Linear Mohr Coulomb Criterion Research Articles

Analysis of the interaction between bolt-reinforced rock and surface support in tunnels based on convergence-confinement method

To investigate the interaction of the bolt-reinforced rock and the surface support, an analytical model of the convergence-confinement type is proposed, considering the sequential installation of the fully grouted rockbolts and the surface support. The rock mass is assumed to be elastic-brittle-plastic material, obeying the linear Mohr-Coulomb criterion or the non-linear Hoek-Brown criterion. According to the strain states of the tunnel wall at bolt and surface support installation and the relative magnitude between the bolt length and the plastic depth during the whole process, six cases are categorized upon solving the problem. Each case is divided into three stages due to the different effects of the active rockbolts and the passive surface support. The fictitious pressure is introduced to quantify the three-dimensional (3D) effect of the tunnel face, and thus, the actual physical location along the tunnel axis of the analytical section can be considered. By using the bolt-rock strain compatibility and the rock-surface support displacement compatibility conditions, the solutions of longitudinal tunnel displacement and the reaction pressure of surface support along the tunnel axis are obtained. The proposed analytical solutions are validated by a series of 3D numerical simulations. Extensive parametric studies are conducted to examine the effect of the typical parameters of rockbolts and surface support on the tunnel displacement and the reaction pressure of the surface support under different rock conditions. The results show that the rockbolts are more effective in controlling the tunnel displacement than the surface support, which should be installed as soon as possible with a suitable length. For tunnels excavated in weak rocks or with restricted displacement control requirements, the surface support should also be installed or closed timely with a certain stiffness. The proposed method provides a convenient alternative approach for the optimization of rockbolts and surface support at the preliminary stage of tunnel design.

Experimental Study on the Dilatancy Characteristics and Permeability Evolution of Sandstone under Different Confining Pressures

It is of practical significance to investigate the dilatancy and seepage characteristics of tight sandstone gas under different confining pressures for its efficient development. Therefore, fluid–solid coupling triaxial loading experiments with gas-bearing sandstone were conducted. The results showed that the gas-bearing sandstone exhibited brittle characteristics with tensile–shear composite failure. The dual logarithmic model can better characterize the sandstone strength (R2 = 0.9952), whereas the fitting effect of the linear Mohr–Coulomb criterion is poor (R2 = 0.9294). The dilatancy capacity of sandstone was negatively correlated with confining pressure, and the dilatancy index decreased by 38.4% in the form of its convex power function with the increasing confining pressure. The sandstone underwent significant damage dilatancy during the yielding stage, resulting in a significant permeability recovery, with an increase of 67.0%~70.4%, which was greater than the decrease of 9.6%~12.6% in the elastic stage. In view of the different dominant factors of permeability reduction induced by pore compaction and recovery induced by crack development, the permeability model was established with volumetric strain and radial strain as independent variables, which could better reflect the whole process of permeability evolution.

Investigation on the change of shear strength of concrete with cold joint under the action of sulfate dry–wet cycles

Large-volume concrete structures in the saline areas of Western China often have cold joint structural surfaces due to the improper handling of construction joints, dividing them into old and new layers of concrete. In this sulfate-rich environment, the shear strength of concrete with cold joint and the evolution pattern of its composition with the sulfate dry–wet cycles are particularly important in assessing the quality of the interface between old and new concrete bond surfaces. Therefore, in this paper, direct shear tests and microscopic scans were performed on concrete specimens with cold joint at different pouring age intervals and numbers of sulfate dry–wet cycles. The mechanism of the effect of both on the shear strength and failure mode of concrete was analyzed. Results show that the sulfate dry–wet cycles of erosion and the age interval have significant effects on the shear strength of the cold with cold joint and their components. As the number of sulfate dry–wet cycles increased, the shear dilation of concrete specimens with cold joints under direct shear action was reduced, the internal friction angle initially increased and then decreased for specimens with a short pouring interval (0–0.25 day), and the reduction of the internal friction angle of the specimens with a long pouring age interval (0.25–28 days) was nonlinear. The shear dilation of the specimen increased, and the internal friction angle decreased nonlinearly with the increase of the pouring interval. In addition, the failure criterion of specimens transformed from the bilinear Mohr–Coulomb criterion before erosion to the linear Mohr–Coulomb criterion under the action of sulfate dry–wet cycles.

