Different Stress Paths Lead to Different Failure Envelopes: Impact on Rock Characterisation and Design

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This article describes, based on a discrete element model of a representative volume element, the importance of applying a stress path close to the in situ stress path during rock characterisation.

Abstract

The strength of rock is a non-intrinsic property, and this means that numerous parameters influence the strength values. In most laboratory experiments, specimens are free of stress at the start of the tests, and the load is increased systematically until failure occurs. Around excavations, the opposite path occurs, i.e., the rock is in equilibrium under a triaxial stress state and at least one stress component decreases while another component may increase. Hence, the stress paths in classic laboratory experiments are different from the in situ stress paths. In the research presented, a first step was made to evaluate with an open mind the effect of these different stress paths on the failure process and failure envelope. The research was based on distinct element models, allowing the simulation of micro-fracturing of the rock, which is essential to correctly model rock failure. The micro-fracturing when loading rock (from zero or low stress state) until failure was different from the micro-fracturing when unloading rock (from the in situ stress state) until failure. And, hence, by this difference in weakening processes, the failure envelopes were significantly different. The conventional loading resulted in the largest strength and, thus, overestimated the rock strength in comparison to the real in situ behaviour. This finding, after being confirmed by further lab experiments, will have a direct effect on how one characterises rock material and on the design of rock excavations.

1. Introduction

One of the most significant challenges in rock mechanics is understanding the mechanisms involved in the behaviour and failure of rock, both in situ and in laboratory experiments. Hence, numerous research projects focus on conducting laboratory experiments and simulating them. One of the key issues in conducting such investigations is to get a better understanding of the variability and uncertainty of rock characteristics, e.g., moduli, uniaxial compressive strength (UCS), and Brazilian tensile strength (BTS). Often a ratio of two is observed between the maximum and minimum measured values for a specific rock and for a specific test. This large variation often is linked to the heterogeneous nature of rock and the presence of micro-defects, pores, voids, flaws, layering, anisotropy, etc. (e.g., [1,2,3,4,5]). The strength of rock is a non-intrinsic property, which means that its value is influenced (1) by the geometric characteristics of the specimen (e.g., its volume, shape, and orientation), (2) by environmental aspects (e.g., humidity and temperature), (3) by the loading rate (e.g., conventional loading rates, dynamic loading, and creep), and (4) by the test set-up. The three most common tests are the Brazilian tensile tests, uniaxial or unconfined compression tests, and conventional triaxial tests. All three types of tests are conducted on cylindrical specimens. Based on the combination of these tests, the failure envelopes are determined, and the latter are used as input for design projects. More and more, true triaxial or poly-axial test equipment has become available to test cubic or rectangular specimens [1,6,7,8,9,10,11,12,13,14]. The main advantage of these true triaxial tests is that three different major principal stress components can be applied to investigate the effect of the intermediate principal stress [9,15,16]. The tests also allow the variation of the three principal components independent of each other as a function of time.

Nearly all such experiments are started from an isotropic stress state, followed by an increase in at least one of the principal stress components (Figure 1c, red and blue arrows). The initial isotropic stress state is either a fully zero-stress state or a non-zero, isotropic stress state. The latter stress state is created by increasing equally the three principal stresses starting from a state of zero stress. No damage should be induced during the initial equal stress increase. From a practical point of view, such loading experiments are entirely logical. Cored or cut specimens are, by definition, not loaded. However, as is well known, this procedure is the opposite of what happens in situ. The rock is in equilibrium under a state of triaxial stress, and at least one stress component decreases due to the excavation, resulting in damage (e.g., micro-fracturing) and, possibly, even in the failure of the rock.

The simplest example to illustrate the actual in situ stress path is by considering an in situ isotropic stress state and the excavation of a circular opening (i.e., a borehole or a tunnel). Before the excavation, the three principal stress components are equal (Figure 1a). After excavation, the radial stress, which is a principal stress component, is decreased to zero or to the supporting pressure, i.e., the pressure of the drilling fluid for a borehole or the pressure that supports the tunnel (Figure 1b). In the latter application, first, the radial stress drops to zero, after which it is increased due to the installation of active support or due to further deformation if passive support is installed. However, in both cases, the final radial stress around a tunnel is small in comparison to the initial in situ stress. The tangential principal stress component increases during excavation. Without any support, it is doubled in comparison to the initial in situ stress. When support pressure is used, its value must be subtracted from the tangential stress. The third principal stress component, i.e., the axial stress, corresponds to the initial in situ stress, and, hence, it is the intermediate principal stress. At least, that is the situation away from the face or borehole bottom and for linear elastic calculations. Hence, as illustrated in Figure 1c (black arrow), the stress path is a combination of an increase in one principal stress component and the decrease in another principal stress component. This is a different path in comparison to conventional uniaxial or triaxial loading experiments. Around the face or borehole bottom, the stress variation is more complex [17,18].

As a function of the initial in situ principal stress ratios, the stress paths are more complex and are different at each point of the circumference. The tangential stress component may decrease for a certain combination of in situ principal stresses and support pressures. This component can even become tensile. When a rectangular cavity is excavated (e.g., in room and pillar workings), all of the stress components of the rock at the sidewall (e.g., in the middle part of the sidewall) may decrease to zero due to the redistribution of the stress. Similar changes may occur in open pit mines depending on the geometry of the slope. In conclusion, at least one stress component decreases around an excavation. When looking at the design of foundations, it could be that the stress state is only increased, but if the volume of the foundation is excavated initially, the stress state decreases first before it continues to increase.

The aim of this paper was to evaluate with an open mind if different stress paths could lead to different failure envelopes when using the distinct element code UDEC (see further). Hereby, it is essential that (micro-)fracturing is integrated into the simulated failure process. At this stage of the research, no specific laboratory experiments are modelled. The study focusses on a small representative volume element (RVE). However, the values of the various input parameters are based on past calibration of laboratory experiments. (See further for more details about the code and model). The behaviour of the RVE model was also verified by modelling published laboratory experiments (See paragraph 5). The loading of the RVEs was not by the use of platens, which undergo a displacement, but by applying directly external stresses. The main conclusion of the presented results is that the chosen stress path has a significant effect on the strength and failure envelopes of rock material. Hence, one should further investigate this finding and complement the research by conducting lab experiments. All this could lead to adapted procedures for characterising rock material.

2. Overview of Unloading Experiments

As mentioned earlier, the fact that unloading stress paths occur around excavations is common knowledge for all rock mechanical engineers. The most illustrative example is the discing phenomenon that occurs when cores are drilled in deep boreholes [19]. The discing or macro-fracturing of the rock is due to the relaxation of stress, and it is an extreme form of damage due to stress relaxation. Holt et al. [20,21] aimed to quantify core damage. Therefore, they manufactured artificial or synthetic rock specimens under a triaxial stress state that was followed by the unloading of one or more stress components. They compared the behaviour of synthetic rocks which were kept under simulated reservoir conditions with the simulated coring process by unloading, i.e., the simulation of the release of stress during core drilling. They observed that the latter could result in a stiffness that was two or three times less than the traditional stiffness [20]. For the unloading due to the coring, the UCS values also were systematically lower. It is clear that damage was induced during unloading, whether or not macro-fractures were observed.

