Rock Bridge Research Articles

Experimental Study of Mechanical and Permeability Behaviors During the Failure of Sandstone Containing Two Preexisting Fissures Under Triaxial Compression

Rock masses are typical inhomogeneous geological materials that contain many fissures and cracks. The coupling effect of the crack propagation and seepage evolution in rocks is very important to the safety of rock engineering. However, hydromechanical coupling behavior during the failure of fissured rocks has rarely been investigated. In this research, hydromechanical coupling tests are performed to fully explore the behaviors in strength, deformation, permeability and failure mode of sandstone samples with two preexisting fissures. The experimental results show that the ratio of crack initiation threshold/peak strength, the ratio of crack damage threshold/peak strength and the value of the elastic modulus decrease by at most 31.8%, 12.2% and 18.4% due to the existence of the two fissures, while the Poisson’s ratio increases by at most 45.6%. Furthermore, the values of permeability before the sudden increase stage range from 2.1% to 17.6% of the maximum permeability value. The influence of bridge length and angle on permeability is more significant under lower confining pressure or higher water pressure. Five failure modes are observed in the double-fissure samples under hydromechanical coupling conditions. Additionally, the “wing cracks + indirect coalescence” failure mode is generated only when the ligament length is shorter than the fissure length. The corresponding strength is lower than that for other failure modes. CT images show that the expansion of cracks inside the samples is more restricted than that at the surface of the samples, especially near the rock bridge region. The effects of failure modes on the mechanical and permeability properties, from greatest to least, are as follows: crack initiation threshold, peak strength, crack damage threshold, elastic modulus, Poisson’s ratio and permeability. This research is contributed to analyze the stability of water-bearing rocks in underground caverns with many preexisting fissures.

New Modified Model for Estimating the Peak Shear Strength of Rock Mass Containing Nonconsecutive Joint Based on a Simulated Experiment

In the Jennings criterion, which is commonly used to estimate the shear strength of nonconsecutive jointed rock masses, the estimated values deviate from the actual values due to the criterion failure to consider the joint undulating angle and the weakening of the mechanical properties of the rock bridge during shearing. Based on the modified Jennings criterion, this paper considers the influence of normal stress and joint continuity on the degree of weakening of the rock bridge’s mechanical parameters and presents a new correction method to further modify the Jennings criterion. The collection of undisturbed rock samples from the study area and the creation of joint(s) for laboratory testing is a cumbersome task. To overcome these limitations, a methodology to simulate the failure of jointed rock mass using the rock-like material, cement mortar, containing a single joint was adopted and its details are presented in this paper. Specifically, the direct shear tests were conducted on artificially prepared jointed rock masses with different joint continuity and undulating angles. The influence of joint continuity and the undulating angle on the shear strength of the nonconsecutive joints was studied. Then, the influence of the normal stress and joint continuity on the deterioration of the mechanical properties of the rock bridge during shearing was analyzed based on the experimental results. A new model for estimating the shear strength of a nonconsecutive jointed rock mass was proposed based on the experimental results and previous studies. The test results show that the smaller the undulating angle and the greater the penetration, the lower the shear strength of the nonconsecutive joint. Besides, the normal stress level and the joint continuity have a significant influence on the weakening of the mechanical properties of the rock bridge. When the normal stress decreases or the joint length increases, the mechanical properties of the rock bridge are significantly weakened. Compared with our estimation of the nonconsecutive joint shear strength based on the original Jennings criterion and the modified Jennings criterion, the calculation results of the newly modified Jennings criterion developed in this study are generally more consistent with the measured values, and thus, they more accurately predict the shear strength of the nonconsecutive jointed rock mass with a regular joint undulating angle.

Fracturing and Damage of 3D-Printed Materials with Two Intermittent Fissures under Compression.

The crack propagation and failure of 3D-printed samples with prefabricated K–S fissures (a kinked fissure and a straight fissure) were observed under uniaxial compression, and the strain and displacement of the sample surface were quantified by the digital image correlation (DIC) method. The experimental results show that the branch inclination angle of the kinked fissure is an important factor affecting the crack initial position, and the evolution of the strain field during the failure process of the sample can better reflect the cracking law of the internal fissures. Furthermore, two coalescence modes are classified: Mode I is a tension–shear composite failure formed by the penetration of the tension–shear composite crack; Mode II is a tensile failure that penetrates the whole samples during the failure process without rock bridge damage. In addition, the numerical simulation results were well consistent with the cracking and failure modes.