Non-linear mechanical characteristics of tailings in large-scale high tailings dams

The non-linear mechanical characteristics of tailings under high pressure are the research foundation of large-scale high tailings dams. Considering the high stress caused by high tailing ponds, consolidated drained triaxial shear tests were carried out. The deterioration mechanism of non-linear mechanics was revealed by particle crushing. The test results show that sample density has a great influence on volumetric strain under low pressure. However, volumetric strain is not related to sample density under high pressure. The shear strength of the tailings is significantly non-linear. The internal friction angle under low pressure can still be obtained by the traditional linear Mohr–Coulomb criterion and the internal friction angle under high pressure by the power function of the Mohr criterion. The particle crushing of tailings occurs not only at high pressure but also at low pressure. The value of the breakage index increases with sample density. The non-linear mechanics of shear strength are affected by particle breakage. The breakage index value increases linearly with increasing shear strength, indicating that the high density of the deep part of the tailings dam is prone to particle crushing, which affects the stability of the large-scale high dam.

Prediction of Seismic Bearing Capacity Considering Nonlinearity and Dilatancy by Sequential Quadratic Programming

Most of the published literature regarding bearing capacity are often focused on linear and associative soils. Concerning the intrinsic strength nonlinearity in dilatancy soils, this study investigates the problem of the seismic bearing capacity in the framework of the kinematic theorem of limit analysis. The conventional linear Mohr–Coulomb criterion is substituted with a nonlinear power law criterion to depict the nonlinearity of the soil strength. The non-associative feature of soil materials is considered by defining a nonlinear dilatancy coefficient. A generalized tangential technique is accordingly introduced to linearize the strength envelope for making the nonlinear criterion tractable in the analysis. A non-symmetrical translational failure mechanism that is comprised of several rigid wedges is used to characterize the failure of the foundation at the limit state. Moreover, the seismic action is considered by the classic pseudo-static method. Based upon the energy equilibrium theory of the upper-bound limit analysis, new analytical solutions are derived from the work-balanced equation with nonlinearity and dilatancy. This rigorous upper-bound solution is formulated as a multivariate optimization problem and is readily addressed by sequential quadratic programming (SQP). To verify the reliability of the new expressions, the present results are compared with already posted solutions and the original pseudo-dynamic solutions. The comparative results show a good agreement with previous works, and the correctness and rationality of the new analytical solutions are validated. The detailed parametric study reveals that, in the non-associative flow soils, the ultimate bearing capacity is significantly decreased with a reduction in the dilatancy coefficient. Particularly in the linear condition, namely m = 1, the larger the internal friction angle is, the more obvious the influence of the non-associative feature on the bearing capacity is.

A novel laboratory investigation to characterize the mechanical behaviors of polyurethane-ballast bonds under complex stress states

The polyurethane has been successfully applied to reinforce the railway ballasted track and the bonding properties of the polyurethane to ballast particles have significant effects on track performance. However, the mechanical behaviors of the polyurethane-ballast bonds(PBBs) haven’t been reported leaving the particle scale mechanism of the polyurethane reinforcement still unclear. A novel laboratory experimental method was developed in this paper to characterize the mechanical behaviors of the PBBs. Using standard test specimens with various specimen size and bond thickness, a series of laboratory tests were conducted to measure the stress–strain responses of the PBBs under tensile, compressive and shear loadings. The measured results show that the stress of a PBB increases nonlinearly with its strain when it subjects to uniaxial tensile or compressive loads. The test results also validate that the strength envelope of a PBB follows the linear Mohr-Coulomb criterion. A modified parallel bond model was proposed to predict the nonlinear stress–strain behaviors of the PBB with high accuracy. Furthermore, it is found that the bond thickness has distinct effects on the tensile and the cohesion strength of the PBB but negligible effect on its friction angle. The initial tangent elastic modulus of the PBB increases approximately linearly with the increase of the bond thickness. The experimental method, results and the modified bond model introduced in this paper are very helpful for the performance evaluation and the numerical simulations of polyurethane reinforced ballast.

