On the Determination of Rock Strength under Uniaxial Compression
Abstract Generated abstract
This paper examines how specimen geometry and end friction influence the measured uniaxial compressive strength of rocks and other brittle granular materials. It proposes a visual mechanical model in which the material is represented as densely packed spheres, using this scheme to explain internal force redistribution, shear development, wedge formation, and characteristic failure shapes under compression. From the assumed failure surfaces of pyramidal or conical wedges, the authors derive relations between relative strength and the height to width or height to diameter ratio of specimens. Comparison with published experimental data for rocks, ores, gypsum, salt, concrete, and related materials suggests that the proposed dependence is useful for recalculating strengths to a standard specimen over ratios of about 0.6 to 1.6.
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On the Determination of Rock Strength under Uniaxial Compression
I. G. Basov, Yu. N. Luzgin
(Presented by the Department of Mining Machines, Mine Transport, and Rock Mechanics)
Rocks, as well as construction materials and frozen soils, consist of solid mineral grains cemented together. When specimens of rocks and construction materials are tested in uniaxial compression, stresses arise in them that can be represented in the form of forces directed in a definite manner. But for this it is necessary to adopt a visual model of the material that is close to reality.
Most materials may be represented in the form of a scheme in which the role of the grains is performed by small spheres. A similar theoretical scheme (B. V. Deryagin, 1963) was used in substantiating the laws of external friction and gave good results. The structural schemes of materials may be represented as consisting of equal-sized or differently sized spheres. For equal-sized spheres, six methods of packing are theoretically distinguished, depending on the mutual arrangement of their centers. In the least dense packing of spheres of radius \(r\), their centers are located at the vertices of a cube, and in the densest packing—at the vertices of a rhombohedron. For differently sized spheres, the number of possible packing methods is considerably greater.
In natural and artificial stone materials, with the chosen scheme of equal-sized spheres, their dense packing appears to be the most probable. It is rather cumbersome to represent the volumetric packing graphically; therefore, for simplicity and for the convenience of the further exposition of the essence of the question, one may adopt a plane image of the arrangement of the spheres (Fig. 1), which in the vertical plane gives a visual representation of the accepted packing method.
In order not to clutter the scheme with the adhesion and friction forces acting between the spheres, let us single out only the forces \(P'\) that arise under the action of the uniformly distributed load \(P\). The distribution of the forces \(P'\) and of their reactions \(R'\) inside the specimen is valid only at small loads, i.e., in those cases when the friction forces at its ends do not yet have a substantial influence on the redistribution of stresses. Since the spheres of individual layers transmit the load through point contacts, the force \(P'\) does not coincide in the direction of action with the load \(P\) and can be found from the expression
\[ P' = \frac{P}{n \cos \alpha}, \tag{1} \]
where \(n\) is the number of contacts between balls in the direction of the action of the force;
\(\alpha\) is the angle at which the force \(P'\) acts.
The magnitude of the angle \(\alpha\) depends on the method of packing the balls when their sizes are equal. With dense packing of equal-volume balls, \(\sin \alpha = 0.5\), i.e., the angle \(\alpha\) does not depend on their sizes and is always approximately equal to \(\dfrac{\pi}{6}\,\text{rad}\) \((30^\circ)\).
For this case, in accordance with formula (1), \(P' = 0.578P\). For specimens consisting of balls of different volumes, the magnitude of the angle \(\alpha\) will also be affected by the sizes of the contacting balls.
Fig. 1. Diagram of the forces acting on a specimen during its compression
Fig. 2. Isolines of the principal shear stresses in a compressed specimen made of optically active material (after V. F. Trumbachev and E. V. Melnikov)
As the load increases and deformation begins, frictional forces arise at the end faces of the specimen, directed toward the center of the bearing plane; these create the cage effect noted by many investigators (Fell, Skramtaev, Heller, Rosch, Taylor, and others). These frictional forces cause a redistribution of forces in the specimen material. As a result, the forces transmitted into the interior of the specimen by the balls located at the periphery of the specimen’s end surfaces will be considerably greater than the forces directed toward the outside. Therefore the stresses at the center of the end surfaces are maximal and decrease in the direction toward the periphery of the ends. This situation agrees well with the results of experimental studies by K. V. Ruppeneit and G. L. Fisenko.
