipsoidal surface of stress distribution;

\(d''\) is the conventional length of the major axis of the ellipsoidal surface of stress distribution in the absence of the influence of the free surface of the specimen.

Since, in cutting at a free surface, the stress distribution will occur over part of a sphere, the value of \(d'\) in the present case is replaced by the value \(d_1'\).

From Fig. 1, \(a\), it is seen that

\[ d_1'=\frac{d'}{2}-d_1=\frac{d'}{2}-d'\sin 32^\circ 30' \simeq 0.23d' \]

Substituting this value into formula (2), we obtain

\[ \sigma=\frac{3F}{2\pi\cdot 0.23d'd''}, \]

where \(F=P_0\) is the horizontal force acting on the cutter in the absence of the influence of the free surface.

Replacing \(d'\) and \(d''\) by values equal to them,

\[ d'=\frac{h}{\cos\alpha}, \qquad d''=b+l, \]

we obtain

\[ \sigma=\frac{3F\cos\alpha}{0.46\pi h(b+l)}=\frac{2F\cos\alpha}{h(b+l)}. \tag{3} \]

If we cut at some depth \(L\) from the surface, under the condition \(0.23(L+d_0)<d'\), then chipping of the rock element will evidently occur at the moment when the total stresses reach the surface of the specimen, and in this case

\[ d_2'=0.23(d_0+L)=0.23\left(\frac{h}{\cos\alpha}+L\right), \]

where \(L\) is the cutting depth (the depth of the groove at the bottom of which cutting takes place).

The stress in the case \(0.23(L+d_0)<d'\) will be equal to

\[ \sigma=\frac{2F}{\left(\dfrac{h}{\cos\alpha}+L\right)(b+l)}. \]

However, experimental data show that stresses propagate in the layer according to a somewhat different law and, consequently, in the first-

in the denominator’s second parenthesis one should add not the quantity \(L\), but the larger quantity \(kL\). Experiments have established that this quantity is \(1.25L\). Substituting this value instead of \(L\), we obtain

\[ \sigma = \frac{2F} {\left(\dfrac{h}{\cos \alpha}+1.25L\right)(b+l)} \tag{4} \]

or

\[ F = 0.5\left(\frac{h}{\cos \alpha}+1.25L\right)(b+l)\sigma . \tag{5} \]

To verify the equations obtained, experiments were carried out on cutting sandstones and siltstones on a special apparatus that recorded the horizontal and vertical forces, as well as the depth of penetration of the cutter when cutting at various depths from the surface of the specimen.

Fig. 2. Comparison of theoretical curves for the dependence of the cutter penetration depth on the cutting depth with experimental points.

In the graph (Fig. 2), the points show the experimental data, while the lines show the results of calculations by formulas (3), (4), and (5). The agreement between the theoretical and experimental data confirms the validity of the conclusions.

Suppose now that the cutting of the rock takes place along the edge of the specimen. Considering in this case the axial force as a force applied to a rectangular wedge in an asymmetric direction, on the basis of conclusions analogous to the preceding ones we obtain

\[ d_1''=\frac{d''}{2}-d_2''=\frac{d''}{2}-d''\sin 32^\circ 30' \simeq 0.23d''. \]

Substituting this expression into formula (1),

\[ \sigma=\frac{3F}{2\pi\,0.23d'\,0.23d''} =\frac{9.0F\cos\alpha}{h(b+l)} . \tag{6} \]

At some distance from the edge of the specimen \(L\) (Fig. 1, c), analogously to formula (4), we have

\[ \sigma=\frac{9F}{\left(\dfrac{h}{\cos\alpha}+\dfrac{5}{4}L\right)(b+l)} \tag{7} \]

or

\[ F=\frac{\left(\dfrac{h}{\cos\alpha}+\dfrac{5}{4}L\right)(b+l)}{9}. \tag{8} \]

To verify this formula, the materials of V. I. Solod on the cutting of anthracite [3] were used.

Fig. 3. Comparison of theoretical curves with experimental data:
a) dependence of cutting forces on the distance of cutting from the free lateral surface of the specimen: 1 — for \(b=4\) mm; 2 — for \(b=8\) mm; 3 — for \(b=16\) mm;
b) dependence of cutting forces on the depth of cutting of anthracite: 1 — for coarse cutters, 2 — for medium cutters.

In the graph (Fig. 3, a), the points denote experimental data; the lines were drawn on the basis of calculation by formulas (6), (7), (8).

To illustrate the foregoing, we present a graph constructed from the materials of E. Z. Pozin (Fig. 3, b), who carried out coal cutting directly at the face [4]. It is evident from the graph that, as the cutting depth increases, the cutting forces increase by a factor of 4, which corresponds to our calculations.

Moreover, it is evident from the graph that the first slotting cutters are under the most unfavorable conditions: the cutting forces acting on them are 4 times greater than those acting on the middle cutters. This also confirms our calculations.

Conclusions

1. When cutting rock along the surface and under face conditions, the forces acting on the cutter at the same depth of cutter penetration will be different. The cutter experiences minimal forces when cutting along the surface of the specimen, and maximal forces when cutting at the face. Moreover, the forces arising when cutting rock “at the face” are 4 times greater than the cutting forces along the surface of the specimen. Knowing the laws governing the distribution of forces on the cutter as the cutting depth increases, we can, on the basis of laboratory studies of cutting along a free surface, judge the character of the cutting process at the bottom of a borehole.

2. The cutters operating at the edge of the specimen are under the most favorable conditions. In this case, at the same depth of cutter penetration, the forces at the edge of the specimen are reduced by a factor of 4 in comparison with the forces of rock cutting at a sufficiently large distance from the edge.

REFERENCES

1. M. M. Frokht. Photoelasticity. Part II, GITTL, 1950.

2. N. I. Bezukhov. Theory of Elasticity and Plasticity. GITTL, 1963.

3. Scientific Transactions of the Moscow Mining Institute, Collection No. 21, Issue 1, 1957.

4. A. I. Beron, A. S. Kazanskii, B. M. Leibov, E. Z. Pozin. Coal Cutting. Gosgortekhizdat, 1962.