A generalized nonlinear three-dimensional Hoek‒Brown failure criterion

To understand the strengths of rocks under complex stress states,a generalized nonlinear threedimensional(3D)Hoek‒Brown failure(NGHB)criterion was proposed in this study.This criterion shares the same parameters with the generalized HB(GHB)criterion and inherits the parameter advantages of GHB.Two new parameters,b,and n,were introduced into the NGHB criterion that primarily controls the deviatoric plane shape of the NGHB criterion under triaxial tension and compression,respectively.The NGHB criterion can consider the influence of intermediate principal stress(IPS),where the deviatoric plane shape satisfies the smoothness requirements,while the HB criterion not.This criterion can degenerate into the two modified 3D HB criteria,the Priest criterion under triaxial compression condition and the HB criterion under triaxial compression and tension condition.This criterion was verified using true triaxial test data for different parameters,six types of rocks,and two kinds of in situ rock masses.For comparison,three existing 3D HB criteria were selected for performance comparison research.The result showed that the NGHB criterion gave better prediction performance than other criteria.The prediction errors of the strength of six types of rocks and two kinds of in situ rock masses were in the range of 2.0724%-3.5091%and 1.0144%-3.2321%,respectively.The proposed criterion lays a preliminary theoretical foundation for prediction of engineering rock mass strength under complex in situ stress conditions.

Three-dimensional limit variation analysis on the ultimate pullout capacity of the anchor cables based on the Hoek-Brown failure criterion

Only simplified two-dimensional model and a single failure mode are adopted to calculate the ultimate pullout capacity(UPC)of anchor cables in most previous research.This study focuses on a more comprehensive combination failure mode that consists of bond failure of an anchorage body and failure of an anchored rock mass.The three-dimensional ultimate pullout capacity of the anchor cables is calculated based on the Hoek-Brown failure criterion and variation analysis method.The numerical solution for the curvilinear function in fracture plane is obtained based on the finite difference theory,which more accurately reflects the failure state of the anchor cable,as opposed to that being assumed in advance.The results reveal that relying solely on a single failure mode for UPC calculations has limitations,as changes in parameter values not only directly impact the UPC value but also can alter the failure model and thus the calculation method.

A failure criterion for shale considering the anisotropy and hydration based on the shear slide failure model

A failure criterion fully considering the anisotropy and hydration of shale is essential for shale formation stability evaluation.Thus,a novel failure criterion for hydration shale is developed by using Jaeger’s shear failure criterion to describe the anisotropy and using the shear strength reduction caused by clay minerals hydration to evaluate the hydration.This failure criterion is defined with four parameters in Jaeger’s shear failure criterion(S_(1),S_(2),a andφ),three hydration parameters(k,ω_(sh)andσ_(s))and two material size parameters(d and l0).The physical meanings and determining procedures of these parameters are described.The accuracy and applicability of this failure criterion are examined using the published experimental data,showing a cohesive agreement between the predicted values and the testing results,R^(2)=0.916 and AAREP(average absolute relative error percentage)of 9.260%.The error(|D_(p)|)is then discussed considering the effects ofβ(angle between bedding plane versus axial loading),moisture content and confining pressure,presenting that|Dp|increases whenβis closer to 30°,and|D_(p)|decreases with decreasing moisture content and with increasing confining pressure.Moreover,|D_(p)|is demonstrated as being sensitive to S1and being steady with decrease in the data set whenβis 0°,30°,45°and 90°.

Prediction and Verifcation of Forming Limit Diagrams Based on a Modifed Shear Failure Criterion

The forming limit diagram plays an important role in predicting the forming limit of sheet metals.Previous studies have shown that,the method to construct the forming limit diagram based on instability theory of the original shear failure criterion is efective and simple.The original shear instability criterion can accurately predict the left area of the forming limit diagram but not the right area.In this study,in order to improve the accuracy of the original shear failure criterion,a modifed shear failure criterion was proposed based on in-depth analysis of the original shear failure criterion.The detailed improvement strategies of the shear failure criterion and the complete calculation process are given.Based on the modifed shear failure criterion and diferent constitutive equations,the theoretical forming limit of TRIP780 steel and 5754O aluminum alloy sheet metals are calculated.By comparing the theoretical and experimental results,it is shown that proposed modifed shear failure criterion can predict the right area of forming limit more reasonably than the original shear failure criterion.The efect of the pre-strain and constitutive equation on the forming limits are also analyzed in depth.The modifed shear failure criterion proposed in this study provides an alternative and reliable method to predict forming limit of sheet metals.