The redistribution of forces at the ends of the specimen, in turn, causes a redistribution of forces in its central part as well. In each row following from the end (Fig. 1), an ever smaller number of balls will be in a triaxial state of stress. At the center, under the action of compressive forces, a zone of maximum stresses is formed, and since, as experiments have shown (B. G. Skramtaev, N. A. Popov, N. A. Gerlivanov, G. G. Mudrov, and others), the strength of the grains is considerably higher than the strength of the cementing substance, wedging of some grains between others occurs. In this process all the balls participating in the rearrangement,
are partially deformed, and some of them tend to be displaced laterally. In connection with this, considerable shear stresses arise in the specimen (Fig. 2).
In the process of compression, cylindrical specimens made of viscous materials assume a barrel-like shape, while, as a result of the failure of cubic specimens made of brittle materials, two truncated pyramids are formed, joined at their vertices (Fig. 3). This difference in the shape of the failed specimens is obtained because in brittle materials the bond between particles is rather rigid and is disrupted even with small displacements of the particles relative to one another. In plastic materials this bond permits a certain displacement of particles without disrupting the integrity of the specimen.
Thus, when specimens made of brittle materials fail under uniaxial compression, fairly stable wedges are formed in the shape of truncated and untruncated pyramids and cones, depending on the height of the specimen and its shape (prism, cylinder). Cementing substances also have a great influence on the residual failure shape. The higher their strength, the more strongly the mineral particles are deformed and the greater the regrouping they undergo at the junction of the truncated pyramids or cones.
In the case when the height of a specimen made of brittle material exceeds the sum of the heights of two cones or pyramids, these, penetrating like wedges into the central part of the specimen, split it.
Fig. 3. Characteristic forms of specimen failure under uniaxial compression: a) cylindrical, of viscous materials; b) cubic, of brittle materials; c) cubic, of brittle materials when the end surfaces are lubricated with paraffin
Fig. 4. Calculation scheme
The smaller the ratio of the specimen height \(h\) to the side of its base \(a\) or to the diameter \(d\), the larger the portions of the entire volume represented by stable wedges in the form of truncated pyramids or cones, and the stronger the specimen will be. M. M. Protodyakonov (1963) arrives at approximately the same conclusion: at constant specimen height and with an increase in its diameter, the presence of radial friction forces at the specimen ends leads to an increase in strength in the central part of the ends and to an increase in the average strength of the specimens.
Analyzing the mechanism of failure of specimens under uniaxial compression and the data of experimental investigations on their failure as a function of the ratio \(\frac{h}{a}\) or \(\frac{h}{d}\), we come to the conclusion that the strength of specimens \(R\), MN/m\(^2\) (kg/cm\(^2\)), is inversely proportional to the lateral surface of truncated or untruncated pyramids or cones, i.e., to the compacted and fairly stable wedges. The indicated surface corresponds to the newly formed surface in the spec-
…of brittle materials and can be expressed in terms of the known quantities $\alpha(d)$ and $h$.
The lateral surface of a regular truncated pyramid is equal to (Fig. 4)
\[ F_{\mathrm{p}} = 2(a+b)L, \tag{2} \]
where $b$ is the side length of the smaller base of the pyramid;
$L$ is the apothem.
