Critical Height Research Articles

How Does the Critical Torus Instability Height Vary with the Solar Cycle?

The ideal magnetohydrodynamic torus instability can drive the eruption of coronal mass ejections. The critical threshold of magnetic field strength decay for the onset of the torus instability occurs at different heights in different solar active regions, and understanding this variation could therefore improve space weather prediction. In this work, we aim to find out how the critical torus instability height evolves throughout the solar activity cycle. We study a significant subset of Helioseismic and Magnetic Imager (HMI) and Michelson Doppler Imager Space-Weather HMI Active Region Patches (SHARPs and SMARPs) from 1996 to 2023, totaling 21,584 magnetograms from 4436 unique active-region patches. For each magnetogram, we compute the critical height averaged across the main polarity inversion line, the total unsigned magnetic flux, and the separation between the positive and negative magnetic polarities. We find the critical height in active regions varies with the solar cycle, with higher (lower) average critical heights observed around solar maximum (minimum). We conclude that this is because the critical height is proportional to the separation between opposite magnetic polarities, which in turn is proportional to the total magnetic flux in a region, and more magnetic regions with larger fluxes and larger sizes are observed at solar maximum. This result is noteworthy because, despite the higher critical heights, more coronal mass ejections are observed around solar maximum than at solar minimum.

A frictional arch model for pile-cap-beam-supported embankment

The pile-cap-beam-supported (PCBS) system can strength the soil arching effect of embankment, increase the lateral stiffness, bending resistance and vertical bearing capacity of the rigid pile, however there is no frictional soil arch model of PCBS embankment. In this paper, first a frictional arch model for PCBS embankment modified from Russell’s frictional arching model was proposed. The proposed model in this paper considers the algorithm of lateral pressure coefficient k and a changing critical height of soil arch. In this new method, the influence of pile spacing, filling properties, height and pile spacing on critical height soil arch was comprehensively considered. Second, a series of numerical cases were performed to verify the effectiveness of the proposed model and study the arching effect of PCBS embankment. By comparing the vertical stress and settlement between the theoretical and simulation results, the rationality of the proposed method to estimate the stress and critical height of arch was validated. The effectiveness of the proposed method was further validated by comparing loading efficacy to a reported case. Last, the stress and deformation of PCBS and pile-cap-supported (PCS) embankment were analyzed and the superiority of PCBS system in improving the performance of embankment was observed finally.

A Wind Tunnel Experiment Study on Splash Functions During Sand Saltation

AbstractSplash functions delineate the intricate interaction between wind‐driven particles and the bed. However, due to limitations in measurement methods, achieving a comprehensive understanding of splash functions remains challenging. In this study, we utilized a high‐speed system and particle trajectory tracking algorithm to reconstruct 857 collision events involving natural sand particles obtained from field samples and artificial glass microspheres interacting with a bed composed of analogous particles in a wind tunnel. During the experiments, the ratio of the wind shear velocity (u*) to the impact threshold (u*ti) consistently ranged between approximately 1.22 and 1.79. Our findings indicate that the impact angle remains independent of both impact velocity and particle size, maintaining an approximate value of 10.5 ± 6.5°. The evaluation of splash functions depends on the criterion used to define ejection, that is, the ratio of the centroid's height of the liftoff particles to their particle size (H/d). Additionally, at the same critical height (H/d = 1.5), our splash functions show differences of varying degrees from previous experiments and theoretical studies performed under no wind conditions. We believe that the main reason for these differences may be that the energy exchange between the bed surface and the airflow increases the looseness of the large‐particle (>0.1 mm) bed surface. These findings hold significant implications for accurately modeling sand‐bed collisions in natural environments.

Changes with Time in the Relation between the Critical Frequency and Height of the F 2 Layer

Changes with Time in the Relation between the Critical Frequency and Height of the F 2 Layer

Effect of temperature anisotropy formed by fast electron beams moving in the flare loop on its excited electron-cyclotron maser instability

