Some Remarks on the Theory of Rock Cutting

V. I. Zvarygin

(Presented by the scientific seminar of the Department of Exploration Engineering)

Many researchers regard the destruction of rocks during drilling as a process consisting of two heterogeneous processes acting one after the other: indentation and shearing. Therefore, in theoretical studies the influence of axial \((P)\) and circumferential \((F)\) forces is considered either separately from one another [3], or one of the forces is not taken into account at all [1, 4].

But since the forces under consideration act on the cutter simultaneously, in our opinion they cannot be separated (i.e., treated as the action first of the axial force and then of the circumferential force). In that case the cutting process may be regarded as the continuous penetration of a moving wedge directed at an angle into the rock being destroyed. Shearing, then, will be only a consequence of the indentation of the wedge. The wedge faces will be the front face and the wear flat of the cutter (the cutting edge). The forces \(P\) and \(F\), equal to the forces of rock resistance and to the friction forces along the faces of the wedge (cutter), constitute the resultant \(R\).

It is also important to note that destruction is a very complex process. The stress distribution in the rock has a complex three-dimensional character; the shearing of rock particles occurs along a curvilinear surface that is difficult to define mathematically, and so on. Therefore, for theoretical studies, in our opinion, one should use the theory of elasticity, which “sets as its aim the solution of the problem as accurately as possible” [6]. The use of the theory of elasticity will make it possible to avoid a number of inaccuracies in determining the forces and depth of cutter penetration required for rock destruction, and, when the optical method of elasticity is used, to reveal more correctly the mechanism of rock destruction. The question of applying the theory of elasticity in theoretical studies in drilling is not being raised for the first time. Yu. M. Pariiskii [5], using formulas of the theory of elasticity, constructed lines of equal principal and reduced stresses for static indentation of a punch and for indentation of a punch during its displacement. However, it does not appear possible to use the equal-stress patterns obtained by Yu. M. Pariiskii for the analysis of cutting conditions. First, he used formulas for the plane problem of the theory of elasticity (whereas in cutting a spatial problem must be considered); second, he did not

the forces of resistance of the rock and the friction forces along the front face of the cutter, which occurs during cutting, have been taken into account.

In the present article an attempt is made, using the theory of elasticity and the photoelasticity method, to reveal the mechanism of rock failure and to determine the magnitude and character of the stress distribution in the rock in the vicinity of the applied load. Knowing the magnitude of the stresses at any point of the body, it is easy to calculate the forces required for cutting and the depth of penetration (the latter will be considered in the following article).

Turning to the consideration of the problem, let us note the following important point.

In the process of cutting rock, the magnitude of the contact of the cutter with the rock gradually changes owing to the increase in the wear area. In fact, the cutter in this case may have various shapes: a double-sided wedge, a single-sided wedge, a rectangular shape, an oval, etc.; consequently, the contact area will also have the corresponding shape.

Photography and analysis of isochromatics obtained in polarized light during the indentation, into cubes of Plexiglas and epoxy resin, of cutters of various shapes show that the character of the stress distribution for different cutter shapes (a single-sided wedge with an included angle \(>60^\circ\); a double-sided wedge with an included angle \(>120^\circ\); a cutter of rectangular shape) is the same.

The character of the distribution of tangential stresses depends only on the so-called reduced contact area of the cutter with the rock, i.e., the area obtained by cutting the cutter with a plane perpendicular to the resultant forces on the cutter in the contact zone. In the present case this is the section of the cutter by the surface of the specimen. In cutting, the reduced area will be the projection of the contact areas along the front and rear faces of the cutter onto a plane perpendicular to the resultant. The lines of equal maximum tangential stresses and the lines of equal total stresses (tangential \(+\) normal) have an analogous form; therefore, from the character of the distribution of tangential stresses we can judge the character of the distribution of total stresses. Consequently, the distribution of total stresses does not depend on the shape of the cutter, but depends on the reduced area.

Taking into account (because of the small magnitude of the contact) that the intensity of the load \(P\) over the entire reduced contact area is uniform, it is easy to calculate the total stress at any point of the infinite half-space on which the cutter acts.

To calculate the stresses at a point, we shall use the formula of the classical problem of the theory of elasticity. The total stress \(\sigma_e\), obtained as a result of the action of a concentrated force on the surface bounding an infinite half-space, at any point \(A\), is equal to

\[ \sigma_e=-\frac{3P\cos^2\Theta}{2\pi l^2}, \tag{1} \]

where \(P\) is the concentrated force,
\(\Theta\) and \(l\) are polar coordinates.

