Principal Stress Components Research Articles

Factor of safety analysis for mine pillar considering the influence of the intermediate principal stress component

Failure of mine pillars, especially in deep underground mines, significantly threatens the safety of miners and equipment. Previous studies on mine pillar stability design have used classical constitutive models that ignore the intermediate principal stress component when determining the factor of safety. In this study, we develop and implement a three-dimensional modified Hoek–Brown (HB) constitutive model that incorporates the intermediate principal stress component into the numerical simulation tool FLAC3D. Furthermore, we propose and apply a strength-reduction technique to determine a more accurate factor of safety for mine pillars. This novel approach provides a more comprehensive and realistic method for geomechanical analysis and pillar design, enhancing our understanding of pillar stability. Through numerical analysis, we illustrate the impact of the intermediate principal stress component on mine pillar plasticity. The factor of safety is calculated via the strength reduction method, revealing a substantial improvement from 1.7 with the classical HB model to 2.0 with the 3D HB model. Including the intermediate principal stress component reduces the evolution of plasticity in the mine pillar. For instance, the volume of plastic zones diminishes, and the factor of safety increases as the width-to-height ratio increases. Exemplary simulations show that ignoring the effect of the intermediate principal stress component, including underestimating safety levels, designing suboptimal pillar design, and misinterpreting in situ observations and measurements, can lead to severe consequences.

Residual stress determination by blind hole drilling and local displacement mapping in aluminium alloy aerospace components

The determination of residual stresses by combining blind hole drilling and optical interferometric measurement of relief deformation is re-visited to evaluate its applicability to aerospace structures. The experimental methodology involves drilling a deep blind hole and evaluating the hole diameter increments in the principal strain directions using electronic speckle-pattern interferometry (ESPI). This is followed by the determination of the principal residual stress components via the solution of the inverse correlation problem. The study presents the pathway to overcoming one of the primary obstacles in residual stress determination, namely, the optimization of the measurement and interpretation procedures to obtain reliable results. It can be concluded from the analysis of the problem that the formulae connecting the raw experimental data and to the sought residual stress component values lead to a well-posed inverse problem. This makes it possible to obtain estimations of the measurement uncertainty. High density fringe patterns from ESPI provide a rapid and reliable method for residual stress determination, as illustrated using examples of approximately 160 MPa stresses in irregular zones of thick-walled structures.

Azimuthal pore pressure response to teleseismic waves: effects of damage and stress anisotropy

SUMMARY Pore pressure oscillations induced by stress variations, including propagating seismic waves from remote earthquakes, have been widely observed in various groundwater systems. The monitored pressure change in wells shows significant water-level oscillations to volumetric strain as well as to S and Love waves. Recent observations demonstrated azimuthal dependence of the pore pressure oscillations with respect to stress indicators and fault zone orientation. Within the fault zone, damage-induced anisotropy is the result of the alignment and orientation of cracks and other internal flaws within the rock. In this work, we provide a complete quantitative description of the pore pressure changes induced by passing seismic waves associated with different orientations and values of principal stress and damage tensor components. The model quantifies the azimuthal dependence of the pore pressure response by a non-dimensional ratio defined as the amplitude of the pressure oscillations induced by a shear strain normalized to the volumetric strain. Three angles and two values are needed to calculate the azimuthal dependence of the pore pressure response: the angle between the directions of the maximum horizontal stress and the seismic event; fault zone orientation; microcrack orientation within the fault zone; and damage and stress values. The model predicts that maximum pore pressure response occurs when microcracks and maximum horizontal stress are in the same orientation, high damage and high stress anisotropy. By adjusting these quantities, we recalculate results of recent seismological studies in the Arbuckle disposal well, Osage County, Oklahoma. The presented model successfully predicts the observed azimuthal dependence in wave-induced fluid pressure response and relates the anisotropic response to tectonic indicators such as the orientations of the maximum horizontal stress, fault zone, and microfractures.