Context. The electron-cyclotron maser instability (ECMI) is a significant coherent radio emission mechanism widely utilized in various astrophysical radio phenomena. It is well known that the velocity anisotropic distribution of energetic electrons, which leads to an inverted perpendicular population in the vertical direction with ∂fb/∂v⊥ > 0, can provide the free energy necessary for the ECMI. Aims. The initial velocity distribution of energetic electrons leaving the acceleration region is typically isotropic or beam-like. However, as these energetic electrons travel along the magnetic field as fast electron beams (FEBs) in magnetic plasma, various velocity anisotropic distributions can emerge. In this paper, we examine the impact of temperature anisotropy formed by beam electrons traveling along a flare loop on the ECMI. Methods. By neglecting the energy loss of energetic electrons as they traverse the corona and invoking the conservation of energy and magnetic moments, we established the relationship between momentum dispersion and the magnetic field. Utilizing the magnetic field model of the flare loop, we calculated the evolution of momentum dispersion and the growth rates of the ECMI as FEBs precipitate along the flare loop. Results. The results demonstrate that the temperature anisotropy arising as FEBs descend along the flare loop significantly impacts the ECMI. The maximum growth rates of the excited modes exhibit a gradual increase initially and then decline rapidly after reaching a critical height for β0 = 0.2c and 0.15c. The results also show that the growth rates of the O2 mode are one order of magnitude smaller than those of the O1 and X2 modes. This indicates that the harmonic radiation is X-mode polarized. Notably, the temperature anisotropy of FEBs as they precipitate along the flare loop with different magnetic field models or at different heights has similar effects on the ECMI.

Relationships between Precipitation and Elevation in the Southeastern Tibetan Plateau during the Active Phase of the Indian Monsoon

The precipitation gradient (PG) is a crucial parameter for watershed hydrological models. Analysis of daily precipitation and elevation data from 30 stations in the southeastern Tibetan Plateau (SETP) during the active phase of the Indian monsoon reveals distinct patterns. Below 3000 m, precipitation generally decreases with increasing altitude. Between 3000 and 4000 m, precipitation patterns are more complex; in western regions, precipitation increases with elevation, whereas in eastern regions, it decreases. Above 4000 m, up to the highest observation point of 4841 m, precipitation continues to decrease with elevation, with a more pronounced decline beyond a critical height. In the SETP, PGs for LYR and NYR are positive, at 11.3 ± 2.7 mm/100 m and 17.3 ± 3.8 mm/100 m, respectively. Conversely, PLZB exhibits a negative PG of −22.3 ± 4.2 mm/100 m. The Yarlung Zangbo River (YLZBR) water vapor channel plays a significant role in these PGs, with the direction and flux of water vapor potentially influencing both the direction and magnitude of the PG. Additional factors such as precipitation intensity, the number of precipitation days, precipitation frequency, and station selection also significantly impact the PG. Notable correlations between elevation and variables such as the number of precipitation days, non-precipitation days, and precipitation intensity. The precipitation intensity gradients (PIGs) are 0.06 ± 0.02 mm/d/100 m, 0.11 ± 0.04 mm/d/100 m, and −0.18 ± 0.04 mm/d/100 m for the three catchments, respectively. Future research should incorporate remote sensing data and expand site networks, particularly in regions above 5000 m, to enhance the accuracy of precipitation–elevation relationship assessments, providing more reliable data for water resource simulation and disaster warning.

Spreading of droplet impact on ribbed superhydrophobic surfaces with varying structure height

Hypothesis:Quantifying droplet spreading characteristics on ribbed superhydrophobic surfaces is challenging due to the complex fluid dynamics, limiting the practical applications. Droplet spreading dynamics are significantly influenced by the diameter ratio (rib diameter to droplet diameter) and height ratio (structure height to rib diameter) during impact. Experiment:Ribbed superhydrophobic surfaces with diameter ratios (0.18 to 0.75) and height ratios (0.125 to 0.5) underwent impact experiments with droplets at varied Weber numbers. Through simulations and analysis, we summarized the patterns of the maximum ribwise spreading coefficient, elucidated the droplet spreading mechanism on ribbed superhydrophobic surfaces. Findings:The findings revealed asymmetry in impact dynamics on ribbed superhydrophobic surfaces in the ribwise and radial directions. The maximum ribwise spreading coefficient initially decreases, followed by increase with increasing diameter ratio, decreasing in a slowing down trend with increasing height ratio. These observations indicated that maximum ribwise spreading coefficient can be minimize with critical ratios. Comparatively, the maximum spreading coefficient at critical height ratios is 72.7% to 92.3% of that on flat surfaces. To precisely forecast maximum ribwise spreading coefficient, two models are introduced for diameter ratios below and above critical range when height ratios below the critical range.

