Effects of Confining Stress on the Post-Peak Behaviour and Fracture Angle of Fauske Marble and Iddefjord Granite

Figure 3 illustrates a typical differential stress–strain curve of a Class I rock in compression test. The curve becomes non-linear after the stress is beyond the yield point, namely the linear elastic portion ends. The stress either remains at the peak level or slowly descends after it reaches the peak. The stress descending suddenly accelerates after a certain amount of displacement and the stress finally stabilizes at the residual strength level. The start point of the stress descending acceleration is associated with the onset of the creation of the major shear rupture plane, or called the onset of faulting. The plastic deformation and micro fractures induced by stresses are uniformly distributed in the material until the onset of faulting. After that, micro fractures coalescence in a narrow band which develops to a major shear rupture plane to the end. At the residual strength level, no more fracturing would occur in the material except for frictional grinding along the rupture plane. The input energy during testing is used for damaging the rock and for elastically deforming the rock. In other words, the input energy is the sum of the fracture energy W_r, dissipated for plastic deformation and rock fracturing, and the elastic energy W_e, used for elastically deforming the rock, as illustrated in Fig. 3. The fracture energy is essentially a more appropriate parameter than the input energy to describe the damage degree of the rock material.

The fracturing of rock involves other energy parameters in addition to the elastic and fracture energies, as mentioned by Hauquin et al. (2018), Tarasov and Stacey (2017), Huang and Li (2014). However, the elastic and fracture energies are directly associated with the stress–strain characteristic of the rock and thus they are considered in this study.

It is seen in Fig. 1a that the residual strength is zero for the specimen tested under confining stress 0 MPa, implying that the input energy is entirely dissipated for the damage of the rock under the uniaxial compression. In general, more fracture energy is dissipated under triaxial compression than under uniaxial compression. In other words, the rock is more damaged in the post-peak stage under triaxial compression than under uniaxial compression. It is easier to figure out how different the damage generated under triaxial compression is from the damage under uniaxial compression by using a relative value instead of the absolute quantities of fracture energy. Therefore, a damage index is constructed below for this purpose: where DI is the damage index expressing the relative damage extent with respect to the damage generated under uniaxial compression; W_r represents the fracture energy at a given axial strain under triaxial compression; and W_r0 the total fracture energy dissipated under uniaxial compression. This index can be used to describe the relative damage of a rock under any other loading condition, such under true-triaxial loading, as long as the fracture energy can be calculated on the stress–strain curves.

The onset of faulting did not show up on the stress–strain curves of the marble specimens that were tested under confining stresses 85 and 120 MPa when the tests were terminated at the axial strain of approximately 4%. In Table 2, therefore, presented are only the fracture energies of the marble specimens tested under confining stresses 10, 30 and 55 MPa. The DI values of those specimens, calculated according to Eq. (1), are also presented in the table. The damage degrees in the specimens tested under confining stresses 10, 30 and 55 MPa are 6, 17 and 26 times the damage in the uniaxial tested specimen, respectively. The DI value increases approximately linearly with the confining stress.

The Iddefjord granite exhibits Class II behaviour, as shown in Fig. 1b. A typical Class II rock behaves as depicted in Fig. 4 under the servo-controlled triaxial loading condition. The differential stress drops from the peak (point P) to a lower level (point A) without any change in the axial strain. After that, the axial strain reverses with a decrease in the stress for an amount (until point B) and then turns to increase. Finally, the stress stabilizes at the level of the residual strength, which is equal to the frictional resistance on the shear rupture planes. One or more than one shear rupture planes are formed to the end. Based on the experimental results, it seems that the length of the vertical segment PA increases with the confining stress (Fig. 1b). The energy, represented by the area bounded by ABCA and denoted as W_a in Fig. 4, is released from the rock specimen in this process of stress dropping. This excess energy would be converted to kinetic energy to eject rock. It is often released through reversing loops at different levels in the post-peak stage as shown in the stress–strain curves of the specimens tested under confining stresses 10 and 30 MPa (Fig. 1b). The reason for the multi reversing loops may be the shear rupturing in the post-peak stage is a progressive and stepwise process. The excess energy would disappear when the confining stress is high enough beyond which the behaviour of the rock becomes Class I.