In [13,22], cubic specimens of crinoidal limestone were tested in a true triaxial or poly-axial cell, and loading and unloading schemes were applied. The crinoidal limestone is composed mainly of calcite. Two stress paths were followed. First, the conventional loading was applied. The specimens were loaded until a pre-defined isotropic stress state was reached, followed by an increase in two stress components until failure occurred, and the minor principal stress remained constant. Second, for the unloading scheme, the minor principal stress was decreased from the pre-defined isotropic stress state until failure occurred. A small difference was applied between the major and intermediate principal stress, allowing a two-dimensional approach for describing the fracture patterns. It also facilitated the numerical simulations of the experiments. For some of the specimens, the loading and unloading schemes were stopped in a controlled way prior to the macro-failure. Figure 2a shows an example for the unloading scheme of a micro-photograph of a petrographic thin section. Some approximate vertical micro-fractures were induced that corresponded to some of the cleavage planes of the calcite crystals, i.e., approximately perpendicular to the minor principal stress. Figure 2b shows an example of a failed specimen (again for the unloading scheme). Relatively wide fractured zones were induced, and the widths of these zones generally ranged between 0.05 and 0.2 mm. Their orientation corresponded to the overall angle of 60° to 70°. In addition to the wide macro-fractures, some thin fractures were present, e.g., activated cleavage planes. Their orientation was for the failed specimen not only perpendicular to the minor principal stress, as was observed when the unloading scheme was stopped in a controlled way prior to the macro-failure. No clear difference was observed between the loading and unloading schemes in the macro-fracture pattern. Two-dimensional numerical simulations also were conducted, using the boundary element code DIGS (Discontinuity Interaction and Growth Simulation). This code allows for explicit fracture modelling [23]. The cracks are modelled as displacement discontinuities. As for the laboratory experiments, no significant effect was observed on the macro-fracturing pattern between the loading and unloading schemes. A global shear type of failure was observed in both schemes. However, the simulations showed that more tensile cracks occurred during the unloading scheme [13,22].

In the literature, there is a large amount of test results of a single loading-unloading cycle (e.g., [24]) or of multiple loading–unloading cycles (e.g., [25,26,27]). Such research is interesting and worthwhile, but its aim is different from the aim of the stress paths presented in this paper. Micro-damage may be induced in the specimens during both the loading and unloading parts of the successive cycles.

Song et al. [28] conducted specific experiments and compared the strength for conventional triaxial loading with the strength for a three-stage loading and unloading. Song et al. used the conventional triaxial cell and conducted five experiments per loading path for three different confinement stresses. Each of the five experiments corresponded with a different inclination angle of the bedding planes in bedded sandstone. Although the overall aim of the research was the storage of energy and the energy flow under rockburst conditions in a coal mine, the comparison of the measured strength values is extremely interesting. Overall, the strength or failure envelope for the three-stage loading and unloading was smaller than the one associated with conventional triaxial loading. Their work is discussed in more detail below.

Chen et al. [29] applied different loading–unloading methods and rates for different confining pressure tests of marble specimens. They tried to approximate the in situ stress path as well as possible, resulting in relatively complex stress paths. They studied the influence of the initial confining pressure; loading and unloading rates; and the stress paths on rock dilatancy, failure, energy, and strength characteristics. Among other findings, they observed that the peak stress and the residual stress were smaller with larger unloading rates. The increase in unloading rates accelerates the deformation and failure of the rock.

Zhang et al. [16] used a true triaxial rock testing machine to conduct loading and unloading tests of sandstone specimens. In comparison to the results of the loading tests, the peak strength of the unloading path was reduced. They presented their results with a Mogi–Coulomb criterion, but looking at the cohesion and friction angle, they obtained cohesion values of about 22.0 MPa for the true-triaxial loading and about 16.5 MPa for the true-triaxial lateral unloading with friction angles of 57.6° and 48.3°, respectively. This means that the cohesion was reduced by about 25%.

Xu et al. [14] conducted minimum principal stress unloading tests of marble specimens under true triaxial compression conditions and with different initial stresses and unloading rates. They also investigated the effect of the middle principal stress. Although considering a large number of different values is very interesting, it also makes it difficult to interpret the failure values for the loading vs. the unloading paths and to judge whether the failure criteria of the unloading paths were indeed stronger than of the loading path as indicated on their graphs of the largest differential stresses (major principal stress minus minor principal stress) vs. the minor principal stress.

When studying the effect of various stress paths, it is essential that one observes the occurrence of micro-fractures during the entire loading and/or unloading process. The recording of acoustic emission is hereby a useful tool (e.g., [8,30,31,32,33,34,35,36,37,38,39,40,41]). The determination concerning whether the emissions are in shear mode, in tension mode, or in mixed mode (e.g., [31,42,43,44,45,46]) allows for a proper interpretation of the fracturing mechanisms involved and how they evolve as a function of time or of the load level. An additional verification by the visualisation of micro- and macro-fractures provides a better understanding of the failure process. The use of micro-photographs of petrographic thin sections or of fracture surfaces (e.g., [2,10,22,47,48]), SEM micrographs (e.g., [14,26,28,35,39,40,41,42,48,49,50]), reflection microscopy images (e.g., [48]), and (micro-)CT images (e.g., [2,11,28,35,51,52,53]) are hereby useful.

Apart from the unloading experiments in a true-triaxial cell as mentioned above [13,22], we studied various stress paths, including the effect of the principal stress orientations. In Lavrov et al. [34], cyclic Brazilian tensile tests were conducted without reaching final macro-failure by rotating the disks between successive cycles. The rotation angle varied between 0° and 90°. The final aim was to investigate how sensitive the Kaiser effect is towards the principal stress orientation. The Kaiser or memory effect gradually became less pronounced as the rotation angle increased, but it remained detectable for angles less than 10°. Larger rotations resulted in complete disappearance of the Kaiser effect [34]. The overall conclusion of this study confirmed that it is essential to consider the micro-fracturing during the entire loading or unloading process. In another study, the damage around a circular opening was studied [32,47]. Two configurations were investigated, i.e., in one set of experiments the damage was induced mainly by shear stresses (only macro-compressive stresses were present), and in the other set, the tangential stresses were tensile stresses at the macro-scale. Some specimens were tested by a succession of first macro-compressive stresses, followed by a tangential macro-tensile stress, and other specimens were tested the other way. In the latter case, i.e., specimens first damaged by a tensile macro-stress, followed by macro-compressive stresses, more damage was observed than in the other sequence. The rock material studied was crinoidal limestone, and it was clear that the damage that was observed was affected by the structure of the crinoidal limestone, e.g., its typical cleavage planes. Hence, one should be careful in extrapolating the conclusions to other types of rocks. However, the sequence of the compressive and tensile stress states within a given experiment is an important factor of the amount of (micro-)damage. In other words, the discontinuous fracturing process always should be considered.

In conclusion, this literature overview clearly highlights the importance of the occurrence of micro-fractures during loading or unloading experiments. This is visualised in lab experiments, but also in numerical simulations of some of these experiments using codes which allow for the modelling of the initiation and growth of individual (micro-)fractures. In [16,20,28], data of lab experiments show that the conventional loading of rock specimens result in larger strength values than stress paths containing at least partly some unloading.

