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.

Nonlinear empirical failure criterion for rocks under triaxial compression

Based on existing triaxial compression experimental data,a new empirical failure criterion with wide applicability was proposed considering hydrostatic pressure,second stress invariance,and maximum shear stress.Four fitting evaluation indicators were used to verify the consistency of the new failure criterion,and the differences with the other 6 failure criteria were discussed.The characteristics of the new failure criteria in the principal stress space were finally analyzed.The results indicate that(1)the new failure criterion exhibits strong predictive ability for triaxial experiments and has good applicability for both intact and jointed rocks;(2)the influence of hydrostatic pressure on the failure surface exhibits a non-linear trend,and different hydrostatic pressure also exhibits different distribution patterns on the deviatoric stress plane,with a distribution characteristic pattern of hexagonal snowflake-regular hexagon.The maximum shear stress has a torsional effect on the new criterion,in the three-dimensional failure surface.The parameters a and b of the rock have an impact on the failure surface morphology of the new criterion function on the offset surface.

Estimating shear strength of high-level pillars supported with cemented backfilling using the HoekeBrown strength criterion

Deep metal mines are often mined using the high-level pillars with subsequent cementation backfilling(HLSCB)mining method.At the design stage,it is therefore important to have a reasonable method for determining the shear strength of the high-level pillars(i.e.cohesion and internal friction angle)when they are supported by cemented backfilling.In this study,a formula was derived for the upper limit of the confining pressure σ3max on a high-level pillar supported by cemented backfilling in a deep metal mine.A new method of estimating the shear strength of such pillars was then proposed based on the Hoek eBrown failure criterion.Our analysis indicates that the horizontal stress σhh acting on the cemented backfill pillar can be simplified by expressing it as a constant value.A reasonable and effective value for σ3max can then be determined.The value of s3max predicted using the proposed method is generally less than 3 MPa.Within this range,the shear strength of the high-level pillar is accurately calculated using the equivalent MohreCoulomb theory.The proposed method can effectively avoid the calculation of inaccurate shear strength values for the high-level pillars when the original HoekeBrown criterion is used in the presence of large confining pressures,i.e.the situation in which the cohesion value is too large and the friction angle is too small can effectively be avoided.The proposed method is applied to a deep metal mine in China that is being excavated using the HLSCB method.The shear strength parameters of the high-level pillars obtained using the proposed method were input in the numerical simulations.The numerical results show that the recommended level heights and sizes of the high-level pillars and rooms in the mine are rational.

A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams

This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved.A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams.The proposed algorithm,named as two-parameter multiple shooting method,is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions.The impact of the axial force and additional point masses is also taken into account.Due to the fact that the method is based directly on the fourth-order ordinary differential equation,the structures do not have to be divided into many small elements to obtain an accurate enough solution,even though the geometry is very complex.To verify the proposed method,three different examples are considered,i.e.,a three-segment non-prismatic beam,a prismatic column subject to non-uniformly distributed compressive loads,and a two-segment beam with an additional point mass.Numerical analyses are carried out with the software MATHEMATICA.The results are compared with the solutions computed by the commercial finite element program SOFiSTiK.Good agreement is achieved,which confirms the correctness and high effectiveness of the formulated algorithm.

Rockburst criterion and evaluation method for potential rockburst pit depth considering excavation damage effect

Excavation-induced disturbances in deep tunnels will lead to deterioration of rock properties and formation of excavation damaged zone(EDZ).This excavation damage effect may affect the potential rockburst pit depth.Taking two diversion tunnels of Jinping II hydropower station for example,the relationship between rockburst pit depth and excavation damage effect is first surveyed.The results indicate that the rockburst pit depth in tunnels with severe damage to rock masses is relatively large.Subsequently,the excavation-induced damage effect is characterized by disturbance factor D based on the Hoek-Brown criterion and wave velocity method.It is found that the EDZ could be further divided into a high-damage zone(HDZ)with D=1 and weak-damage zone(WDZ),and D decays from one to zero linearly.For this,a quantitative evaluation method for potential rockburst pit depth is established by presenting a three-element rockburst criterion considering rock strength,geostress and disturbance factor.The evaluation results obtained by this method match well with actual observations.In addition,the weakening of rock mass strength promotes the formation and expansion of potential rockburst pits.The potential rockburst pit depth is positively correlated with HDZ and WDZ depths,and the HDZ depth has a significant contribution to the potential rockburst pit depth.