Experimental Study and Failure Criterion Analysis of Rubber Fibre Reinforced Concrete under Biaxial Compression-Compression

In order to examine the biaxial compression-compression properties of rubber fibre reinforced concrete(RFRC),an experimental study on RFRC under different lateral compressive stresses was carried out by considering different rubber replacement rates and polypropylene fibre contents.The failure modes and mechanical property parameters of different RFRC working conditions were obtained from the experiment to explore the effects of rubber replacement rate and polypropylene fibre content on the biaxial compression-compression properties of RFRC.The following conclusions were drawn.Under the influence of lateral compressive stress,the biaxial compression-compression failure mode gradually developed from a columnar pattern to a flaky pattern,suggesting that the incorporation of rubber and polypropylene fibres into the concrete resulted in a significant change in the development of cracks.For different rubber replacement rates and polypropylene fibre contents,the vertical compressive stress exhibited the same developing trend under the influence of lateral compressive stress.Specifically,the lateral compressive stress imposed the minimum effect on the vertical compressive stress when the rubber replacement rate and polypropylene fibre content were 20%and 0.4%,respectively,and imposed the maximum effect when the rubber replacement rate and polypropylene fibre content were 20%and 0%,respectively.With the increase of rubber replacement rate,the vertical peak stress was significantly reduced,which implies that an appropriate amount of polypropylene fibres can increase the vertical peak stress to a certain extent.Then,the biaxial compression-compression mechanism of RFRC was analysed from the microscopic level by using scanning electron microscope(SEM).Meanwhile,based on Kupfer’s biaxial compression-compression failure criterion and the octahedral stress space,a biaxial compression-compression failure criterion for RFRC was proposed,which was proven to have good applicability.The research results of this study provide important theoretical basis for the engineering application and development of RFRC.

A generalized three-dimensional failure criterion for rock masses

The smooth convex generalized failure function, which represents 1/6 part of envelope in tile deviatoric plane, is proposed. The proposed function relies on four shape parameters （L, a, b and c）, in which two parameters （a and b） are dependent on the others. The parameter Ls is called extension ratio. The proposed failure function could be incorporated with any two-dimensional （2D） failure criteria to make it a three-dimensional （3D） version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion tbr geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering （1） constant extension ratio and （2） variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio （L~） that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters （c~. and m） are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameterfis introduced. The modified extension ratio is related tofand extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range off varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression betweenfand Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures.

Validation of Hoek-Brown failure criterion charts for rock slopes

Although stability charts suggested by Hoek and Bray on the basis of Mohr-Coulomb criterion are used for rock slopes,but complete and precise recognition is required for distinguishing cohesive strength and Mohr-Coulomb equivalent internal friction angle for rock mass.The paper by Lia et al.[6]is the only one that introduced rock slope charts according to Hoek-Brown failure criterion.In this paper,at first,this type of charts is introduced.Then,Mohr-Coulomb failure criterion charts[2]are compared and validated with Hoek-Brown failure criterion ones[6].Next,Bishop method utilizing Slide software is compared with Hoek-Brown failure criterion stability charts.Average standard deviation(ASD),root mean square error(RMSE)and variance account for(VAF)were used for the comparison.According to the results,because of high distribution and very low correlation among the comparisons,Hoek-Brown failure criterion charts are not efficient.

Failure criterion for soft rocks considering intermediate principal stress

The significant differences between hard rocks(more brittle)and soft rocks(more ductile)may suggest the use of different failure criteria.A strength criterion for soft rocks that includes intermediate principal stress was proposed.The new criterion includes two independent parameters:the uniaxial compressive strength(σ_(ci)),which can be obtained from common laboratory tests or indirectly estimated by alternative index tests in the laboratory or field;and f(joint),which is used to characterize the rock mass quality and can be easily estimated.The authors compared the predictive capabilities of the new criterion with other criteria using the database of soft rocks under two conditions:with and without triaxial data.For the estimation of triaxial and true-triaxial strengths,the new criterion generally produced a better fit.The proposed criterion is practical for an approximate first estimation of rock mass strength,even without triaxial data,as it balances accuracy(lower prediction error)and simplicity(fewer independent parameters).

