Introduction

In the past few decades, when the mining operation goes deeper, the mechanical response of rock around deep excavation has been observed different from that at the shallow depth. In particular, the dynamic effect induced by excavation would be a very important factor affecting the development excavation damaged zone (EDZ) (Zhu et al. 2014a). During the underground excavations such as mining opening, tunnel, nuclear waste repository and underground protective engineering, the surrounding rock masses may undergo a complex loading and unloading process, especially under high in situ stress. Deep rock is subjected to high in situ stress (quasi-static triaxial stress) before excavation and the unloading owing to the formation of free surface which gives rise to stress change from original three-dimensional to two-dimensional or one-dimensional state. In addition, the excavation surface may experience dynamic disturbance such as rock blasting, which may trigger the occurrence of unstable failure of surrounding rock mass. Therefore, the failure characteristic of rock sequentially subjected to the quasi-static triaxial loading, unloading of confining pressure and dynamic disturbance defines one of the most important mechanisms that excavation might induce the unstable rock failure such as rock bursts (Wu and Liu 2009; Zhao and Wang 2010).

Currently, more attention is paid to study the influence of stress unloading on the failure of rock as a result of excavation activities. Extensive studies have been performed to investigate the unloading mechanism and predict the extent of excavation damaged zone (Falls and Young 1998; Backblom and Martin 1999; Diederichsa et al. 2004; Read 2004; Martino and Chandler 2004). In this respect, many theoretical models, laboratory and in situ experiments and numerical simulations are conducted. Cai and Kaiser (2005) presented a method to quantify the degree of damage and to characterize the excavation damaged zone (or excavation disturbed zone) based on microseismic monitoring data. Guan et al. (2007) reported two categories of theoretical methods for ground reaction analyses in conventional tunneling excavation, in which one of them the unloading process of ground is taken into account. The non-Euclidean dynamic model is proposed to investigate the zonal disintegration mechanism of isotropic rock masses around a deep circular tunnel subjected to dynamic unloading (Zhou and Shou 2013). Yang et al. (2014) simplified the analysis on the possible excavation damaged zone around a tunnel where the constitutive curve under loading and unloading is adopted. An exact closed-form analytical solution for the ground response around cylindrical and spherical openings unloaded from isotropic and uniform stress states was proposed by Vrakas and Anagnostou (2014). Moreover, a series of representative unloading experiments (Hua and You 2001; Corkum and Martin 2007b; Malmgren et al. 2007; Zhou et al. 2008) on rocks were carried out to clarify the fracture features observed from the underground workshop excavation zone. Based on many physical and numerical model tests, Sellers and Klerck (2000) investigated the behavior of the rock surrounding circular excavations during loading and unloading. Bossart et al. (2002) analyzed the geological and hydraulic characterization of the excavation disturbed zone. Fakhimi et al. (2002) performed a biaxial compression test on a sandstone specimen with a circular opening to simulate the failure around an underground excavation in brittle rock. The mechanism of crack initiation, propagation and interaction was also studied in deep tunneling in brittle rock with continuous excavation (Diederichsa et al. 2004). Conventional triaxial unloading tests were also conducted to reveal the characteristics of stress–strain relation, deformation, damage, modulus and strength under unloading conditions (Huang et al. 2001; Xie and He 2004). The rock-burst process under true triaxial unloading was monitored by using the acoustic emission monitoring technique (He et al. 2010; He and Zhao 2013). The experiment of the thick-walled hollow cylindrical specimens subjected to a mechanical unloading to mimic a gallery excavation was carried out to investigate the excavation damaged zone (EDZ) around Opalinus Clay and the nuclear waste disposal in clay formations (Labiouse and Vietor 2014). Axisymmetric triaxial loading–unloading tests are conducted on marble specimens with initial different confining pressures and unloading rates (Huang and Li 2014). Using a true triaxial state test condition, one side of the specimen was unloaded to model the excavation and study the phenomenon of splitting failure in the rock mass adjacent to an underground excavation in a region of high initial stresses (Zhu et al. 2014b).