Transient Stability Analysis of Converter-based Islanded Microgrids Based on Iterative Equal Area Criterion Considering Reactive Power Loop Dynamics and Varying Damping

With the rapid increase in the installed capacity of renewable energy in modern power systems,the stable operation of power systems with considerable power electronic equipment requires further investigation.In converter-based islanded mi-crogrid(CIM)systems equipped with grid-following(GFL)and grid-forming(GFM)voltage-source converters(VSCs),it is chal-lenging to maintain stability due to the mutual coupling effects between different VSCs and the loss of voltage and frequency sup-port from the power system.In previous studies,quantitative transient stability analysis was primarily used to assess the active power loop of GFM-VSCs.However,frequency and voltage dy-namics are found to be strongly coupled,which strongly affects the estimation result of stability boundary.In addition,the vary-ing damping terms have not been fully captured.To bridge these gaps,this paper investigates the transient stability of CIM consid-ering reactive power loop dynamics and varying damping.First,an accuracy-enhanced nonlinear model of the CIM is derived based on the effects of reactive power loop and post-disturbance frequency jump phenomena.Considering these effects will elimi-nates the risk of misjudgment.The reactive power loop dynamics make the model coefficients be no longer constant and thus vary with the power angle.To evaluate quantitatively the effects of re-active power loop and varying damping on the transient stability of CIM,an iterative criterion based on the equal area criterion theory is proposed.In addition,the effects of parameters on the stable boundary of power system are analyzed,and the dynamic interaction mechanisms are revealed.Simulation and experiment results verify the merits of the proposed method.

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.

Strength criterion for crystalline rocks considering grain size effect and tensile-compressive strength ratio

The macroscopic mechanical properties of rocks are significantly influenced by their microstructure.As a material bonded by mineral grains,the grain morphology of crystalline rock is the primary factor influencing the strength.However,most strength criteria neglect the strength variations caused by different grain characteristics in rocks.Furthermore,the traditional linear criteria tend to overestimate tensile strength and exhibit apex singularity.To address these shortcomings,a piecewise strength criterion that considers the grain size effect has been proposed.A part of an ellipse was employed to construct the envelope of the tensive-shear region on the meridian plane,to accurately reproduce the low tensile-compressive strength ratio.Based on the analysis of experimental data,both linear and exponential modification functions that account for grain size effects were integrated into the proposed criterion.The corresponding finite element algorithm has been implemented.The accuracy and applicability of the proposed criterion were validated by comparing with the experimental data.

Maximum Correntropy Criterion-Based UKF for Loosely Coupling INS and UWB in Indoor Localization

Indoor positioning is a key technology in today’s intelligent environments,and it plays a crucial role in many application areas.This paper proposed an unscented Kalman filter(UKF)based on the maximum correntropy criterion(MCC)instead of the minimummean square error criterion(MMSE).This innovative approach is applied to the loose coupling of the Inertial Navigation System(INS)and Ultra-Wideband(UWB).By introducing the maximum correntropy criterion,the MCCUKF algorithm dynamically adjusts the covariance matrices of the system noise and the measurement noise,thus enhancing its adaptability to diverse environmental localization requirements.Particularly in the presence of non-Gaussian noise,especially heavy-tailed noise,the MCCUKF exhibits superior accuracy and robustness compared to the traditional UKF.The method initially generates an estimate of the predicted state and covariance matrix through the unscented transform(UT)and then recharacterizes the measurement information using a nonlinear regression method at the cost of theMCC.Subsequently,the state and covariance matrices of the filter are updated by employing the unscented transformation on the measurement equations.Moreover,to mitigate the influence of non-line-of-sight(NLOS)errors positioning accuracy,this paper proposes a k-medoid clustering algorithm based on bisection k-means(Bikmeans).This algorithm preprocesses the UWB distance measurements to yield a more precise position estimation.Simulation results demonstrate that MCCUKF is robust to the uncertainty of UWB and realizes stable integration of INS and UWB systems.

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.

Prestack migration velocity analysis based on simplified two-parameter moveout equation

Stacking velocity V_（C2）,vertical velocity ratio γ_0,effective velocity ratio γ_（eff）,and anisotropic parameter x_（eff） are correlated in the PS-converted-wave（PS-wave） anisotropic prestack Kirchhoff time migration（PKTM） velocity model and are thus difficult to independently determine.We extended the simplified two-parameter（stacking velocity V_（C2） and anisotropic parameter k_（eff）） moveout equation from stacking velocity analysis to PKTM velocity model updating and formed a new four-parameter（stacking velocity V_（C2）,vertical velocity ratio γ_0,effective velocity ratio γ_（eff）,and anisotropic parameter k_（eff）） PS-wave anisotropic PKTM velocity model updating and process flow based on the simplified twoparameter moveout equation.In the proposed method,first,the PS-wave two-parameter stacking velocity is analyzed to obtain the anisotropic PKTM initial velocity and anisotropic parameters;then,the velocity and anisotropic parameters are corrected by analyzing the residual moveout on common imaging point gathers after prestack time migration.The vertical velocity ratio γ_0 of the prestack time migration velocity model is obtained with an appropriate method utilizing the P- and PS-wave stacked sections after level calibration.The initial effective velocity ratio γ_（eff） is calculated using the Thomsen（1999） equation in combination with the P-wave velocity analysis;ultimately,the final velocity model of the effective velocity ratio γ_（eff） is obtained by percentage scanning migration.This method simplifies the PS-wave parameter estimation in high-quality imaging,reduces the uncertainty of multiparameter estimations,and obtains good imaging results in practice.

