Brown Criterion Research Articles

New cycle counting method for multiaxial fatigue

Multiaxial fatigue is a very important physical phenomenon in several mechanical components. Fatigue life study under cyclic stresses is of utmost importance to avoid unexpected failure of equipments, vehicles or structures. Among several fatigue characterization tools, a correct definition of a loading cycle under multiaxial fatigue loading conditions shows to be crucial to estimate multiaxial fatigue life.The aim of this work is to achieve a correct definition for a multiaxial fatigue loading cycle and accomplish a multiaxial fatigue model to estimate block’s fatigue life under multiaxial loading conditions. To reach this goal, several loading paths were carried out using the 42CrMo4 low alloy steel under different loading conditions. Sequential, proportional, non-proportional and asynchronous loading effects were modulated through eleven loading blocks. Furthermore, two models were proposed: a cycle counting method and a fatigue life evaluation criterion. The results from the proposed models were correlated with the fatigue data and compared with two well known cycle counting models: the Bannantine and Socie and the Wang and Brown criteria. The proposed models were successfully validated by experimental data. Results show that the new proposals lead to an improved multiaxial fatigue characterization under complex loading conditions.

Modification of Hoek–Brown Criterion and Its Application Based on Fuzzy Synthetic Evaluation

The importance of mechanical parameters of rock mass is presented, and some methods to determine these mechanical parameters are introduced, whereas its accuracy is questionable and controversial. As there is difficulty in determining the disturbance factor D relayed on the Hoek–Brown strength criteria, fuzzy comprehensive evaluation is employed. Considering the factors influencing on disturbance and relevant engineering experience, the calculation of the value of D is 0.9379, which is more corresponded to the actual situation. Additionally, the rationality, that the bounds characterized by slip surface for dividing disturbed and undisturbed zone, was discussed and it overcome deficiency of other expressions. Relayed on mechanical parameters of rock mass from Liberia Bone Iron Mine, calculation of the factor D was presented by means of the modified parameters and other methods respectively. On the basis of the comparative analysis, the results showed that the former was more suitable to mirror the disturbance degree in-suit. What’s more, it verified the reasonability and significance of partition of disturbed and non-disturbed zone and provided important reference in the future research.

Three-dimensional numerical analysis for rock slope stability using shear strength reduction method

Existing numerical modeling of three-dimensional (3D) slopes is performed mainly by using the shear strength reduction (SSR) technique based on the linear Mohr–Coulomb (MC) criterion, whereas the nonlinear failure criterion for rock slope stability is seldom used in slope modeling. However, it is known that rock mass strength is a nonlinear stress function and that, therefore, the linear MC criterion does not agree with the rock mass failure envelope very well. In this research, a nonlinear SSR technique is proposed that can use the Hoek–Brown (HB) criterion to represent the nonlinear behavior of a rock mass in the FLAC3D program to analyze 3D slope stability. Extensive case studies are carried out to investigate the influence of the convergence criterion and boundary conditions on the 3D slope modeling. Results show that the convergence criterion used in the 3D model plays an important role, not only in terms of calculation of the factor of safety (FOS), but also in terms of the shape of the failure surface. The case studies also demonstrate that the value of the FOS for a given slope will be significantly influenced by the boundary condition when the slope angle is less than 50°.

Stress determination and geomechanical stability analysis of an oil well of Iran

In this paper, a numerical model using three-dimensional finite difference code FLAC3D is proposed for analysing the stability of an oil well drilled in four formations. Normalized Yielded Zone Area (NYZA, i.e., the ratio of surrounding yielded cross-sectional area to initial area of well) has been determined for different mud pressures. In each formation, by interpolating of obtained NYZA equal to one, the optimized mud pressure was determined using MATLAB software. Practical data including geomechanical parameters along with drilling data from one of Iranian oilfields, Mansouri-54 well have been utilized in this analysis. In situ stress was determined using stress polygon method and conducting hydraulic fracturing data in the field. Analytical solution using the Mogi–Coulomb and the Hoek–Brown failure criteria has been carried out and results are compared with the presented model. The results demonstrated that the NYZA and Hoek–Brown criteria might underestimate and overestimate the drilling mud pressure, respectively, and should be used cautiously. In the inclined section of the well, plastic zone showed more extension in the lower part than upper part of the well because of the high stress concentration.

Plastic flow and state-dilatancy for geomaterials

In this paper, the stress-dilatancy hypothesis is revisited and a stress-dilatancy framework is established such that it can be compatibly used for different yield functions. The influence of density–pressure and other state variables are systematically accommodated. The framework is further evaluated by obtaining and extending some flow rules for the Mohr–Coulomb criterion and novel stress-dilatancy formalism for the Hoek–Brown criterion.

