True Triaxial Strength Research Articles

True-triaxial strength characteristics of cement stone subjected to sulfuric acid corrosion: An experimental and theoretical study

True triaxial tests were conducted to investigate the effect of sulfuric acid corrosion on the true triaxial strength of cement stone under different acid immersion times, including the common loading path and novel loading path with constant Lode angle. The chemical composition and microstructure of cement stone under different acid immersion times (0, 7, 14, and 28 d) were examined via X-ray diffraction and scanning electron microscopy. As the duration of acid immersion increases, the concentrations of portlandite (Ca(OH)2) and calcium silicate hydrate (C-S-H) within the cement stone exhibit a gradual decline, while the presence of calcium sulfate (CaSO4) progressively rises. The microstructure of the cement stone was primarily affected by two factors: the damaging and strengthening effects of sulfuric acid and CaSO4, respectively. The damaging effects of the acid were dominant for days 0–14 of immersion, with continuous increase in microstructure damage. Subsequently, the strengthening effect of CaSO4 became apparent for days 14–28 of immersion, with continuous improvement in microstructure integrity. Consequently, the strength of cement stone exhibited a diminishing trend during the initial immersion period of 0–14 days, followed by an enhancement in strength from 14 to 28 days of immersion. Furthermore, a three-dimensional strength criterion was proposed by combining the Hoek–Brown and Matsuoka–Nakai criteria. The failure envelope first shrank and then expanded with increasing immersion time, accurately describing the variation in strength in the three-dimensional principal stress space.

A true triaxial strength criterion for rocks by gene expression programming

A true triaxial strength criterion for rocks by gene expression programming

The modification of Mohr-Coulomb criteria based on shape function and determination method of undetermined parameters

Rock strength criteria is a key step to estimate the stability of rock engineering, especially, the effect of intermediate principal stress on rock failure should be considered in formation at great depth. However, the mostly used Mohr-Coulomb criterion don't consider the effect of intermediate principal stress, besides, most rock mechanics laboratories do not have the ability to do true triaxial strength, the strength parameters in poly-axial strength criteria are difficult to be determined. In order to solve these problems, based on the least absolute deviation method, the square of the correlation coefficient, absolute difference and mean difference are adopted to verify the fitting effect of 10 strength criteria on 32 groups of true triaxial strength data. On this basis, the elliptical Lode angle shape function, hyperbolic Lode angle shape function, and the Lode angle shape function based on spatial slip plane are used to establish the modified MC strength criteria, which not only meets the requirements of smoothness and convexness, but also can be used widely in many geomaterials. Besides, strength parameters of the established criteria can be decided by conventional triaxial strength experiment, and the prediction accuracy of the modified MC strength criteria for the 32 groups of true triaxial strength data is better than or close to the lowest fitting error of the existing poly-axial strength criteria.

Modeling of true triaxial strength of rocks based on optimized genetic programming

The strength of a rock is the main factor affecting the stability of an engineered rock mass. As laboratory testing requires sophisticated equipment and considerable time to determine rock strength, prediction models are needed for establishing rock strength criteria. Genetic programming (GP) is a soft computing technology often used to address rock mechanics and engineering challenges. However, GP also has limitations, such as a long running time, complex individual growth without a corresponding fitness improvement, and difficulty in finding the optimal solution. Therefore, we conducted this study by applying a dynamic restriction on individual size, local search of the neighborhood of the optimal individual, and multithreaded evaluation to optimize GP and guarantee the accuracy of the results and to build a prediction model for the true triaxial strength involving different rock types. The results showed that the restriction dynamically changes to restrict the redundant bloat of strength individuals without a corresponding fitness improvement; using local search rules can effectively find individuals with high fitness, so the strength predicted by the system was in good agreement with the measured strength. We also found the predicted strength was suitable for fitting the rock strength criteria. Using this multithreaded evaluation sped up the operation of the algorithm and produced accurate predictions; and for complex problems, increasing the threads had a more pronounced effect on the runtime and fitness improvements. Based on the Sobol global sensitivity analysis, we analyzed the influence of each prediction parameter on the true triaxial strength of rocks. Combined with the statistical assessment indices involving sum of the absolute error, mean, a10-index, and regression determination coefficient, the predictions of the optimized GP model that we established in this study were more accurate than those of multiple regression analysis.