Phase-field fracture modelling of crack nucleation and propagation in porous rock

In this work, we suggest a modified phase-field model for simulating the evolution of mixed mode fractures and compressive driven fractures in porous artificial rocks and Neapolitan Fine Grained Tuff. The numerical model has been calibrated using experimental observations of rock samples with a single saw cut under uniaxial plane strain compression. For the purpose of validation, results from the numerical model are compared to Meuwissen samples with different angles of rock bridge inclination subjected to uni-axial compression. The simulated results are compared to experimental data, both qualitatively and quantitatively. It is shown that the proposed model is able to capture the emergence of shear cracks between the notches observed in the Neapolitan Fine Grained Tuff samples as well as the propagation pattern of cracks driven by compressive stresses observed in the artificial rock samples. Additionally, the typical types of complex crack patterns observed in experimental tests are successfully reproduced, as well as the critical loads.

Numerical Brazilian split test of pre-cracked granite with randomly distributed micro-components

PurposeThe purpose of this paper is to investigate the mechanical behavior and propagation of cracks of numerical granite samples through the Brazilian split test and to provide a reference for predicting the behavior of real granite samples.Design/methodology/approachThe numerical models of granite containing two fissures are established using the parallel bond model (PBM) and the smooth joint model (SJM) in PFC2D. The peak stresses, number of cracks and anisotropic ratios are obtained to study the influence of the mineral composition and the angle of inclination of rock bridge on the strength, failure mode and deformation characteristics.FindingsThe numerical results obtained show that the mineral composition has a marginal influence on the peak stress. When the angle of inclination of rock bridge β increases, the peak stress drops to its minimum value at β = 90° and then gradually increases to a relatively low level. The behavior of cracks falls into three categories based on the distribution of cracks. By analyzing the stress–strain curve and the process of crack propagation for sample No. 4 with β = 60°, it is found that the process of failure can be divided into four stages and tensile cracks dominate. The anisotropic ratios of peak stress and a number of cracks obtained show that the peak stress is low anisotropic and the number of cracks is medium anisotropic.Originality/valueThis paper presents a numerical simulation method to analyze mechanical behavior and propagation of cracks under different conditions. The proposed method and the results obtained are useful for predicting the behavior of real granite samples in laboratory and engineering projects.

Analysis of Progressive Failure Mechanism of Rock Slope with Locked Section Based on Energy Theory

Progressive failure in rock bridges along pre-existing discontinuities is one of the predominant destruction modes of rock slopes. The monitoring and prediction of the impending progressive failure is of great significance to ensure the stability of the rock structures and the safety of the workers. The deformation and fracture of rocks are complex processes with energy evolution between rocks and the external environment. Regarding the whole slope as a system, an energy evolution equation of rock slope systems during progressive failure was established by an energy method of systemic stability. Then, considering the weakening effect of joints and the locking effect of rock bridges, a method for calculating the safety factor of rock slopes with a locked section was proposed. Finally, the energy evolution equation and the calculation method of safety factor are verified by a case study. The results show that when the energy dissipated in the progressive failure process of rock bridges is less than the energy accumulated by itself, the deformation energy stored in the slope system can make the locked section deform continuously until the damage occurs. The system energy equal to zero can be used as the critical criterion for the dynamic instability of the rock slope with locked section. The accumulated deformation energy in the slope system can promote the development of the cracks in the locked section, and the residual energy in the critical sliding state is finally released in the form of kinetic energy, which is the main reason for the progressive dynamic instability of rock slopes.

Experimental study on crack propagation and the coalescence of rock-like materials with two preexisting fissures under biaxial compression

Rock masses consist of rock and fissures, and fissures in rock masses play an important role in rock mass stability. In this paper, rock-like materials with preexisting fissures were prepared, and then, the specimens were subjected to biaxial compression. The lateral stress contained two categories: 1.0 MPa and 2.0 MPa. During the experiments, strain gauges were distributed on the surface of the specimens, and a digital camera was used to record the process of crack propagation. Through the experiments, it can be found that the rock bridge coalescence mode can be generally classified into three categories; meanwhile, the influence of the preexisting fissure angle, rock bridge angle, and lateral stress due to crack propagation was specifically analyzed. Consequently, the strain concentration of the tips of preexisting fissures was calculated and discussed.

