ON THE QUESTION OF ROCK FAILURE DURING CUTTER INDENTATION

V. I. Zvarygin, S. S. Sulakshin

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

The cutting of rocks may be regarded as the chipping off of rock elements under the continuous indentation of a moving cutter. Therefore, the study of the penetration of a cutter into rock, which is the fundamental element of the cutting process in drilling boreholes, is of definite interest.

In the present article an attempt is made to consider the process of rock failure during cutter indentation from the standpoint of the theory of elasticity and to determine the forces required for the penetration of cutters into rock.

As is known from the theory of elasticity, when a concentrated load P acts on the edge of a semi-infinite plate, stresses arise in the latter that are characterized by lines of equal tangential (and also total) stresses in the form of circles tangent to the point of application of the load and lying on the line of direction of the applied force.

When a uniformly distributed load acts over a segment \(b\), the lines of equal stresses (tangential and total) in the plate also have the form of circles with a common chord \(b\). In both cases the circles are closed and do not emerge onto the upper edge of the plate (Fig. 1, \(a\)).

When, however, some indenter is pressed into the surface of an infinite half-space, such a picture is not observed. The conditions of stress distribution in a body of sufficiently large dimensions (an infinite half-space) are nonuniform: the closer to the surface, the more favorable they are. Therefore the pattern of distribution of lines (more precisely, surfaces) of equal stresses has a different character.

In the immediate vicinity of a punch acting on some elastic body, the stresses are distributed analogously to the stress distribution in a plate. Moreover, the maximum stress arises along a hemisphere or semi-ellipse (in a plate, along a semicircle) adjacent to the base of the punch [4, 5, 6, 10].

At considerable loads around the zone of maximum pressure there arises a second deformed zone, which is, as it were, the shell of the first zone and, during indentation of the punch, serves as a kind of tip.

This phenomenon was noted in his work by L. A. Schreiner [3]. The most stable sphere will be the “skeleton” sphere (the inscribed triangle), which—

the second is an equilateral triangle with a side equal to the width of the punch; therefore the height of the second zone, which L. A. Schreiner described, is equal to the height of the equilateral triangle, i.e., \(0.866\,v\). After the formation of the spherical “tip,” the pressure will propagate

Fig. 1.

along the lower part of the sphere, normal to the tangent planes of the sphere. If the conditions for the propagation of stresses in the body were the same (as in a plate), the pressure distribution would follow a sinusoidal law. But since such conditions are absent for bodies of considerable dimensions, as a result of the subsequent transfer of pressure to the spherical layers, the latter, as it were, unfold. The sphere, closed at the base of the punch, at some distance from the punch again turns into a hemisphere. Fig. 1, b shows the character of the distribution of lines of equal stress in glass (according to observations of isochromes in polarized light and traces of deformations developing in the glass after spalling).

As a result of the horizontal thrust stresses arising in the tested material (the numerical value of which, according to the investigations of M. M. Frocht [4], for thick plates and blocks is \(P/\pi\)), at the initial moment of indentation a crack arises in the body, having

the form of a semicircle. The plane of the crack coincides with the axis of symmetry on the larger side of the punch. The crack is readily observed in an optically transparent material (glass, plexiglass, etc.). With a further increase in pressure, when the total stresses emerging on the surface of the body reach the magnitude of the stress characterizing the strength of the given material, a chip of the rock element will occur along the lines \(AB\), \(A_1B_1\), which are the trajectories of the maximum total stresses (Fig. 1, b).

Let us dwell in greater detail on indentation into rock by wedge-shaped punches (cutters).

When a wedge-shaped punch acts on the surface of an infinite half-space, stresses arise in the latter which, as is known, are distributed over spherical surfaces.

In the immediate vicinity of the wedge faces, spherical surfaces of equal stresses will be concentrated, and the axes of the centers of the spheres will be normals to the wedge faces. At some distance from the faces, the closed spheres will turn into hemispheres embracing the wedge on both sides (Fig. 1, c). Chipping will occur along a line connecting the apex of the wedge with the points lying at the intersection of the hemisphere of equal total stresses with the surface of the specimen.

