Enhanced Hoek-Brown(H-B)criterion for rocks exposed to chemical corrosion

Underground constructions often encounter water environments,where water–rock interaction can increase porosity,thereby weakening engineering rocks.Correspondingly,the failure criterion for chemically corroded rocks becomes essential in the stability analysis and design of such structures.This study enhances the applicability of the Hoek-Brown(H-B)criterion for engineering structures operating in chemically corrosive conditions by introducing a kinetic porosity-dependent instantaneous mi(KPIM).A multiscale experimental investigation,including nuclear magnetic resonance(NMR),X-ray diffraction(XRD),scanning electron microscopy(SEM),pH and ion chromatography analysis,and triaxial compression tests,is employed to quantify pore structural changes and their linkage with the strength responses of limestone under coupled chemical-mechanical(C-M)conditions.By employing ion chromatography and NMR analysis,along with incorporating the principles of free-face dissolution theory accounting for both congruent and incongruent dissolution,a kinetic chemical corrosion model is developed.This model aims to calculate the kinetic porosity alterations within rocks exposed to varying H+concentrations and durations.Subsequently,utilizing the generalized mixture rule(GMR),the kinetic porositydependent mi is formulated.Evaluation of the KPIM-enhanced H-B criterion using compression test data from 5 types of rocks demonstrated a high level of consistency between the criterion and the experimental results,with a coefficient of determination greater than 0.96,a mean absolute percentage error less than 4.84%,and a root-mean-square deviation less than 5.95 MPa.Finally,the physical significance of the porosity-dependent instantaneous mi is clarified:it serves as an indicator of a rock’s capacity to leverage the confining pressure effect.

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.

Analysis of plane strain bending of a strain hardening curved beam based on unified yield criterion

The analysis of plane strain elastic-plastic bending of a linear strain hardening curved beam with a narrow rectangular cross section subjected to couples at its end is conducted based on a unified yield criterion. The solutions for the mechanical properties of plane strain bending are derived, which are adapted for various kinds of non-strength differential materials and can be degenerated to those based on the Tresca, von Mises, and twin-shear yield criteria. The dependences of the two critical bending moments, the radii of the interfaces between the elastic and plastic regions and the radial displacements of the points at the symmetrical plane on different yield criteria and Poisson’s ratios are discussed. The results show that the influences of different yield criteria and Poisson’s ratio on the two critical bending moments, the radii of the interfaces between the elastic and plastic regions and the radial displacements of the points at the symmetrical plane of the curved beam are significant. Once the value of bis obtained by experiments, the yield criterion and the corresponding solution for the materials of interest are then determined.

Limit load and failure mechanisms of a vertical Hoek-Brown rock slope

The problem considered in this short note is the limit load determination of a vertical rock slope.The classical limit theorem is employed with the use of adaptive finite elements and nonlinear programming to determine upper and lower bound limit loads of a Hoek-Brown vertical rock slope.The objective function of the mathematical programming problem is such as to optimize a boundary load,which is known as the limit load,resembling the ultimate bearing capacity of a strip footing.While focusing on the vertical slope,parametric studies are carried out for several dimensionless ratios such as the dimensionless footing distance ratio,the dimensionless height ratio,and the dimensionless rock strength ratio.A comprehensive set of design charts is presented,and failure envelopes shown with the results explained in terms of three identified failure mechanisms,i.e.the face,the toe,and the Prandtl-type failures.These novel results can be used with great confidence in design practice,in particularly noting that the current industry-based design procedures for the presented problem are rarely found.

A new three-dimensional Hoek-Brown strength criterion

The Hoek-Brown （HB） strength criterion has been widely applied to the estimation of strength of intact rock and rock mass, while evolving ever since. However, negligence of the effect of the intermediate principal stress still remains in the criterion＇s latest version. At the same time, several three-dimensional （3D） HB strength, which can takes into account the influence of the intermediate principal stress, have already been proposed, among which the 3D HB criterion proposed by Zhang and Zhu seems to be the most reasonable one. However, the Zhang 3D HB criterion may have problems with some stress path close to triaxial extension state because of the non-convexity characteristic of its failure surface. In this paper, a new 3D HB strength criterion is presented based on a generalized form of the HB criterion, which also considers the effect of the intermediate principal stress and inherits all the merits of the original version of the HB criterion. In addition, this new criterion can remedy to some extent the shortcomings observed in the Zhang 3D HB criterion. Polyaxial tests for five different rocks from pub- lished literatures are used for evaluating this new criterion and comparing it with the Zhang 3D HB criterion. The re- sults show that this new criterion may over-predict or under- predict the polyaxial strength of rocks but the errors are rela- tively small, and similar results are also found for the Zhang 3D HB criterion, which one is better depends on the type of the rock under estimation.