From Fig. 4 we find that
\[ b = a - h \tg \alpha; \qquad L = \frac{0.5h}{\cos \alpha}. \]
Substituting these values into formula (2), we obtain
\[ F'_{\mathrm{p}} = \frac{(2a - h \tg \alpha)h}{\cos \alpha}. \]
The total newly formed surface upon failure of a cubic specimen under uniaxial compression, for $\alpha = \frac{\pi}{6}\ \mathrm{rad}\ (30^\circ)$, can be found from the expression
\[ F_{\mathrm{p}} = 2.3(2a - 0.577h)h \tag{3} \]
and for cylindrical specimens
\[ F_{\mathrm{k}} = 1.81(2d - 0.577h)h. \tag{4} \]
Taking various ratios $\frac{h}{a}$ and $\frac{h}{d}$ and expressing $h$ in terms of $a$ and $d$, from formulas (3) and (4) we find the surfaces $F_{\mathrm{p}}$ and $F_{\mathrm{k}}$ (Table). For convenience in comparing the calculated data with experimental data, we express these surfaces in relative units, taking the values $F_{\mathrm{p}1}$ and $F_{\mathrm{k}1}$ at $h=a$ and $h=d$, respectively, as 100% (Table). It also follows from formulas (3) and (4) that, when $\frac{h}{a}=\frac{h}{d}$,
\[ \frac{F_{\mathrm{k}}}{F_{\mathrm{k}1}} = \frac{F_{\mathrm{p}}}{F_{\mathrm{p}1}} . \]
Table
| $h = fd$ | $F_{\mathrm{k}}$ | $F_{\mathrm{p}}$ | $\dfrac{F_{\mathrm{k}}}{F_{\mathrm{k}1}} = \dfrac{F_{\mathrm{p}}}{F_{\mathrm{p}1}}$ |
|---|---|---|---|
| $h = 0.25d$ | $0.84d^2$ | $1.07a^2$ | 3.06 |
| $h = 0.35d$ | $1.14d^2$ | $1.25a^2$ | 2.28 |
| $h = 0.4d$ | $1.28d^2$ | $1.63a^2$ | 2.02 |
| $h = 0.5d$ | $1.55d^2$ | $1.97a^2$ | 1.66 |
| $h = 0.6d$ | $1.79d^2$ | $2.28a^2$ | 1.43 |
| $h = 0.8d$ | $2.22d^2$ | $2.82a^2$ | 1.16 |
| $h = 1.0d$ | $2.57d^2$ | $3.27a^2$ | 1.00 |
| $h = 1.2d$ | $2.83d^2$ | $3.6a^2$ | 0.91 |
| $h = 1.4d$ | $3.02d^2$ | $3.84a^2$ | 0.85 |
| $h = 1.6d$ | $3.12d^2$ | $3.97a^2$ | 0.825 |
A comparison of the dependences
\[ \frac{F}{F_1} = \frac{\sigma}{\sigma_1} = f\!\left(\frac{h}{a}, \frac{h}{d}\right), \]
obtained experimentally by various investigators (Fig. 5), with the calculated ones (Table, Fig. 5, dashed lines) shows their quite complete agreement. It follows from Fig. 5 that the change in the relative strength of specimens as a function of the ratio of their geometrical parameters $\left(\frac{h}{a}\ \text{and}\ \frac{h}{d}\right)$ follows the same pattern for all the materials considered.
Some authors (S. P. Timoshenko, 1965) believe that the true compressive resistance of brittle materials, for example concrete, can be obtained by eliminating or reducing the influence of friction on the bearing surfaces of the specimen. For this purpose A. Föppl covered the bearing surfaces…
...with paraffin and found that the strength limit in this case decreases considerably.
B. G. Skramtaev, N. A. Popov, N. A. Gerlivanov, and G. G. Mudrov (1952) believe that under such testing conditions the strength of cubes decreases by 50%. If we again turn to the spherical scheme for representing the specimen being tested, then the mechanism of failure of the latter under uniaxial compression and in the absence of friction forces on the supporting surfaces can be represented as follows. The mineral grains—spheres of the end layers, under the action of compressive forces, wedge themselves
Fig. 5. Dependence of the relative strength and of the ratio of newly formed surfaces during crushing of specimens on \(\frac{h}{a}\) or \(\frac{n}{a}\) for:
a) ferruginous quartzite (according to L. I. Baron et al.);
b) stone materials (according to L. I. Baron et al.);
c) anthracite (according to V. I. Zamtaradze);
d) apatite ore (according to L. I. Baron et al.);
e) rock salt (according to G. N. Kuznetsov);
f) gypsum (according to L. I. Baron et al.);
g) concrete (according to P. F. Shubenkin)
between the spheres of adjacent layers. A stressed state is created along the specimen, and its splitting occurs in the direction of action of the compressive forces. Therefore, when the specimen fails, separate plates are obtained (Fig. 3), parallel to one of the lateral faces. Since a brittle material resists splitting much more poorly than compression, the compressive strength limit under such a testing condition will be lower. And its value will depend on the height of the specimen being destroyed.
From the material presented in this paper, the following conclusions may be drawn:
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For a visual representation of the mechanism of failure of brittle materials under uniaxial compression, one may use the spherical scheme for substituting the specimen. The picture of the failure mechanism described with the aid of the proposed substitution scheme is confirmed by the results of experiments obtained by various researchers on real materials.
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The revealed dependence between the relative strength of the material and the geometric parameters of the tested specimens on
uniaxial compression gives good results for \(\frac{h}{a} = \frac{h}{d} = 0.6 \div 1.6\). This makes it possible to compare the strengths of materials when testing specimens of different heights by recalculating the strength to a standard specimen.
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