Moreover, chipping occurs along the trajectory of the maximum total stresses. It is obvious that chipping will occur at the moment when the total stresses reach the critical value for the given rock \((\sigma_{\mathrm{cr}})\). The stress at point \(B\) will be equal to the stress at point \(C\). The total stresses at point \(C\) can be determined from the formula of the theory of elasticity for a concentrated force acting on a plane bounding an infinite space [5].

\[ \sigma = -\frac{3R}{2\pi d^2} = -\frac{3R}{2\pi}\cdot \frac{1}{d'} \cdot \frac{1}{d''}, \tag{1} \]

where \(d'\) is the distance from the point of application of the load to point \(C\);

\(d''\) is the conditional length of the axis of the elliptic stress lines in the plane of symmetry of the cutter, which would exist in the absence of the influence of the free surface of the specimen (Fig. 1, c);

\(R\) is the force acting on the edge of the cutter, equal to the forces acting on the faces of the cutter.

From the conditions of the distribution of forces on the wedge,

\[ R = \frac{P\cos(90-\gamma)}{2}. \]

As can be seen from Figs. 1, c and 1, d, constructed from the materials of photographing isochromatics in polarized light and the traces of developing deformations during chipping:

\[ d' = AB = \frac{h}{\cos \alpha}; \]

\[ d'' = b + 2a = b + e \simeq b + l, \]

here \(h\) is the depth of penetration of the cutter;

\(\alpha\) is the chipping angle of the rock element (the angle between the vertical and the chipping surface);

\(b\) is the width of the cutter;

\(a\) is the height of the elliptic segment of the stress surface which would exist in the absence of the influence of the free surface;

$e$ — the height of the elliptical segment, taking into account the influence of the free surface;

$l$ — the distance from the lateral face of the cutter to the edge of the cavity formed.

Substituting the obtained values of $d'$ and $d''$ into formula (1), we obtain

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

whence

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

The chipping angle $\alpha$ may be determined as follows. Chipping of a rock element will occur when the total stresses on the surface, as we noted above, are equal to $\sigma_{\mathrm{cr}}$. For simplicity of reasoning, let us assume that the force acting on the infinite half-space is concentrated.

The stress from a concentrated force acting on an infinite half-space is determined by the formula

\[ \sigma=-\frac{3R}{2\pi}\cdot\frac{\cos^2\Theta}{r^2}. \]

Substituting

\[ r=\frac{h}{\cos\alpha}, \]

we have

\[ \sigma=-\frac{3R}{2\pi}\cdot\frac{\cos^2\Theta\cos^2\alpha}{h^2}. \]

The maximum stress (the stress at which chipping will occur) will be under the condition

\[ -\cos^2\Theta\cdot\cos^2\alpha=\max, \quad \text{i.e., when } \Theta=\alpha. \]

Thus the chipping angle $\alpha$ will be equal to the angle $\Theta$ (the angle between the direction of the force and the chipping line).

If the angle between the vertical and the direction of the force is denoted by $\alpha_0$ (Fig. 1, c), then

\[ \alpha=\frac{\alpha_0}{2}, \qquad \text{but} \qquad \alpha_0=90+\gamma, \]

where $\gamma$ is the angle between the vertical and the faces of the cutter. Therefore,

\[ \alpha=\frac{\alpha_0}{2}=\frac{90+\gamma}{2}. \tag{5} \]

Thus, we have arrived at a somewhat modified formula of K. D. Zvorykin [7], which finds application in the theory of metal cutting,

\[ \beta_1=45^\circ+\frac{\gamma-\varphi_{\mathrm{int}}}{2}. \tag{6} \]

This formula was somewhat simplified by A. M. Rosenberg [8]:

\[ \beta_1=45^\circ+\frac{\gamma-\rho}{2}, \tag{7} \]

where $\beta_1$ is the angle between the plane of shear of the metal elements and the horizontal plane;

\(\varphi\) is the angle of friction of the metal against the cutter;
\(\varphi_{\text{in}}\) is the angle of internal friction.

But since

\[ \alpha = 90 - \beta_1, \]

then

\[ \alpha = 90^\circ - 45^\circ - \frac{\gamma - \varphi}{2}. \]

The angle \(\gamma\), which we take as positive during cutting, has a negative value here, and therefore

\[ \alpha = 45^\circ + \frac{\gamma + \varphi}{2} = \frac{90 + \gamma + \varphi}{2}. \tag{8} \]

As we see, the difference between the formula (5) obtained by us and the transformed formula (8) consists in the fact that we did not take account of the frictional force under compression (here, obviously, static friction will act).

To verify this proposition, experiments were carried out in which cutters with different included angles were indented into sandstone.

After each indentation of the cutter, the dimensions of the cavity were measured. Then, using the formula \(\alpha = \operatorname{arc tg}\frac{l'}{h}\), the values of the fracture angles were calculated (here \(l'\) is the distance from the center to the edge of the cavity, and \(h\) is the depth of the cavity). The results of the experiments are summarized in Table 1.