Combined finite-discrete element method for modeling the interaction between single PDC cutter and brittle rock

This paper describes a combined finite-discrete element method for simulating the interaction between single PDC cutter and brittle rock. The proposed method is capable of capturing the nucleation, propagation, branching, and coalescence of cracks as well as the contact between fragments during the cutting process. A detailed description of the method is provided. A linear Mohr-Coulomb criterion with tension cutoff is applied to identify which edge is failure based on the normal and shear stresses estimated from the triangular elements abutting from the edge. The Munjiza-NBS algorithm and Simple-Common-Plane (SCP) contact algorithm are used for the contact detection and interaction, respectively. The method is then validated against the theoretical results of Brazilian disc test and uniaxial compressive strength test. The results demonstrate that the proposed method is of high accuracy, especially before rock failure, because the cracks in the model are introduced on the fly by splitting the nodes where and when needed. Finally, several typical cases of the interaction between single PDC cutter and brittle rock are performed. The numerical results indicate that the wedging effect is significant, leading to the formation of large size chippings for brittle rock cutting.

A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts

The use of fully grouted passive bolts as a reinforcement technique has been widely applied to improve the stability of tunnels. To analyze the behaviors of passive bolts and rock mass in a deep circular tunnel, a new semi-analytical solution is presented in this work based on the finite difference method. The rock mass was assumed to experience elastic–brittle–plastic behavior, and the linear Mohr–Coulomb criterion and the nonlinear generalized Hoek–Brown criterion were employed to govern the yielding of the rock mass. The interaction and decoupling between the rock mass and bolts were considered by using the spring–slider model. To simplify the analysis process, a bolted tunnel was divided into a bolted region and an unbolted region, while the contact stress at the bolted–unbolted interface and the rigid displacement of the bolts were obtained using two boundary conditions in combination with the bisection method. Comparisons show that the results obtained using the proposed solution agree well with those from the commercial numerical software and the in situ test. Finally, parametric analyses were performed to examine the effects of various reinforcement parameters on the tunnel’s stability. The proposed solution provided a fast but accurate estimation of the behavior of a reinforced deep circular tunnel for preliminary design purposes.

A hybrid simulation approach to evaluate stimulation micro-seismic events and production potential of fractured geothermal reservoirs

This study presents an integrated approach to simulate fluid flow and to predict the micro seismic events during stimulation and circulation of cold water over a longer term in geothermal reservoirs. The integrated approach based on new three dimensional fully coupled thermo-poroelastic numerical model for evaluation of energy recoverable. In the presented approach, the fracture aperture due to fracture slippage is calculated by shear and dilation. The shear slippage is controlled by the concept of shear failure using linear Mohr-Coulomb criterion. The numerical model is validated against an analytical Oda’s model for permeability tensor calculation and against an analytical solution for thermo-poroelastic model. The heat transfer between the rock and fluid is modelled by using the conductive heat transfer within the reservoir rock and convective heat transfer in discrete fractures. The thermal stress changes are included in the model to be studied by using roughness induced shear displacement principle in a poro-thermo-elastic environment. The fracture aperture changes are estimated by using an analytical model based on the distributed dislocation technique. The roughness of fracture surfaces is used in the calculation of residual fracture aperture. The presented approach is used to study the potential of permeability enhancement for Habanero geothermal reservoir at a depth of 3600 m. The result show that the increasing in tensile effective stress tend to increase the fracture aperture within the zone of cooling. This increasing in fracture aperture led to significant changes in pressure distribution (decrease in impedance) and hence, increase in the flow rate.

Experimental investigation on post-peak strength and deformation behavior of cracked sandstone

A series of triaxial compression experiments of intact and cracked sandstone with different degrees of damage were carried out using a rock mechanics servo-controlled system. Based on the experimental results, the influence of the confining pressure and damage on the peak strength, residual strength parameters, deformation parameters, and failure modes of cracked sandstone was investigated in detail. The results show that the strength and deformation behavior depends on not only the confining pressure but also the damage. The peak and residual strength of intact and cracked specimens increases linearly with increasing confining pressure, which is in good agreement with the linear Mohr-Coulomb criterion. The peak strength and equivalent cohesion of cracked specimens decreases linearly and exponentially with increasing damage, respectively. However, the internal friction angle is independent of damage. The elastic and secant modulus of cracked specimens increases linearly with increasing confining pressure but decreases with increasing damage. Then the effects of the confining pressure and damage on the peak strain and failure modes of cracked specimens are also analyzed in detail, indicating the transition of the failure modes from single shear failure to multiple shear failure with increasing damage.