In addition to these theoretical models and experimental studies, numerical simulation is also conducted to analyze the failure mechanism and pattern under excavation-induced unloading. Three extensions to the discontinuous deformation analysis (DDA) method, as suggested by Kim et al. (1999), consisted of hydromechanical coupling between rock blocks and steady water flow in fractures, sequential loading or unloading, and rock reinforcement, were used to simulate underground excavation in fractured rock. A study of bifurcation in the problem of unloading a circular excavation in a strain-softening rock mass is investigated with the FDM-based code (Varas et al. 2005). The unloading processes of rocks under high initial triaxial stress were analyzed by the finite element method with continuous surface cap model (CSCM) (Tao et al. 2012, 2013) and damage-based model (Zhu et al. 2014a), respectively, in which the influence of in situ stresses, the unloading rate and path on the dynamic effects of excavation is examined. The similar numerical simulation is also conducted by the use of discrete element code PFC2D (Li et al. 2014a).

In reality, underground rock also experiences dynamic disturbance caused by mechanical shock and drilling and blasting operation. Using the self-invented SHPB device, Li et al. (2008) and Zhou et al. (2014) investigated the dynamic compressive and tensile behavior of rocks under different coupled static and dynamic loads. Yin et al. (2012a) analyzed the failure mechanism and failure surface morphology of highly stressed rocks subjected to unloading and dynamic disturbance, and they observed tensile failure or combined shear failure under low impact energy and tensile failure under high impact energy. The compression strength of sandstone subjected to dynamic disturbance after excavation was studied with the coupled dynamic and static loading split Hopkinson pressure bar system (Ye et al. 2013). The damage and failure process of rock under combined static and dynamic loading was also simulated by using the RFPA-Dynamics (Zhu et al. 2012).

The analytical solution and physical experiments are very important for the rock failure analysis under dynamic loading, especially for the excavation damage zone, but these boundary and initial conditions for the numerical simulation on rock failure are too complex for the analytical and experimental approach to tackle. Therefore, it is difficult to give the mathematical model of rock failure in differential form and conduct the physical experiments under complex loading and unloading path. Compared with the experimental observation, failure process of rock can be easily examined with the numerical simulation; thus, the advantage of numerical simulation is highlighted. In this work, the failure process of rock sequentially subjected to the quasi-static triaxial loading before excavation, unloading of confining pressure and dynamic disturbance after excavation is simulated by the simulator RFPA-Dynamics, which makes it possible to reveal the damage and fracture mechanism of rock during this complex loading path.

Model setup for rock under confined compression and unloading of confining pressure

Briefly, the simulator used in this work is a two-dimensional (2D) code that can simulate the fracture and failure process of rock under static, dynamic and combined static and dynamic loading conditions (Zhu et al. 2012). In RFPA-Dynamics, the rock specimen is assumed to be composed of many elements with the same size, and the mechanical properties of these elements are assumed to conform to a given Weibull distribution (Weibull 1951). The finite element method is employed to obtain the stress fields in the mesoscopic elements. Elastic damage mechanics is used to describe the constitutive law of the mesoscale elements when the maximum tensile strain criterion and the Mohr–Coulomb criterion that incorporate the effect of stress rate are utilized as damage thresholds (Zhu and Tang 2004). Flowchart for the coupled static and dynamic analysis of RFPA-Dynamics code has been given in the paper (Zhu et al. 2010).

More descriptions of the theoretical basis, capabilities and validations of RFPA and RFPA-Dynamics have been given in previous publications (Zhu and Tang 2004, 2006; Zhu et al. 2010). Therefore, detailed descriptions of RFPA-Dynamics, as well as its validation, are not presented here. Rather, the applicability of RFPA-Dynamics to simulate the failure process of rock under unloading of confining pressure and dynamic disturbance will be considered.

In this section, the numerical model is established according to previous experiments with sandstone conducted by Yin et al. (2012b) and Huang et al. (2011). The mass density of the specimen is 2500 kg/m3. The uniaxial compressive strength (UCS) and tensile strength of the rock are 90.0 and 9.0 MPa, respectively. The Young’s modulus, Poisson ratio and internal frictional angle of the specimen are 21.6 GPa, 0.25° and 35°, respectively.

The Young’s modulus and uniaxial compressive strength can be simulated with a Weibull distribution with the parameters listed in Table 1 when the rock specimen is assumed heterogeneous. The corresponding means of the Young’s modulus and strength are obtained from a trial-and-error simulation process in order to fit the physical–mechanical parameters of this rock. In this numerical simulation, no heterogeneity is incorporated in the mass density, Poisson’s ratio and internal frictional angle, and the ratio of uniaxial compressive/tensile strength is assigned a constant of 10.