Mechanical characterization and torsional buckling of pediatric cardiovascular materials

In complex cardiovascular surgical reconstructions, conduit materials that avoid possible large-scale structural deformations should be considered. A fundamental mode of mechanical complication is torsional buckling which occurs at the anastomosis site due to the mechanical instability, leading surgical conduit/patch surface deformation. The objective of this study is to investigate the torsional buckling behavior of commonly used materials and to develop a practical method for estimating the critical buckling rotation angle under physiological intramural vessel pressures. For this task, mechanical tests of four clinically approved materials, expanded polytetrafluoroethylene (ePTFE), Dacron, porcine and bovine pericardia, commonly used in pediatric cardiovascular surgeries, are conducted (n = 6). Torsional buckling initiation tests with n = 4 for the baseline case (L = 7.5 cm) and n = 3 for the validation of ePTFE (L = 15 cm) and Dacron (L = 15 cm and L = 25 cm) for each are also conducted at low venous pressures. A practical predictive formulation for the buckling potential is proposed using experimental observations and available theory. The relationship between the critical buckling rotation angle and the lumen pressure is determined by balancing the circumferential component of the compressive principal stress with the shear stress generated by the modified critical buckling torque, where the modified critical buckling torque depends linearly on the lumen pressure. While the proposed technique successfully predicted the critical rotation angle values lying within two standard deviations of the mean in the baseline case for all four materials at all lumen pressures, it could reliably predict the critical buckling rotation angles for ePTFE and Dacron samples of length 15 cm with maximum relative errors of 31% and 38%, respectively, in the validation phase. However, the validation of the performance of the technique demonstrated lower accuracy for Dacron samples of length 25 cm at higher pressure levels of 12 mmHg and 15 mmHg. Applicable to all surgical materials, this formulation enables surgeons to assess the torsional buckling potential of vascular conduits noninvasively. Bovine pericardium has been found to exhibit the highest stability, while Dacron (the lowest) and porcine pericardium have been identified as the least stable with the (unitless) torsional buckling resistance constants, 43,800, 12,300 and 14,000, respectively. There was no significant difference between ePTFE and Dacron, and between porcine and bovine pericardia. However, both porcine and bovine pericardia were found to be statistically different from ePTFE and Dacron individually (p < 0.0001). ePTFE exhibited highly nonlinear behavior across the entire strain range [0, 0.1] (or 10% elongation). The significant differences among the surgical materials reported here require special care in conduit construction and anastomosis design.

Variations in the present-day stress field and its implications for hydrocarbon development in the Hassi Terfa field, Hassi Messaoud, Algeria

Understanding the contemporary stress field and its variations is important in hydrocarbon production. These fluctuations are significant for borehole stability and trajectory, hydraulic stimulation, fault reactivation, and optimal well completion. In this study, we provide the first geomechanical model to determine the orientation and magnitude of the present-day stress of the Hassi Terfa field using geophysical logs to understand the origin of the spatial variation of the stress state. The breakouts from acoustic image logs provide an average maximum horizontal stress (SHmax) orientation of N115°, consistent with the Africa–Eurasia plate motion, which imposes the largest first-order stress source. However, our analysis demonstrates local stress perturbations near faults. This may mean that the stress field in the Hassi Terfa is superimposed by a third-order source of stress (<100 km). Our model demonstrates variations in stress magnitude with depth interpreted to be lithology-dependent, which act as a natural barrier against fracture propagation. We propose a new approach to quantify these perturbations via the stress regime index R′, using the magnitude of the principal stress components. The results demonstrate different layer-to-layer faulting regimes. These perturbations will limit the vertical propagation of hydraulic fracturing to the overlying formation, which enables it to access a larger reservoir volume and improve production. In general, a strike–slip regime with a slight normal slip component is interpreted for the study area. We develop the first-ever quantitative stress map using the relative principal stress magnitude parameter (Aϕ) to present the lateral variation of the stress regime throughout the Saharan platform. Results demonstrate a consistent strike–slip to normal/strike–slip regime with a uniform SHmax orientation. This map will help operators achieve optimal well direction and optimize hydraulic stimulation. Our results have considerable implications for hydrocarbon production in the central and southeastern Saharan platform.