Derivation of Plastic Work Rate Done per Unit Volume for Mean Yield Criterion and Its Application

In Haigh Westergaard stress space linear combination of twin shear stress and Tresca yield functions is called the mean yield （MY） criterion. The mathematical relationship of the criterion and its plastic work rate done per unit volume were derived. A generalized worked example of slab forging was analyzed by the criterion and its corresponding plastic work rate done per unit volume. Then, the precision of the solution was compared with those by Mises and Twin shear stress yield criterions, respectively. It turned out that the calculated results by MY criterion were in good agreement with those by Mises criterion.

Limit analysis for the seismic stability of three-dimensional rock slopes using the generalized Hoek-Brown criterion

The parameters that influence slope stability and their criteria of failure are fairly understood but over-conservative design approaches are often preferred,which can result in excessive overburden removal that may jeopardize profitability in the context of open pit mining.Numerical methods such as finite element and discrete element modelling are instrumental to identify specific zones of stability,but they remain approximate and do not pinpoint the critical factors that influence stability without extensive parametric studies.A large number of degrees of freedom and input parameters may make the outcome of numerical modelling insufficient compared to analytical solutions.Existing analytical approaches have not tackled the stability of slopes using non-linear plasticity criteria and threedimensional failure mechanisms.This paper bridges this gap by using the yield design theory and the Hoek-Brown criterion.Moreover,the proposed model includes the effect of seismic forces,which are not always taken into account in slope stability analyses.The results are presented in the form of rigorous mathematical expressions and stability charts involving the loading conditions and the rock mass properties emanating from the plasticity criterion.

Unified strength model based on the Hoek-Brown failure criterion for fibre-reinforced polymer-confined pre-damaged concrete columns with circular and square cross sections

Fibre-reinforced polymer(FRP)has the advantages of high strength,light weight,corrosion resistance and convenient construction and is widely used in repairing and strengthening damaged concrete columns.Most of the existing strength models were built by regression analysis of experimental data;however,in this article,a new unified strength model is proposed using the Hoek-Brown failure criterion.To study the strength of FRP-confined damaged and undamaged concrete columns,900 test data were collected from the published literature and a large database that contains the cross-sectional shape of each specimen,the damage type,the damage level and the FRP-confined stiffness was established.A new strength model using the Hoek-Brown failure criterion was established and is suitable for both circular and square columns that are undamaged,load-damaged and fire-damaged.Based on the database,most of the existing strength models from the published literature and the model proposed in this paper were evaluated.The evaluation shows that the proposed model can predict the compressive strength for FRP-confined pre-damaged and undamaged concrete columns with good accuracy.

A simplified three-dimensional extension of Hoek-Brown strength criterion

The Hoek-Brown(HB)strength criterion has been applied widely in a large number of projects around the world.However,this criterion ignores the intermediate principal stress s2.Many evidences have demonstrated that the rock strength is dependent on s2.Thus it is necessary to extend the HB criterion into a three-dimensional(3D)form.In this study,the effect of s2 on the strength of rocks is identified by reviewing the true triaxial tests of various rock types reported in the literature.A simple 3D strength criterion is developed.The modified criterion is verified by the true triaxial tests of 13 rock types.The results indicate that the modified criterion can achieve a good fit to most of rock types.It can represent a series of criteria as b varies.For comparisons,several existing 3D versions of the HB criterion are selected to predict the strengths of these rock types.It is indicated that the proposed criterion works better than other criteria.A substantial relationship between parameter b and the unconfined compressive strength is established,which guarantees that the proposed criterion can still work well even in the absence of true triaxial test data.

Elasto-plastic Analysis of Circular Tunnels in Jointed Rock Masses Satisfy the Hoek-Brown Failure Criterion

The relationship between the Hoek-Brown parameters and the mechanical response of circular tunnels is il-lustrated. Closed-form and approximate solutions are given for the extent of the plastic zone and the stress and dis-placement fields under axisymmetrical and asymmetric stress conditions. For the same rock masses and under axisym-metrical stress conditions,the radius of the plastic zone in terms of Hoek-Brown criterion is generally an approximation of the radius in terms of the Mohr-Coulomb criterion. The radius in terms of the Hoek-Brown criterion is larger under low stress conditions. For poor quality rock masses (GSI<25),measures (such as grouting,setting rock bolts,etc.) that improve the GSI of rock masses are effective in improving the stability of tunnels. It is not advisable to improve the sta-bility of the tunnels by providing a small support resistance p through shotcrete,except for very poor quality jointed rock masses. Without reference to the quality of the rock mass,the disturbance factor D should not less than 0.5. Meas-ures which disturb rock masses during tunnel construction should be taken carefully when the tunnel depth increases.