Revisiting Seismic Active/Passive Earth Pressure in Nonuniform Cohesive–Frictional Backfill

Abstract A limit analysis of rigid retaining walls is presented to predict active/passive earth pressures for a flat backfill with a linear Mohr-Coulomb (MC) and nonlinear power-law yield criterion...

Fracture evolution mechanism of hollow sandstone under conventional triaxial compression by X-ray micro-CT observations and three-dimensional numerical simulations

The knowledge of the fracture evolution mechanism of hollow rock under conventional triaxial compression is very important to evaluate the stability and safety in deep tunnel engineering. In this research, a series of X-ray micro-CT observations and three-dimensional (3D) numerical simulations were conducted on hollow sandstone specimens with various borehole diameters (d = 0, 15 and 26 mm) under different confining pressures (0, 8, 16, 24 and 35 MPa). A 3D realistic failure process analysis method (RFPA3D) was established for the simulation of the deformation failure behavior of hollow sandstone under conventional triaxial compression. First, a group of micro-parameters used in RFPA3D, which reflect the macro mechanical behavior of the tested sandstone material, was calibrated using the experimental triaxial compression results obtained for intact specimens. Subsequently, a series of 3D numerical simulations was conducted on hollow sandstone specimens with various borehole diameters under different confining pressures. The quantitative comparison of the axial deviatoric stress-strain curve and peak strength of hollow sandstone obtained from the experiments and simulations indicates that the 3D numerical results agrees very well with the experimental results. The nonlinear Hoek-Brown criterion better reflects the peak strengths of intact and hollow sandstone materials than the linear Mohr-Coulomb criterion. The effect of the borehole diameter on the peak strength of hollow sandstone depends on the confining pressure, which is interpreted by radial and tangent stress distribution around the borehole. In addition, the surface and internal fracture characteristics of the hollow sandstone obtained by X-ray micro CT observations and simulations were compared. Generally, the internal cracks of the intact and hollow sandstone specimens determined using numerical simulations are very similar to those observed using X-ray micro CT. The influence of confining pressure on the fracture evolution mechanism of hollow sandstone with various borehole diameters was analyzed in detail based on the results of numerical simulations. The conclusions drawn in this study are significant for the prediction of the instability around deep well-bores in petroleum engineering and to ensure the safety of deep excavation damage zones in tunnel engineering.

A Macroscopic Strength Criterion for Isotropic Metals Based on the Concept of Fracture Plane

Although the linear Mohr–Coulomb criterion is frequently applied to predict the failure of brittle materials such as cast iron, it can be used for ductile metals too. However, the criterion has some significant deficiencies which limit its predictive ability. In the present study, the underlying failure hypotheses of the linear Mohr–Coulomb criterion were thoroughly discussed. Based on Mohr’s physically meaningful concept of fracture plane, a macroscopic strength criterion was developed to explain the failure mechanism of isotropic metals. The failure function was expressed as a polynomial expansion in terms of the stresses acting on the fracture plane, and the quadratic approximation was employed to describe the non-linear behavior of the failure envelope. With an in-depth understanding of Mohr’s fracture plane concept, the failure angle was regarded as a generalized strength parameter in addition to the failure stress (i.e., the conventional basic strength). The undetermined coefficients of the non-linear failure function were calibrated by the strength parameters obtained from the common uniaxial tension and compression tests. Theoretical and experimental assessment for different types of isotropic metals validated the effectiveness of the proposed criterion in predicting material failure.