Indications, Protocols, and Outcomes of Drug Desensitizations for Chemotherapy and Monoclonal Antibodies in Adults and Children

Advances in the understanding of various malignancies and chronic inflammatory diseases has led to the development of better treatment options for prolonging patient survival and minimizing morbidity. The recognition of "first-line" chemotherapy and monoclonal agents for these conditions has given more urgency to the need to re-administer these drugs in cases of drug hypersensitivity reactions. Therefore, in these cases, not only is desensitization considered when there is no alternative therapy available but also when alternative treatments are considered therapeutically inferior and/or more toxic. In this article, we describe the steps involved in the evaluation of these patients, factors to consider before making a decision to desensitize, the implementation of desensitization protocols, and the outcomes of such procedures.

Practical application of failure criteria in determining safe mud weight windows in drilling operations

Wellbore instability is reported frequently as one of the most significant incidents during drilling operations. Analysis of wellbore instability includes estimation of formation mechanical properties and the state of in situ stresses. In this analysis, the only controllable parameter during drilling operation is the mud weight. If the mud weight is larger than anticipated, the mud will invade into the formation, causing tensile failure of the formation. On the other hand, a lower mud weight can result in shear failures of rock, which is known as borehole breakouts. To predict the potential for failures around the wellbore during drilling, one should use a failure criterion to compare the rock strength against induced tangential stresses around the wellbore at a given mud pressure. The Mohr–Coulomb failure criterion is one of the commonly accepted criteria for estimation of rock strength at a given state of stress. However, the use of other criteria has been debated in the literature. In this paper, Mohr–Coulomb, Hoek–Brown and Mogi–Coulomb failure criteria were used to estimate the potential rock failure around a wellbore located in an onshore field of Iran. The log based analysis was used to estimate rock mechanical properties of formations and state of stresses. The results indicated that amongst different failure criteria, the Mohr–Coulomb criterion underestimates the highest mud pressure required to avoid breakouts around the wellbore. It also predicts a lower fracture gradient pressure. In addition, it was found that the results obtained from Mogi–Coulomb criterion yield a better comparison with breakouts observed from the caliper logs than that of Hoek–Brown criterion. It was concluded that the Mogi–Coulomb criterion is a better failure criterion as it considers the effect of the intermediate principal stress component in the failure analysis.

Abels’s groups revisited

We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown's criterion due to Martin Fluch and the author.

Excavation-induced zonal disintegration of the surrounding rock around a deep circular tunnel considering unloading effect

A new 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, in which the defect parameter R is introduced to describe effects of micro-defects on the deformation and failure of deep rock masses. On the basis of the deformation incompatibility condition, the non-Euclidean dynamic equation is established. Then Laplace transformation and inverse Laplace transformation theory are applied to solve the non-Euclidean dynamic equations. The stress fields are obtained from the non-Euclidean dynamic equations and the boundary conditions. The number and size of fractured and non-fractured zones are determined using the Hoek–Brown criterion. Numerical computation is carried out. It is found from numerical results that the number and size of fractured and non-fractured zones significantly depend on unloading rate, in-situ stress and dynamic mechanical parameters of deep rock masses.

Chart-based slope stability assessment using the Generalized Hoek–Brown criterion

We present new stability charts for the analysis of rock mass slopes satisfying the Generalized Hoek–Brown (GHB) criterion. Firstly, charts for calculating the factor of safety (FOS) of a slope for a specified slope angle β=45° are proposed. Secondly, a disturbance weighting factor fD is introduced to illustrate the effect of disturbance factor D upon the stability of rock slopes. Thirdly, a slope angle weighting factor fβ is proposed to show the influence of slope angle β on slope stability. Combined with stability charts based on β=45°, the weighting factors fD and fβ allow the calculation of the FOS of a slope assigned various slope angle under different blasting damage and stress relief conditions. The reliability of the proposed charts is tested against numerical solutions. The results show that FOS from the proposed charts exhibits only 3.1% average discrepancy from numerical solutions using 1680 sets of data. The proposed charts are simple and straightforward to use and can be adopted as useful tools for the preliminary rock slope stability analysis.