A modified failure criterion for transversely isotropic rocks

A modified failure criterion is proposed to determine the strength of transversely isotropic rocks. Me-chanical properties of some metamorphic and sedimentary rocks including gneiss, slate, marble, schist, shale, sandstone and limestone, which show transversely isotropic behavior, were taken into consider-ation. Afterward, introduced triaxial rock strength criterion was modified for transversely isotropic rocks. Through modification process an index was obtained that can be considered as a strength reduction parameter due to rock strength anisotropy. Comparison of the parameter with previous anisotropy in-dexes in literature showed reasonable results for the studied rock samples. The modified criterion was compared to modified Hoek-Brown and Ramamurthy criteria for different transversely isotropic rocks. It can be concluded that the modified failure criterion proposed in this study can be used for predicting the strength of transversely isotropic rocks.

Seismic bearing capacity of strip footings on rock masses using the Hoeke Brown failure criterion

In this paper,the bearing capacity of strip footings on rock masses has been studied in the seismic case.The stress characteristics or slip line method was used for analysis.The problem was analyzed in the plane strain condition using the Hoeke Brown failure criterion.First,the equilibrium equations along the stress characteristics were obtained and the rock failure criterion was applied.Then,the equations were solved using the finite difference method.A computer code has been provided for analysis.Given the footing and rock parameters,the code can calculate the stress characteristics network and obtain the stress distribution under the footing.The seismic effects have been applied as the horizontal and vertical pseudo-static coefficients.The results of this paper are very close to those of the other studies.The seismic bearing capacity of weightless rock masses can be obtained using the proposed equations and graphs without calculating the whole stress characteristics network.

A simplified approach to directly consider intact rock anisotropy in Hoek——Brown failure criterion

Many rock types have naturally occurring inherent anisotropic planes, such as bedding planes, foliation,or flow structures. Such characteristic induces directional features and anisotropy in rocks＇ strength anddeformational properties. The HoekeBrown （HeB） failure criterion is an empirical strength criterionwidely applied to rock mechanics and engineering. A direct modification to HeB failure criterion toaccount for rock anisotropy is considered as the base of the research. Such modification introduced a newdefinition of the anisotropy as direct parameter named the anisotropic parameter （Kb）. However, thecomputation of this parameter takes much experimental work and cannot be calculated in a simple way.The aim of this paper is to study the trend of the relation between the degree of anisotropy （Rc） and theminimum value of anisotropic parameter （Kmin）, and to predict the Kmin directly from the uniaxialcompression tests instead of triaxial tests, and also to decrease the amount of experimental work. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.

Definition of failure criterion for frozen soil under directional shear-stress path

A series of directional shear tests on remolded frozen soil was carried out at 10°C by using a hollow cylinder apparatus to study failure criterion under a directional shear-stress path.Directional shear tests were conducted at five shear rates(10,20,30,40,and 50 kPa/min)and five intermediate principal stress coefficients(b=0,0.25,0.5,0.75,and 1),with the mean principal stress(p=4.5 MPa)kept constant.The results show that the torsional strength and the generalized strength both increase with the increase of the shear rates.According to the failure modes of frozen soil under different shear rates,the specimens present obvious plastic failure and shear band;and the torsional shear component dominates the failure modes of hollow cylindrical specimens.A shear rate of 30 kPa/min is chosen as the loading rate in the directional shear tests of frozen soil.The shape of the failure curve in theπplane is dependent on the directional anglesαof the major prin cipal stress.It is reasonable to use the strain-hardening curves to define the deviatoric stress value atγg=15%(generalized shear strain)as the failure criterion of frozen soil under a directional shear-stress path.

Strength Regularity and Failure Criterion of High-Strength High-Performance Concrete under Multiaxial Compression

Multiaxial compression tests were performed on 100 mm×100 mm×100 mm high-strength high-performance concrete （HSI-IPC） cubes and normal strength concrete （NSC） cubes. The failure modes of specimens were presented, the static compressive strengths in principal directions were measured, the influence of the stress ratios was analyzed. The experimental results show that the ultimate strengths for HSHPC and NSC under multiaxial compression are greater than the uniaxial compressive strengths at all stress ratios, and the multiaxial strength is dependent on the brittleness and stiffness of concrete, the stress state and the stress ratios. In addition, the Kupfer-Gersfle and Ottosen＇s failure criteria for plain HSHPC and NSC under multiaxial compressive loading were modified.