Persistent high-latitude ionospheric response to solar wind forcing

The solar wind continuously transfers energy into the Earth’s thermosphere-ionosphere system and variations in the solar wind properties modify the state of the system. The modifications are best visible during storm conditions when the ingestion of extreme amounts of solar wind energy into the thermosphere-ionosphere system causes global changes in thermosphere as well as large deviations in the ionospheric electron density from its quiet conditions. This study shows that there exists a persistent impact of the solar wind on the high-latitude electron density. A data set of 22 years of Total Electron Content (TEC) and 15 years of ionosonde data (critical frequency foF2 and height of maximum electron density hmF2) at Tromsø (70°N, 19°E) are used for correlation analyses with different solar wind parameters derived from measurements of the Advanced Composition Explorer (ACE). The results show that the ionospheric parameters systematically respond with an increase or decrease depending on local time, season and solar cycle. TEC and foF2 increase with solar wind energy during winter night conditions and decrease with increasing solar wind energy during summer daytime. The summer negative ionospheric response is more intense during solar maximum conditions, while the winter positive ionospheric response is stronger during solar minimum. An anomaly is observed around 10 UT (noon), when TEC and foF2 respond with an increase during solar minimum conditions. Cross polar cap plasma convection, particle precipitation and Joule heating are considered to be the main drivers of the electron density changes at Tromsø. Local time, season and solar cycle changes in the background ionosphere-thermosphere conditions lead to different effects of theses driving processes. The results help to better understand the variability of the high-latitude electron density and show that solar wind forcing causes a systematic and persistent response of the ionosphere which alternates depending on local time, season and solar cycle.

Variations in Diffusion Coefficients on Equinox Days at Specified Critical Heights of the Ionosphere at the Equator

This study investigates the local time diffusion coefficient for stable and unstable ( , ) states at the equator (00), "the geographical latitude where the first peak of the magnetic equatorial trough in the ionosphere occurs," during the spring and fall equinoxes (March 21 and September 23). The findings show that the diffusion tensor in a steady state is completely real and has a magnitude equal to the speed of light in a steady state. But in the unsteady state, the diffusion tensor consists of two parts, real and imaginary. The diagonal elements of the real part tensor are of the size of the conductivity in the ionosphere and the elements of the imaginary part are of the size of the speed of sound with about the same magnitude. Furthermore, for all assumed conditions, the diffusion elements form the first peak of the magnetic equatorial trough at 6.00 am local time.

Aerodynamic Optimization Design of a Supergravity Centrifuge: A Low-Resistance Strategy

Wind resistance optimization is crucial for enhancing the rotational speed of supergravity centrifuges. We conducted a study using computational fluid dynamics on the Centrifugal Hypergravity and Interdisciplinary Experiment Facility (CHIEF) under construction at Zhejiang University and validated it experimentally using a ZJU400gt centrifuge. Our findings indicate significant reductions in wind resistance through structural modifications of the CHIEF. Reducing the outer radius from 4650 to 4150 mm decreased wind resistance by 16%, primarily due to reduced effective viscosity in the wake region’s gases. More substantial reductions were achieved by lowering the height of the outer wall from 2200 to 1400 mm, which cut wind resistance by 25%. This height reduction suppressed vortex shedding and Kármán vortex street development via the Venturi effect. Adjustments to the roughness height of wall surfaces further decreased wind resistance, with minimal impact from arm roughness. A critical roughness height was identified, below which no further reductions in wind resistance could be attained. Notably, using disc-shaped arms reduced wind resistance by approximately 73% because of their minimal pressure–resistance components and predominant frictional resistance, highlighting their potential in future high-speed centrifuge designs.

Scaling Transition of Active Turbulence from Two to Three Dimensions.

Turbulent flows are observed in low-Reynolds active fluids, which display similar phenomenology to the classical inertial turbulence but are of a different nature. Understanding the dependence of this new type of turbulence on dimensionality is a fundamental challenge in non-equilibrium physics. Real-space structures and kinetic energy spectra of bacterial turbulence are experimentally measured from two to three dimensions. The turbulence shows three regimes separated by two critical confinement heights, resulting from the competition of bacterial length, vortex size and confinement height. Meanwhile, the kinetic energy spectra display distinct universal scaling laws in quasi-2D and 3D regimes, independent of bacterial activity, length, and confinement height, whereas scaling exponents transition in two steps around the critical heights. The scaling behaviors are well captured by the hydrodynamic model wedevelop, which employs image systems to represent the effects of confining boundaries. The study suggests a framework for investigating the effect of dimensionality on non-equilibrium self-organizedsystems.

