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• The maximum mesh size and maximum loading rate that can be adopted for different specimens are proposed. • The maximum mesh size and calculation method for tunnel excavation simulation of different diameters are proposed. • The method for determining the critical kinetic energy of the system during excavation simulation is proposed. • The reduction function of the Young's modulus of the core material is proposed. The combined finite-discrete element method (FDEM) is extensively employed to model rocks and rock-like materials, in which calibration against the results from uniaxial/triaxial compression tests, Brazilian tests and shear tests have been widely carried out. However, since different element sizes and loading rates were used, it is difficult to assess the numerical results of these studies if the effects of the element size and loading rate are ignored. This paper discusses the effect of the element size, loading/unloading rate and unloading mode on the cracking processes in laboratory-scale (uniaxial compression tests and Brazilian tests) and field-scale (circle tunnel excavations). The results indicate that the element size and loading rate should not be less than 27–28 meshes in a diameter zone and larger than 0.5 m/s, respectively, in laboratory-scale. In field-scale models, four different tunnel diameters (3, 4, 5 and 6 m) are used in tunnel excavation simulations. The results reveal that the mesh number around the tunnel wall should not be less than 120, meanwhile the element size should not be longer than the length of fracture process zone (l FPZ). The unloading rate for the field-scale model is influenced by the model size and rock density. It cannot be determined directly but a new method is proposed to determine the unloading rate based on crack development curve. To compare the fracture patterns well to those observed from the tunnel simulations, the unloading mode of an exponential formula is apposite. [ABSTRACT FROM AUTHOR]

The large deformation mechanisms and predictions of soft rock tunnels have always been important but difficult to solve problems in the field of geotechnical engineering. A combined finite–discrete element numerical simulation method (FDEM) was used to study large deformation mechanism, classification and prediction. The deformations or failure mechanisms of tunnel surrounding rock were revealed, and the failure modes and displacement value prediction of surrounding rock with different strength-stress ratios were also investigated. The following conclusions were obtained: (1) Critical hysteresis damping can be adopted to simulate the progressive large deformation process of soft rock tunnels, which can obtain the final deformation of unsupported tunnels and avoid a dynamic response. (2) Under concentrated tangential stress, the surrounding rock undergoes X-shaped conjugate shear fracture, and this type of fracture network continues to propagate toward the depth of the surrounding rock until the model reaches a stable state; the reduction in tunnel cross-section is mainly caused by the macroscopic movement and volume expansion of rock fragments, and the latter is due to the generation of a large number of macroscopic voids. (3) As the strength-stress ratio decreases, the deformation or failure modes of the surrounding rock can be divided into four categories: elastic–plastic deformation, closed fracturing, shear dilation and broken expansion. (4) Finally, a prediction equation for isotropic and homogeneous soft rock unsupported tunnel deformation with general size driven by hydrostatic in situ stress is obtained, which indicates that with an increasing strength-stress ratio, the surrounding rock displacement decreases as an exponential function, with a correlation coefficient of R 2 = 0.997. In addition, the robustness and reliability of the prediction equation is verified. [ABSTRACT FROM AUTHOR]

Rock materials are typically heterogeneous. Traditional digital image processing (DIP) technology and the grain-based model (GBM) method are difficult to apply to engineering-scale problem studies, such as tunnel excavation. Based on the combined finite-discrete element method (FDEM), a random parameter assignment method is proposed to simulate the mechanical properties and failure behavior of heterogeneous rock. The elastic moduli of triangular elements, as well as the strength parameters of quadrilateral joint elements, assigned by this method obey a Weibull distribution. The simulation results of uniaxial compression, triaxial compression and Brazilian disc show that with the increase in the heterogeneity m value, the uniaxial compressive strength, elastic modulus, triaxial compressive strength, equivalent cohesion, equivalent internal friction angle and tensile strength all increase exponentially and approach those of the homogeneous rock sample. In addition, the heterogeneous rock samples mainly shear fail along their corresponding theoretical failure angle under uniaxial and triaxial compression. The simulation results of tunnel excavation indicate that for heterogeneous rock with different m values, the overall fracture morphology of the surrounding rock remains unchanged, and X-shaped conjugate shear failures mainly occur followed by a small amount of tensile failures, which are similar to the homogeneous surrounding rock. However, with the increase in the m value, the failure degree and the maximum fracture propagation range of the surrounding rock decrease exponentially. Moreover, the simulation results of uniaxial compression and tunnel excavation with different element sizes suggest that the random parameter assignment method proposed in this paper is reliable. [ABSTRACT FROM AUTHOR]