3. Distinct Element Model for Simulating Loading and Unloading Paths

The aim of the study that is presented was to try to quantify the effect of different loading and unloading paths both on the evolution of the induced (micro-)damage and on the final strength of rock. The choice was made to study the topic by numerical simulations and to consider a small RVE only. So, at this stage of the research, the aim was not to simulate entire specimens under laboratory conditions, or the rock mass surrounding an excavation, but to focus entirely on the single effect of different stress paths. The RVE model should be considered to represent a black box rock. Before applying a stress path, one does not know the strength of the RVE model under a specific stress path. However, the choice of the values for the various input parameters is based on past experience, whereby both observed fracture patterns and measured stress–strain curves were used to calibrate distinct element models [54,55]. The readers are referred to these publications to learn more about the set-up of a distinct element model and about the calibration of the input parameters. One of the main benefits of the approach by numerical simulations is that one can repeat loading or unloading the exact same model, which is impossible in laboratory experiments because a new core or block must be used for each test.

To study the initiation and growth of fractures, my research team at the KU Leuven has used with success in the past the two-dimensional Universal Distinct Element Code (UDEC), which was developed for the simulation of rock blocks and their deformation and relative movements [56,57]. The latter is still the main application of this code. However, an intact rock can be approximated by an assembly of individual blocks, whereby initially these blocks are glued together along all contact lines. These contact lines are possible future fracture paths. A contact between two adjacent blocks does not represent a physical crack as long as it is not activated or has passed the pre-defined failure criterion. In my opinion, the fact that future fractures are composed of relatively straight lines is an advantage in comparison to some other discrete element codes. Thus, the contact lines or elements are given strength properties and hence can fail in shear and/or tension, simulating the occurrence of (micro-)fractures. After activation, the contact elements can deform, slide, and open. The blocks also are able to deform, e.g., in a linear elastic way. Both the individual blocks, as the contact elements within a single model, can have different property values.

Debecker and Vervoort [54] used UDEC to study transversely isotropic rock slate, which has a very large strength anisotropy. The distinction was made between contacts that represent the schistosity and the contacts in other directions. The blocks were allowed to undergo plastic behaviour. Although a block cannot break internally, the plastic deformation of a block could physically be interpreted as the occurrence of very small fractures on a sub-grid scale. Both uniaxial compressive and Brazilian tests were simulated and compared to observed fracture patterns in laboratory experiments. The simulations of the diametric load tests provided further insight concerning the fracture mechanisms. The simulations confirmed that millimetre-scale strength anisotropy resulted in a strength anisotropy on the specimen scale. Moreover, analysis of the stress distribution and of the contact strength properties explained the failure modes of the different fractures.

Van Lysebetten et al. [55] studied the heterogeneous nature of soil-mix specimens and more specifically the effect of soft inclusion on the induced fracture patterns. The starting point was the distribution of such soft inclusions in real specimens, but the simulations allowed them to conduct a sensitivity analysis of the number of inclusions, their sizes, and their relative positions on the strength and fracture pattern. Different property values were given to the inclusions and the surrounding material or matrix material. The blocks only deformed elastically. Three different contact characteristics were considered, i.e., contacts within the soft inclusions, contacts within the cemented matrix material, and contacts matrix-inclusions. Mohr–Coulomb criteria were applied. The simulated fracture patterns were comparable to those observed in real soil-mix specimens that were loaded uniaxially.

For the study presented here, a much smaller model was created, allowing for a larger number of simulations and facilitating the interpretation of failure types. The entire model contained 273 elements. The number of elements was small but sufficient for studying a RVE, as is shown by the results presented further. Figure 3 is a view of part of the model. The thick green lines are the contact lines and possible future fracture paths. These lines form the boundaries of the elastically deformable blocks, which are subdivided further in triangular elements (Figure 3, thin grey lines). The chosen values for the properties are based on earlier studies. The most relevant property is the failure criterium of the contacts. Van Lysebetten et al. [55] calibrated the model with laboratory experiments, resulting in a ratio of 2 between the cohesion and the tensile strength of the contacts within the matrix material. For very weak rock, i.e., slate, the tensile strength was 75% of the cohesion for the various types of contacts [54]. For the black box rock in this study, the contacts were characterised by a Mohr–Coulomb criterium with a cohesion of 20 MPa, a friction angle of 30°, and a tensile strength of 10 MPa. These values are for the contacts and should not be confused with the macro-behaviour of an entire rock specimen. The orientations of the contact lines were chosen in such a way that the orientations were distributed equally over individual classes at 30° intervals. Hence, the black box rock that was chosen behaves on a macro-scale as an isotropic rock.

As explained above, no specific test set-up was being modelled. The RVE of the black box rock model had a square shape. Its size can be assumed to be dimensionless. No platens were integrated into the model, and normal stresses were applied directly (and changed) on the specimen boundaries. So, the specimen can deform freely. The consequence is that no stress–strain curve was recorded and that the testing was stress driven rather than strain driven. Various percentages of failed contact elements were evaluated, but the most stable percentage to indicate full failure was the activation of 50% of all contact elements. For some simulations, a larger percentage would result in a loss of coherence between the individual blocks. A smaller percentage does not clearly represent the full failure process. During the simulations, the type of failure or activation of a contact element was recorded, i.e., tensile, shear, or mixed mode.

The focus was on three basic types of stress paths (Figure 4). All three stress paths started from an isotropic stress state, including a zero-stress state. The first type was characterised by an increase in successive steps of the major principal stress until failure, for a constant minor principal stress. This stress path is similar to the conventional compressive testing, but it is not exactly the same (i.e., square specimen, stress driven, etc.). This type of stress path is abbreviated as S1(loa),S3(=). The second type was just the other way, S1(=),S3(unl), i.e., the minor principal stress was decreased and the major principal stress remained constant. The third type was a combination of the previous two types. It corresponded to the example presented above (Figure 1). The major principal stress was increased, and the minor principal stress was decreased with the same stress increments. This type of stress path is abbreviated as S1(loa),S3(unl). In comparison to the previous two types, the change in deviatoric stress was larger for the third type. Further, some additional stress paths were analysed with some specific aims, e.g., effect of an anisotropic initial stress state.

4. Results for a Medium-to-Strong Black Box Rock

4.1. Failure Envelopes for the Three Basic Types of Stress Paths

First, a set of uniaxial loading experiments were simulated (S1(loa),S3(=)). Four different confining stresses (S3) were applied, i.e., 0, 5, 10, and 15 MPa (Figure 5; vertical stress paths). The failure load was determined for each simulation. As explained above, this was assumed to correspond to 50% of failed or activated contact elements. Figure 5 presents the failure envelope in a major vs. minor principal stress diagram. The failure envelope of the uniaxial loading was the one corresponding best to the classic failure envelope of laboratory experiments. The second set started from the isotropic stress state, which was equal to the failed stress (i.e., the major principal stress) calculated in the first set. In the second set (horizontal stress paths in Figure 5), the minor principal stress was decreased until failure occurred (again defined as 50% of failed or activated contact elements), and the major principal stress remained constant (S1(=),S3(unl)). The reason to use this procedure was that, if there was no effect of the stress path, the failure loads in the second set of simulations should correspond to the confining stresses applied in the first set. This would be the case if one considers a continuum model, whereby a perfect elasto-plastic behaviour is applied. This procedure allows for a relatively easy comparison of the failure loads between both sets (

Table 1. Strength values for the three types of stress paths. (a) Uniaxial loading (S1(loa),S3(=)), i.e., reference for other stress paths. (b) Uniaxial unloading (S1(=),S3(unl)) and reduction along the stress path in comparison to uniaxial loading. (c) Simultaneous loading and unloading (S1(=),S3(=)) and reduction along the stress path in comparison to uniaxial loading.