Formulation of cross-anisotropic failure criterion for soils

Inherently anisotropic soil fabric has a considerable influence on soil strength. To model this kind of inherent anisotropy, a three-dimensional anisotropic failure criterion was proposed, employing a scalar-valued anisotropic variable and a modified general three- dimensional isotropic failure criterion. The scalar-valued anisotropic variable in all sectors of the deviatoric plane was defined by correlating a normalized stress tensor with a normalized fabric tensor. Detailed comparison between the available experimental data and the corresponding model predictions in the deviatoric plane was conducted. The proposed failure criterion was shown to well predict the failure behavior in all sectors, especially in sector II with the Lode angle ranging between 60° and 120°, where the prediction was almost in accordance with test data. However, it was also observed that the proposed criterion overestimated the strength of dense Santa Monica Beach sand in sector III where the intermediate principal stress ratio b varied from approximately 0.2 to 0.8, and slightly underestimated the strength when b was between approximately 0.8 and 1. The difference between the model predictions and experimental data was due to the occurrence of shear bending, which might reduce the measured strength. Therefore, the proposed anisotropic failure criterion has a strong ability to characterize the failure behavior of various soils and potentially allows a better description of the influence of the loading direction with respect to the soil fabric.

Failure criterion effect on solid production prediction and selection of completion solution

Production of fines together with reservoir fluid is called solid production. It varies from a few grams or less per ton of reservoir fluid posing only minor problems, to catastrophic amount possibly leading to erosion and complete filling of the borehole. This paper assesses solid production potential in a carbonate gas reservoir located in the south of Iran. Petrophysical logs obtained from the vertical well were employed to construct mechanical earth model. Then, two failure criteria, i.e. Mohre Coulomb and Mogi-Coulomb,were used to investigate the potential of solid production of the well in the initial and depleted conditions of the reservoir. Using these two criteria, we estimated critical collapse pressure and compared them to the reservoir pressure. Solid production occurs if collapse pressure is greater than pore pressure. Results indicate that the two failure criteria show different estimations of solid production potential of the studied reservoir. Mohre Coulomb failure criterion estimated solid production in both initial and depleted conditions, where Mogi-Coulomb criterion predicted no solid production in the initial condition of reservoir. Based on Mogi-Coulomb criterion, the well may not require completion solutions like perforated liner, until at least 60% of reservoir pressure was depleted which leads to decrease in operation cost and time.

Reply to Discussion on “A generalized three-dimensional failure criterion for rock masses”

General:Jaiswal and Shrivastva(2012)proposed the mathematical formulations,i.e.the J–S criterion for converting generalized H-B failure criterion into 3D smooth convex failure criterion at deviatoric plane.The J–S strength criterion is in two versions:uniform and variable extension ratio.It has been observed from the analysis that at uniform extension ratio,the required strength parameters are only UCS and m(other parameters such as Ls,a,b and c are related with m).In the case of variable extension ratio,extra parameter f is required along with UCS and m.Thus,it has minimal strength parameters compared to You strength criterion.Furthermore,You strength criterion does not obey the smooth convex condition at deviatoric plane.

Non-probabilistic Hydraulic Turbine Blade Vibration Reliability Research Based on Fuzzy Failure Criterion

In hydraulic turbine engineering,turbine blade vibration reliability assessment is of great significance. Based on the interval mathematical theory, the variables existing in hydraulic turbine blade are described as interval variables. Considering the fuzzy failure criterion of turbine blade distancing from resonance and vibration fatigue stress,fuzzy possibilistic reliability is expressed and analyzed qualitatively taking normal bathtub function as the membership function of blade resonance failure and deflection major type function as the membership function of the intensity failure. As a result,hydraulic turbine blade vibration reliability is analyzed based on the fuzziness of variables and failure criterion. A safer working environment is provided under possibility context by comparing with the qualitative conclusions in the past literature.