Influence of hydrogen ingress on residual stress and strain in pipeline steels

This study examines the effect of hydrogen ingress on the internal stress and strain of steels, using lab-source and synchrotron X-ray diffraction (XRD), and exploring both electrochemical and gaseous hydrogen ingress. The results show that compressive strains were generated after electrochemical hydrogen charging, and the residual stresses on the examined surface changed from being primarily tensile to compressive stresses. The compressive stress and strain were confirmed by a change in direction of the biaxial principal stress components, and a shift of the high-energy XRD peaks to higher 2θ values for the hydrogen-charged steels. The magnitude of the compressive stress and strain generated was dependent on the microstructure of the steel and increased with charging time or more hydrogen ingress into the steel. In-situ synchrotron XRD during gaseous charging at a pressure of 150 psi also showed a peak shift towards higher 2θ values, which slightly increased for the palladium-coated steel sample, when compared to the uncoated steel sample.

Influence of plate thickness on the results of residual stresses determination by through hole drilling in orthotropic composites of different fiber orientation

The main objective of this study is to characterize residual stress in composite orthotropic plates using hole drilling method. The values of hole diameter increment in the principal anisotropic directions, which were obtained from electronic speckle-pattern interferometry on the external surface of the specimen, serve as the initial experimental information. The theoretical foundation of the approach is based on S. G. Lekhnitsky's analytical solution, which describes a stress concentration along the edge of the central open hole in a rectangular orthotropic plate under tension in principal anisotropy directions. The transitional model, that is essential link for deriving residual stress components from initial experimental data, provides the unequivocally solution to the properly posed inverse problem. Firstly, the validation of the transitional model regarding prescription of residual stress values for specific plies within a specimen’s thickness is based on calibration experiments. These experiments yield data for unidirectional and cross-ply composite plates of three different thicknesses. The results demonstrate that the values of principal residual stress components for cross-ply composite material are independent of plate thickness. This means that the transition model, based on the measurements of deformation responses to through hole drilling on the external plate face, can be reliably implemented to characterize the maximal residual stress values present in the middle plane of the composite plate. An investigation into the effect of hole diamter values on the determination of residual stress is conducted for cross-ply and angle-cross-ply composites. It is shown that a hole of diameter of 3.2 mm provides the most reliable results.

Macro-micro Performances of Granular Materials Considering the Influences of Density and Stress Path under True Triaxial Conditions: A DEM Investigation

The mechanical behaviors of granular materials are dominated by internal structure, which are related to fabric evolution during loading. This study investigated the fabric evolution of granular materials with different densities and stress paths under true triaxial conditions. A series of discrete element numerical simulations with different intermediate principal stress coefficient b were carried out along the constant mean stress p and the constant minor principal stress σ3 stress paths for both loose and dense specimens. The results indicated that the constant-p stress path produced a faster increase in stress ratio than the constant-σ3 stress path at the same b. The effects of specimen density on the peak friction angle are greater than that of stress path. The microscopic analyses revealed that the constant-p stress path facilitates a much more preferential distribution of normal contact force network along the major principal direction. The discrepancies in the peak stress ratio under two stress paths were thus interpreted. The dense specimen will rapidly form a higher anisotropic distribution of the normal contact force network upon shearing, and its anisotropic intensity was almost twice that of the loose specimen at the peak stress state. In addition, a unique relationship between the strong deviatoric fabric ratio and stress ratio was presented. The ratio of the two was approximately 1.0 regardless of stress path, density and b value. Finally, an underlying relationship between the stress components and the whole fabric components at the critical state was confirmed by introducing a new stress tensor. The three principal components (F1, F2 and F3) of the whole fabric tensor can be quantitatively represented with the imposed three principal stress components (σ1, σ2 and σ3) by employing a relationship of F1:F2:F3 = σ10.27:σ20.27:σ30.27. It provides a more comprehensive perspective to analyze the macro-micro performances of granular materials at the critical state.