If on the reduced contact area (of the cutter with the rock) \(S\) we select an elementary area \(dS\) (Fig. 1), then the load acting on it will be

\[ p=\frac{R\,dS}{S}. \]

From the drawing it is evident that

\[ dS=dx\cdot dy=\frac{l\cos\alpha\,d\beta\cdot l\,d\alpha}{\cos\beta\cdot\cos\alpha} =\frac{l^2\,d\alpha\cdot d\beta}{\cos\beta}. \]

Then

\[ p=\frac{R l' \, d\alpha \cdot d\beta}{S\cos \xi}. \]

Further, according to the law of cosines,

\[ \cos \Theta=\frac{AO^2+OD^2-AD^2}{2\,AO\cdot OD}. \]

Fig. 1.

substituting the values

\[ AO=\frac{OC}{\cos \alpha}; \qquad OD=\frac{OC}{\cos \beta}; \qquad AD=OC\sqrt{\tan^2\alpha+\tan^2\beta}, \]

we obtain

\[ \cos \Theta=\cos \alpha \cdot \cos \beta. \]

Substituting the values of \(P\) and \(\cos \Theta\) into formula (1), we then obtain, at point \(A\), the total stress \(d\sigma_e\) from the elementary load \(p\) acting on the area \(dS\):

\[ d\sigma_e=-\frac{3R\cos^2\alpha\cdot \cos\beta\, d\alpha\cdot d\beta}{2\pi S}. \]

The total stress at point \(A\) will be

\[ \sigma_e=-\frac{3R}{2\pi S}\int_{\alpha_1}^{\alpha_2}\cos^2\alpha\,d\alpha \int_{\beta_1}^{\beta_2}\cos\beta\,d\beta, \]

\[ \sigma_e=-\frac{3R}{4\pi S} \left(\alpha_2-\alpha_1+\sin\alpha_2\cdot\cos\alpha_2-\sin\alpha_1\cdot\cos\alpha_1\right) \left(\sin\beta_2-\sin\beta_1\right). \tag{2} \]

Using this formula, assigning values of $\beta_2$ and calculating the values of $\beta_1$ (Table 1), we construct a diagram of the distribution of equal total stresses in the plane of action of the resultant, perpendicular to the motion of the cutter, which divides the reduced contact area into 2 equal parts (Fig. 2, a).

Table 1

Table 2

Similarly to the preceding, having calculated the values of $\alpha_1,\beta$ for the assigned $\alpha_2$ (Table 2), we construct lines of equal stresses in the plane of action of the resultant along the motion of the cutter (Fig. 2, b).

But the arguments presented will be valid only for the pair: cutter—infinite half-space, without taking into account the influence of the free surface, i.e., cutting under bottom-hole conditions, when the upper exposed surface is insignificant. When cutting a rock specimen at the surface, the latter will substantially distort the stress distribution.

It is evident that the stresses propagating from the direction of the resultant into the depth of the specimen will obey formula (2), while the stresses emerging at the surface of the specimen will be distributed according to the formula of the plane problem of the theory of elasticity

\[ z_e=-\frac{2P}{\pi}\cdot\frac{\cos\Theta}{l}. \tag{3} \]

Or, replacing the values $P$ and $\Theta$ by values equal to them (analogously to those given above),

\[ p=\frac{Rld\beta}{bt\cos\beta};\qquad \cos\Theta=\cos\beta, \]

we obtain:

\[ dz_e=-\frac{2Rl\cos\beta\,d\beta}{\pi lbt\cos\beta} =-\frac{2R\,d\beta}{\pi bt}, \]

\[ z_e=-\frac{R}{\pi bt}(\beta_2-\beta_1). \tag{4} \]

The lines of equal stresses constructed from these equations have the form shown in Fig. 2, v.

Further, if we suppose that, at a certain value of the forces, a rock element will chip along some curve $AB$ (Fig. 2), then the projections of the lines of equal stresses $\sigma_e$ onto the surface of the specimen at the moment of chipping will have an ellipsoidal form, shown in ...

Fig. 2, d. Moreover, depending on the direction of \(R\), the eccentricity may be equal to from 0 to 1. Comparison of the constructed curves with the experimental ones, obtained by photographing in polarized light the stresses during indentation of cutters into specimens of Plexiglas, epoxy resin, and glass, shows their analogy.

Fig. 2.