Instantaneous Friction Angle and Cohesion of 2-D and 3-D Hoek–Brown Rock Failure Criteria in Terms of Stress Invariants

The Mohr–Coulomb (M–C) failure criterion is one of the most widely used failure criteria in rock mechanics, although it has a number of shortcomings such as neglecting the nonlinear strength observed in rock or the effect of the intermediate principal stress σ 2. Other failure criteria have been proposed to effectively include in the predictions of failure the non-linear response of rock to confinement or the effects of the intermediate principal stress. The M–C criterion is still widely used, and it is arguably the criterion most used in practice. For example, stability evaluations of shallow rock structures such as slopes and foundations are routinely carried out by estimating a friction angle and a cohesion of the rock mass. To include the dependency of cohesion and friction angle on stresses, efforts are being made to estimate equivalent values of the M–C parameters for the range of stresses applicable to a particular design. The paper suggests a new and convenient approach to find the equivalent friction angle and cohesion from any failure criterion that can be expressed in terms of the Nayak and Zienkiewicz’s stress invariants. To demonstrate the capabilities and application of the methodology, the new approach is applied to two failure criteria: the Hoek–Brown (H–B) criterion and the Hoek–Brown and Willam–Warnke (HB–WW) criterion, 2-D and 3-D failure criteria, respectively. Results from the new method, in terms of equivalent friction and cohesion for the H–B criterion, are exactly the same as the results obtained from Balmer’s theory, which confirms the validity of the new method. The predicted equivalent friction and cohesion for the HB–WW criterion show a dependency on σ 2, which does not occur for a 2-D failure criterion.

True-triaxial compressive strength of Maha Sarakham salt

True triaxial compressive strengths of the Maha Sarakham salt are experimentally determined by using a polyaxial load frame. The salt specimens with nominal dimensions of 5.4×5.4×10.8cm3 are loaded to failure with the minimum principal stress, σ3, varying from 0 to 7MPa, and the intermediate principal stress, σ2, varying from 0 to 80MPa. The major principal (axial) stress is increased at a rate of 0.5–1.0MPa/s until failure occurs. Based on the Coulomb criterion the internal friction angle determined from the triaxial loading condition (σ2=σ3) is 50°, and the cohesion is 5.0MPa. The elastic parameters of the salt tend to be independent of σ2 for the applied stress range. The effect of σ2 on the salt strength can be described best by the modified Wiebols and Cook criterion with a mean misfit of 3.5MPa. The (power law) Mogi criterion underestimates the salt strength, particularly under high σ3 values. The modified Lade and 3-D Hoek & Brown criteria overestimate the strength at all levels of σ3. The Coulomb and Hoek & Brown criteria cannot describe the salt strengths beyond the condition where σ2=σ3, as they cannot incorporate the effects of σ2. Both circumscribed and inscribed Drucker–Prager criteria severely underestimate σ1 at failure for all stress conditions, showing the largest mean misfit of 19.5MPa.

Modification of a generalized three-dimensional Hoek–Brown strength criterion

To take account of the influence of the intermediate principle stress, Zhang and Zhu [1,2] proposed a three-dimensional (3D) version of the generalized Hoek–Brown strength criterion. The generalized Zhang–Zhu strength criterion is a true 3D version of the Hoek–Brown criterion, not only inheriting the advantages of the Hoek–Brown strength criterion, but predicts the same strength as the two-dimensional (2D) Hoek–Brown strength criterion at both triaxial compression and extension states. However, the failure surface of the generalized 3D Zhang–Zhu strength criterion is not smooth at either the triaxial compression or extension state and concave at the triaxial extension state, which may have problems with some stress paths and cause inconvenience for numerical applications. In this paper, the reason for the non-smoothness and non-convexity of the generalized 3D Zhang–Zhu strength criterion was first discussed by studying its Lode dependence. Then the criterion was modified by utilizing three different Lode dependences with characteristics of both smoothness and convexity to replace its Lode dependence. Finally the smoothness, convexity and prediction accuracy of the modified criteria were evaluated by applying them to analyze both intact rocks and jointed rock masses. The modified criteria not only keep the advantages of the generalized 3D Zhang–Zhu strength criterion, but solve the non-smoothness and non-convexity problem with no loss of accuracy for strength prediction.

An Empirical Failure Criterion for Intact Rocks

The parameter mi is an important rock property parameter required for use of the Hoek–Brown failure criterion. The conventional method for determining mi is to fit a series of triaxial compression test data. In the absence of laboratory test data, guideline charts have been provided by Hoek to estimate the mi value. In the conventional Hoek–Brown failure criterion, the mi value is a constant for a given rock. It is observed that using a constant mi may not fit the triaxial compression test data well for some rocks. In this paper, a negative exponent empirical model is proposed to express mi as a function of confinement, and this exercise leads us to a new empirical failure criterion for intact rocks. Triaxial compression test data of various rocks are used to fit parameters of this model. It is seen that the new empirical failure criterion fits the test data better than the conventional Hoek–Brown failure criterion for intact rocks. The conventional Hoek–Brown criterion fits the test data well in the high-confinement region but fails to match data well in the low-confinement and tension regions. In particular, it overestimates the uniaxial compressive strength (UCS) and the uniaxial tensile strength of rocks. On the other hand, curves fitted by the proposed empirical failure criterion match test data very well, and the estimated UCS and tensile strength agree well with test data.