In Situ Stress Determination Based on Acoustic Image Logs and Borehole Measurements in the In-Adaoui and Bourarhat Hydrocarbon Fields, Eastern Algeria

The study of in situ stress from image logs is a key factor for understanding regional stresses and the exploitation of hydrocarbon resources. This work presents a comprehensive geomechanical analysis of two eastern Algerian hydrocarbon fields to infer the magnitudes of principal stress components and stress field orientation. Acoustic image logs and borehole measurements were used in this research to aid our understanding of regional stress and field development. The studied In-Adaoui and Bourarhat fields encompass a combined thickness of 3050 m of Paleozoic and Mesozoic stratigraphy, with the primary reservoir facies in the Ordovician interval. The Ordovician sandstone reservoir interval indicates an average Poisson’s ratio (v) of 0.3, 100–150 MPa UCS, and 27–52 GPa Young’s modulus (E). Direct formation pressure measurements indicate that the sandstone reservoir is in a hydrostatic pore pressure regime. Density-derived vertical stress had a 1.1 PSI/feet gradient. Minimum horizontal stress modeled from both Poisson’s ratio and an effective stress ratio-based approach yielded an average 0.82 PSI/feet gradient, as validated with the leak-off test data. Drilling-induced tensile fractures (DITF) and compressive failures, i.e., breakouts (BO), were identified from acoustic image logs. On the basis of the DITF criterion, the maximum horizontal stress gradient was found to be 1.57–1.71 PSI/feet, while the BO width-derived gradient was 1.27–1.37 PSI/feet. Relative stress magnitudes indicate a strike-slip stress regime. A mean SHMax orientation of N130°E (NW-SE) was interpreted from the wellbore failures, classified as B-quality stress indicators following the World Stress Map (WSM) ranking scheme. The inferred stress magnitude and orientation were in agreement with the regional trend of the western Mediterranean region and provide a basis for field development and hydraulic fracturing in the low-permeable reservoir. On the basis of the geomechanical assessments, drilling and reservoir development strategies are discussed, and optimization opportunities are identified.

Electrode pattern definition in ultrasound power transfer systems

We use a high pattern-fidelity technique on piezoelectric electrodes to selectively excite high-order vibration modes, while isolating other modes, in multi-layered through-wall ultrasound power transfer (TWUPT) systems. Physical mechanisms, such as direct and inverse piezoelectric effects at transmitting and receiving piezoelectric elements, as well as wave propagation across an elastic barrier and coupling layers, all contribute to TWUPT. High-order radial modes in a TWUPT system feature strain nodes, where the dynamic strain distribution changes sign in the direction of disks' radii. This study explains theoretically and empirically how covering the strain nodes of vibration modes with continuous electrodes results in substantial cancelations of the electrical outputs. A detailed analysis is given for predicting the locations of the strain nodes. The electrode patterning for creating the transmitter and receiver shapes is determined by the regions where local force and charge cancelation do not occur, i.e., the two modal principal stress components have the same sign. Patterning for creating the electrode shapes is performed by high-fidelity numerical modeling supported by experiments. Using differential excitation on the transmitter side while monitoring transmitted power and efficiency on the reception side at various vibration modes is made possible by the unique nature of TWUPT systems. Due to an improvement in system quality and power factors, it is determined that employing the proposed electrode pattern designs enhances overall device efficiency and active power. The suppression of other modes makes up a filter feature that is paired with the enhancement at the mode under consideration.