The stiffness of the test machine used for the tests is very high up to 15 GN/m; with such stiffness, the energy released from the test machine frame is negligibly small. It was observed that rock specimens “exploded” (rock ejection) even though being tested on such a stiff machine when they were loaded uniaxially under load-controlled condition. The kinetic energy for the rock ejection must come from the rock specimen itself. It is believed that a part of the elastic strain energy stored in the rock specimen is dissipated to fracture rock and the rest excess energy is transformed to kinetic energy to eject rock. The excess energy is called intrinsic potential energy for strain burst in this paper. The excess energy is not zero only in Class II rock. It is illustrated on the stress–strain curve in Fig. 4.

The released excess energy W_a is an absolute measure for the intrinsic potential energy for strain burst in the rock. Another parameter to express the intensity of a burst event is the ejection velocity that is more explicit than the excess energy in the description of a burst event. The intrinsic ejection velocity, denoted as V₀, refers to the velocity of rock ejection in a burst event, which is purely caused by the excess energy W_a released from the ejected rock. Assuming that the excess energy is completely converted to kinetic energy to eject the rock specimen, we obtained the following expression for the ejection velocity: where V₀ is in m/s, W_a in kJ/m³ and ρ is the density of the rock in kg/m³. Figure 5 shows the axial stress–strain curves of Kuru granite and Iddefjord granite under uniaxial compression. The excess energy W_a is marked in the two diagrams, which is 28.3 kJ/m³ for the Kuru granite specimen and 3 kJ/m³ for the Iddefjord granite. Assuming the density of the granites is 2700 kg/m³, the intrinsic ejection velocities of the two granites are 4.6 and 1.5 m/s, respectively. It means that the Kuru granite would be more burst-prone than the Iddefjord granite.

It is known that the angle of the shear rupture plane decreases with the confining stress, but the physics of the phenomenon is not clear to date. A hypothetical model for the formation of the shear rupture plane is proposed below. It has been observed that wing cracks are initiated at the tips of pre-existing micro cracks and they tend to propagate in the direction of the major principal stress σ₁, as shown in Fig. 6 (Horii and Nemat-Nasser 1986; Li and Nordlund 1993; Li et al. 1998; Bobet and Einstein 1998; Wong and Chau 1998; Wong et al. 2001; Yang et al. 2012). Tensile wing cracks would be initiated at the tips of the most shear-favourable micro cracks. Under uniaxial loading, the wing cracks could extend for a relatively long distance before they coalescence with sparsely spaced shear-favourable micro cracks to form a steep oblique fracture, as shown in Fig. 6a. Under triaxial loading, particularly when the confining stress σ₃ is high, the extension of wing cracks is limited by the confining stress and short wing cracks would be initiated on the pre-existing micro cracks. To the end, a less steep oblique fracture would be formed in the rock specimen by short wing cracks and densely spaced pre-existing micro cracks, as shown in Fig. 6b. The reduction in the inclination angle of the final fracture plane may be due to the short wing crack extension and the activation of many shear-favourable pre-existing micro cracks in the triaxial loading condition.

The microscopic observations of the micro cracks in the shear bands by Bésuelle et al. (2000) showed that the micro cracks dominantly oriented in the direction of σ₁ at relatively low confining stress 30 MPa, but the cracks did not show preferential orientations at confining stress 50 MPa. The statistic results support the hypothesis that the wing cracks would be depressed at high confining stresses.

In order to reduce the randomness of the test results, the fracture angles were also estimated on the failure envelope as illustrated in Fig. 7 (the angle in the figure is equal to 2β). The estimated fracture angles agree well with the measured angles shown in Fig. 2.

For the Fauske marble, the following relationship is obtained between the fracture angle and the confining stress by fitting the fracture angles estimated on the failure envelope: where β is in degrees and UCS is the uniaxial compressive strength of the rock.

Similarly, for the Iddefjord granite, the following relationship is obtained:

In general, the dependence of the fracture angle on the confining stress can be expressed as follows: where β₀ stands for the fracture angle of the rock specimen under uniaxial loading (σ₃ = 0). Letter a is a material constant. The value of a is 0.19 for the Fauske marble and 0.36 for the Iddefjord granite.