Experimental study on mechanical and damage characteristics of coal under true triaxial cyclic disturbance

A true triaxial test system was used to study the mechanical and damage characteristics of coal under tiered cyclic maximum principal stress (σ1), intermediate principal stress (σ2), and minimum principal stress (σ3). The influence of each principal stress cycle on the deformation, energy evolution and damage characteristics of coal was systematically analyzed, and the mechanical evolution characteristics of coal under the disturbance action of true triaxial excavation were revealed. In the process of true triaxial tiered cyclic loading, it is found that: the stress-strain curves of different principal stress cycles processes are quite different, but they will all cause plastic deformation or plastic hysteresis loop of principal strain in each principal stress directions of coal, and the principal strain corresponding to the cyclic principal stress direction is the dominant deformation, and its deformation is relatively large. According to the true triaxial strength curve and experimental results, the cyclic loading or unloading σ2 may cause instability and failure of coal. The variation trend of residual deformation caused by σ1 and σ3 is the same: with the increase of the amplitude of true triaxial tiered cyclic loading, the absolute value of the cumulative maximum and minimum residual principal strains increase exponentially and quadratic parabola, respectively. And the absolute value of the relative maximum and minimum residual principal strains decrease first and then increase, showing U-shaped and arch shaped changes, respectively. While the changes of cumulative and relative residual intermediate principal strains are all small. The total dissipation energy of coal increases exponentially with the increase of the amplitude of true triaxial tiered cyclic loading. Combined with the energy dissipation characteristics, the damage variable equation of coal under true triaxial cyclic loading is newly defined. With the increase of the amplitude of true triaxial tiered cyclic loading, the damage variable presents an obvious S-type change trend of deceleration acceleration re-deceleration.

A new analysis model for borehole stability in gas drilling

Gas drilling is one of the main underbalanced drilling techniques, which has a significant effect on the ROP (rate of penetration) improvement in hard formation. However, because the circulating medium in drilling process is gas instead of liquid, the fluid column pressure of gas drilling cannot provide enough support force for the rock around borehole as that of conventional overbalanced drilling. Due to the combination influence of engineering and geological factors, the borehole rock is prone to instability during gas drilling, which can lead to rock blocks falling from the well wall and the formation of irregular borehole shape. To avoid the occurrence of complex underground accidents caused by too many falling rock blocks, such as the borehole instability and pipe-sticking, it is necessary for gas drilling to take into consideration the true triaxial stress state of the rock in the deep formation and the possibility of the borehole breakout. Based on the permitted breakout angle of borehole and true triaxial strength theory of rock, an analysis model of borehole stability has been developed for the engineering of gas drilling. The theory of linear elasticity, involved in this analysis model, is used to analyse the stress state of rock around a cylindrical hole under lower wellbore pressure. The failure of the rock around borehole is determined by the Mogi-Coulomb criterion, which considers the intermediate principal stress effect on rock material. The results show that the model is as simple and practical as the traditional model, which takes no account the effect of the intermediate principal stress or the permitted borehole breakout, and it can provide an accurate analysis for the borehole stability of gas drilling. Therefore, the results of this study, combined with the related work such as the prediction of water formation, can guide the adaptability evaluation for the engineering design of gas drilling.

A numerical study on true triaxial strength and failure characteristics of jointed marble

A numerical study on true triaxial strength and failure characteristics of jointed marble

A Simple Three-Dimensional Failure Criterion for Jointed Rock Masses under True Triaxial Compression

The joint configuration and the intermediate principal stress have a significant influence on the strength of rock masses in underground engineering. A simple three-dimensional failure criterion is developed in this study to predict the true triaxial strength of jointed rock masses. The proposed failure criterion in the deviatoric and meridian planes adopts the elliptic and hyperbolic forms to approximate the Willam–Warnke and Mohr–Coulomb failure criterion, respectively. The four parameters in the proposed failure criterion have close relationships with the cohesion and the internal friction angle and can be linked with the joint inclination angle using a cosine function. Two suits of true triaxial strength data are collected to validate the correctness of the proposed failure criterion. Compared with other failure criteria, the proposed failure criterion is more reasonable and acceptable to describe the strength of jointed rock masses.