Table 1 Weibull distribution parameters of heterogeneous rock specimen
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In order to reduce the computing efforts, the analyses presented here are performed under plane stress condition. As shown in Fig. 1, the rock specimen has a dimension of 135 mm × 135 mm and is composed of 135 × 135 quadrilateral iso-parametric elements. The upper and bottom loading plates are stiff and with a width of 27 mm. The left and right loading plates are soft and with a width of 18 mm. The soft load-bearing plate with a width of 18 mm is sandwiched between the rock specimen and the right loading plate, where the unloading of the rock specimen in horizontal direction becomes possible by removing this plate instantly after the high confined pressure is applied.

Fig. 1
figure 1

Model setup of the numerical simulation

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Validation of numerical simulator RFPA-dynamics

Previous study by Huang et al. (2011) indicated that the failure of rock under confined compression shows intensive dilatation along the unloading direction once the confining pressure is unloaded. In this section, the failure characteristics of rock specimens sequentially subjected to highly confined compression, unloading of confining stress and disturbance of dynamic loading are studied and are compared previous experimental results, in order to validate RFPA-Dynamics in simulating this kind of complex loading paths.

The failure process of rock under quasi-static loading

The numerical simulation on the rock failure under uniaxial quasi-static loading is also achieved with the model setup as shown in Fig. 1, in which, however, the left and right sides of the specimen are free from the confinement of soft load-bearing plates. An external displacement is applied at a constant rate of 0.005 mm/step in the vertical direction until the rock specimen fails.

The stress–strain curve of rock specimen under uniaxial compression is shown in Fig. 2. The stress–strain curve is basically linear until peak loading is attained. Then, the specimen loses its loading capacity substantially because the specimen is relatively brittle. The uniaxial compressive strength of the specimen is 86.98 MPa, and it approximates the experimental result given by Yin et al. (2012b).

Fig. 2
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The stress–strain curve and failure pattern of specimen under uniaxial compression loading

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As shown in Fig. 2, the isolated damages initiate near the surface in the lower left corner of the specimen and then propagate diagonally upwards to the center of the specimen. A shear damage zone is also developed at the lower right corner of the specimen and propagates diagonally upwards to interconnect with the former damage zone. At last, an X-shaped damage zone that looks like a shear failure zone is formed, leading to the failure of this rock specimen.

The failure process of rock under dynamic loading after confining pressure unloading

In this section, the rock specimen is firstly subjected to confined compression. The boundary stresses in the vertical and horizontal directions, denoted with p s and κp s (κ is called lateral pressure coefficient), respectively, are applied incrementally until pre-specified quasi-static stresses are attained. The vertical stress (p s) is 54 MPa and approximately equals to sixty percent of the uniaxial compressive strength. The lateral pressure coefficient (κ) is 0.74. The bottom and right sides of the model domain are prevented from displacement in the vertical and horizontal directions, respectively. Then, the unloading of confining pressure is achieved through removing the soft load-bearing plate. At last, the dynamic incident stress pulse p d(t) as shown in Fig. 3, obtained from the previous published (Yin et al. 2012b; Ye et al. 2013), is applied at the top surface of a model domain after confining pressure unloading. As for the dynamic loading condition, the bottom surface of the domain is fixed in vertical direction.

Fig. 3
figure 3

The incident stress wave applied in numerical simulation and obtained from the previous published (Yin et al. 2012b; Ye et al. 2013)

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The failure patterns, together with the damage zone, elastic modulus and maximum shear stress distribution during the failure process of rock, are shown in Fig. 4. At t = 0 μs, the confining pressure is unloaded, and the dynamic stress wave is applied. The rock specimen does not fail after confining pressure unloading, but there is tensile damage zones appearing near the free surface in the upper and lower right corner. Then, this damage zone at the upper right corner of the specimen develops further inwards into rock after the dynamic disturbance is applied at top of the domain in vertical direction. However, the damage zone at the lower right corner does not develop until the dynamic stress wave travels toward this location around t = 100 μs. When t = 125 μs, the arc-shaped damage zone is observed near the free face. If the rock specimen is subjected to the unloading of confining pressure symmetrically at left and right sides, two arc-shaped damage zones will develop symmetrically at both the left and right sides of the specimen. In that case, finally, the failure mode of the rock specimen would be similar to the experimental observation given by Yin et al. (2012b), as shown in Fig. 5.