Shear Properties of Asphalt Mixtures under Triaxial Compression

In order to study the influences of confining pressure and temperature on the shear properties of asphalt mixtures, triaxial tests were conducted at 40 °C, 50 °C, and 60 °C, with the confining pressure ranges from 0 to 1 MPa for the widely used continuous-graded AC (Asphalt Concrete)-13, open-graded OGFC (Open-Graded Friction Course)-13, and gap-graded SMA (Stone Mastic Asphalt)-13 asphalt mixtures in China. A nonlinear regression/prediction model of triaxial strength for asphalt mixtures was proposed. The results show that confining pressure and temperature had a significant effect on the shear resistance of asphalt mixtures. With increasing temperature, the shear strength of the asphalt mixture gradually decreased due to the decreasing of cohesion strength; the shear strength of the asphalt mixture increased with the increase of confining pressure. Meanwhile, the cohesion strength increased and the friction angle decreased gradually with the increase of confining pressure. When the confining pressure was close to 0.6 MPa, the Mohr–Coulomb failure envelope bended down, so the linear Mohr–Coulomb criterion is not suitable to describe the failure behavior of asphalt mixtures. Therefore, a power function failure envelope was put forward to characterize the nonlinear shear properties of asphalt mixtures. The nonlinear evolutional laws of shear parameters, which includes cohesion strength and friction angle, were also proposed for asphalt pavement material and structure design. Among these asphalt mixtures, the gap-graded SMA-13 asphalt mixture exhibited better performance on the resistance to shear failure, and it was recommended as the upper layer material to improve the shear performance of asphalt pavement.

An approximate nonlinear modified Mohr-Coulomb shear strength criterion with critical state for intact rocks

In this paper, the Mohr-Coulomb shear strength criterion is modified by mobilising the cohesion and internal friction angle with normal stress, in order to capture the nonlinearity and critical state concept for intact rocks reported in the literature. The mathematical expression for the strength is the same as the classical form, but the terms of cohesion and internal friction angle depend on the normal stress now, leading to a nonlinear relationship between the strength and normal stress. It covers both the tension and compression regions with different expressions for cohesion and internal friction angle. The strengths from the two regions join continuously at the transition of zero normal stress. The part in the compression region approximately satisfies the conditions of critical state, where the maximum shear strength is reached. Due to the nonlinearity, the classical simple relationship between the parameters of cohesion, internal friction angle and uniaxial compressive strength from the linear Mohr-Coulomb criterion does not hold anymore. The equation for determining one of the three parameters in terms of the other two is supplied. This equation is nonlinear and thus a nonlinear equation solver is needed. For simplicity, the classical linear relationship is used as a local approximation. The approximate modified Mohr-Coulomb criterion has been implemented in a fracture mechanics based numerical code FRACOD, and an example case of deep tunnel failure is presented to demonstrate the difference between the original and modified Mohr-Coulomb criteria. It is shown that the nonlinear modified Mohr-Coulomb criterion predicts somewhat deeper and more intensive fracturing regions in the surrounding rock mass than the original linear Mohr-Coulomb criterion. A more comprehensive piecewise nonlinear shear strength criterion is also included in Appendix B for those readers who are interested. It covers the tensile, compressive, brittle-ductile behaviour transition and the critical state, and gives smooth transitions.

A Study on the Mechanical Behavior and Statistical Damage Constitutive Model of Sandstone

Triaxial compression test results of sandstone indicate that the peak point strain, elastic modulus, peak deviatoric stress and residual deviatoric stress of the tested sandstone increase with increasing confining pressure, and the variations in them with the confining pressure can be described with a linear function, a logistic function, the generalized Hoek–Brown criterion and the linear Mohr–Coulomb criterion, respectively. Supposing that the rock material can be divided into an elastic part and a damaged part in the rock failure process, the deviatoric stress–strain relationship of the elastic part satisfies Hooke’s law, while the damaged part provides residual deviatoric stress. On this basis, it was assumed the rock meso-element strength follows a composite power function distribution. Then, the damage evolution equation was deduced using a statistical method, and a new damage model, which can reflect the rock residual deviatoric stress, was proposed. The reasonability of the new model was verified using the test results of the sandstone. A comparison of the predicted and test results shows that this damage model can well simulate the deviatoric stress–strain response in the failure process of the tested sandstone. In particular, it can reflect the residual deviatoric stress after rock failure.

A failure criterion for isotropic materials based on Mohr’s failure plane theory

A failure criterion for isotropic materials is developed in the present study. The fundamental hypotheses of the Mohr-Coulomb criterion are re-examined and re-evaluated. The failure function of Mohr’s envelope is firstly expanded into a polynomial in terms of the stress components (σn, τn) on the failure plane, then truncated at the second order due to the fine curve-fitting results of quadratic functions with experimental data. The parameters of the nonlinear failure functions are calibrated by the three basic mechanical properties—the uniaxial tensile and compressive strengths, T and C, as well as the pure shear strength, S. Theoretical and experimental evaluation for various isotropic materials demonstrates that the present failure criterion provides no worse results for ductile materials and far better results for brittle materials compared with the linear Mohr-Coulomb criterion. The strength theory shows good agreement with physical reality and has a wide range of applicability.

