Mohr Failure Envelope Research Articles

A New Discrete Form of Hoek–Brown Criterion and Its Application to Limit Equilibrium Analysis of Rock Slope Stability

The generalized Hoek–Brown criterion (GHB) is recognized as one of the standard failure criteria in rock engineering and its validity extends to a wide range of rock mass quality. A drawback of this criterion is the difficulty of transforming it into an explicit form defining the Mohr failure envelope when its strength parameter a is not equal to 0.5. The information on the functional form of the Mohr envelope for the full range of rock mass conditions enables the implementation of classical engineering approaches, such as the limit equilibrium method and limit analysis, in the framework of the GHB criterion. Knowing that for a≠0.5 the exact closed-form representation of the Mohr envelope is not feasible, an alternative is to express it in an approximate analytical form. The main purpose of this study is to propose a new improved method to define an approximate Mohr envelope of the GHB criterion that is much more accurate compared with the recently published approximations. The idea behind the formulation is to expand the Balmer’s equation, which defines the relationship between the normal stress and minor principal stress at failure, by invoking the finite Taylor series centered at the known solution for a=0.5. The formulation is then completed by substituting this solution into another Balmer’s equation, defining the relationship between the shear strength and the minor principal stress. The Taylor polynomial approximations of up to third degree are considered in the formulation. The accuracy of the shear strength prediction is shown to be much better than that of the approximate formula of Lee and Pietruszczak proposed in 2021. An illustrative example of limit equilibrium analysis of rock slope stability, incorporating the new approximate expression for the Mohr envelope, is provided. The analysis incorporates a modified version of the Bishop approach, which is simpler and more rigorous than the original nonlinear expression. The study confirms that the new approximate representation of the Mohr failure envelope can facilitate the application of the GHB criterion to a range of practical rock engineering calculations.

Comparative Study of Shear Fracture between Fe-based Amorphous and Ultrafine-grained Alloys Using Micro-tensile Testing

Micro-tensile tests were employed to clarify the post-plastic-instability behavior in the shear fractures of specimens with the dimensions of 18×25×50 µm3 made from iron-based amorphous (AM) and ultrafine-grained (UFG) alloys. The AM specimen yielded by localized shear bands with an inclination angle of ~52° with respect to the loading axis, followed by sliding off almost throughout the entire specimen thickness. Micro-tensile and micro-shearing tests revealed that the Mohr failure envelope of the AM specimens could be described by a quadratic equation rather than a linear equation. Therefore, the sliding-off process is assisted by the applied normal stress, which suggests that it is caused by free-volume coalescence. For the UFG specimen, yielding set in by shear band formation with an inclination angle of ~45° with respect to the loading axis, following the Tresca criterion. Necking after shear band diffusion formed a triaxial stress state, which resulted in a final shear fracture plane via void coalescence in the UFG specimens. Voids formed along the intersection of the primary shear bands with secondary shear bands during the necking process. This indicates that the deviation of the shear fracture plane in the UFG specimen was determined by the strain development process. A comparison of the post-plastic-instability behavior between the AM and UFG specimens suggests that the external control of triaxial stress conditions is key to improving the formability of AM specimens.

Near-surface soils: self-supported unconfined drained sand specimens

Knowledge about soil behaviour in the near surface is fundamental to a broad range of military strategic research efforts. The prevailing mathematical representations of near-surface shear strength do not necessarily reflect the observed shear strengths. This paper presents an experimental procedure for testing self-supported, unconfined-drained specimens to investigate the shear strength of a poorly graded sand with the inferred unsaturated behaviour from typical effective stress expressions. The impact of soil suction is quantified from the soil–water characteristic curve obtained from a modified, unsaturated, one-dimensional oedometer device: Fredlund’s device. Seventy-seven unconfined tests illustrate that while individual or discrete states can be explained by effective stress relationships, those expressions do not accurately quantify the observed shear strength from one state to the next. Saturation failure is observed when the degree of saturation is between 70.5% and 76.5%, well below the assumed threshold. The results show that the dry shear resistance is a function of the initial density state, but as the degree of saturation increases, the shear resistance becomes independent of the initial density, and converges to the saturation failure. These findings expose the limitations of effective stress expressions and suggest a revised nonlinear, three-dimensional, unconfined Mohr failure envelope based on an applied external stress.