A Generalized Yield Criterion

A generalized yield criterion is proposed based on the metai plastic deformation mechanics and the fundamental formula in theory of plasticity. Using the generalized yield criterion, the reason is explained that Mises yield criterion and Tresca yield criterion do not completely match with experimental data. It has been shown that the yield criteria of ductile metals depend not only on the quadratic invariant of the deviatoric stress tensor J2, but also on the cubic invariant of the deviatoric stress tensor J3 and the ratio of the yield stress in pure shear to the yield stress in uniaxial tension κ/σs. The reason that Mises yield criterion and Tresca yield criterion are not in good agreement with the experimental data is that the effect of J3 and κ/σs is neglected.

A modified Hoek-Brown failure criterion considering the damage to reservoir bank slope rocks under water saturation-dehydration circulation

After water is impounded in a reservoir, rock mass in the hydro-fluctuation belt of the reservoir bank slope is subject to water saturation- dehydration circulation （WSDC）. To quantify the rate of change of rock mechanical properties, samples from the Longtan dam area were measured with uniaxial compression tests after different numbers （1, 5, 10, 15, and 20） of simulated WSDC cycles. Based on the curves derived from these tests, a modified Hock- Brown failure criterion was proposed, in which a new parameter was introduced to model the cumulative damage to rocks after WSDC. A case of an engineering application was analyzed, and the results showed that the modified Hock-Brown failure criterion is useful. Under similar WSDC-influenced engineering and geological conditions, rock mass strength parameters required for analysis and evaluation of rock slope stability can be estimated according to this modified Hoek-Brown failure criterion.

Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek-Brown criterion

The stability analysis of passive bolt-reinforced rock slopes under seismic loads is investigated within the framework of the kinematic approach of limit analysis theory.A pseudo-static method is adopted to account for the inertial forces induced in the rock mass by seismic events.The strength properties of the rock material are described by a modified Hoek-Brown strength criterion,whereas the passive bolts are modeled as bar-like inclusions that exhibit only resistance to tensile-compressive forces.Taking advantage of the ability to compute closed-form expressions for the support functions associated with the modified Hoek-Brown strength criterion,a rotational failure mechanism is implemented to derive rigorous lower bound estimates for the amount of reinforcement strength to prevent slope failure.The approach is then applied to investigating the effects of relevant geometry,strength and loading parameters in light of a preliminary parametric study.The accuracy of the approach is assessed by comparison of the lower bound estimates with finite element limit analysis solutions,thus emphasizing the ability of the approach to properly predict the stability conditions and to capture the essential features of deformation localization pattern.Finally,the extension of the approach to account for slipping at the interface between reinforcements and surrounding rock mass is outlined.

Three-Dimensional Analysis of Rolling by Twin Shear Stress Yield Criterion

Using the twin shear stress yield criterion, the surface integral of the co-line vectors, and the integration depending on upper limit, Kobayashi＇s three-dimensional velocity field of rolling was analyzed and an analytical expression of rolling torque and single force was obtained. Through redoing the same experiment of rolling pure lead as Sims, the calculated results by the above expression were compared with those of Kobayashi and Sims formulae. The results show that the twin shear stress yield criterion is available for rolling analysis and the calculated results by the new formula are a little higher than those by Kobayashi and Sims ones if the reduction ratio is less than 30%.

Equal perimeter yield criterion and its specific plastic work rate: Development, validation and application

In order to overcome the nonlinearity of Mises criterion, a new linear yield criterion with a dodecagon shape of the same perimeter as Mises criterion was derived by means of geometrical analysis. Its specific plastic work rate expressed as a linear function of the yield stress, the maximum and minimum strains was also deduced and compared with that of Mises criterion. The physical meaning of the proposed yield criterion is that yielding of materials begins when the shear yield stress τs reaches the magnitude of 0.594σs. By introducing the Lode parameter, validation of evolution expressions of the proposed yield criterion with those based on Tresca, Mises and TSS criteria as well as available classical yield experimental results of various metals shows that the present results intersect with Mises results and coincide well with experimental data. Moreover, further application to the limit analysis of circle plate as an example is performed to demonstrate the effectiveness of the proposed yield criterion, and the subsequent comparison of limit loads with the Tresca analytical solutions and Mises numerical results shows that the present results are higher than the Tresca analytical results, and are in good agreement with the Mises numerical results.

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.

New double yield surface model for coarse granular materials incorporating nonlinear unified failure criterion

A new double-yield-sarface （DYS） model was developed to characterize the strength and deformation behaviors of coarse granular materials （CGMs）. Two kinds of deformation mechanisms, including the shear and compressive plastic deformation, were taken into account in this model, These two deformation mechanisms were described by the shear and compressive yield functions, respectively. The Lode angle dependent formulations of proposed model were deduced by incorporating a 3D nonlinear unified failure criterion. Some comparisons were presented between the numerical predictions of proposed model and test data of true triaxial tests on the modeled rockfills. The model predictions are in good agreement with the test data and capture the strain hardening and plastic volumetric dilation of CGMs. These findings verify the reasonability of current DYS model, and indicate that this model is well suited to reproduce the stress-strain-volume change behavior of CGMs in general.