Investigation of the Mechanical Behavior of 3D Printed Polyamide-12 Joints for Reduced Scale Models of Rock Mass

This study presents the experimental results of the mechanical characterization of artificial rock joints constructed by 3D-printing (3DP). The mechanical behavior of rock masses is controlled by the presence of joints. Understanding the mechanical behavior of rock joints is essential to predict their influence on a rock mass. The application of innovative 3DP technology in rock mechanics to model artificial rock-like joints allows strict control of joint geometry (orientation, roughness, number of rock bridges, etc.), and thus of its mechanical behavior. The 3DP technology used in this work is selective laser sintering, and the material is Polyamide 12. Geometric characterization shows that this technology gives high dimensional precision for details smaller than 0.4 mm. More than 30 discontinuous samples were printed to investigate the global mechanical properties of a joint relative to its geometric features including Young’s Modulus (E), shear stiffness (ks), cohesion (cj), friction angle (φj) and dilation (i). The results show that the number of rock bridges (Nrb) and the roughness significantly influence the mechanical properties. A failure criterion that considers these parameters is proposed. These 3D-printed joints can be used in physical modeling of rock mass to understand the influence of the fractures on its stability by applying scaling laws. The application of scale factors to the experimental results shows the possibility of representing actual rocks with artificial 3DP joints.

Numerical Study of Anisotropic Weakening Mechanism and Degree of Non-Persistent Open Joint Set on Rock Strength with Particle Flow Code

The strength of a rock cut by non-persistent open joint set is controlled by complex interactions of the joint network and intact rock bridges. An extensive series of uniaxial numerical tests of the particle models are carried out under different joint fabrics and rock compression-tension strength ratios. The test results show joint dip angle is the main factor controlling the overall anisotropic strength characteristics. With joint dip angle increasing, the main failure modes successively appear as splitting failure, block rotation failure, step-path failure, planar failure, and splitting failure. Accordingly, the strength generally shows a trend of decreasing first and then increasing, and the joint dip angle corresponding to the minimum strength is biased towards 0°, usually at 15°–30°. The damage of rock bridges is mainly controlled by tensile failure, except in planar failure mode. Joint spacing has an impact on the local anisotropic strength characteristics. The joint spacing and persistence ratio jointly control the degree of penetration difficulty of the rock bridges. The compression-tension strength ratio of intact rock uniformly affects the absolute magnitude of the jointed rock strength and does not affect its anisotropic characteristics. Based on the weakening mechanism, an empirical strength formula is established to realize the practical equivalent characterization of the anisotropic weakening degrees of the non-persistent open joint set on the rock strength.

Evaluation of Maximum Rockfall Dimensions Based on Probabilistic Assessment of the Penetration of the Sliding Planes into the Slope

There is intrinsic difficulty in the investigation of the largest volume of rockfalls that is expected in an area, which lies in the small number of large events, in registrable times. The maximum credible rockfall size has been associated with the properties of the rock mass discontinuities, as they delimit detachable rock blocks, and in particular with the penetration of those discontinuities that comprise rockfall sliding planes. In highly fractured rock masses, the evaluation of the penetration remains an issue. A probabilistic methodology is proposed, to measure the penetration of potential sliding planes into the interior of a rocky slope. The main hypothesis of the method is that the sliding plane persistence is interrupted along its two directions, at the intersection with two lateral discontinuity sets, as the latter displaces the former. Due to the displacement, the sliding planes are formed by quasi-planes that contain a maximum number of spacings of the intersecting joints, hence their size is restricted. The methodology requires as an input the spacing of the intersecting joint sets. Its application to a granodiorite slope confirms that for the study site, there is a maximum volume of rockfalls, excluding the possibility of large stepped failures and rupture of rock bridges. The maximum calculated persistence for the two existing sliding planes in the study site is, respectively, 28.0 m and 48.5 m. The maximum calculated sliding plane surfaces are, accordingly, 282.5 m2 and 289.3 m2. These results are compared against the observed scar dimensions at the study site, which have been retrieved alternatively, by processing a LiDAR point cloud. The results from the two alternative sources are similar, indicating that the methodology can be efficiently used to assess the sliding plane persistence and the expected maximum rockfall magnitude at the study site.