Effect of Rock Particle Content on the Mechanical Behavior of a Soil‐Rock Mixture (SRM) via Large‐Scale Direct Shear Test

The mechanical strength of the landslide deposits directly affects the safety and operation of the roads in the western mountainous area of China. Therefore, the research is aimed at studying the mechanisms of a landslide deposit sample with different rock particle contents by analyzing its characteristics of the stress‐strain behavior, the “jumping” phenomenon, the volumetric strain, and the shear strength parameters via a large‐scale direct shear test. Stress‐strain results show that stress‐strain curves can be divided into 3 different stages: liner elastic stage, yielding stage, and strain‐hardening stage. The shear strength of SRM behaves more like “soil” at a lower rock particle content and behaves more like “rock joints” at a higher rock particle content. Characteristics of the “jumping” phenomenon results show that the “intense jumping” stage becomes obvious with the increasing rock particle content and the normal stress. However, the lower the rock particle content is, the more obvious the “jumping” phenomenon under the same normal stress is. Volumetric strain results show that the sample with a lower rock particle content showed a dilatancy behavior under the low normal stress and shrinkage behavior under the high normal stress. The dilatancy value becomes smaller with the increasing normal stress. The maximum shear stress value of the rock particle content corresponds to the maximum value of dilatancy or shrinkage. We also conclude that the intercept of the Mohr failure envelope of the soil‐rock mixture should be called the “equivalent cohesion,” not simply called the “cohesion.” The higher the normal stress and rock particle content are, the bigger the equivalent cohesion and the internal friction angle is.

Analytical representation of Mohr failure envelope approximating the generalized Hoek-Brown failure criterion

The latest version of the Hoek-Brown criterion, referred to as the generalized Hoek-Brown (GHB) criterion, is an empirical failure condition for rock mass that correlates the rock mass strength with the data collected from the laboratory tests and field observations by use of GSI values. A shortcoming of the GHB criterion is that it does not provide an explicit relation between the shear and normal stresses on the failure plane, which limits its application in many geotechnical design calculations. In order to extend the application of the GHB failure criterion to limit equilibrium analysis and/or the upper-bound approach of limit analysis, the analytical approximations of the Mohr failure envelope corresponding the GHB criterion are derived. At present, the exact formulation is not available except for a=0.5, i.e. for a rock mass with GSI=100. In the formulation proposed here, the 1st- and 2nd- order Taylor approximations are employed to define the relation between the dimensionless normal stress and the sine of the instantaneous friction angle. Given the latter, the corresponding shear stress at failure is calculated from the established explicit relationship between the shear stress and the instantaneous friction angle. The Taylor expansion is carried out by taking the sine of the instantaneous friction angle as the expansion variable. A series of examples are given which demonstrate that the approximated Mohr envelope, derived by invoking the 2nd-order Taylor approximation with the expansion center of 0.5, is nearly identical to the exact analytical form. Even the 1st-order Taylor approximation with the expansion center of 0.5 is proven to be satisfactory. Furthermore, it is shown that the formulation is valid for a broad range of values of GSI. The latter has been verified by simulating the conventional triaxial compression tests within the framework of the critical plane approach (CPA) adopting the proposed form of Mohr failure envelope as a failure function. The results confirm that the calculated axial stresses at failure and the orientation of the failure plane are almost identical to the values stipulated by the original GHB criterion.