Fig. 4
figure 4

Distribution of damage, elastic modulus and maximum shear stress during the failure of rock specimen under dynamic loading after confining pressure unloading

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Fig. 5
figure 5

Typical failure pattern of sandstone under dynamic loading after confining pressure unloading (Yin et al. 2012b)

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In general, the acceptable agreement between the experimental and numerical observations confirms the capability of RFPA-Dynamics in reproducing the failure process of rock sequentially subjected to the quasi-static triaxial loading, unloading of confining pressure and disturbance of dynamic loading.

Further simulation on the rock specimen under complex loading paths

In this section, numerical simulation of rock failure under different loading paths in terms of dynamic loading after confined compression and confining pressure unloading is discussed. The axial pressures (p s) of 16.0, 32.0 and 48.0 MPa are, respectively, applied step by step at the top of the rock specimen in vertical direction, while the confining pressure coefficient κ = 0.3, 1.0 and 1.5 is specified. After a specific static stress level is attained, the confining pressure unloads, and then, a dynamic incident stress wave is applied at the top surface. The development of excavation damaged zone is mainly influenced by the dynamic disturbance under the low in situ stress conditions. The rectangle wave and triangular wave can be simplified as impacting loading during the impacting tests and are regarded as the special trapezoidal wave through adjusting the rising time, falling time and constant time. In order to clarify the effect of waveforms of incident waves on the rock failure, the incident stress waveform, as shown in Fig. 6, is applied. The amplitudes of the incident stress pulses p dm = 100, 200, 300 and 400 MPa are specified, while the time durations of the stress pulses t dm = 4, 9 and 13 μs are selected.

Fig. 6
figure 6

The incident stress waves applied at the top surface of the model domain with different amplitudes and durations

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Effect of the axial pressure

Figure 7 shows the failure process of the rock specimen subjected to complex loading paths with various axial pressures while κ = 1.5. The axial pressures p s = 16.0, 32.0 and 48.0 MPa are applied, based on which the corresponding overburden depth is 640, 1280 and 1920 m, respectively. The incident stress wave II in Fig. 6 is applied, where p dm = 200.0 MPa and t dm = 9 μs, and the failure process of rock specimen under different in situ stress conditions and a specific dynamic disturbance is investigated as shown in Fig. 7.

Fig. 7
figure 7

The failure processes of rocks subjected to complex loading paths with various axial loading

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From Fig. 7, at t = 0 μs, the damage initiates near the free surface in the upper and lower right corner of the specimen under the higher axial pressure of p s = 48.0 MPa; however, it does not when p s = 16.0 and 32.0 MPa. This indicates that the initiation of this initial damage after confining pressure unloading is closely related to the axial pressure. The effect of the confining pressure unloading becomes more pronounced under the initial high axial pressure, since more energy is stored during the quasi-static loading stage. At t = 28 μs, the dynamic stress wave propagates downloads until the half height of the specimen. The damage zone is developed near the upper end of the specimen, and it propagates downwards, forming some vertical cracks, which is due to the tensile damage induced by high compressive stress in axial direction. This kind of axial crack is restrained under the higher axial stress, because the more pronounced confinement occurs under the higher axial pressure, as shown in Fig. 8. Figure 8 presents the distribution of stress (σ X ) in X-direction along the horizontal line crossing the center of the specimen. Stress σ X becomes higher with axial pressure until the specific static stress level is attained, as shown in Fig. 8a. When the dynamic loading is applied, the stress σ X decreases along the X-direction; however, its confinement on the left part of the specimen becomes more pronounced under the higher axial pressure. In general, as shown in Fig. 7a, the tensile crack initiates parallel to the axial loading direction, especially at the left half of the specimen, followed by the arc-shaped spalling cracks initiated at the right unloading surface after confining pressure unloading. In this regard, the volume of arc-shaped failure zone increases and the number of axial cracks decreases with the rising axial pressure, denoting that the high axial pressure tends to promote the arc-shaped failure zone but restrain axial cracks. It is worth mentioning that under the low axial pressure, the axial cracks are connected with the arc-shaped spalling cracks, reflecting the interaction between these two kinds of cracking mechanisms. This denotes that the failure mode of the rock specimen under this complex loading path depends on the axial pressure under dynamic disturbance after confining pressure unloading.