Numerical study on stress paths in grounds reinforced with long and short CFG piles during adjacent rigid retaining wall movement

Abstract Ensuring the safety of existing structures is an important issue when planning and executing adjacent new foundation pit excavations. Hence, understanding the stress state conditions experienced by the soil element behind a retaining wall at a given location during different excavation stages has been a key observational modelling aspect of the performance of excavations. By establishing and carrying out sophisticated soil–structure interaction analyses, stress paths render clarity on soil deformation mechanism. On the other hand, column-type soft ground treatment has recently got exceeding attention and practical implementation. So, the soil stress–strain response to excavation-induced disturbances needs to be known as well. To this end, this paper discusses the stress change and redistribution phenomena in a treated ground based on 3D numerical analyses. The simulation was verified against results from a 1 g indoor experimental test conducted on composite foundation reinforced with long and short cement–fly ash–gravel (CFG) pile adjacent to a moving rigid retaining wall. It was observed that the stress path for each monitoring point in the shallow depth undergoes a process of stress unloading at various dropping amounts of principal stress components in a complex manner. The closer the soil element is to the wall, the more it experiences a change in principal stress components as the wall movement progresses; also, the induced stress disturbance weakens significantly as the observation point becomes farther away from the wall. Accordingly, the overall vertical load-sharing percentage of the upper soil reduces proportionally.

Rock Failure Envelope and Behavior Using the Confined Brazilian Test

AbstractComplete rock strength envelopes spanning both the tensile and compressive stress domains are essential to engineering design in Rock. However, the exact shape of the envelope in the tensile zone is not known. The major impediment to establishing the true profile of the envelope is the inherent difficulties in conducting tension tests under confinement. In this study, we use the confined Brazilian test to investigate the impact of confining pressure on tensile strength, the failure mechanism, and the brittle‐ductile transition. By applying a constant confining pressure, a state of triaxial stress is created in the disc such that the disc center is under three nonzero and unequal principal stress components. By increasing the confining pressure, the least principal stress changes from tensile to compressive so that rock failure can be observed over a wide range of stress conditions. Four lithologies have been tested in this study, including limestone and three sandstones. Experimental results suggest that the complete strength envelope resembles an offset parabola opening to the right with its vertex above the horizontal axis in the domain. The tensile strength shows a pronounced increase under confinement. For all lithologies tested, tensile fracturing disappears as the confining pressure increases, with the stress‐displacement curves exhibiting typical brittle‐ductile transitions and a change in failure mode from extension to shear. Observation of the progressive increase in the fracture angle with confining pressure also provides laboratory evidence for the continuous transition of failure from extension to shear.

The Photoelastic Constant of (0001) 4H Silicon Carbide Determined by Scanning Infrared Polariscopy

The piezo‐optic coefficients (π11‐π12) and the photoelastic constant Cσ of on‐axis (0001) 4H silicon carbide at λ = 1.3 μm are determined to (−3.3 ± 0.5) × 10−13 and (−2.8 ± 0.4) × 10−12 Pa−1, respectively. The experiment is conducted using a scanning infrared depolarization imager (SIRD) that uses a rotational mode for wafer scanning equipped with a special calibration setup for diametrical loading of 150 mm diameter wafers. The loading force is varied by gradually setting the wafer rotation frequency. (π11–π12) and Cσ are determined by analyzing the dependence of the shear stress equivalent on the loading force in the wafer center. This calibration procedure is made difficult by the pronounced inhomogeneous defect distribution on the wafer under study. Furthermore, the unloaded wafer is characterized using the multipolarization analysis on the base of an eccentric measurement mode. The resulting map of the shear stress maximum τmax completed by the direction of the large principal stress component reveals that the outer wafer part is tensile stressed in circumferential direction like a ring with magnitudes up to 10 MPa.

Machine Learning Based Methods for Obtaining Correlations between Microstructures and Thermal Stresses