Geostructural and Geomechanical Study of the Piastrone Quarry (Seravezza, Italy) Supported by Photogrammetry to Assess Failure Mode

The use of non-contact-techniques for rock mass characterization has been growing significantly over the last decade. However, their application to stability assessment of ornamental stone has not yet received much attention from researchers. This study utilizes rock mass data both in terms of slope orientations and degree of fracturing obtained from a point cloud, a set of three-dimensional (3D) points representing a rock mass surface, to (1) investigate the influence of geostructures at different scales and (2) assess quarry stability by determining areas susceptible to different failure types. Multi-resolution point clouds are obtained through several photogrammetric survey techniques to identify important structural elements of the site. By integrating orientation data of discontinuity planes, obtained with a traditional survey, and of traces, outlined on point clouds, several joint sets were identified. Kinematic tests revealed various potential failure modes of the rock slope. Moreover, an analysis of the influence of the discontinuity strength determined by the presence of rock bridges was carried out. The study revealed that the strength of the quarry face is governed by the presence of rock bridges that act to improve the stability condition of the rock fronts.

Investigation on Crack Coalescence Behaviors for Granite Containing Two Flaws Induced by Cyclic Freeze-Thaw and Uniaxial Deformation in Beizhan Iron Mining, Xinjing, China

This work is aimed at investigating the effect of freeze-thaw (F-T) cycle on the crack coalescence behavior for granite samples containing two unparallel flaws under uniaxial compression. The flaw geometry in the samples was a combination of an upper inclined flaw with a horizontal flaw underneath. After the uniaxial compression experiments, macroscopic crack pattern description and the mesoscopic posttest CT imaging were used to reveal the effects of F-T cycle on the crack coalescence morphology at the rock bridge area. Results show that the stress–strain curves present a fluctuating growth trend and stress drop phenomenon becomes weaker with increasing F-T cycles. In addition, three different kinds of cracks (tensile-wing cracks, oblique shear cracks, and antiwing cracks) were observed, and the crack coalescence pattern was influenced by the F-T cycles and approach angle. A mix of tensile and shear failure occurs for the sample subjected to weak F-T treatment, and simple tensile failure occurs for the sample subjected to high F-T treatment. Moreover, CT imaging reveals a crack network pattern at the rock bridge area, and it is found that the fracture degree deceases with increasing F-T cycles and increases with the increasing approach angle. It suggests that the rock bridge area can be easily fractured for the sample subjected to high F-T cycles. Results of this study can provide theoretical foundation for the instability predication of fractured rock structures in cold regions.

Investigating the Effect of Rock Bridge on the Stability of Locked Section Slopes by the Direct Shear Test and Acoustic Emission Technique.

As a portion of intact rock separating joint surfaces, rock bridge plays a significant role in the stability of rock slopes. This paper aims to investigate the effect of different rock bridges on the mechanical properties and failure mode of rock slope by means of the direct shear test and acoustic emission technique. Field conditions were simulated in direct shear tests which were carried out on specimens with rock bridges at different continuity rates, normal stress, arrangements, and joint angles. Experimental results indicate that the strength of specimens is controlled by the rock bridge and the structural plane. The rock bridge contributes to the strength of the specimen, while the through plane weakens the strength of the specimen. The increase of normal stress can weaken the stress concentration near the tip of the rock bridge and improve the shear resistance of the specimen. The different arrangement of rock bridge has little effect on the normal displacement of the specimen, and has a great influence on the shear strength. The shear capacity of the specimen is related to the angle of the crack, and the angle of the crack is approximately proportional to the peak shear strength. For the specimens with different joint occurrence, the mode of crack propagation at the initial stage is basically the same, and the specimen is finally damaged due to the generation of through cracks in the core area of rock bridge. The instantaneous release of the huge energy generated during the experiment along the shear direction is the root cause of the sudden failure of the rock bridge. The formation, aggregation, and transfixion process of rock bridge is of concern and has been experimentally investigated in this paper for the prevention and control of the locked section rock slope with sudden disasters.