Triaxial Test on the Construction-Aid Soil Stabilizer in Stabilizing Fine-Grained Soil

Based on the airport pavement construction features and utilization requirements, we stabilize the fine-grained soil in Jining district of Shandong province by using the construction-aid (con-aid) soil stabilizer.We test the stress-strain curve and Mohr failure envelope of the triaxial shear of the fine-grained soil before and after being stabilized under various confining pressure by using the unconsolidated-undrained (UU) triaxial test method. We also analyze the change rules of the cohesive force Cu and internal friction angle φu to evaluate the effect of stabilizing fine-grained soil by using the con-aid soil stabilizer. In addition, we discuss the feasibility of applying the stabilizer in constructing the airport pavement and roads.

Change of stress magnitudes during the polyphase tectonic history of the Cretaceous Gyeongsang basin, southeast Korea

Abstract In order to evaluate the change of stress magnitudes in the Gyeongsang basin during its tectonic history, we analyzed multiple faulting episodes in the Barremian-Aptian Hasandong Formation at the Yusu site. As elsewhere in southeast Korea, the recorded sequence consists of a succession of more than fourteen faulting episodes, and the relative chronology shows that a strike-slip faulting episode usually coexisted with a coaxial extensional episode. Likewise, seven couples of synchronous coaxial episodes recognised in the Gyeongsang basin are assigned to seven tectonic events (T_1 to T_7 events). The friction line (in the sense of Byerlee) allows us to determine the ratios between principal stress magnitudes as well as the origin of the dimensionless Mohr diagram. This line can be deduced from tension fractures on fault planes affected by friction and from the lower limit of scattered distribution of the normal stresses vs. shear stresses of faults. Dimensionless failure envelopes drawn for coaxial strike-slip and extensional episodes are adjusted to the experimental Mohr failure envelope derived from rock mechanic tests to determine the complete stress tensors. The maximum principal stress magnitudes of strike-slip episodes show a transition from 169 MPa in the Barremian-Coniacian T_1 Event through 263 MPa and 246 MPa in the T_2 and T_4 events, respectively to 235 MPa in the Quaternary T_7 Event; additional horizontal extension (ΔσT) have changed from −6 MPa in the T_1 Event through −8 MPa in the T_2 Event to −17 MPa in the T_7 Event. Because the studied site is currently exposed, the determined overburden (1.9 km) for the T_7 Event seems to be important, indicating the presumable occurrence of this event during the early Quaternary rather than at the present day.

The Bearing Capacity of Circular Footings in Sand: Comparison between Model Tests and Numerical Simulations Based on a Nonlinear Mohr Failure Envelope

This paper presents the results of a series of triaxial tests with dry sand at confining pressures varying from 1.5 kPa to 100 kPa at relative densities of 0.20, 0.59, and 0.84. The results, which are in reasonable accordance with an equation given by Bolton, show that the friction angle is strongly dependent on the stress level and on the basis of the test results, a nonlinear Mohr failure criterion has been proposed. This yield criterion has been implemented in a finite element program and an analysis of the bearing capacity of a circular shaped model foundation, diameter 100 mm, has been conducted. Comparisons have been made with results from 1g model scale tests with a foundation of similar size and a good agreement between numerical results and test results has been found.

Transitional–tensile fracture propagation: a status report

One model for the development of hybrid shear fractures is transitional–tensile fracture propagation, a process described as the in-plane propagation of a crack subject to a shear traction while held open by a tensile normal stress. Presumably, such propagation leads to a brittle structure that is the hybrid of a joint and a shear fracture. Crack–seal veins with oblique fibers are possible candidates. While these veins clearly show shear offset, this is not conclusive evidence that a shear traction was present at the time of initial crack propagation. Many recent structural geology textbooks use a parabolic Coulomb–Mohr failure envelope to explain the mechanics of transitional–tensile fracturing. However, the laboratory experiments cited as demonstrating transitional–tensile behavior fail to produce the fracture orientation predicted by a parabolic failure envelope. Additional attempts at verification include field examples of conjugate joint sets with small acute angles, but these conjugate joints form neither simultaneously nor in the stress field required by the transitional–tensile model. Finally, linear elastic fracture mechanics provides strong theoretical grounds for rejecting the notion that individual cracks propagate in their own plane when subject to a shear traction. These observations suggest that transitional–tensile fracture propagation is unlikely to occur in homogeneous, isotropic rock, and that it is not explained by a parabolic Coulomb–Mohr failure envelope as several recent structural geology textbooks have suggested.