• A novel joint element calibration procedure is proposed for FDEM. • A fracture energy calculation method with different element sizes is proposed to realize cross-scale simulation. • More accurate results can be obtained for tunnel excavation simulation using the proposed joint element calibration technique. • More accurate values of the rock fracture energies G I and G II can be calibrated by the modified Brazilian disc model and variable-angle shear model. The combined finite–discrete element method (FDEM) has been widely used to simulate the fracture process of rock materials. The FDEM simulation requires many material parameters which should be calibrated through laboratory tests or directly cited from previous conclusions. However, it is difficult to obtain the joint element parameters, including the joint element penalty P f and the type I and type II fracture energies G I and G II. Therefore, a novel calibration method for the above three parameters must be proposed. First, P f can be obtained by uniaxial compression simulation, and P f = 30 E (E is Young's modulus) is applicable for homogeneous and isotropic rock materials. Second, an accurate G I value in the static loading state can be obtained by Brazilian disc simulation with a central preprepared failure path. Third, an accurate G II value can be calibrated by the modified variable-angle shear model. Finally, the G I and G II values with different element sizes can be calculated by the proposed linear equations. In addition, normal uniaxial and triaxial compression simulations can be used to verify the reliability of the calibrated G I and G II values. Moreover, the tunnel excavation simulation results show that the joint element parameters calibrated by the modified method can achieve cross-scale simulation, and the simulation results are more accurate. [ABSTRACT FROM AUTHOR]

The failure mechanism and failure mode of anisotropic rock masses are very different from those of isotropic rock masses. For unsupported tunnels, the failure modes of anisotropic rock masses can be summarized as composite failure, V-shaped notch failure and spalling of bedding planes. Different failure modes are controlled by the mechanical parameters of the rock masses, the environmental occurrences and the excavation conditions. The combined finite-discrete element method (FDEM) was employed to investigate the failure mechanism and failure process of layered rock masses and the influence of different factors on the failure mode. The study results indicate that (1) composite failure is the basic failure mode of the layered rock masses and that it is caused by the combined action of shear-slip fractures F1 parallel to the bedding plane, tensile fractures F2 perpendicular to the bedding plane, and conjugate shear fractures F3; the rotation movement of slab-like rock fragments induced by fractures F1 and F3 into the tunnel causes significant deformation of the surrounding rock masses; (2) on the basis of composite failure, when fractures F1 disappear and fractures F2 obliquely intersect with the bedding plane, V-shaped notch failure occurs; furthermore, spalling of the bedding plane occurs when the propagation of fractures F3 is blocked at the bedding plane; and (3) with increasing rock mass strength, lateral pressure coefficient of in situ stress and layer thickness, or decreasing tunnel span, the surrounding rock masses change from composite failure to V-shaped notch failure and then to spalling of the bedding plane. However, the effect of the elastic modulus on the failure mode of anisotropic rock masses is weak, and the influence of the dip angle of the rock layer on the failure mode is complicated. [ABSTRACT FROM AUTHOR]