A GENERAL FORM OF THE INCREMENTS OF A TWO-PARAMETER WIENER PROCESS

In this paper, we consider a general form of the increments for a two-parameter Wiener process. Both the Csorgo-Revesz's increments and a class of the lag increments are the special cases of this general form of increments. Our results imply the theorem that have been given by Csorgo and Revesz (1978), and some of their conditions are removed.

Buckling analysis of shear deformable composite conical shells reinforced by CNTs subjected to combined loading on the two-parameter elastic foundation

The main objective of this study is to investigate the buckling analysis of CCSs reinforced by CNTs subjected to combined loading of hydrostatic pressure and axial compression resting on the twoparameter elastic foundation(T-P-EF).It is one of the first attempts to derive the governing equations of the CCSs reinforced with CNTs,based on a generalized first-order shear deformation shell theory(FSDST)which includes shell-foundation interaction.By adopting the extended mixing rule,the effective material properties of CCSs reinforced by CNTs with linear distributions are approximated by introducing some efficiency parameters.Three carbon nanotube distribution in the matrix,i.e.uniform distribution(U)and V and X-types linear distribution are taken into account.The stability equations are solved by using the Galerkin procedure to determine the combined buckling loads(CBLs)of the structure selected here.The numerical illustrations cover CBLs characteristics of CCSs reinforced by CNTs in the presence of the T-P-EF.Finally,a parametric study is carried out to study the influences of the foundation parameters,the volume fraction of carbon nanotubes and the types of reinforcement on the CBLs.

A New Two-Parameter Heteromorphic Elliptic Equation: Properties and Applications

The ellipse and the superellipse are both planar closed curves with a double axis of symmetry. Here we show the isoconcentration contour of the simplified two-dimensional advection-diffusion equation from a stable line source in the center of a wide river. A new two-parameter heteromorphic elliptic equation with a single axis of symmetry is defined. The values of heights, at the point of the maximum width and that of the centroid of the heteromorphic ellipse, are derived through mathematical analysis. Taking the compression coefficient θ = b/a = 1 as the criterion, the shape classification of H-type, Standard-type and W-type for heteromorphic ellipse have been given. The area formula, the perimeter theorem, and the radius of curvature of heteromorphic ellipses, and the geometric properties of the rotating body are subsequently proposed. An illustrative analysis shows that the inner contour curve of a heteromorphic elliptic tunnel has obvious advantages over the multiple- arc splicing cross section. This work demonstrates that the heteromorphic ellipses have extensive prospects of application in all categories of tunnels, liquid transport tanks, aircraft and submarines, bridges, buildings, furniture, and crafts.

Relation between Fresnel transform of input light field and the two-parameter Radon transform of Wigner function of the field

This paper proves a new theorem on the relationship between optical field Wigner function＇s two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field ψ （x＇） propagates through an optical [D （-B） （-C） A] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D, B. It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A, C.

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°.

A New Two-Parameter Formula for Rotational Spectra

A new two-parameter formula for the rotational spectra of well deformed nuclei isproposed. The formula is deduced from experimental level systematics and alternatively fromnuclear hydrodynamics. Comparisons with a great number of rotational spectra of even-even nu-clei in rare-earth and actinides region show that the formula is the best one among all two-pa-rameter formulas. It is pointed out that this formula can be applied to the spin assignment forsuperdeformed band.

Description of Rotational Molecular Spectra by the Two-Parameter Universal Equation

The n-power two-parameter universal equation for rotational spectra which we deduced recently is appliedto the description of the rotational bands of several diatomic and tetra-atomic molecules. Excellent agreement withexperimental data can be obtained with small n values. The relation between our equation and the famous Dunhamformula is discussed.

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.

BOUNDARY INTEGRAL EQUATIONS FOR BENDING PROBLEM OF REISSNER'S PLATES ONTWO-PARAMETER FOUNDATION

Two fundamental solutions for bending problem of Reissner's plates on twoparameter foundation are derived by means of Fouier integral transformation of generalized function in this paper.On the basis of virtual work principles, three boundary integral equations which fit for arbitrary shapes, loads and boundary conditions of thick plates are presented according to Hu Haichang's theory about Reissner's plates. It provides the fundamental theories for the application of BEM. A numerical example is given for clamped, simply supported and free boundary conditions. The results obtained are satisfactory as compared with the analytical methods.