Unloading Mechanics and Energy Characteristics of Sandstone under Different Intermediate Principal Stress Conditions

The TRW‐3000 true triaxial rock testing machine was used to conduct loading and unloading tests of sandstone under different σ2, and the true triaxial lateral unloading mechanics and energy characteristics of sandstone under different σ2 were studied. The experimental results show the following: (1) compared with the results of the loading test, the peak strength of the sandstone under the unloading σ3 path is reduced, the unloading direction has obvious expansion and deformation, and the amount of expansion increases significantly with the increase of σ2; sudden brittle failure occurs at the end of unloading. E gradually decreases with the increase of H, and it performs well to use the cubic polynomial to fit the curve of E-H. (2) The Mogi–Coulomb strength criterion can accurately describe the true triaxial strength characteristics of sandstone under loading and unloading conditions. Compared with the results of the loading test, the values of c and φ obtained based on this criterion under the unloading σ3 path are reduced. (3) Under the condition of unloading σ3, U, Ue, and Ud, when the specimen is broken, are all linearly positively correlated with σ2. Ud increases nonlinearly with the increase of H, and as σ2 increases, the slope of the Ud -H curve becomes larger, and the specimen consumes more energy under the same unloading amount. Most of the energy absorbed by the specimen under the unloading σ3 path is converted into Ue, but as σ2 increases, Ud /U increases, and the energy consumed when the specimen is broken is greater.

Mathematical Model of Constitutive Relation and Failure Criteria of Plastic Concrete under True Triaxial Compressive Stress.

To establish the mathematic model of the constitutive relation and failure criteria of plastic concrete under true triaxial compressive stress, uniaxial compressive strength and true triaxial compressive strength of plastic concrete under three kinds of confining pressures with a size of 150 × 150 × 150 mm3 and a curing age of 540 days were tested, and the elastic modulus of plastic concrete with a size of 150 × 150 × 300 mm3 and a curing age of 90 days was tested. Based on the database, under uniaxial compressive stress tests and true triaxial compressive stress tests, the mathematic model of constitutive relation and the failure criteria of plastic concrete were investigated. It was observed that the strength of plastic concrete increased with confining stress. The mathematic model of constitutive relation in the form of the quartic polynomial is in good agreement with measured data. The general equations of failure criteria based on the octahedral stress-space under true triaxial compressive stress in the form of quadratic polynomial are well-fitting with experimental data. The mathematic model of constitutive relation and failure criteria of plastic concrete could provide the basis for a numerical simulation analysis of plastic concrete under true triaxial compressive stress, as well as promote the engineering application of plastic concrete.

True triaxial strength and failure characteristics of cubic coal and sandstone under different loading paths

In deep underground coal mining, engineering activities are performed within anisotropic in-situ stress fields due to engineering disturbances and tectonic stress. Many such activities involve the development of excavations in soft rock and anisotropic coal. Accordingly, studying the mechanical properties of soft rocks is important for the stability of deep underground excavations. In this study, the deformation, strength, and failure characteristics of soft sandstone and raw coal under two different true triaxial loading paths were investigated using a self-developed true triaxial test apparatus. The results indicated that the inelastic strain in the pre-peak stage of sandstone and coal gradually increased with increasing intermediate principal stress. Also, the strength-drop in the post-peak stage increased. The crack initiation stress, crack damage stress, and peak strength of sandstone and coal first increased and then decreased with increasing intermediate principal stress for a given σ3. Moreover, with increasing intermediate principal stress, the failure mode of sandstone and coal changed from shear to tensile shear, and from brittle to semi-brittle. The linear Mogi criteria as found to characterize the true triaxial strength of coal well, while the modified Lade criteria was more applicable to soft sandstone. Owning to the symmetry assumption, the linear Mogi criteria predicted low strength when the intermediate principal stress coefficient exceeded to 0.5. In addition, the peak strength curve of rock on the π plane and the influence of weak structures on the failure mode of anisotropic coal was discussed. Weak structures have an important influence on the failure mode of coal, which depends on the strength difference between the structural plane and the coal rock mass. The strength envelope on the π plane had a significant stress Lode angle effect, which gradually decreased as the mean stress increased.