(a)
Initial Stress State	Stress State at Failure
Isotropic Stresses, MPa	Minor Principal Stress (S3), MPa	Major Principal Stress (S1), MPa
0.0 5.0 10.0 15.0	0.0 5.0 10.0 15.0	79.4 104.2 137.6 157.3
(b)
Initial stress state	Stress state at failure		Strength reduction for S3, MPa (%) (along stress path)
Isotropic stresses, MPa	Minor principal stress (S3), MPa	Major principal stress (S1), MPa
79.4 104.2 137.6 157.3	30.1 46.6 63.6 75.6	79.4 104.2 137.6 157.3	30.1 (38%) 41.6 (40%) 53.6 (39%) 60.6 (39%)
(c)
Initial stress state	Stress state at failure		Strength reduction for S3 and S1, MPa (%) (along stress path)
Isotropic stresses, MPa	Minor principal stress (S3), MPa	Major principal stress (S1), MPa
39.7 54.6 73.8 86.2	8.2 15.4 28.6 35.5	71.2 93.8 119.0 136.8	11.6 (21%) 14.6 (21%) 26.3 (29%) 29.0 (29%)

(b)). Even if the orientation of the contact elements was distributed equally in the RVE, the major and minor principal stresses were in the same direction in all simulations, i.e., major corresponds to the vertical direction in Figure 3. So, in case a small anisotropy existed in the model set-up, its effect would be similar for all simulations. The third set of simulations, i.e., S1(loa),S3(unl), also aimed to reach the failure loads in the first set of simulations and to do so for the same reasons. The initial stress state was again isotropic. The amount of increase in the stress was equal to the amount of decrease in the stress in absolute terms for the major and minor principal stresses (stress paths under an angle of 45° in Figure 5).

The uniaxial loading experiment (S1(loa),S3(=)) without a confinement stress resulted in a failure load of 79.4 MPa (Figure 5, Table 1 (a)). This was the reason the rock was called medium to strong. As explained above, one should not really call this value the UCS-value, as the model set-up was different from the ISRM suggested method of a UCS test (i.e., different shape, different loading, different size, etc.). However, it approximately gives an idea about the uniaxial strength of the black box rock. As pointed out earlier, no calibration took place to reach a specific failure load. In other words, this was the consequence of working with a black box rock. When the same rock was initially under an isotropic load, equal to 79.4 MPa, the unloading of the minor principal stress reached failure for a value of 30.1 MPa (Table 1 (b)). This was about 38% earlier (30.1 MPa vs. 79.4 MPa) than if there would be no effect of the stress path. For the latter assumption, failure should be reached for a zero minor principal stress. The reduction by 38% is significant. For the third type of stress paths, i.e., S1(loa),S3(unl), and aiming at a failure load of (0.0, 79.4) MPa, failure was reached for (8.2, 71.2) MPa. The initial isotropic stress state was 39.7 MPa. So, along the stress path or the 45° line, the failure occurred about 21% earlier, i.e., the distance between (0.0, 79.4) MPa and (8.2, 71.2) MPa vs. the distance between (0.0, 79.4) MPa and (39.7, 39.7) MPa. The conclusion for these three simulations is that the stress path had a significant effect on the failure load.

The other simulations (Figure 5, Table 1) showed the same trend. The unloading of the minor principal stress only, i.e., S1(=),S3(unl), resulted in the weakest failure criterium. By simultaneously increasing the major principal stress and decreasing the minor principal stress, i.e., S1(loa),S3(unl), its failure criterium was situated between the two other criteria. The three failure criteria widened for larger stress levels. Or, in other words, they were not parallel. However, the relative effect remained similar. For example, for a confinement of 10 MPa, the failure for uniaxial loading was reached at 137.6 MPa. Failure for a uniaxial unloading from an initial isotropic stress level of 137.6 MPa was reached for a minor principal stress of 63.6 MPa instead of 10 MPa. In other words, the failure occurred about 39% earlier (53.6 MPa vs. 137.6 MPa). This percentage was 38% for a zero confinement (See Table 1 for other cases).

Figure 6 shows the circles of Mohr for all simulations in a shear stress–normal stress diagram. Per type of stress path, a linear failure criterium was fitted by hand. These fits were acceptable, but the uniaxial loading would benefit most from a non-linear failure criterium. The cohesion (and friction angle) values for the three fitted criteria were about 22 MPa (38°), 10 MPa (15°), and 20 MPa (22°) for S1(loa),S3(=), S1(=),S3(unl), and S1(loa),S3(unl), respectively. As for the presentation in a major–minor principal stress diagram, the three failure criteria are separated clearly.

4.2. (Micro-)Fracturing for the Three Basic Types of Stress Paths

In the literature review above, it was stressed that micro-fracturing plays an important role when loading or unloading rock specimens. Most likely, it is the occurrence of these micro-fractures during the entire period that induces a difference in the final macro-strength and macro-fracture pattern. The Universal Distinct Element Code (UDEC) records the type of contact activation or failure [56,57]. These contacts can fail in tension, shear, or a combination of both. For example, a contact can open (tensile failure), followed in a later calculation step by a closure and/or by a shear displacement. The latter two situations are labelled as tensile failure in the past and tensile failure in the past combined with shear failure, correspondingly. The latter is an example of a mixed failure or activation mode. Figure 7 presents the variation of the contact failure modes as a function of the stress path, i.e., from the initial isotropic stress state until the load level of failure, corresponding to 50% failed contacts. As an example, the three types of stress paths are presented for the case, starting in the loading experiment with a confining pressure of 10 MPa (i.e., the purple stress paths in Figure 5). In Figure 7, the horizontal axis is presented as a relative scale between 0% and 100% from the initial load until the failure load. At the top of each graph, the latter two are indicated as absolute values. As one can observe, different load intervals are needed to realise 50% of failed or activated contact elements. (Also see Table 1).

For all uniaxial loading simulations, i.e., S1(loa),S3(=), it was typical that the fracturing started in tension only and that this was at an early stage of the increase in the major principal stress (Figure 7a). When the major principal stress was increased by 5 MPa (i.e., from isotropic stress state of 10 MPa to (10, 15) MPa), four contact elements (about 1.5% of all contact elements) failed in tension. It was only at about 23% of the failure load (relative to the initial isotropic stress state of 10 MPa) that some of the contact elements started to undergo a shear displacement (mixed mode). At about 40% of the failure load, the first contacts failed in shear only. When the specimen was assumed to have failed completely (

Table 2. Percentage of activation modes (i.e., tensile only, mixed mode, or shear only) for all activated contact elements at moment of failure. (a) uniaxial loading stress path (S1(loa),S3(=)); (b) uniaxial unloading stress path (S1(=),S3(unl)); (c) simultaneous loading and unloading stress path (S1(=),S3(=)).