Artificial excitation and propagation of ultra-low frequency signals in the polar ionosphere

This paper has established a relatively comprehensive model for ultra-low frequency (ULF) current induced by thermal pressure gradients and its propagation. In the ULF current excitation model, we decomposed the current into a constant term unaffected by altitude and a product with a function significantly influenced by altitude. Combining this with the EISCAT background, we determined that for modulation frequencies below 5 Hz, the optimal height for ULF current excitation corresponds to the critical frequency height. We calculated the ionospheric currents at heating altitudes of 332 km for modulation frequencies of 5 Hz; the corresponding maximum currents were 1.03 × 10−10 A·m−2. By incorporating the current into the ULF waves propagation model based on magnetoionic theory, we found that the electromagnetic field energy is mainly concentrated in the horizontal direction, indicating that the energy primarily propagates outward through magnetosonic waves. The dominant components are the electric field component Ey and the magnetic field component Bz, whose maximum values reached 1.1 μV·m−1 and 1.5 pT. Unfortunately, magnetosonic waves cannot propagate downward due to the sharp variation in the real part of the refractive index between 200 and 300 km. However, the shear Alfvén waves component By can propagate downward, and there is still an intensity of approximately 0.1 pT at the bottom of the ionosphere, which is because the refractive index of shear Alfvén waves is most uniform in the parallel magnetic field direction, allowing By to propagate parallel to the magnetic field effectively.

Streamwise-Elongated Sinusoidal Roughness Elements with Enhanced Laminarizing Effect on Three-Dimensional Boundary Layer

As a laminar flow control device for delaying the crossflow-induced transition of a three-dimensional boundary layer, sinusoidal roughness elements (SREs) are placed in a Falkner–Skan–Cooke boundary layer, and the resultant laminarizing effect is numerically investigated in comparison with discrete roughness elements (DREs). Because SREs are elongated in the streamwise direction and designed to avoid flow tripping, the critical height of SREs is much higher than that of DREs. Moreover, the wake flow behind SREs efficiently generates and sustains crossflow vortices that are not dangerously unstable against secondary instabilities but able to strongly distort the mean crossflow profile into a less unstable one. By measuring this mean flow distortion by SREs and DREs, the laminarizing effect is compared among them. It is shown that the effect of SREs is higher than that of DREs and can be enhanced by choosing the appropriate height, angle, and wavelength depending on the local boundary-layer profile.

A simple predictive three-step chemical model for gaseous stoichiometric hydrogen–oxygen detonation quenching

Two types of three-step chain-branching kinetics for gaseous hydrogen–oxygen detonation quenching are proposed. The reaction rates dependency on density and pressure, as well as the change in molecular weight are included and fitted to results of detailed chemistry not only along the Hugoniot curve but also behind transverse waves. The trend of induction and reaction times and reduced activation energy of zero-dimensional constant pressure combustion process as well as the induction, reaction lengths, and peak thermicity of the ZND model are focused on in the calibration procedure. One model (3SM-I) is designed to model the ideal detonation whereas the other (3SM-N) focuses on the Dn−κ curve during calibration, i.e. the detonation velocity-mean front curvature curves, aiming to capture the non-idealities. The predictive ability using the proposed models is evaluated in a detonation transmission configuration, a layer of ▪ mixture being confined by N2, in determining the critical height, reactive layer height at which detonation transmission is not possible. Both approaches get closer to the critical height from detailed chemistry. With the present set of model parameters, 3SM-I is better than 3SM-N for prediction of critical height. Moreover, borrowing analytical ignition criteria and wave hierarchy approach, the comparison of various chemical models that are able to recover the critical height suggests that the slope of the Dn−κ curve, linked with the chemical response to losses, is the first key parameter for the critical height determination. In addition, the dynamics of the extinction near criticality such as appearance of transverse detonation and quenching distance from 3SM-I are in very good agreement with that of detailed chemistry due to better estimation of the chemical features along the Hugoniot curve and behind transverse waves. The present study provides a guideline to get simplified chemical models from detailed chemistry for predictive simulations.Novelty and significance statementThe novelty of research is two-fold. The first is to provide general guideline to get simplified models from detailed chemistry for the simulation of detonation extinction with improved predictive ability. The tested configuration was that of a stoichiometric hydrogen–oxygen mixture semi-confined by nitrogen. The evolution of the induction, reaction times, and the maximum thermicity were used as fitting targets for our 3-step model. Furthermore, they were evaluated not only at the post-shock states along the Hugoniot curve, but also behind the transverse waves, keystones of the cellular structure. Second, borrowing analytical ignition criteria and wave hierarchy approach, the comparison of different chemical models that were able to recover the critical height suggested that the slope of the normal velocity–curvature curve was the first key parameter for the critical height determination. Moreover, the dynamics of extinction near criticality, and quenching distance depended on the reaction rate dependency on density and pressure.