Experimental and numerical study on the interaction between holes and fissures in rock-like materials under uniaxial compression

Holes and fissures, regarded as typical defects, dominate a critical role in the mechanical behavior of rock. To investigate the effect of different horizontal distances L or L′ between the centroids of holes and fissures on the mechanical properties of pre-flawed rock-like material, two types of rock-like material specimens with holes and fissures are fabricated and subjected to uniaxial compression using Digital image correlation (DIC), Acoustic emission (AE) monitoring, Computerized tomography (CT) scanning, and two-dimensional Particle flow code (PFC2D). These results demonstrate that the peak strength, peak strain and elastic modulus all show a monotonically nonlinear decrease with increasing L or L′. Based on the DIC and CT methods, three rock bridge coalescence modes can be distinguished: tensile-tensile, shear-shear and tensile-shear modes. Moreover, the damage of the rock-like specimens, in the entire loading process, exhibits notable progressive failure characteristics in the strain localization zones. The crack initiation and coalescence cause the deformation curve to undergo a stress drop with a surge in the AE events and in the accumulated AE counts from AE monitoring. There is good agreement between the numerical simulation results and the experimental testing results, which further heighten the instability mechanism of defective rock engineering structures.

A precursor of bedding rockslide: rock spalling in the key block triggered by tensile cracks

A rock mass sliding along one controlling bedding plane is a shear failure that occurs under compressive stress. The failure mechanism is relevant to the propagation and coalescence of cracks in the rock bridge. A direct shear test was used to examined samples containing one preexisting crack to investigate the initiation, propagation, and coalescence of cracks. These cracking behaviors were analyzed by a charge-coupled device camera, the acoustic emission (AE), and a resistance strain gauge. Linear elastic fracture mechanics were applied to analyze the crack initiation in theory. In the study, the deformation and AE count rate were increased before failure. Simultaneously, shear spalling occurs in this process. This phenomenon shows that shear spalling is a precursor to failure. The spalling triggered by tensile cracks is similar to the in situ rock mass failure that explained the failure mechanism of the key block in the Jiweishan rockslide. The mechanical behavior shows that rockfalls induced by tensile cracks are a precursor to failure. Additionally, the degree of spalling is affected by the continuity factor of the preexisting crack. The failure modes are affected by normal stress and the preexisting crack length. This research provides theoretical support for predicting the stability of a bedding rock mass containing one key block.

Microfracture propagation in gneiss through frost wedging: insights from an experimental study

Ice-driven mechanical weathering in mountainous environment is considered as an efficient process for slow but cyclical mechanical preconditioning of rockfall events. In this study, we simulate subcritical microfracture propagation under frost wedging conditions along pre-existing mechanical weaknesses of intact rock bridges with an innovative experimental approach. Two series of freeze–thaw experiments conducted in an environmental chamber were carried out to investigate and monitor the propagation of artificially induced fractures (AIF) in two twin gneiss samples. A displacement sensor recorded the sample’s in situ deformation in an environmental chamber during the experiments. 3D X-ray CT scans, performed before and after the experiments, as well as thin sections showing the post-experiment state of the deformed samples allowed tracking and quantification of fracture propagation. Our results demonstrate that frost wedging propagated the AIFs 1.25 cm2 and 3.5 cm2 after 42 and 87 freeze–thaw cycles, respectively. The experiments show that volumetric expansion of water upon freezing, cooperating with volumetric thermal expansion and contraction of the anisotropic rock, plays a key role in fracture widening and propagation. Based on these results, this study proposes that: (1) frost wedging exploits intrinsic pre-existing mechanical anisotropies of the rock; (2) the fracturing process is not continuous but alternates between stages of fast propagation and more quiet stages of stress accumulation; and (3) downward migration of “wedging grains,” stuck between the walls of the fracture, increases the tensile stress at the tip, widening and propagating the fractures with each freeze–thaw cycle. The experimental design developed in this study offers the chance to visualize and quantify the long-term efficiency of frost wedging in near-natural scenarios.