(a)
Stress State at Failure		Percentage of All Activated Contacts
Minor Principal Stress (S3), MPa	Major Principal Stress (S1), MPa	Tension Only	Mixed Mode	Shear Only
0.0 5.0 10.0 15.0	79.4 104.2 137.6 157.3	36% 26% 22% 18%	28% 26% 22% 30%	36% 48% 56% 52%
(b)
Stress state at failure		Percentage of all activated contacts
Minor principal stress (S3), MPa	Major principal stress (S1), MPa	Tension only	Mixed mode	Shear only
30.1 46.6 63.6 75.6	79.4 104.2 137.6 157.3	42% 36% 32% 28%	40% 36% 46% 40%	18% 28% 22% 32%
(c)
Stress state at failure		Percentage of all activated contacts
Minor principal stress (S3), MPa	Major principal stress (S1), MPa	Tension only	Mixed mode	Shear only
8.2 15.4 28.6 35.5	71.2 93.8 119.0 136.8	45% 29% 32% 24%	31% 42% 32% 38%	24% 29% 36% 38%

(a)), about 56% of the failed contacts (i.e., 28% of all contacts) had failed in shear only and about 22% in a mixed mode. Hence, at the failure load, about 22% of the failed contacts had failed in tension only. For the other uniaxial loading simulations with a non-zero initial stress state, the variation of the failure mode as a function of the stress path was similar, i.e., at the start of loading, the tensile failure was predominant, but at the end, the shear failure was the most important failure or activation mode. For an initial zero-stress state, the number of failed contacts in tension was larger. In the first part of loading (up to 40% of the loading interval until failure), 10 to 20% of all contacts failed in tension, without any failure in shear. At the moment of failure, about 36% of all failed contacts had failed in shear only and about the same percentage had failed in tension only (Table 2 (a)). For the case of 10 MPa initial stress, these percentages were about 56% and 22%, respectively. These observations seem logical. Due to a zero-confinement or a small confinement, the extension type of failure already occurred at the start, and shear bands were induced only for the larger deviatoric stress levels. The presence of more extension type of fractures with zero confinement also is logical.

For all uniaxial unloading simulations, i.e., S1(=),S3(unl), a typical characteristic in comparison to the previous type of simulations is that the (micro-)fracturing started later (Figure 7b). For the first 20% of the total stress path until failure, no contact elements were activated. This clearly was the effect of the large confining stress, which was present at the start of each experiment (i.e., large initial isotropic stress state). Similar to the previous type of stress paths, the first contact activations also were in tension. For the situation of Figure 7b, only tensile activation took place until about 50% of the total load interval. At that moment, about 11% of all contacts had failed. For the other initial isotropic stress states, the first shear activation occurred at lower unloading percentages (between 35% and 45%). During further unloading of the minor principal stress, shear failure continued, but the pure shear activation remained limited in comparison to the uniaxial loading simulations. For the case of Figure 7b, about 22% of all failed contact elements had failed in shear only mode at the moment of the macro-failure (Table 2 (b)). At that moment, 32% of all failed contact elements had undergone a pure tensile activation. The remaining 46% of all failed contact elements were characterised by a mixed mode failure. For the other cases of uniaxial unloading, these percentages were similar (Table 2 (b)). These results seemed to be realistic, certainly when taking into account the simplicity of the model. As illustrated in Figure 2, extension fractures were observed during the unloading phase and before the full failure or macro-failure was reached [13,22].

Finally, for the stress path, where simultaneous loading and unloading took place, i.e., S1(loa),S3(unl), observations of both previous types of stress paths were observed (Figure 7c). For the purple case in Figure 5, the first contact failed in tension only, and this at about 30% of the total load interval until failure. The first activation in shear took place at 55% of the total load interval. At that moment, about 4% of all contact elements had failed in tension only. Both observations were similar for the larger stress state (i.e., blue case in Figure 5). For the two cases with lower stress states (i.e., the red and green cases in Figure 5), the first activations in tension took place at about 15% and 23%, respectively. The difference between the two lower stress levels and the two larger stress levels could be the reason for the slight deviation in the failure envelope of Figure 5. For the case of Figure 7c, at failure, 32% of all failed contacts were the result of tensile-only activation, 36% of them were due to pure shear activation, and the remaining 32% had failed in mixed mode (Table 2 (c). For the two cases with lower stress states (i.e., the red and green cases in Figure 5), the pure shear activations were less, i.e., 24% and 29%, respectively.

4.3. Effect of an Initial Anisotropic Stress State for Uniaxial Unloading Stress Paths

The main conclusion of the 12 simulations presented above was that, due to micro-fracturing during a change in stress state, the material was weakened, and this affected the final strength of the material. For example, the micro-fracturing in the beginning of a loading experiment was different from the micro-fracturing at the beginning of an unloading experiment. Without doubt, in situ rock can undergo many different stress paths (See also further). The initial stress state also was very different between various sites. Above, the initial stress state always was taken to be isotropic. It is a stress state which occurs, but, of course, more often than not the initial stress state is anisotropic. One of the reasons to assume an initial isotropic stress state is to make sure that no activation of contact elements takes place in all cases when the initial stresses are applied to the model. Below is an evaluation of how an initial anisotropic stress state affects the failure envelope. The earlier simulations of uniaxial unloading, i.e., S1(=),S3(unl), were repeated but now for initial minor principal stresses equal to 75% and 50%, respectively, of the initial major principal stress. For these combinations, it was checked that no contacts were activated when applying these initial stress levels. In the simulations presented, it was assumed that the initial minor principal stress was decreased, and, in this way, it remained the minor principal stress. In situ, the initial major principal stress can also decrease, but at a certain moment, this component becomes the minor principal stress.

Figure 8 (

Table 3. Strength values for three anisotropy values of the initial stress state when applying the uniaxial unloading (S1(=),S3(unl)) stress path. The load interval in absolute terms between the initial stress and the failure load also is presented.

Major Principal Stress (=), MPa	Initial Stress Anisotropy (Initial Minor Principal Stress vs. Major Principal Stress)
	100%			75%			50%
	Initial Stress, MPa	Failure Load, MPa	Unloading Interval, MPa	Initial Stress, MPa	Failure Load, MPa	Unloading Interval, MPa	Initial Stress, MPa	Failure Load, MPa	Unloading Interval, MPa
79.4 104.2 137.6	79.4 104.2 137.6	30.1 46.6 63.6	49.3 57.6 74.0	59.6 78.2 103.2	23.4 34.7 47.4	36.2 43.5 55.8	39.7 52.1 68.8	13.5 14.8 31.2	26.2 37.3 37.6

) presents the various stress paths (horizontal lines), whereby the crosses refer to the cases with an initial isotropic stress state, presented and discussed in the previous paragraph (Figure 5). Its failure envelope (S1(=),S3(unl)) is presented in Figure 8 and intersects the set of left crosses. No new failure envelopes were drawn for the two anisotropic initial stress states, but the two other failure envelopes of Figure 5 were reproduced for comparison (i.e., S1(loa),S3(=) and S1(loa),S3(unl)). The reader can easily visualise the new failure envelopes for the two anisotropic initial stress states by looking at the set of left circles and squares, for initial anisotropies of 0.75 and 0.50, respectively.

For the new cases presented, the lower the initial horizontal or minor principal stress was, the stronger the material. The difference between the three anisotropy values (1.00, 0.75, and 0.50) was significant, but it also was relatively consistent. This observation can be explained in that, if one starts from a larger minor principal stress level, one must overall cover a longer path, and thus there is a larger possibility of inducing micro-fractures and of weakening the rock material. It is interesting to note that the total unloading interval decreased significantly with the lower initial minor stress level (Table 3). For example, for a major principal stress equal to 137.6 MPa, the unloading intervals until failure were about 74 MPa, 56 MPa, and 38 MPa for the initial minor principal stresses of 137.6 MPa, 103.2 MPa, and 68.8 MPa, respectively. When starting the uniaxial unloading from an anisotropy of 0.50, the failed load levels were situated around the earlier determined failure envelope of simultaneous loading and unloading (S1(loa),S3(unl)). This meant that the material remained significantly weaker in comparison to a stress path of loading in one direction only (S1(loa),S3(=)), which is the classic failure envelope. In other words, also for these initial anisotropic stresses, the classic failure envelopes overestimated the real in situ strength.