Critical Drop Height Prediction of Loquat Fruit Based on Some Engineering Properties with Machine Learning Approach

The lowest height at which a product can fall without suffering severe harm is known as the “critical drop height” for agricultural products. It is a crucial factor to take into account for crops like loquats that are prone to bruising or damage upon impact. By establishing the minimum altitude at which the product can be dropped without experiencing substantial harm, suitable processing procedures may be established from harvest to the end consumer, thereby preserving product quality and worth. The critical drop height can be ascertained through swift, affordable, non-destructive, and non-traditional methods, rather than time-consuming and expensive laboratory trials. In the study, we aimed to estimate the critical drop height for loquat fruit using machine learning methods. Three different machine learning methods with different operating principles were applied. R2, MAE, RMSE, and MAPE metrics were used to assess the models. There were no obvious differences in both the comparisons within the models, namely the training and test results and the mutual comparisons of the models. However, with a slight difference, the SVMs model performed better in the training data set, and the ETs model performed better in the test data set. Plots were drawn to visualize model performances, and the results obtained from the plots and metrics support each other.

Probing Velocity Dispersion Inside Coronal Mass Ejections: New Insights on Their Initiation

This work studies the kinematics of the leading edge and the core of six coronal mass ejections (CMEs) in the combined field of view of Sun Watcher using Active Pixel System detector and Image Processing (SWAP) on board PRoject for On-Board Autonomy (PROBA-2) and the ground-based K-Cor coronagraph of the Mauna Loa Solar Observatory. We report, for the first time, on the existence of a critical height hc , which marks the onset of velocity dispersion inside the CME. This height for the studied events lies between 1.4 and 1.8 R ⊙, in the inner corona. We find the critical heights to be relatively higher for gradual CMEs, as compared to impulsive ones, indicating that the early initiation of these two classes might be different physically. We find several interesting imprints of the velocity dispersion on CME kinematics. The critical height is strongly correlated with the flux-rope minor radius and the mass of the CME. Also, the magnitude of the velocity dispersion shows a reasonable positive correlation with the above two parameters. We believe these results will advance our understanding of CME initiation mechanisms and will help provide improved constraints on CME initiation models.

Stability of the coronal magnetic field around large confined and eruptive solar flares

Context. The coronal magnetic field, which overlies the current-carrying field of solar active regions, straps the magnetic configuration below. The characteristics of this overlying field are crucial in determining if a flare will be eruptive and accompanied by a coronal mass ejection (CME), or if it will remain confined without a CME. Aims. In order to improve our understanding of the pre-requisites of eruptive solar flares, we study and compare different measures that characterize the eruptive potential of solar active regions – the critical height of the torus instability (TI) as a local measure and the helicity ratio as a global measure – with the structural properties of the underlying magnetic field, namely the altitude of the center of the current-carrying magnetic structure. Methods. Using time series of 3D optimization-based nonlinear force-free magnetic field models of ten different active regions (ARs) around the time of large solar flares, we determined the altitudes of the current-weighted centers of the non-potential model structures. Based on the potential magnetic field, we inspected the decay index, n, in multiple vertical planes oriented alongside or perpendicular to the flare-relevant polarity inversion line, and estimated the critical height (hcrit) of TI using different thresholds of n. The critical heights were interpreted with respect to the altitudes of the current-weighted centers of the associated non-potential structures, as well as the eruptive character of the associated flares, and the eruptive potential of the host AR, as characterized by the helicity ratio. Results. Our most important findings are that (i) hcrit is more segregated in terms of the flare type than the helicity ratio, and (ii) coronal field configurations with a higher eruptive potential (in terms of the helicity ratio) also appear to be more prone to TI. Furthermore, we find no pronounced differences in the altitudes of the non-potential structures prior to confined and eruptive flares. An aspect that requires further investigation is that, generally, the modeled non-potential structures do not really reside in a torus-instable regime, so the applicability of the chosen nonlinear force-free modeling approach when targeting the structural properties of the coronal magnetic field is unclear.