Brown’s criterion in Bredon homology

We translate Brown's criterion for homological finiteness properties to the setting of Bredon homology.

Modified Hoek?Brown criterion for nonlinear strength of frozen soil

Abstract The strength of frozen soil presents obvious nonlinearity, which makes many theoretical criteria unable to accurately predict failure. The Hoek–Brown criterion, which is one of the most widely used strength criteria in geotechnical engineering, could predict the strength of frozen soil subjected to low confining pressure. For high values of confining pressure, the predicted results do not agree well with experimental data. In the present study, the criterion was modified by incorporating a new parameter to account for the nonlinearity of frozen soil under high confining pressures. The parameter and instantaneous friction angle for the nonlinear failure envelope were also analyzed. To describe the failure mechanism of frozen soil under complicated stress state, the deviatoric section of strength criterion for frozen soil, in which the Mohr–Coulomb criterion forms the low bound, and the Drucker–Prager criterion forms the upper bound, was obtained, and a united strength criterion was established to take the effect of intermediate principal stress on strength behavior into account. The strength criterion could well reflect the nonlinear strength characteristic of frozen soil under the influence of pressure melting and crushing phenomena.

Sensitivity analysis of influencing parameters in cavern stability

In order to analyze the stability of the underground rock structures, knowing the sensitivity of geomechanical parameters is important. To investigate the priority of these geomechanical properties in the stability of cavern, a sensitivity analysis has been performed on a single cavern in various rock mass qualities according to RMR using Phase 2. The stability of cavern has been studied by investigating the side wall deformation. Results showed that most sensitive properties are coefficient of lateral stress and modulus of deformation. Also parameters of Hoek–Brown criterion and σc have no sensitivity when cavern is in a perfect elastic state. But in an elasto-plastic state, parameters of Hoek–Brown criterion and σc affect the deformability; such effect becomes more remarkable with increasing plastic area. Other parameters have different sensitivities concerning rock mass quality (RMR). Results have been used to propose the best set of parameters for study on prediction of sidewall displacement.

Zonal disintegration mechanism of cross-anisotropic rock mass around a deep circular tunnel under dynamic unloading

A mechanical model is proposed to investigate zonal disintegration mechanism of cross-anisotropic rock masses around a deep circular tunnel subjected to dynamic unloading under hydrostatic pressure condition as well as the total elastic stress-field distributions. Effects of unloading rate and dynamic mechanical parameters on the zonal disintegration phenomenon of the surrounding rock masses around a deep circular tunnel as well as the total elastic stress field distributions are taken into account. Numerical computation is carried out. The location, number and size of fractured and non-fractured zones are determined using the Hoek–Brown criterion. It is found from numerical results that the location, number and size of fractured and non-fractured zones significantly depend on unloading rate and dynamic mechanical parameters of deep rock masses.

Three-Dimensional Failure Criteria Based on the Hoek–Brown Criterion

List of symbols mb Hoek–Brown material constant s Hoek–Brown material constant a Hoek–Brown material constant Co Uniaxial compressive strength I0 1 First invariant of the effective stress tensor r1 Major principal effective stress r2 Intermediate principal effective stress r3 Minor principal effective stress soct Octahedral shear stress r1hb Major principal effective stress at failure for the 2D Hoek–Brown criterion r3hb Minor principal effective stress at failure for the 2D Hoek–Brown criterion a Simplified Priest material constant b Simplified Priest material constant w Simplified Priest material constant

Stability analysis of rock slopes subjected to seepage forces using the modified Hoek–Brown criterion

The paper deals with the stability analysis of a water-saturated rock slope by means of the kinematic approach of limit analysis theory. Particular emphasis is first given to the effects of pore water pressure on the global stability of geotechnical structures. The rock strength properties being formulated in terms of effective stresses, it is shown how the effect of seepage flow may be accounted for such an analysis through driving body forces derived from the gradient of excess pore pressure distribution is shown. The latter is obtained as the solution to a hydraulic boundary value problem, and then incorporated as external loading in the subsequent stability analysis. The rock strength properties are modeled by a modified Hoek–Brown failure criterion, for which closed-form expressions of the support functions have been derived in a previous paper. The approach is then applied to investigate the stability of rock slopes. Computational results are given, providing ample evidence of the destablizing effects induced by the seepage forces. The influence of relevant geometrical, strength and loading parameters is also discussed.