Figure 9 presents graphs such as those in Figure 7b for the purple case of Figure 8, i.e., a constant major principal stress of 137.6 MPa. For all three initial stress levels, no or a few contacts were failed during the first 20% of the entire unloading interval (Figure 7b and Figure 9a,b). As explained above, 20% of the entire stress path interval corresponded to different absolute values, expressed in MPa. In other words, 20% corresponded for the initial isotropic stress state to about 15 MPa (i.e., 20% of 74.0 MPa) and for the initial stress anisotropy of 50% to about 7.5 MPa (i.e., 20% of 37.6 MPa). For the initial stress anisotropy of 75%, it was about 11 MPa (i.e., 20% of 55.8 MPa). However, the largest difference between the three graphs was that until halfway, only tension activations were observed in the entire unload interval, i.e., about 10% of all contact elements, for the initial isotropic stress state (Figure 7b). However, for the two cases of an initial anisotropic stress state, contacts already had begun to fail in shear mode after reaching 20% of the entire unloading interval. So far, this was the first time in all of the simulations that, at the start of fracturing, the contacts were activated in both tensile and shear modes. It was probably logical that with confinement, a larger stress anisotropy led more towards the shear type of failure. At the moment of failure, i.e., activation of 50% of all contact elements, the percentage of the tensile-only mode activation was similar for the six simulations of the initial anisotropic stress states (

Table 4. Percentage of activation modes (i.e., tensile only, mixed mode, or shear only) for all activated contact elements at moment of failure when applying the uniaxial unloading (S1(=),S3(unl)) stress path. (a) Initial stress anisotropy of 100%; (b) initial stress anisotropy of 75%; (c) initial stress anisotropy of 50%.

(a)
Stress State at Failure		Percentage of All Activated Contacts
Minor Principal Stress (S3), MPa	Major Principal Stress (S1), MPa	Tension Only	Mixed Mode	Shear Only
30.1 46.6 63.6	79.4 104.2 137.6	42% 36% 32%	40% 36% 46%	18% 28% 22%
(b)
Stress state at failure		Percentage of all activated contacts
Minor principal stress (S3), MPa	Major principal stress (S1), MPa	Tension only	Mixed mode	Shear only
23.4 34.7 47.4	79.4 104.2 137.6	30% 28% 25%	46% 39% 40%	24% 33% 35%
(c)
Stress state at failure		Percentage of all activated contacts
Minor principal stress (S3), MPa	Major principal stress (S1), MPa	Tension only	Mixed mode	Shear only
13.5 14.8 31.2	79.4 104.2 137.6	30% 31% 26%	42% 38% 34%	28% 31% 40%

(b,c)). It was found that 26% to 31% of the failed contacts were in the tension only mode. For the initial isotropic stress states, this percentage varied between 32% and 42% (Table 4 (a)). This larger percentage was most likely the result of a larger number of tensile contact activations at the start of the fracturing.

5. Comparison between the RVE Model and Published Laboratory Experiments

Prior to compiling this paper, numerous simulations were conducted with alternative set-ups, such as other spatial distributions of contact elements, other property values of the contacts and of the blocks, different initial stress states, different stress paths, and others. All of them provided a picture that was similar to the one presented and discussed in detail above for the so-called medium-to-strong black box rock. Incorporating in the paper all the other simulations would not provide any additional value. However, it was thought useful, mainly to validate the approach of using a RVE model, to apply a complex stress path, published recently [28]. The stress paths presented by Song et al. [28] corresponded to a complex in situ path occurring in the roof of a coal seam during mining. Laboratory specimens of bedded sandstone were loaded by these complex in situ stress paths, and the results were compared to classic loading tests [28]. Their research project aimed to investigate the stability of roadways and the prevention and control of rockburst in deep coal mines. First, cylindrical specimens were tested by applying the conventional triaxial loading path, whereby three different confining stresses were applied (i.e., 5, 10, and 20 MPa). Since the rock can be classed as transversally isotropic, five different inclination angles of the bedding planes were tested per confining stress, i.e., 0°, 30°, 45°, 60°, and 90°. In other words, 15 specimens were tested in this way. Second, the same number of cylindrical specimens were loaded following a three-step stress path. The zero-stress specimens were loaded in a way that was approximately isotropic until a stress state of about 25 MPa (circumference) to 27 MPa (axial) was reached. In the next step, the circumferential stress was decreased until a value of 5, 10, or 20 MPa was reached. In the final step, the axial load was increased until failure occurred. The circumferential stress was then kept constant. The provided data [28] allow for a comparison between the strength for both stress paths. The average of the five reduction values, i.e., for the five different inclination angles, showed that the specimens with a confinement stress of 5 MPa were about 24% weaker when the three-step stress path of loading–unloading–loading was applied in comparison to the conventional triaxial tests. For larger confinement stresses, these reductions were 11% and 6% on average for 10 MPa and 20 MPa of confinement stress, respectively.

The aim of the simulations presented below was not to simulate the tests conducted by Song et al. [28]. There are various reasons why that is not opportune for this study, e.g., the shapes and volumes of the specimens were different, calibration was needed, and the transversally isotropic property must be integrated in the model. However, by applying the more complex stress paths on the RVE model, one could learn something about the merits of the RVE model and the approach presented in this paper, i.e., when similar percentages in the strength reductions are observed between the laboratory experiments [28] and the RVE model.

The same input parameters for the medium-to-strong black box rock were considered, as applied in the presented simulations above. The initial stress state corresponded to an isotropic stress of 25 MPa, and, in a first step, one principal stress component was reduced to 10 MPa, 12.5 MPa, and 15 MPa (Figure 10a). Afterwards, the major principal stress was increased until failure occurred. For the black box model, a reduction to 5 MPa in the first step would already result in a full failure. A reduction to 20 MPa would not induce the activation of a contact element. The reason for both findings is that the black box rock is different from the bedded sandstone [28].

For a confining stress of 10 MPa in the loading part, the strength reduction along the final stress path was 47% in comparison to the failure criterium of uniaxial loading (73 MPa (Figure 10b,

Table 5. Percentage of activation modes (i.e., tensile only, mixed mode, or shear only) for the unloading of the minor principal stress, followed by the loading of the major principal stress, whereby three stress levels were considered for the change from unloading to loading, i.e., 10 MPa, 12.5 MPa, and 15 MPa. (a) Percentage at change from unloading to loading for contact elements activated at that moment; (b) percentage at failure for activated contact elements.