Mechanical Properties of Layered Composite Coal–Rock Subjected to True Triaxial Stress

Underground mining or tunnelling activity is always associated with the composite geological formations. The mechanical properties of layered composite coal–rock subjected to true triaxial stress conditions are significantly different from those under conventional triaxial or uniaxial stress conditions. In this work, we conducted a series of true triaxial tests using the self-developed true-triaxial apparatus to investigate the mechanical response (e.g., deformation, strength, and failure characteristics) of the layered composite coal–rock (CCR). The results show that the uniaxial strength of CCR lies between the strength of pure sandstone and coal, and the direction of the bedding affects the overall strength of the samples. The true triaxial strength of both the pure rock and CCR increases first and then decreases with the increase of the intermediate principal stress. Moreover, for a given loading direction, as the thickness of the sandstone layer increased, the strength of the CCR increases. The deformation of the CCR shows more obvious plasticity than that of the pure sandstone due to the coordinated deformation of the coal and sandstone layers. In addition, a new true triaxial strength criterion expressed by the first and third equivalent principal stress invariants was proposed, which can well describe the strength characteristics of different coal rocks. The stress states, weak structural planes, and localized stress have a great influence on the failure modes of CCR. The local stress concentration near the contact surface promotes the development of the secondary failure fractures. These findings are of great significance in stability designing in deep underground engineering.

Calculation of Silo Wall Pressure considering the Intermediate Stress Effect

The reasonable determination of wall pressure is critical for the design of silo structures. In this study, the primary objective is to present four novel wall pressure coefficients based on four true triaxial strength criteria in the quasiplane strain state. These four strength criteria are the Drucker‐Prager (D‐P) criterion, the Matsuoka‐Nakai (M‐N) criterion, the Lade‐Duncan (L‐D) criterion, and the unified strength theory (UST), and they all consider the effect of the intermediate stress yet to different extent. These coefficients have a wide application range and are readily used to predict the distribution of wall pressure for deep and squat silos. Comprehensive comparisons are made between the predictions from the wall pressure coefficients described herein and experimental data reported in the literature as well as the results from the European, American, and Chinese silo standards or the Rankine and the modified Coulomb theories. It is found that the effect of the intermediate stress on the wall pressure is very significant for both deep and squat silos; the wall pressure of the D‐P criterion is underestimated, whereas that of the Mohr‐Coulomb (M‐C) criterion is overestimated; the L‐D criterion is recommended to be adopted to calculate the soil wall pressure.

True Triaxial Strength and Failure Modes of Cubic Rock Specimens with Unloading the Minor Principal Stress

True triaxial tests have been carried out on granite, sandstone and cement mortar using cubic specimens with the process of unloading the minor principal stress. The strengths and failure modes of the three rock materials are studied in the processes of unloading σ 3 and loading σ 1 by the newly developed true triaxial test system under different σ 2, aiming to study the mechanical responses of the rock in underground excavation at depth. It shows that the rock strength increases with the raising of the intermediate principal stress σ 2 when σ 3 is unloaded to zero. The true triaxial strength criterion by the power-law relationship can be used to fit the testing data. The “best-fitting” material parameters A and n (A > 1.4 and n < 1.0) are almost located in the same range as expected by Al-Ajmi and Zimmerman (Int J Rock Mech Min Sci 563 42(3):431–439, 2005). It indicates that the end effect caused by the height-to-width ratio of the cubic specimens will not significantly affect the testing results under true triaxial tests. Both the strength and failure modes of cubic rock specimens under true triaxial unloading condition are affected by the intermediate principal stress. When σ 2 increases to a critical value for the strong and hard rocks (R4, R5 and R6), the rock failure mode may change from shear to slabbing. However, for medium strong and weak rocks (R3 and R2), even with a relatively high intermediate principal stress, they tend to fail in shear after a large amount of plastic deformation. The maximum extension strain criterion Stacey (Int J Rock Mech Min Sci Geomech Abstr 651 18(6):469–474, 1981) can be used to explain the change of failure mode from shear to slabbing for strong and hard rocks under true triaxial unloading test condition.

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

There are many methods to construct true triaxial strength criteria for rocks. Jaiswal and Shrivastva (2012) proposed a strength criterion, named J–S criterion, in the deviatoric plane, which provides nearly the same misfits for true triaxial test data as the exponential criterion. It is difficult to calculate the strength at given σ2 and σ3 using the J–S criterion, and the multiple solutions to the nonlinear equation may induce confusion and mistake. Strength envelopes in deviatoric planes are not geometric similar; therefore, true triaxial test data cannot be grouped in the mean stress to check strength criteria in the deviatoric plane.