Fig. 8
figure 8

Stresses in X-direction σ X along the horizontal line crossing the center of the specimen subjected to complex loading paths with various axial loadings. a The final step under static loading, b t = 0 μs, c t = 56 μs, d t = 112 μs

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Before the dynamic stress wave is applied, the rock specimen is intact even though the confining pressure is unloaded, as shown in Fig. 7 (t = 0 μs), thus confirming the contribution of dynamic stress wave in triggering the arc-shaped spalling. In this respect, because quasi-static axial pressure is not provide sufficient energy for the rock failure during confining pressure unloading, the further inpouring of energy due to dynamic disturbance is an important factor to trigger the rock failure.

Effect of the lateral pressure coefficient

In this section, the axial pressure is a constant of p s = 48 MPa, and the lateral pressure coefficients of κ = 0.3, 1.0 and 1.5 are applied. As for dynamic disturbance, the incident stress wave II as shown in Fig. 6 is also applied.

As shown in Fig. 9, only at the lateral pressure coefficient of κ = 1.5, the unloading of confining pressure leads to the damage zone initiation near the free surface in the upper and lower right corner of the specimen. This denotes that, similar to the contribution of high axial pressure, the high confining pressure is also a main factor to induce the failure after excavation, because both the axial pressure and confining pressure may contribute to the accumulation of elastic strain energy in the rock prior to unloading.

Fig. 9
figure 9

The failure processes of rocks subjected to complex loading paths with various confining pressures (lateral pressure coefficient)

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Under different lateral pressure coefficients, the arc-shaped spalling cracks near the right free surface are developed, and the area of them increases with the lateral pressure coefficient. However, the axial cracks are developed more evident under the low lateral pressure coefficients, i.e., κ = 0.3, which may also interconnect with the arc-shaped spalling crack at t = 112 μs. This kind of axial crack is restrained under the higher lateral pressure coefficient, as shown in Fig. 10, because the stress σ X at the left of the specimen becomes higher under the higher lateral pressure coefficient.

Fig. 10
figure 10

Stresses in X-direction σ X along the horizontal line crossing the center of the specimen subjected to complex loading paths with various confining pressures. a The final step under static loading, b t = 0 μs, c t = 56 μs, d t = 112 μs

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In general, in situ stress dominates the quasi-static stress distribution around excavation surface before unloading, while the dynamic disturbance applied in this manner may trigger the arc-shaped spalling zone at the right of the rock specimen and enhance the axial crack propagation at the left half of the specimen. Under the small lateral pressure coefficient, the axial cracks propagate mainly due to the triggering action of the dynamic disturbance. With the increase in lateral coefficient, the higher compressive stresses at the left of the specimen that are induced by in situ stress may restrain the tensile stress triggered by dynamic disturbance, thus preventing the initiation of axial cracks.

Effect of the duration of the incident stress wave

The maximum shear stresses during failure process of the rock specimen under different durations of dynamic disturbances are shown in Fig. 11. In this section, the axial pressure p s = 48 MPa and κ = 1.5. The incident stress wave as shown in Fig. 6 is applied at the top surface of the model domain, while p dm = 200 MPa and t dm = 4, 9 and 13 μs.

Fig. 11
figure 11

The failure processes of rocks subjected to complex loading paths with various durations of stress wave

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When t = 0 μs, the stress distributions of rock specimens are same, since they are subjected to the same in situ stress and confining pressure unloading. As shown in Fig. 11, at t = 28 μs, the failure mode of rock specimens is similar although the duration of the incident waves is different. Under the same amplitude of the incident stress wave (p dm), the more energy is input into the rock specimen if the duration of the incident stress wave (t dm) is prolonged. So the area of arc-shaped spalling zone increases with the prolonged duration of the incident stress wave. But the arc-shaped spalling zones all initiate at the upper right corner near the free surface and terminate at the lower righter corner. It indicates that the shape and area of the arc-shaped spalling zone are influenced by the duration of the incident wave. Under these loading conditions, as shown in Fig. 11, the axial crack is restrained in despite of the duration of stress wave, because the stresses σ X at the left part of specimen are so high that the rock is horizontally confined, as shown in Fig. 12. This also indicates that the initiation of axial crack is mainly dominated by in situ stress conditions before excavation.