Idealized branch sections and hydrodynamic analysis characterize rainfall partitioning

AbstractWater flow on plant canopies determines the partitioning of intercepted rainfall between stemflow and throughflow, yet understanding of these flow processes remains minimally developed. Plant canopies may concentrate intercepted water into rivulets that flow beneath branches. If the rivulets remain attached to branches until they encounter the main tree trunk, they form stemflow. If the rivulets detach from branches, however, they form ‘pour points’, regions of concentrated throughfall. Here, rivulet flow below uniform cylinders was studied experimentally. Experimental observations were interpreted using hydrodynamic theory to predict the likelihood of a rivulet detaching based on a critical rivulet height (). Predictions of rivulet detachment were explored for a simplified case where water is introduced at known flow rates at a discrete point above a uniform, cylindrical ‘branch’. Where the branch was straight, the percentage of trials forming a rivulet, or the rivulet formation rate, increased with increasing branch inclination angle to the horizontal axis. Rivulet formation rates did not vary with flow rate. However, the volume of water collected at the end of the branch, analogous to stemflow, decreased with a higher flow rates due to the formation of flow instabilities. Rivulets forming on straight branches did not detach. Instead, the rivulet stream accelerated and reduced in height as it flowed along the cylinder. Rivulets formed on curved branches detached only when the branch axis after the curve approached the horizontal. Future theoretical and experimental studies can extend this work to improve understanding of more complex surfaces and branch architectures to mechanistically understand canopy interception.

Study on the critical stable height of vertical excavation in rocky foundation pit within layered structural plane

The structural plane characteristic was the most critical factor for determining the self-stability ability of deep foundation pit vertical-rock-wall in layered rock stratum. Multiple methods such as model testing, numerical calculation, and theoretical calculation were utilized comprehensively in this paper. The self-stabilizing control effect on the deep foundation pits vertical-rock-wall that under the different structural plane inclination angle (α) and under the different structural plane strength was systematically studied. The results indicated that the overall variation trend of "Sharp decrease ~ Slow decrease ~ Slow increase ~ Sharp increase" in the symmetrical distribution for the self-stability critical height (Hcr) varied with the gradually increasing of α was presented. Meanwhile, the variation trend of "continuously decreasing and rapidly decreasing first, and then slowly decreasing and tending to stabilize" with the structural plane strength reduction coefficient (k). The key factor to control the self-stability of the deep foundation pit vertical-rock-walls lied in fully grasping and utilizing the basic characteristics of rock structural planes. The research results of this paper provided the theoretical basis for scientifically determining the safety level and designing reasonable support structures of the deep foundation pit vertical-rock-walls in layered rock stratum.

Seismic analysis of coastal slope stability considering strength nonlinearity and tension cut-off

Tensile stresses may lead to cracks forming at the crest of coastal slopes and subsequent slope failure, particularly under seismic load conditions. Traditional shear strength failure criteria often overestimate the tensile strength of geomaterials, and in turn the factor of safety against sliding. This paper presents an improved nonlinear analysis method for assessing the stability of slopes, while considering tension cut-off in the failure criterion as well as the detrimental effects of inertial seismic loading and pore water pressure. The method employs the principle of energy consumption to obtain solutions for the geometry of the potential sliding surface and stress distribution, while the critical slope height and the stability factor are determined by means of a particle swarm optimization algorithm. Comparison with existing methods is conducted to validate the effectiveness and accuracy of this method. Results showed that the tensile strength of geomaterials had a significant impact on the stability of the slope and its associated failure mechanism, particularly for the vertical slope. This study could avoid the assumptions of the dilatancy angle on the velocity discontinuity and critical slope sliding surface, and reflect the tensile and nonlinear characteristics of the geomaterials well.