The microstructure–property relationship is critical for parts made using the emerging additive manufacturing process where highly localized cooling rates bestow spatially varying microstructures in the material. Typically, large temperature gradients during the build stage are known to result in significant thermally induced residual stresses in parts made using the process. Such stresses are influenced by the underlying local microstructures. Given the extensive range of variations in microstructures, it is useful to have an efficient method that can detect and quantify cause and effect. In this work, an efficient workflow within the machine learning (ML) framework for establishing microstructure–thermal stress correlations is presented. While synthetic microstructures and simulated properties were used for demonstration, the methodology may equally be applied to actual microstructures and associated measured properties. The dataset for ML consisted of images of synthetic microstructures along with thermal stress tensor fields simulated using a finite element (FE) model. The FE model considered various grain morphologies, crystallographic orientations, anisotropic elasticity and anisotropic thermal expansion. The overall workflow was divided into two parts. In the first part, image classification and clustering were performed for a sanity test of data. Accuracies of 97.33% and 99.83% were achieved using the ML based method of classification and clustering, respectively. In the second part of the work, convolution neural network model (CNN) was used to correlate the microstructures against various components and measures of stress. The target vectors of stresses consisted of individual components of stress tensor, principal stresses and hydrostatic stress. The model was able to show a consistent correlation between various morphologies and components of thermal stress. The overall predictions by the model for all the microstructures resulted into R2≈0.96 for all the stresses. Such a correlation may be used for finding a range of microstructures associated with lower amounts of thermally induced stresses. This would allow the choice of suitable process parameters that can ensure that the desired microstructures are obtained, provided the relationship between those parameters and microstructures are also known.

A new methodology for measuring residual stress using a modified Berkovich nano-indenter

In this paper, a new methodology, based on nanoindentation, is proposed to estimate the magnitude and orientation of the principal residual stress components, in a non-equi-biaxial residual stress field at the nanoscale. The proposed method exploits the force-penetration response provided by a modified Berkovich tip that, compared to the most traditional one, is characterized by a preferential elongated size. This feature allows in estimating non-equibiaxial residual stress field since the force-penetration response changes according to the indenter orientation. No observation of the residual imprint is required. Principal stress components and orientations are estimated thanks to theoretical and finite element analysis, establishing a linear relation between stress components and load variations respect to the stress-free sample. Finite element analysis is used to calibrate the model coefficients and to verify the accuracy of this approach for isotropic metals. Obtained results revealed that the method is well suitable in estimating both tensile and compressive residual stresses with an average error lower than 6% for the first case and lower than 10% for the second one.

Drained deformation characteristics of granular soil under pure principal stress axis rotation: impact of sample preparation

The distinct initial fabrics of sand may give rise to dramatically different responses to the applied loading, as evidenced by numerous laboratory tests reported in the literature. In this study, both dry deposition (DD) and moist tamping (MT) methods were employed to prepare sand specimens to generate different initial fabrics. Drained tests were conducted on these specimens under pure principal stress axis rotation while keeping the principal stress components constant. The factors that may influence the deformation characteristics of sand, such as deviatoric stress q, coefficient of intermediate principal stress b, and mean principal stress p, were also taken into consideration in the tests. Test results show that the developments of three normal strain components with number of cycles are very different between DD and MT specimens, which is also closely related to the factors of q, b, and p. It is seen that DD specimen tends to generate larger contractive volumetric strains than MT specimen, which is more significant when q and b increase or p decreases. The shear stiffness of DD specimen is greater than that of MT specimen. The shear modulus ratio, Gj/G1, increases with the increase of q and b or the decrease of p, which is more prominent in DD specimen than in MT specimen. It is also shown that DD specimen appears to exhibit stronger non-coaxiality than MT specimen, in which such difference of non-coaxiality tends to decrease with the increase in number of cycles.

A semi-analytical solution to spherical cavity expansion in unsaturated soils

This paper presents a rigorous solution for spherical cavity expansion in unsaturated soils under constant suction condition. The hydraulic behavior that describes the saturation-suction relationship is modeled by a void ratio-dependent soil-water characteristic curve, which allows the hydraulic behavior to fully couple with the mechanical behavior that is described by an extended critical state soil model for unsaturated soil through the specific volume. Considering the boundary condition and introducing an auxiliary coordinate, the problem is formulated to a system of first-order differential equations with three principal stress components and suction as basic unknowns, which is solved as an initial value problem. Parameter analyses are conducted to investigate the effects of suction and the overconsolidation ratio on the overall expansion responses, including the pressure-expansion response, the distribution of the stress components around the cavity, and the stress path of the soil during cavity expansion. The results reveal that the expansion pressures and the distribution of the stress components in unsaturated soils are generally higher than those in saturated soils due to the existence of suction.