Numerical Analysis of Fracture Behaviour on Marble Samples Containing Two Flaws

Uniaxial compression tests were conducted on marble specimens containing two flaws. There are coplanar flaws and noncoplanar flaws. The inclination angle and spacing of flaws were considered of the coplanar flaws model, and the step angle and spacing of flaws were considered of the noncoplanar flaws model. Strength failure and crack coalescence behaviour were analysed in the paper. The crack evolution process containing microcrack initiation, coalescence, and failure is focused on the rock bridge coalescence and the extent of the pre‐existing flaws. There are four forms of rock bridge coalescence: tensile crack coalescence, shear crack coalescence, mixed tensile and shear crack coalescence, and no coalescence. Also, there are four forms of the rock failure mode: tensile failure, shear failure, mixed tensile and shear failure, and split fracture. The outer end of the critical stress values were used to compare with the crack initiation strengths, and the crack initiation strengths were slightly larger than the critical stress. In addition, energy dissipation laws were analysed during the model fracturing process. The crack evolution mechanisms around the pre‐existing flaw in the model were revealed by the distribution of microcrack and energy dissipation.

Brittle failure of rockslides linked to the rock bridge length effect

A rock bridge along the potential sliding surface of a rockslide has a relatively large bearing capacity and governs the stability of the rock slope. The physical and mechanical properties of specimens representing four rock bridge lengths were researched by direct shear experimentation. The micromechanism associated with the change in shear strength parameters was analyzed by PFC2D numerical simulations that corresponded to the experiments. The results revealed a rock bridge length effect, through which the peak shear strength increases with rock bridge shortening, accompanied by an increase in cohesion and a decrease in internal friction angle and main fracture surface roughness. Rock bridge shortening decreased the density of the microcracks at the shear stress peak point, changing the cohesive strength and frictional strength. The decrease in the internal friction angle could change the critical stress intensity factors to make shear cracks preferentially initiate over tensile cracks. Fracture roughness therefore decreased, inducing a lower residual shear strength. As a result, the difference in the peak and residual strengths increased with rock bridge shortening, which would promote brittle failure of the corresponding rock slope.

Geometric and mechanical properties of a shear-formed fracture occurring in a rock bridge between discontinuous joints

Rock bridges are commonly encountered in rock engineering practices and play a significant role in the stabilization of rock masses. Under shear loading, they are supposed to provide resistance not only at before-failure stages, but also at after-failure stages, and this is not fully understood yet by now. To promote the solving of this problem, the present study tried to discover the geometric and mechanical properties of a shear-formed fracture occurring in a rock bridge between discontinuous open joints through laboratory experiments. After careful preparation of sandstone specimens containing discontinuous open joints, a series of direct shear tests were carried out under constant normal load (CNL) conditions. Once the rock bridge failed, another two sets of cyclic loading were subsequently applied to the specimen to examine the after-failure properties. The geometric properties of the newly formed fracture in rock bridge area, including undulating angle, fracture roughness, and fracture aperture, were systematically examined and analyzed with respect to the applied normal stress and joint persistence. Furthermore, the failure mode was dependent on the normal stress and joint persistence and also influenced the properties of the shear-formed fracture. Accordingly, different fracture types occurring to the rock bridges were summarized and described in terms of geometric morphology. The findings in present study would enhance our understanding of the influence of rock bridges, especially at after-failure stages, on engineering practices such as rock slopes.

Experimental Study on the Effects of Unloading Normal Stress on Shear Mechanical Behaviour of Sandstone Containing a Parallel Fissure Pair

To gain deeper insight into the effects of unloading normal stress on shear mechanical behaviour, laboratory tests are carried out on the red-sandstone specimens containing a parallel fissure pair under the constant shear stress and the unloading normal stress. The results reveal that the trace of the entire rupture surface is mainly controlled by the rock bridge, and the impacts of the rupture surfaces of the unloading tests are narrower than those of the direct shear tests. Tensile failure, tension–shear failure, shear failure, and two-stage failure are observed, the failure rules of rock bridges are further summarized according to the ranges of the length and inclination of the rock bridge. The shear strength of the normal unloading test has a slight increase compared with that of the direct shear test. With an increase in the initial normal/shear stress, the shear strength of rock specimens increase; under the low to medium initial stress conditions, the shear strength increment has a linear growth tendency, but under the medium to high initial stress conditions, the growth trend slows down. The ratio of shear deformation on rupture surface increases with the increase of the initial shear stress but decreases with the increase of the initial normal stress. The ratio of shear deformation on rupture plane increases with the increase of the initial shear stress, and decreases with the increase of the initial normal stress, There is a little difference in the deformation ratio (shear damage deformation divided by tensile damage deformation, ΔDrs/ΔDrn) between the observed tensile failure and tension–shear failure. The dominate damage deformation under different geometric conditions is closely related to the failure patterns.