Fig. 12
figure 12

Stresses in X-direction σ X along the horizontal line crossing the center of the specimen subjected to complex loading paths with various durations of stress wave. a t = 56 μs, b t = 112 μs

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The incident waves III, IV and V, as shown in Fig. 6, are applied in order to study the dynamic response of rock specimen under different durations of dynamic disturbance. Here, the axial pressure p s = 48 MPa and κ = 1.5. The less energy is input into the rock specimen if the rising time is prolonged under the same amplitude and duration of the incident wave. It indicates that shapes of the arc-shaped spalling zones are similar and areas of those increase with decreasing rising time, as shown in Fig. 13.

Fig. 13
figure 13

The failure processes of rocks subjected to complex loading paths with various rising times of stress wave

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Effect of the amplitude of the incident stress wave

In this section, the axial pressure is a constant of p s = 48 MPa while κ = 1.5. The incident stress wave VI, II, VII and VIII as shown in Fig. 6 is applied where its duration is a constant of t dm = 9 μs while p dm = 100, 200, 300 and 400 MPa, respectively.

Under the same duration of the incident stress wave, the energy input into the rock specimen increases with the amplitude of the incident stress wave (p dm) at the same time range. As shown in Fig. 14, the arc-shaped spalling zone becomes larger under longer duration of the incident stress wave, because more energy is input.

Fig. 14
figure 14

The failure processes of rocks subjected to complex loading paths with various amplitudes of stress wave

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As a whole, developments of shape and area of the arc-shaped damage zones are similar under different durations and amplitudes of dynamic disturbance as shown in Figs. 11, 13 and 14. It indicates that the arc-shaped damage zone is dominated by the energy input into the rock specimen under the dynamic disturbance after the same confining stress unloading, and influences of the duration and amplitude of incident stress wave on the failure pattern are similar because they all contribute to the energy input.

Conclusions

The failure of rock sequentially subjected to the complex quasi-static and dynamic loading paths is numerically simulated by using the numerical simulator RFPA-Dynamics. The influences of the quasi-static in situ stress conditions before excavation and the waveform of the dynamic disturbance on the rock failure are examined. This kind of work is of great significance to clarify the rock failure mechanism during this complex loading path of rock such as the excavation of underground opening. Based on the numerical simulations, the following conclusions can be drawn from the numerical simulations:

  1. 1.

    The numerical model established by using RFPA-Dynamics is used to simulate the failure of rock sequentially subjected to the quasi-static triaxial loading, confining pressure unloading and dynamic loading. The acceptable agreement between experimental and numerical results shows that RFPA-Dynamics is capable of reproducing the failure process of rock, thus confirming RFPA-Dynamics as an eligible candidate to simulate the failure process of rock triggered by dynamic disturbance after confining pressure unloading.

  2. 2.

    When the confining pressure is unloaded, the arc-shaped spalling damage zone is prone to develop with the increase in the axial pressure and lateral pressure coefficient, but the axial crack tends to be restrained by the horizontal stress in the rock specimen.

  3. 3.

    After rock specimen subjected to the same in situ stress and confining stress unloading, the effect of the duration and amplitude of dynamic disturbance on the energy input is similar and the area of the arc-shaped spalling zone of rock specimen is dominated by the energy input into the rock specimen. Altogether, during the excavation of underground opening, the shape and area of the excavation damaged zone may be influenced by in situ stress condition and waveform of dynamic disturbance.

Compared to the experiment conducted by Yin et al. (2012b), the more complex loading and unloading path can be carried out, and the failure process of rock can be observed and clarified by using the numerical simulation. But it is still difficult to be used to solve the engineering problems, such as the failure of rock mass induced by the blasting (Lapčević et al. 2014) and earthquake (Kostić et al. 2013). The application of numerical simulation method for the engineering problem would be studied further.

Although the numerical simulation is two-dimensional and the engineering problem is three-dimensional, the rock failure mechanism during this the excavation process is fairly similar. Accordingly, the 2D numerical and experimental results, it is really helpful for understanding different mechanisms associated with the excavation-induced damage zone. Moreover, since underground openings are always 3D geometries and undergo 3D geo-stress condition, in the long run, the 3D simulations should be done to characterize the more realistic failure patterns.