After a chip of material is split off during the cutting of Plexiglas and glass (and also during indentation of a cutter into the indicated material), traces of the development of deformations remain in the specimen. Comparison of the deformation traces with the constructed lines also indicates their similarity.

All this confirms the correctness of the theoretical conclusions and the idea that the deformations developing in a solid body are a consequence of stresses.

Thus, analyzing the theoretical and experimental data, it may be concluded that at the initial moment of the cutting cycle, when the interaction of the cutter with the rock occurs along the entire length of the cutter blade, as a result of the stresses of the compression zone (Fig. 2) in front of the cutter

rock elements are spalled out. Behind the cutter a tensile zone arises. Owing to this zone, the face is weakened by microcracks.

If rock cutting takes place under conditions favorable to the development of deformations, then after the spalling of an ellipsoidal element the cutter, moving farther, bears against the curvilinear spall surface. In this position the load is transmitted to the rock only through the corners of the cutter and, with an insignificant contact area (a sharp cutter), the action of the cutter forces may be regarded as concentrated at two points \(A, B\) (Fig. 2, \(d\)), and the radial stress can be calculated from the formula

\[ \sigma_e = - \frac{3P}{2\pi K} \cdot \frac{\cos^2 \Theta}{l^2}, \]

where \(l\) and \(\Theta\) are the polar coordinates of the point relative to the plane of the resultant. \(K\) is a coefficient allowing for the influence of the free surface.

Substituting the value

\[ P = \frac{R \cos \delta}{2}, \]

(here \(\delta\) is the angle between the direction of motion of the cutter and the direction of action of the concentrated force), we obtain

\[ \sigma_e = - \frac{3R \cos \delta}{4\pi K} \cdot \frac{\cos^2 \Theta}{l^2}. \tag{5} \]

As is evident from the construction of the diagram (Fig. 2, \(d\)), the maximum stress will be distributed along the contour \(ABCD\), and therefore the spall surface has the corresponding shape.

At point \(K\) there is unbroken rock (a ridge). Having spalled the volume \(ABCD\), the cutter encounters ridge \(K\). In this position the force of the cutter will be concentrated on a small area, which in practice may be regarded as a point. In this case the maximum stresses will propagate over a sphere; consequently, the spall contour will have a round shape (Fig. 2, \(e\)). Experiments on rock cutting confirm these conclusions.

Under face conditions, obviously, the spall will have only a round and an ellipsoidal shape.

Conclusions

1. The application of the method of the theory of elasticity makes it possible to calculate with sufficient accuracy the forces arising during cutting and the depth of penetration of the cutter, while the photoelastic method makes it possible to reveal the mechanism of rock failure. For this it is sufficient to know the magnitude and the nature of the distribution of the stresses developing in the rock during cutting.

2. The nature of the distribution does not depend on the form of sharpening of the cutter, but depends only on the reduced contact area.

3. The lines of equal total stresses during cutting under face conditions are distributed over the surface of an ellipsoid close in shape to a sphere.

4. When cutting rock along its free surface, the form of the lines of equal stresses is distorted owing to the influence of the surface.

5. As a result of the stressed compression zone in front of the cutter, spalling of a rock element occurs, and owing to the stresses of the tensile zone, the face is weakened by cracks.

1. Comparison of the patterns of bands and traces of deformations developing during shearing confirms the idea of the uniformity of the nature of the distribution of stresses and deformations.

2. When cutting rock under conditions favorable for the development of deformations, an ellipsoidal chip is first formed, and then a trapezoidal and a round one.

References

1. E. F. Epshtein. Theory of drilling—cutting of rocks by hard alloys. GONTI, 1939.

2. I. A. Ostroushko. Destruction of rocks during drilling. Gostoptekhizdat, 1952.

3. I. A. Ostroushko. Face processes and tools in drilling rocks. 1963.

4. V. S. Vladislavlev. Destruction of rocks during well drilling. Gostekhizdat, 1958.

5. Yu. M. Pariiskii. On the question of the destruction of rocks during drilling. Proceedings of the Leningrad Mining Institute, vol. XI, issue 2, Gosgortekhizdat, 1961.

6. N. I. Bezukhov. Theory of elasticity and plasticity. State Publishing House for Technical-Theoretical Literature, 1953.

7. M. M. Filonenko-Borodich. Theory of elasticity. Fizmatgiz, 1959.

8. M. M. Frokht. Photoelasticity. Vols. I and II, State Publishing House for Technical-Theoretical Literature, 1950.