Characterizing the Failure Envelope of a Conductive Adhesive

This paper focuses on mechanical testing designed to determine the static failure envelope for a conductive adhesive. Samples were made by bonding copper pegs together with the conductive adhesive. The samples were then tested at various loading conditions including tension, tension shear, and compression shear. Results were analyzed in order to check for correlations between the data and the testing procedures. The statistical distribution of the data was also analyzed. Furthermore, a finite element model of the test sample was constructed and used to verify the assumptions made with respect to the interpretation of the data. The data from various loading conditions was then used to construct the static failure envelope of the material. A modified Coulomb–Mohr failure criterion was used to model the failure envelope of the conductive adhesive. This criterion contains four material constants to be determined experimentally. Once these parameters are determined, a failure envelope can be easily constructed. The envelope can then be used to predict failure at any combination of shear and normal stresses. The test results showed that the empirical data are well characterized by the modified Coulomb–Mohr failure envelope.

Comparison and Verification of Electric-Log-Derived Rock Stresses and Rock Stresses Determined From the Mohr Failure Envelope

Summary This paper evaluates the correlation between values of minimum principal in-situ stress derived from sonic logs and those obtained from triaxial core tests and coefficient for earth at rest correlations. The minimum principal in-situ stress is then compared to actual field mini-frac test data, which accurately determine the minimum principal in-situ stress and are used to verify the accuracy of the correlations. The cores and the mini-frac stress test were obtained from two wells, the Gas Research Inst.'s (GRI's) Staged Field Experiment (SFE) #1 well through the Travis Peak Formation in the East Texas Basin, and the U.S. Dept. of Energy's Multiwell Experiment (MWX) wells located west-southwest of the town of Rifle, Colorado, near the Rulison gas field. Results from this study indicate that of the two approaches used in obtaining the minimum principal in-situ stress, the normalized Mohr failure envelope approach is in better agreement with the measured stress values (from mini-frac tests) in sandstone, while the sonic log approach gives closer match in shale.

Is internal friction friction?

Mogi [1974] proposed a simple model of the incipient rupture surface to explain the Coulomb failure criterion. We show here that this model can plausibly be extended to explain the Mohr failure criterion. In Mogi's model the incipient rupture surface immediately before fracture consists of areas across which material integrity is maintained (intact areas) and areas across which it is not (cracks). The strength of the incipient rupture surface is made up of the inherent strength of the intact areas plus the frictional resistance to sliding offered by the cracked areas. Although the coefficient of internal friction (slope of the strength versus normal stress curve) depends upon both the frictional and inherent strengths, the phenomenon of internal friction can be identified with the frictional part. The curvature of the Mohr failure envelope is interpreted as a consequence of differences in damage (cracking) accumulated in prefailure loading at different confining pressures.

Failure Paths and Safety Factors for Multiaxial Stress in Concrete

Although mass concrete structures, such as dams are governed by multiaxial stress patterns, their safety is commonly evaluated in terms of a uniaxial safety measure. To assist such a structure's safety more realistically a multiaxial failure criterion and a relevant stress path to failure could be chosen in a reformulation of the safety measure. This paper compares the Mohr failure envelope with the principal stress failure envelope in terms of feasible paths, and their effect on the safety factor for a medium-strength concrete. Different values of the commonly defined safety factor are shown to be obtained for identical states of stress because of different assumptions for the initial stress state. To evaluate the safety of existing structures, the most likely failure path should be considered, and a distinction made between new and existing structures in the assumption of the initial state of stress.

Utilizing the torsional shear test to determine soil strength-properties relationships

The draft force on a tillage tool is dependent on the soil strength. The relationship between soil strength and the soil's density and moisture content would be useful in determining the draft forces generated during a tillage operation. Torsional shear tests were performed on a Ruston loamy sand to investigate the relationship between the soil strength parameters (soil cohesion and friction angle) and soil properties (soil density and moisture content). These soil strength-properties relationships are discussed. The effect of linearizing the Mohr failure envelope at different normal pressures is investigated. Multiple linear regression equations were developed, defining the soil strength-properties relationships for this soil type. The resulting R-square values ranged from 0.89 to 0.96.