(a)
Stress State at Change from Unloading to Loading		Percentage of Activated Contacts at That Moment
Minor Principal Stress (S3), MPa	Major Principal Stress (S1), MPa	Tension Only	Mixed Mode	Shear Only
10.0 12.5 15.0	25.0 25.0 25.0	86% 100% 100%	11% 0% 0%	3% 0% 0%
(b)
Stress state at failure		Percentage of all activated contacts
Minor principal stress (S3), MPa	Major principal stress (S1), MPa	Tension only	Mixed mode	Shear only
10.0 12.5 15.0	72.8 116.1 153.0	55% 38% 26%	19% 22% 18%	26% 40% 56%

(b)) vs. 138 MPa (Table 2 (a))). This reduction decreased to 21% (116 MPa vs. 147 MPa) and 3% (153 MPa vs. 157 MPa) for confining stresses of 12.5 MPa and 15 MPa, respectively. The trend in reduction is similar to that in [28]. Less unloading, i.e., a larger confining stress, resulted in less reduction of the strength. By taking the differences into account between the laboratory experiments and specimens, as well as the model, it can be concluded that similar reduction percentages were obtained. Above, the averages over five inclination angles were given for the laboratory experiments on bedded sandstone (i.e., 6%, 11%, and 24%). The strength reduction for the same inclination angle and confining stress varied more [28]. Sometimes, no reduction was observed, and sometimes the reduction values were larger. For example, a reduction of 41% was observed for the two specimens with an inclination of 45° and a low confining stress (5 MPa) [28]. Or, in other words, the 47% reduction that was calculated for the simulations with the smallest confining stress certainly seemed to be realistic.

Figure 11 presents the variation of the mode of contact element activations for the three simulations, i.e., the unloading of the minor principal stress, followed by the loading of the major principal stress. The total stress path interval is considered in these graphs, i.e., the sum in absolute terms of the unloading interval of the minor principal stress and of the loading interval of the major principal stress until failure. In other words, two different stress directions were added, which is unconventional, but it was justifiable for these specific graphs. The change from unloading to loading or the change from horizontal to vertical stress paths is indicated by the coloured vertical lines. By referring to the basic uniaxial unloading simulations, i.e., S1(=),S3(unl), it was observed that the contact elements only failed in tension during the first part of the unloading interval (Figure 7b). Thus, for the simulations of the unloading, followed by loading in the other direction, it was logical that tensile activation was the predominant mode during the first phase, i.e., the unloading interval. When unloading until 12.5 MPa and 15 MPa, only tensile activation was observed (Figure 11b,c, Table 5 (a)). When the unloading covered 15 MPa (Figure 11a, light blue colour), i.e., until a stress of 10 MPa, a limited number of contacts also failed in shear at the end of this phase, but the large proportion of failed contacts was in the tensile-only mode, i.e., 86% of all failed contacts at that moment (Table 5 (a)). For this unloading interval, i.e., over 15 MPa, the weakening of the rock material was the largest in comparison to the two other unloading intervals. At the end of this interval, about 23% of all available contact elements had failed, i.e., for a final unloaded stress of 10 MPa (Figure 11a). For the shorter unloading intervals, these percentages were about 14.5% and 9.5%, i.e., until 12.5 MPa and 15 MPa, respectively.

In the successive loading phase, the amount of tensile activation remained large for the confining stress of 10 MPa. For this case (Figure 11a) and at the moment of full failure, 55% of all failed contacts were in tension-only mode, 26% were in shear-only mode, and the remaining 19% were in mixed mode (Table 5 (b)). For the basic uniaxial loading simulations for a confining stress of 10 MPa (Figure 7a, Table 2 (a)), i.e., S1(loa),S3(=), these percentages were 22% tension, 56% shear, and 22% mixed, respectively. The difference for both stress paths was significant. The reason for this difference was most likely the relatively large number of tensile activated contacts at the moment that the uniaxial loading phase started, in comparison to zero activated contact elements when an initial isotropic stress state of 10 MPa was applied, as shown in Figure 7a. In other words, the effect of the preceding unloading phase from (25, 25) MPa to (10, 25 MPa) was large, and this interval weakened the rock significantly.

When looking at the case of an unloading interval of 10 MPa, until a stress level of 15 MPa occurred at the start of the successive loading interval, the effect of the unloading interval was smaller. As mentioned above (Figure 11c), at the moment of the change from unloading to loading in the other direction, 9.5% of all available contact elements had failed. At the moment of failure, the distribution of the different failure modes was more similar to what was observed for a uniaxial loading only from an initial isotropic stress state of 15 MPa (Table 2 (a) and Table 5 (b)). For the latter stress path, 18% of the failed contacts were in tension only, 52% were in shear only, and 30% were in mixed mode. For the unloading that was followed by loading in the other direction, these percentages were 26%, 56%, and 18%, respectively. In other words, in comparison to the large unloading interval of 15 MPa (Figure 11a), now there was far less activation in tension and much more activation in shear (Figure 11c), which resembled the stress path with only uniaxial loading more. The case with a change from unloading to loading at a minor principal stress of 12.5 MPa (Figure 11b) clearly showed the transition from the behaviour for 10 MPa to the one for 15 MPa.

These additional simulations, referring to the stress paths applied in the laboratory [28], indicated that the modelling approach to quantify the effect of stress paths indeed has merits. The importance of (micro-)fracturing during a specific interval in a stress path is of the upmost importance when analysing the failure of rock. Even though the RVE model is relatively small and simple, the interpretation of the data shows a logic behaviour, e.g., when analysing the different failure modes, and it provides useful insights concerning the behaviour of the rock. The simulations also show that the behaviour and failure load are sensitive to the various stress levels. Unloading the minor principal stress over an interval of 10 or 15 MPa makes a significant difference in the micro-fracturing and thus in the final strength.

6. Discussion: Consequences of Results Obtained

At the end of this paper, one should reflect on the consequences of the above results. It is well known that rock evolves from one stress state to another stress state under in situ conditions, e.g., around an excavation. The initial stress state is always under compression. Close to the excavation, where the probability of failure normally is the largest, at least one stress component decreases. In laboratory experiments, one starts from a free specimen, i.e., a specimen that is under zero stress. For all compression tests (e.g., uniaxial, triaxial, and true triaxial), one externally increases the compressive stress(es). Or, in other words, one loads the specimen until failure occurs. And these data are used to determine the failure envelope and other characteristics. So, in laboratory experiments for a basic characterisation of rock, the stress paths are different from the in situ stress paths. As illustrated by the discrete simulations for a RVE model, the followed stress paths had a non-negligeable effect on the failure envelope or strength. The reason for this was the different micro-fracturing over the entire period of the stress paths. The question remains as to how this difference affects the calculations and design applications.

First, if one would assume the rock to be a perfect continuum, i.e., without any micro-fissures and no occurrence of such features during the stress variation, although we know that this is a theoretical (and incorrect) model, we still apply it frequently. If we conduct linear elastic or perfect elasto-plastic calculations for such a continuum model, the results presented above indicate a systematic error. Let us consider a simple set-up, i.e., the in situ stresses are isotropic, a circular opening is excavated or drilled, and no support is present. In this case, the major principal stress, i.e., the tangential stress, is twice the initial stress, and the minor principal stress, i.e., the radial stress, is equal to zero. For the medium-to-strong black box rock (Figure 5), the conclusion is that, for in situ stresses smaller than 39.7 MPa (half the uniaxial strength of 79.4 MPa; Table 1 (a)), no failure would occur around the circular opening, i.e., when the rock is characterised by a classic loading experiment. However, if one were to characterise the rock by a combination of simultaneous loading and unloading, i.e., stress path S1(loa),S3(unl), the conclusion would be that the surrounding rock has already failed at an in situ stress of about 15 MPa. The intersect of the failure envelope S1(loa),S3(unl) with the vertical axis was situated at about 30 MPa (Figure 5). The difference in failure prediction was large, i.e., an isotropic, in situ stress of about 40 MPa vs. 15 MPa. Of course, this information must be compared to other uncertainties that are present in rock mechanical studies. However, the difference with other uncertainties is that the effect of the stress path seems to be a systematic overestimation of the strength in classic laboratory experiments, while the other uncertainties are linked to spatial variations and the heterogeneous nature of the rock material. Sometimes the latter results in larger values, and sometimes it results in smaller values. In other words, they are situated around an average. And, of course, one could criticise the choice of a continuum model and linear elastic or perfect elasto-plastic calculations, but that is not the point in this discussion. For various reasons, especially the simplicity of the calculations, continuum and elastic models are still used extensively. However, more and more discontinuum models are being used. They show that incorporating flaws or weaknesses in the model has an important effect on when fractures are induced around a circular opening, both at the laboratory scale (e.g., [38,48]) and in situ (e.g., [58]).