• A new constitutive model is proposed to eliminate the dependence of parameters on element size. • The influence laws of different parameters on the selection of fracture energy values are proposed, including E , ν , c , φ i , f t and h. • The reliability of the selection of fracture energy values is verified by uniaxial and triaxial compression simulations. The combined finite-discrete element method (FDEM) has been widely used in the numerical study of rock mechanics and geotechnical engineering. Selection of the correct parameter values is a prerequisite to ensure the accuracy of the simulation results. However, current calibration procedures are relatively cumbersome, and the parameters are strongly dependent on the element size. To eliminate the dependence of parameter values on the element size, this paper proposes a new constitutive model based on the stress–strain relationship. In addition, uniaxial compression and direct tension simulations are used to investigate the sensitivities of type II and type I fracture energies, i.e., G II and G I , including the influence of the G I (or G II) value on the simulation results of direct tension (or uniaxial compression) and the influence of different parameters (such as Young's modulus E , Poisson's ratio ν , cohesion c , internal friction angle φ i , tensile strength f t and element size h) on the selection of the values of G II and G I. The following results are obtained. (1) As G I or G II increases, the strength of the rock sample increases, and the plastic characteristics are more obvious, but the failure mode tends to be stable. (2) As the E value increases, the value of G I or G II decreases as a power function. As the value of f t or c increases, the value of G I or G II increases as a power function. However, other parameters, including ν , φ i and h , have no effect on the selection of the values of G II and G I. (3) Actual uniaxial compression and triaxial compression simulations prove that the calibrated G II and G I values are reasonable. In this study, only two fracture energies, G II and G I , must be calibrated; other macroparameters can be obtained through laboratory tests, and other microparameters can be obtained from previous empirical or theoretical values, which greatly improves the calibration efficiency of FDEM input parameters. [ABSTRACT FROM AUTHOR]

• A new calculation equation for normal contact stiffness is proposed. • The value of the coefficient in the new equation is proposed and the robustness of this value is verified. • The influence of different normal contact stiffness on uniaxial compression and tunnel excavation simulations are studied. The combined finite-discrete element method (FDEM) has been widely used in numerical studies in the fields of rock mechanics and geotechnical engineering. The normal contact stiffness between triangular elements is an important influencing parameter, but there is currently no effective method of measuring it. First, an equation for normal contact stiffness is proposed (P b = αP f , where P b is the basic stiffness, α is the coefficient, and P f is the joint penalty; the specific stiffness of each contact couple is determined by P b and the geometric sizes of the triangular elements together); then, the compression-shear failure of a single joint element is used to study the value range and robustness of α ; finally, uniaxial compression and tunnel excavation simulations are used to study the influence of different α values on rock crack propagation. The study results show that (1) an α value of 0.1448 is optimal and robust for a single joint element simulation and is therefore suitable for all simulation conditions; in other words, the value of α should not be too low to avoid decreasing the rock mass stiffness of existing natural fractures and deviating from the actual situation; in addition, the value of α should also not be too large to avoid the development of additional cracks, especially for hard rock simulation, in which the value of α is more sensitive; and (2) the crack topologies of the surrounding rock obtained by tunnel excavation simulation with a large α value deviate from the actual rock core (i.e., when α = 0.1448, the simulation results coincide well with the actual fractures). Through this study, the reliability of FDEM numerical simulation results is improved. [ABSTRACT FROM AUTHOR]

• The long-term degradation behavior of rock strength was systematically summarized. • The long-term degeneration laws of cohesion and internal friction angle were proposed. • FDEM was used to study the long-term crack propagation of tunnel surrounding rock. • The conclusion of this paper was verified by engineering case. The time-dependence of rock strength is an issue that cannot be ignored in the rock engineering field. This paper summarizes previous study results on the degradation of the uniaxial compressive strength of rock (including component cohesion and internal friction angle), tensile strength and Young's modulus over time. Based on previous studies, the degeneracy formulas of intact rock cohesion and internal friction angle are derived. Taking circular tunnel excavation under hydrostatic pressure as an example, the surrounding rock response is studied under different degradation degrees of rock strength and stiffness, including the radius of the excavation damage zone, damage degree of surrounding rock, lining support pressure and tunnel boundary displacement. The results show that with the increasing degree of degradation of the surrounding rock compressive strength, the above four parameters increase in the form of an exponential function and that weakening of the tensile strength had little effect on the surrounding rock response. In addition, as the degradation degree of rock stiffness increases, the radius of the excavation damage zone remains unchanged, whereas the other three parameters values are reduced. An actual engineering case simulation using the combined finite-discrete element method shows that results consistent with field observations can be obtained if the rock strength degradation law proposed in this paper is adopted. [ABSTRACT FROM AUTHOR]