Comparison of two true-triaxial strength criteria

Comparison of two true-triaxial strength criteria

True triaxial strength, deformability, and brittle failure of granodiorite from the San Andreas Fault Observatory at Depth

True triaxial strength, deformability, and brittle failure of granodiorite from the San Andreas Fault Observatory at Depth

Consistent trends in the true triaxial strength and deformability of cores extracted from ICDP deep scientific holes on three continents

An extensive true triaxial testing program was carried out on core samples from three ICDP-sponsored deep scientific boreholes, KTB (Germany), SAFOD (United States), and TCDP (Taiwan). The three rocks differ in the processes that formed them and in many of their mechanical properties. However, all three rocks exhibited similar failure mechanism, in which induced or reopened microcracks are primarily aligned with the σ 1-σ 2 plane, and the developed fault is steeply inclined in the σ 3 direction. Rock strength in all tested rocks increases with σ 2 when σ 3 is kept constant. Thus, the common Mohr-type criteria, which ignore the effect of σ 2, typically underestimate rock strength. Rather a 3D criterion that involves all three principal stresses represents well experimental results. Fracture plane slope for the same σ 3 steepens as σ 2 rises, contrary to Mohr-type criteria. With respect to deformation, the onset of dilatancy increases with σ 2. In conclusion, true triaxial tests conducted on cores from three scientific boreholes, revealed important details of mechanical behavior not otherwise observed in conventional triaxial tests. In addition, they show mechanical behavior similarities as related to σ 2 effect regardless of rock type.

Integrating borehole-breakout dimensions, strength criteria, and leak-off test results, to constrain the state of stress across the Chelungpu Fault, Taiwan

The paper describes the computation of the maximum horizontal stress ( σ H) magnitude in the vicinity of the Chelungpu Fault, Taiwan, host of the slip zone during the Chi-Chi earthquake ( M w 7.6; 1999). The scientific hole B intercepts the Chelungpu Fault at 1136 m. At the depths of logged breakouts (940–1310 m), the vertical stress ( σ v) as estimated from density logs increases linearly with depth from 22 to 31 MPa. A series of leak-off tests yielded two reliable shut-in pressures, 23.7 MPa at 1085 m and 29.8 MPa at 1279 m, which are lower than the estimated σ v, albeit by only 2.1 and 0.6 MPa, respectively. In our analysis the shut-in pressures were considered to represent estimates of the least horizontal principal stresses ( σ h) at the respective depths, and consequently the test-induced fractures were assumed to have been vertical. Principal stress directions had been previously determined by others (105°–155° for the maximum horizontal stress, σ H, except in the immediate vicinity of the Chelungpu Fault). The contribution of this paper is the estimation of the σ H magnitude by considering that the state of stress at the points of intersection between breakout and borehole wall is in a state of limit equilibrium with the true triaxial strength criterion. The resulting σ H in the range of logged breakouts increases with depth from 55 MPa at 940 m to 59 MPa at 1310 m. Thus, the estimated state of stress prevailing across the Chelungpu Fault is compatible with strike-slip, but marginally also with thrust faulting. However, the likelihood that the shut-in pressures actually represent σ v magnitudes, and that the leak-off test-induced fractures were sub-horizontal, cannot be ignored. In that case the state of stress would clearly favor thrust faulting.

True triaxial strength and deformability of the siltstone overlying the Chelungpu fault (Chi‐Chi earthquake), Taiwan

We conducted true triaxial compression tests to examine the effect of the intermediate principal stress (σ2) on brittle failure and deformability of the siltstone overlying the Chelungpu fault, Taiwan. For constant minimum stress (σ3) a pattern of increased strength (σ1) as a function of σ2, unpredicted by common strength criteria, was observed. Strength data are well fitted by a Mogi‐modified Nadai criterion relating the octahedral shear stress at failure to the mean normal stress on the induced shear fracture or fault. The induced fault strikes subparallel to σ2 and dips steeply in the direction of σ3. Significantly, fault dip angle steadily increases as σ2 rises for unchanged σ3. SEM inspection of tested specimens reveals only sporadic localized intergranular microcracks associated with the creation of the fault. The modulus of elasticity and the onset of dilatancy exhibit a similar behavior to that of σ1, as they increase with σ2 when subjected to a constant σ3.