Second, the effect of the results presented also should be discussed if one applies discontinuous or discrete models to estimate the stress redistribution and the fracturing around excavations. One could claim that if one would be able to incorporate correctly all natural weak elements into a model of the in situ situation, and the numerical method would perfectly simulate the real behaviour of a rock, the results presented above would not be relevant, as the effect of the stress paths would be implemented correctly. However, first, the characteristics and location of the weak elements are not known, and second, each model is based on assumptions. Rather, most rock specimens are a black box, and they are far different from a white box. This is why one must calibrate a model before applying it in other conditions. Most of the calibrations of discontinuous models use uniaxial compressive tests and Brazilian tensile tests. Referring to the results presented above for different loading schemes, it is probably more advisable to calibrate a discontinuous model by using laboratory experiments that apply a stress path, closer to the in situ stress variations which one expects. Or, at least to use it in addition to classic laboratory tests, i.e., uniaxial compressive tests and Brazilian tensile tests.

An additional problem highlighted by applying various stress paths on black box rocks is that the derived failure envelopes are different for each different stress path. The only conclusion that can be linked to this observation is that the rock behaves in an even more complex way than one normally assumes, and that micro-fracturing plays an important role. From a practical perspective, it is impossible to test all possible stress paths. For example, again, for the relatively simple case of a circular excavation, the stress state in each location along the circumference is different if the initial stress state is anisotropic. So far, the simulations have shown that the various stress paths result in failure envelopes that are situated between the one whereby the major principal stress is increased and the one whereby the minor principal stress is decreased. So, at least by conducting experiments for these two sets of stress paths, one gets a relatively good idea about the possible effect of a stress path on the failure envelope. And then this information could be translated into an uncertainty factor. As artificial intelligence (AI) finds its way more and more in the field of geosciences [59,60], one should learn a lot by working with black box models and repeating simulations a large number of times to evaluate the impact of different scenarios, e.g., different in situ stress paths around an excavation. The impact of some of these scenarios has certainly to be verified by in situ or laboratory experiments, but from a practical and financial point of view, the amount of such experiments will always remain (too) small. As has been shown by a comparison with published data of lab experiments [16,20,28], using a RVE model to study the rock behaviour provides good results. The aim of modelling is not to use the most complex model or the largest feasible model, but to use a model that helps us to better understand the process of rock failure. Or, as Corkum and Board [61] pointed out, the philosophy should be to apply certain simplifications to focus on the dominant behaviour and, at the same time, avoid the use of overly sophisticated methods. So, a good compromise is needed between enough details, but not too many. For the study of rock failure and rock strength, incorporating the occurrence and growth of (micro-)fractures in the model is such an essential “detail”.

7. Conclusions and Summary

The main conclusion for the results presented is that the type of stress path influences the failure envelope, whereby the classic laboratory experiments overestimate the rock strength in comparison to the real in situ stress paths. This conclusion is derived for a RVE model, using distinct element simulations. The latter were earlier calibrated at laboratory scale [54,55]. The simulations clearly indicate the importance of (micro-)fracturing during the entire interval of stress changes. This (micro-)fracturing influences the macro-fracture and the strength. The simulations also indicate that the effect is significant. Strength reduction values of up to 40% were calculated. The strength reduction values in the simulations seemed to be realistic when compared to some published laboratory experiments [16,20,28].

In situ rock can undergo very complex stress paths. In this study, three basic stress paths were analysed. First, the conventional loading experiments, whereby the minor principal stress was kept constant, and the major principal stress was increased till failure. These simulations were close to the conventional compressive tests. The other two stress paths were close to the in situ change of stress around an excavation. The second stress path was rather the opposite of the first ones. For the second stress path, the major principal stress was kept constant, and the minor principal stress was decreased until failure occurred. The simulations that started from an initial isotropic stress state, followed by unloading one stress component, resulted in the weakest failure envelope. The simulations close to the conventional loading resulted in the strongest failure envelope and, thus, significantly overestimated the real in situ strength. The third basic type of stress path was the simultaneous increase in the major principal stress and decrease in the minor principal stress with the same absolute stress increments. The failure envelope of this third type was situated between the two previous envelopes. If the second type of stress paths was applied, but starting from an anisotropic stress state, the failure envelope for a large anisotropy moved closer to the one of the conventional loading paths. For these additional simulations, the initial minor principal stress was decreased.

The variation of the fracture mode (i.e., tensile, shear, or mixed) during the stress changes showed some interesting observations. The start of all (micro-)fracturing in the simulations was characterised by tensile failure. This tensile (micro-)fracturing started at small load increases of the major principal stress for the stress paths close to the conventional loading experiments. At the end of the stress path, the predominant fracturing mode was shear. At the point of macro-failure, the sum of the proportion of shear and mixed modes varied between 64% (unconfined loading) and 82% (largest confinement stresses) of all activated contact elements. It was difficult to conclude if this was also what happens in the laboratory when studying acoustic emissions. For cored specimens, the Kaiser effect could play a role [34], and this could incorrectly result in an absence of acoustic emission for low stress levels in laboratory experiments. The black box models were not affected by the memory effect, as no coring, i.e., stress relaxation, took place. Chang and Lee [43] clearly observed acoustic emissions from the start of axial loading in a triaxial test set-up, i.e., similar to the simulations. The analysis of the recorded signals indicated that the predominant failure mechanism was in shear mode, also at the start of loading. However, non-negligeable amounts of tensile and mixed mode cracks were recorded from the initial cracks until full failure occurred. A similar observation was made for uniaxial [46] and biaxial compression tests [31]. For the simulations of the two other basic stress paths with an unloading component (with and without an increase in the major principal stress), no activation occurred in the first 20% to 30% of the entire stress interval until failure. At the point of macro-failure, no activation mode of the three was predominant.

The effect of the (micro-)fracturing during the entire stress path on the strength was illustrated best by the example of a decrease in the minor principal stress followed by an increase in the major principal stress until failure occurred, i.e., a stress path which was based on the study by Song et al. [28]. During the interval when unloading occurred, the predominant mode of (micro-)fracturing was tensile. For the largest unloading interval simulated (i.e., 15 MPa), many contact elements already were activated at the end of this interval, i.e., about 23% of all contact elements. These activated contact elements significantly weakened the rock and had a significant effect on the final strength. The tensile activation remained predominant over the entire stress path.

The original aim of the simulations certainly was not to suggest a revolutionary change in how rock specimens should be best tested in the laboratory. At the end of this initial study, the need for such a change is not the conclusion. However, I am now more convinced than at the start of the study that, in addition to the conventional tests, it would be advisable for certain projects and from time to time to also apply stress paths in laboratory, which are closer to the in situ stress paths. This will help the rock mechanical community to better understand the behaviour of in situ rock. A change in the way rock is characterised has a direct impact on the design of rock excavations in a high-stress environment.