Stability analysis of intermittently jointed rock slopes based on the stepped failure mode

In practical engineering,due to the noncontinuity characteristics of joints in rock slopes,in addition to plane failure,stepped sliding failure may occur for intermittently jointed rock slopes.Especially for intermittently bedding jointed rock slopes,the correlation and difference in strength parameters between joints and rock bridges,along with the various failure modes and intermittency of rock bridges,contribute to the complexity of stepped failure modes and the unpredictability of failure regions.Based on the upper-bound limit analysis method and multi-sliders step-path failure mode,considering the shear and tensile failure of rock bridges and the weakened relationship between the strength parameters of rock bridges and jointed surfaces,by introducing the modified M-C failure criterion and the formula for calculating the energy consumption of tensile failure of rock bridges,two failure mechanisms are constructed to obtain the safety factor(F_(s))of intermittently jointed rock slopes.The sequential quadratic programming method is used to obtain the optimal upper-bound solution for F_(s).The influence of multiple key parameters(slope height H,horizontal distance L,Slope angleβ,shear strength parameters of the rock bridgeφr and cr,Dimensionless parameter u,weakening coefficients of the internal friction angle and cohesion between the rock bridges and joint surfaces Kφand Kc)on the stability analysis of intermittently jointed rock slopes under the shear failure mode of rock bridges as well as under the tensile failure mode is also explored.The reliability of the failure mechanisms is verified by comparative analysis with theoretical results,numerical results,and landslide cases,and the variation rules of F_(s)with each key parameter are obtained.The results show that F_(s) varies linearly withφr and cr of the rock bridge and with K_(φ)and K_(c),whereas F_(s)changes nonlinearly with H and L.In particular,with the increase in Kφand Kc,Fs increases by approximately 52.78%and 171.02%on average,respectively.For rock bridge tensile failure,F_(s) shows a nonlinearly positive correlation withφr,cr,Kφand Kc.In particular,with the increase in Kφand Kc,Fs increases by approximately 13%and 61.69%on average,respectively.Fs decreases rapidly with increasing slope gradientβand decreasing dimensionless parameterμ.When Kφand Kc are both less than 1.0,the stepped sliding surface occurs more easily than the plane failure surface,especially in the case of tensile failure of the rock bridge.In addition,rock slopes with higher strength parameters,taller heights,and greater weakening coefficients are prone to rock bridge tension failure with lower Fs,and more attention should be given to the occurrence of such accidents in actual engineering.

Study on triaxial test method and failure criterion of asphalt mixture

Asphalt mixture is the most widely used pavement material all over the world. In China, more than 90% of service expressways are asphalt pavement. However, current asphalt pavement design method still has irrationality. Even though maximum tensile stress theory is used as failure criterion, pavement structure under the effects of wheel load is in three-dimensional complex stress state. Obviously, one-dimensional strength theory cannot reflect the failure characteristics and the resistance of pavement structure. So it is necessary to study the failure criterion of asphalt mixture under three-dimensional com- plex stress state. Due to limitations of test equipment, there are almost no studies in related area. Under this background, this paper develops a new triaxial test method, ac- cording to the investigation of strength characteristics of asphalt mixture under complex stress state through plane isobaric/axial tensile test, plane isobaric/axial compression test, plane tensile and compression/axial tensile test, to reveal the general rules of asphalt mixture＇s strength failure. The failure mode is divided into three types： tensile failure, shear failure and rheological failure. The tensile meridian and compression meridian in the stress space and strength envelope in the π plane where hydrostatic pressure is greater than zero are obtained, and the failure criterion of asphalt mixture under complex stress state is established, providing theoretical method and scientific basis for structure design as well as strength check of asphalt pavement under three-dimensional stress state.

A new stress-based multi-scale failure criterion of composites and its validation in open hole tension tests

A new stress-based multi-scale failure criterion is proposed based on a series of off-axis tension tests, and their corresponding fiber failure modes and matrix failure modes are determined at the microscopic level. It is a physical mechanism based, three-dimensional damage analysis criterion which takes into consideration the constituent properties on the macroscopic failure behavior of the composite laminates. A complete set of stress transformation, damage determination and evolution methods are established to realize the application of the multi-scale method in failure analysis. Open-hole tension（OHT） specimens of three material systems（CCF300/5228, CCF300/5428 and T700/5428） are tested according to ASTM standard D5766, and good agreements are found between the experimental results and the numerical predictions. It is found that fiber strength is a key factor influencing the ultimate strength of the laminates, while matrix failure alleviates the stress concentration around the hole. Different matchings of fiber and matrix result in different failure modes as well as ultimate strengths.