Limit Analysis Method for Evaluating Soil Resistance to Lateral Motions of Pipelines Partially Embedded in a Seafloor

The lateral soil resistance available to pipelines partially embedded in a seafloor is of interest in evaluating pipeline lateral buckling capacity, stability under forces resulting from current loads, or route pullout loads. An upper-bound limit analysis method, including equilibrium requirements in a manner common to lower-bound theories, was used to evaluate lateral soil resistance to pipe displacements. The method is applicable to sand and clay foundations with homogeneous or layered conditions. Comparisons with results from physical tests and other limit analysis and finite-element investigations are presented. The numerical limit analysis procedure was implemented through numerical-computing software. An optimization procedure was used to identify the minimum resistance satisfying the governing objective equations. The seafloor surface geometry may include soil berms or other features. The soil strength may be described by a linear Mohr-Coulomb criterion or a curved criterion representing a varying material friction angle.

An experimental study on deformation and failure mechanical behavior of granite containing a single fissure under different confining pressures

Fissures in natural rocks play an important role in determining the strength, deformability and failure behavior of rock mass. However in the past, triaxial compression experiments have rarely been conducted for rock materials containing three-dimensional (3-D) fissures and the failure mechanical behavior of fissured rocks is not well known due to the difficulty of conducting triaxial experiments on fissured rocks. Therefore in this research, conventional triaxial compression experiments were performed to study the strength, deformability and failure behavior of granite specimens with one preexisting open fissure. Thirty-one specimens were prepared to perform conventional triaxial compression tests for intact and fissured granite. First, based on the experimental results, the effects of the confining pressure and the fissure angle on the elastic modulus and the peak axial strain of granite specimens are analyzed. Second, the influence of the confining pressure on the crack damage threshold and the peak strength of granite with respect to various fissure angles are evaluated. For the same fissure angle, the crack damage threshold and the peak strength of granite both increase with the confining pressure, which is in good agreement with the linear Mohr–Coulomb criterion. With increasing fissure angle, the cohesion of granite first increases and later decreases, but the internal friction angle is not obviously dependent on the fissure angle. Third, nine crack types are identified to analyze the failure characteristics of granite specimens containing a single fissure under conventional triaxial compression. Finally, a series of X-ray microcomputed tomography (CT) observations were conducted to analyze the internal damage mechanism of granite specimens with respect to various fissure angles. Reconstructed 3-D CT images indicate obvious effects of confining pressure and fissure angle on the crack system of granite specimens. The study helps to elucidate the fundamental nature of rock failure under conventional triaxial compression.

Experimental study on deformation, peak strength and crack damage behavior of hollow sandstone under conventional triaxial compression

Based on a series of triaxial compression experiments using hollow sandstone specimens with various hole diameters (d=0, 11, 15 and 26mm), the deformation, peak strength and crack damage behavior of hollow sandstone specimens under different confining pressures are investigated. The experimental results show that the Young's modulus of hollow sandstone only depends on the confining pressure and is not affected by hole diameter in the tested range of d=0–26mm. Two types of methods used to confirm the elastic modulus and Poisson's ratio of rock material are proposed to evaluate the triaxial deformation characteristics of hollow sandstone. The effects of confining pressure and hole diameter on the Poisson's ratio and peak strain of hollow sandstone are analyzed. The peak strength and crack damage parameters of hollow sandstone depend on not only the confining pressure (σ3) but also the hole diameter. Under uniaxial compression, the peak strength and crack damage threshold (σcd) of hollow sandstone are independent of hole diameter, whereas the triaxial compressive strength of hollow sandstone decreases linearly with increasing hole diameter. The peak strength and crack damage threshold of hollow sandstone increase with increasing confining pressure, which can be better described by the nonlinear Hoek-Brown criterion than by the linear Mohr-Coulomb criterion. Furthermore, the sensitivity of the crack damage threshold of hollow sandstone to hole diameter is lower than that of peak strength on hole diameter. The concluding remarks can be used to improve the stability and safety of deep underground engineering.