Analytical Solution for Three-dimensional Forging Taking into Account Bulging of Sides by Mean Yield Criterion

Currently, for some complex plastic deformations, the analytical solution can not be obtained by using Mises yield criterion, because Mises yield criterion is nine dimensions, the velocity field is complex, and the solving methods are not innovative. Corresponding solutions of these problems are that yield criterion is linearized to reduce the variable numbers, and the velocity field and the solving methods are reasonably simplified, respectively. In this paper, a new linear yield criterion--mean yield（MY） criterion and inner-product of strain rate vector are used to analytically solve 3D forging taking into account bugling of sides. The velocity field is expressed as a vector in three dimensions, and rotation and divergence are applied to confirm that the velocity field is kinematically admissible. Then, the corresponding strain rate tensor of the velocity field is transformed into principal one by making the determinant of coefficients of the tensor cubic equation be zero. By using MY criterion, the plastic power is term by term integrated and summed according to inner-product of strain rate vector. An upper bound analytical solution is obtained for the forging, and verified by a pure lead press test. The test result turns out that the total pressure calculated by MY criterion is higher by 2.5%-15% than measuring value. In addition, a measuring formula of bulging parameter （a） is proposed, but the values of a measured by the formula are lower than those optimized by the golden section search. The total pressure calculated by MY criterion is compared with the ones by twin shear, Trasca yield, and Mises yield criterion. The comparing result shows that the total pressure calculated by MY criterion is slightly higher than the mean value of that by twin shear and Trasca yield criterion, and lower than that by Mises yield criterion, but more close to that by Mises yield criterion compared with that by other two. The proposed analytical solving methods can be effectively used to other complex plastic deformation, simplifying the solving process and obtaining the reasonable results.

Two-dimensional face stability analysis in rock masses governed by the Hoek-Brown strength criterion with a new multi-horn mechanism

The face stability problem is a major concern for tunnels excavated in rock masses governed by the Hoek-Brown strength criterion.To provide an accurate prediction for the theoretical solution of the critical face pressure,this study adopts the piecewise linear method(PLM)to account for the nonlinearity of the strength envelope and proposes a new multi-horn rotational mechanism based on the Hoek-Brown strength criterion and the associative flow rule.The analytical solution of critical support pressure is derived from the energy-work balance equation in the framework of the plastic limit theorem;it is formulated as a multivariable nonlinear optimization problem relying on 2m dependent variables(m is the number of segments).Meanwhile,two classic linearized measures,the generalized tangential technique(GTT)and equivalent Mohr-Coulomb parameters method(EMM),are incorporated into the analysis for comparison.Surprisingly,the parametric study indicates a significant improvement in support pressure by up to 13%compared with the GTT,and as expected,the stability of the tunnel face is greatly influenced by the rock strength parameters.The stress distribution on the rupture surface is calculated to gain an intuitive understanding of the failure at the limit state.Although the limit analysis is incapable of calculating the true stress distribution in rock masses,a rough approximation of the stress vector on the rupture surface is permitted.In the end,sets of normalized face pressure are provided in the form of charts for a quick assessment of face stability in rock masses.

An improved yield criterion characterizing the anisotropic and tension-compression asymmetric behavior of magnesium alloy

A novel yield criterion based on CPB06 considering anisotropic and tension-compression asymmetric behaviors of magnesium alloys was derived and proposed(called M_CPB06).This yield criterion can simultaneously predict the yield stresses and the Lankford ratios at different angles(if any)under uniaxial tension,compression,equal-biaxial and equal-compression conditions.Then,in order to further describe the anisotropic strain-hardening characteristics of magnesium alloy,the proposed M_CPB06 criterion was further evolved to the M_CPB06ev model by expressing the parameters of the M_CPB06 model as functions of the plastic strain.As the model was developed,the stresses and Lankford ratios of AZ31B and ZK61M magnesium alloys at different angles under tensile,compressive and through-thickness compressive conditions were used to calibrate the M_CPB06/M_CPB06ev and the existing CPB06ex2 model.Calibration results reveal that compared with the CPB06ex2 yield criterion with equal quantity of coefficients,the M_CPB06 criterion exhibits certain advancement,and meanwhile the M_CPB06ev model can relatively accurately predict the change of the yield locus with increase of the plastic strain.Finally,the M_CPB06ev model was developed through UMAT in LS-DYNA.Finite element simulations using the subroutine were conducted on the specimens of different angles to the rolling direction under tension and compression.Simulation results were highly consistent with the experimental results,demonstrating a good reliability and accuracy of the developed subroutine.