Pressure‐to‐Depth Conversion Models for Metamorphic Rocks: Derivation and Applications

AbstractPressure‐to‐depth conversion is a crucial step toward geodynamic reconstruction. The most commonly used pressure‐to‐depth conversion method assumes that pressure corresponds to the lithostatic pressure. However, deviatoric stresses can cause pressure to deviate from the lithostatic case strongly, thus adding considerable uncertainty to pressure to‐depth conversion. First, we rederive formulas of pressure‐to‐depth conversion that take into account deviatoric stresses. Then, we estimate the range of possible depth independently for each point in a data set containing peak and retrograde metamorphic pressure data (one‐point method). In a second time, we use both the peak and retrograde pressure of a rock sample together, assuming that both pressures were recorded at the same depth (two‐point method). We explore different cases to explain the transition from peak to retrograde pressure by varying the direction and magnitude of stresses. This alternative model is consistent with all data points but for a more restricted range of stress state and depth than the one‐point model. Our results show that (1) even small deviatoric stresses have a significant impact on depth estimates, (2) the second principal stress component σ2 plays an essential role, (3) several models can explain the pressure evolution of the data but lead to different depth estimates, and (4) strain data offer a mean to falsify our proposed two‐point pressure‐to‐depth conversion. The maximum predicted depth at peak pressure is 170 km using the assumption that pressure is lithostatic, compared to &lt;75 km for our two‐point model, which could correspond to the crustal root Moho's depth.

A novel yield criterion and its application to calculate the rolling force of a thick plate during hot rolling

In order to solve the problem of establishing the rolling force model, which is caused by the nonlinear Mises specific plastic power, a novel yield criterion, called mean slope yield criterion, is constructed by averaging the slopes of yield loci of the Tresca criterion and Mises criterion. The yield criterion is a linear combination of the principal stress components, and its locus on the π-plane is an irregular dodecagon which intersects the Mises circle. For verification, the yield criterion was rewritten by introducing the Lode stress parameter and compared with the experimental data, and a good consistency is achieved. Meanwhile, a three-dimensional velocity field whose horizontal component satisfies the elliptic distribution from the entrance to the exit is proposed. Based on these achievements, the energy analysis of the velocity field is carried out with the derived yield criterion, and the expression of the internal deformation power is derived. Also, the friction power and shear power are derived in terms of the velocity field. Ultimately, the analytical solutions of rolling torque, rolling force, and the stress state coefficient are obtained through the minimization of the total power. By comparison, it is shown that the theoretical rolling torques and rolling forces coincide well with the measured ones since both the maximum errors are only 13.77%.

Bayesian analysis for uncertainty quantification of in situ stress data

Estimates of in situ stress state may be unreliable due to the inherent variation of stresses in rock masses coupled with the usual lack of sufficient stress data. This renders uncertainty quantification critically important for stress estimation, as it both permits quantitative assessment of the reliability of estimated stresses and facilitates application of reliability-based design in rock engineering. This paper presents a Bayesian approach that can probabilistically quantify uncertainty in both mean stress estimation and predicted stresses. We show that the quantified uncertainty supports our general understanding in that (i) more stress data tend to result in more reliable estimates, and (ii) the usual case of small numbers of stress data is liable to yield highly uncertain estimates of the mean stress. The results reveal that large uncertainty may exist in estimates of both the magnitudes and orientations of the principal mean stress, and this suggests that in practice the estimation reliability should be considered for not only stress magnitudes but also stress orientations. The results also show that the three principal mean stress components may display different degrees of uncertainty, thereby highlighting the importance of identifying which mean stress components are of most interest for the specific engineering objective. Finally, we demonstrate how, within the Bayesian framework, the large uncertainty in mean stress estimates arising from limited stress data can be effectively reduced by means of informative priors.