Faulting mechanisms in high-porosity sandstones; New Red Sandstone, Arran, Scotland

Summary Faults in the ‘New Red’ aeolian sandstones of Arran are unusual, firstly for occurring as closely-spaced (less than 1 m) often conjugate sets affecting large volumes of rock, and secondly for forming upstanding fault zones with numerous anastomosing strands of granulated rock, each preserving a small increment of slip. Anisotropy, such as bedding and cross-bed sets, has no discernible effect on fault behaviour. In contrast to the underlying Carboniferous rocks, large displacements are rarely concentrated on a single fault plane within the high-porosity sandstones. The proposed cause is slip-hardening of each fault after a very small displacement (less than 10 mm) causing the next slip increment to be taken up through undeformed rock rather than on the original plane. The common factor in recent records of similar faults elsewhere is their occurrence in high-porosity sandstones. Because of the low grain-contact strength, these rocks are partly analogous to unconsolidated sediment. The high porosity promotes high grain-contact stresses which induce rapid cataclasis during initial slip. Grain fracture and spalling of iron oxide coatings and quartz overgrowths produce a seam with reduced grain size, poorer sorting, higher angularity and lower porosity than the unfaulted rock. These factors collectively strengthen the seam because the coefficient of friction is increased, even though cohesion is reduced. This results in a Mohr failure envelope that lies outside the envelope of the undeformed rock for most stress states. A transient pore pressure increase in the fault seam may be important during slip. Rocks deformed by this slip-hardened faulting preserve a record of each increment of strain. If the displacement on each individual fault seam is the same, the geometry of the total fault systems is directly related to the bulk strain. Quadrimodal systems observed by us in Arran, and by others elsewhere, are probably a response to triaxial strain and show that bimodal ‘Andersonian’ fault systems are only special plane strain cases. If the bulk strain is irrotational, both the orientation and relative magnitude of the principal strains might be estimated.

Shear strength of Athabasca Oil Sands

Previously published data are inadequate to explain the high natural shear strength of oil sand. Dissolved gas comes out of solution when confining stresses are removed rapidly, and this results in an internal pressure that expands the oil sand specimens disrupting their fabric. Geophysical logs indicate that in situ densities are much higher than those determined from conventionally cored specimens.Although the behavior of slopes in oil sands suggests that the shear strength is high, the source of strength of the oil sand has not been explained. Therefore detailed oil sand strength testing was undertaken on samples obtained in a special manner. Down-hole refrigeration of cored sections resulted in relatively high-quality specimens, and these were shaped on a lathe to provide triaxial and shear-box test samples.Strength tests on dense Ottawa sand, oil sand tailings and densely recompacted oil sand were performed: standard behavior was observed throughout. A series of triaxial and shear-box tests on undisturbed oil sands demonstrated a Mohr failure envelope that is highly curved, which displays no cohesion intercept and which is extremely steep for the initial portion of the envelope. Optical and scanning electron microscope investigations have revealed a dense interpenetrative structure and a considerable degree of grain surface rugosity. These factors give rise to a very high dilation rate before failure, and the dilation rate is suppressed as normal stress increases. The suppression of dilation results in shear of grains and grain asperities, giving rise to an apparent cohesion intercept at higher normal stresses. The curvilinear failure envelopes may be conveniently expressed as power-law relationships, and this form of expression will prove useful in stability analysis.

Tensile-Shear Bond Strength and Failure Between Aggregate and Mortar

Abstract The purpose of this study was to investigate the tensile-shear bond strength and failure of the mortar-aggregate interface. The techniques used are reported. The results indicate that the Mohr failure envelope in the tension-shear quadrant is almost a straight line and that the failure is essentially tensile in nature.