Table 1

As can be seen from this, the actual fracture angles have values greater than the calculated ones, the deviations amounting to \(1.5 \div 5^\circ\).

Apparently, when cutters are indented into rock, the action of the friction force also manifests itself. Indeed, with the coefficient of friction for the tested specimens (sandstones) \(f = 0.1 \div 0.2\) (according to the data of V. S. Fedorov and E. F. Epstein [11]),

\[ \frac{\varphi}{2} = \frac{6 \div 12}{2} = 3 \div 6 \text{ deg}, \]

i.e., the deviations of the actual angles from the calculated ones are approximately equal to the value \(\frac{\varphi}{2}\), which appears in Zvorykin’s formula. Consequently, Zvorykin’s formula will also be applicable to rocks.

Let us return again to formula (1). Let us consider how the value of \(l\) will change with increasing penetration depth. Suppose that the cutter has penetrated by some amount \(h\) (Figs. 1, 2). If the upper layer of thickness \(t\) were absent, fracture would probably occur along the trajectory of the maximum stresses \(AB\). The fracture angle in the plane of symmetry in this case would be equal to \(\alpha_1\). In the presence of a layer of thickness \(t\), the lines of equal stresses, on emerging to the surface, will again come together along a circumference, as is shown in Fig. 1, v, g (Fig. 1, v, g was drawn from materials recording the lines of equal stresses in epoxy resin in polarized light). As a result, the line of the trajectory of maximum stresses \(AD\) will make with the vertical an angle \(\alpha_1^1\) smaller than the angle \(\alpha_1\). The value of \(e\), however, will differ very little from the value \(e_1\), and for a small depth of cutter penetration it may be considered constant, and then

\[ e = e_1 = l = l_1. \]

The change in the fracture angle \(\alpha_1\) with increasing depth of indentation of a punch into rosin was noted by V. S. Vladislavlev [9].

Below are data on the change in \(\alpha_1\) and \(l\) (Table 2) during cutting of coals, according to the research data of A. I. Beron [8].

Table 2

The value of \(l\) was calculated by the formula \(l = h \tan \alpha\).

It is seen from Table 2 that, for cutting depths from 5 to 25 mm, the values of \(l\) vary within small limits. Moreover, the variations are associated not only with an increase in depth but, apparently, with the nonuniformity of the cutting conditions.

Thus, as was noted above, the value of \(e\) when cutters are indented to a small depth may be taken (with very small error) as a constant quantity.

Fig. 2. Comparison of theoretical straight lines for the dependence of indentation forces on cutter width and cutter penetration depth.

To check the obtained formula (3), we shall use the data obtained by V. S. Vladislavlev in indenting cutters of different widths into marble.

In the graph (Fig. 2), the points show the results of V. S. Vladislavlev’s studies; the straight lines are constructed according to formula (3) for

\[ \sigma = 1.31\ \text{kg/mm}^2 \quad \text{and} \quad l = 4\ \text{mm}. \]

As can be seen from the graph, the theoretical conclusions agree well with the experimental data, and the obtained formula (3) makes it possible

with a sufficient degree of accuracy to determine the values of the forces required for the indentation of cutters into rock, for example, during the operation of a roller-cone bit.

REFERENCES

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

3. L. A. Shreiner, N. N. Pavlova, V. P. Yakushev, B. V. Baidyuk. Application of the indentation method for establishing the resistance of rocks to destruction during drilling and for the qualitative evaluation of the plasticity of rocks. Article in the collection Experimental Studies in the Field of Development of Deep Oil and Gas Fields, Moscow, 1964.

4. M. M. Frokht. Photoelasticity, Vol. II. Moscow–Leningrad, GITTL, 1950.

5. N. I. Bezukhov. Theory of Elasticity and Plasticity. GITTL, Moscow, 1953.

6. M. M. Filonenko-Borodich. Theory of Elasticity. Fizmatgiz, 1959.

7. M. I. Klushin. Cutting of Metals. Mashgiz, 1958.

8. M. M. Rozenberg and A. A. Eremin. Elements of the Theory of the Metal-Cutting Process, 1956.

9. V. S. Vladislavlev. Destruction of Rocks During Well Drilling. Gostoptekhizdat, 1958.

10. S. S. Sulakshin. Fundamentals of the Theory of Rock Destruction and the Transport of Destruction Products from Wells. Tomsk Polytechnic Institute Press, Tomsk, 1964.