Core material softening method has been widely used in two-dimensional tunnel excavation simulation by the combined finite-discrete element method (FDEM). In numerical studies, softening stress paths (generally described by critical kinetic energy (CKE), softening time step (STS) and softening curve (SC)) usually have a significant influence on simulation results and computational efficiency. There is not a direct way to determine these three parameters. In the present study, a series of numerical studies by the FDEM are performed to discuss the effects of these three parameters on the simulation results and calculation efficiency in a circular tunnel excavation project with a 4 m diameter. The results show that the CKE is closely related to the model size and rock density. The CKE can be obtained from the model kinetic energy curve, and a selection method is proposed. The STS value of 50 000 steps is optimal. To obtain a uniform and stable propagation process of the surrounding rock crack, the SC should be in an exponential function. The engineering case shows that the simulation results are consistent well with the field observations as the softening stress paths are calibrated by the proposed method. [ABSTRACT FROM AUTHOR]

• A new phenomenological tensile failure criterion with an anisotropic coefficient was proposed. • Direct tensile test of the Kangding slate was carried out to verify the accuracy of the new criterion. • The tensile strength data of eight typical layered rocks were collected and compared with the other six typical tensile strength criteria to verify the universality and superiority of the new criterion. • Based on the sensitivity analysis of tensile rate, the loading rate of 0.01 m/s was recommended for the samples with various foliation angles. • Layer thickness has little effect on tensile strength, but a significant effect on tensile failure modes. The tensile strength is a key parameter for the design of many geotechnical engineerings such as tunnel support, slope support, and petroleum drilling. The tensile strength in layered rocks, however, is anisotropic due to the existence of bedding planes. In this paper, to study the anisotropic tensile behaviors, a series of direct tensile tests were conducted using Kangding slate with five different foliation angles (β = 0°, 30°, 45°, 60°, and 90°). Based on the N-Z criterion, a new tensile failure criterion with the anisotropic coefficient was proposed. The tensile strength data of nine typical layered rocks were collected to evaluate the prediction ability of seven anisotropic tensile failure criteria including the new criterion. Embedding the new criterion into FDEM, the loading rate-dependent and layer thickness-dependent tensile mechanism of layered rock under direct tensile test were simulated. The experimental and numerical simulation results showed that the foliation angle significantly affected the tensile strength and failure modes of the slate. With the increase of foliation angle (β), the tensile strength of nine typical layered rocks, including Kangding slate, presented a non-linear increase trend. Compared with the other six typical anisotropic tensile failure criteria, the new criterion has the best prediction capability on the tensile strength of rocks with four different anisotropic degrees, and therefore the new criterion was recommended. The new criterion was embedded in FDEM to carry out direct tensile simulation of slate with different β. The numerical simulation results were in good agreement with the experimental results, further verifying the accuracy of the new criterion. The loading rate sensitivity analysis indicated that a tensile rate of 0.01 m/s was recommended to assure quasi-static loading of the specimen. Through the sensitivity analysis of layer thickness, it is concluded that layer thickness has very little influence on tensile strength, but has a very significant influence on the final failure modes of the specimens. [ABSTRACT FROM AUTHOR]

The combined finite-discrete element method (FDEM) is an advanced finite element and discrete element coupling method, which is commendably suitable for simulating the slope's entire evolution process from cracking, expansion, penetration, sliding, collision to deposition under seismic load. The original FDEM has apparent shortcomings in simulating the dynamic instability process of earthquake-induced landslides, and some targeted improvements are made to the original program. Numerical tests are carried out to prove the accuracy and robustness of the improved FDEM, and the influence of the structural plane on the dynamic evolution mechanism and process of the slope is studied. The results show that the improved FDEM can accurately reproduce the entire dynamic evolution processes and failure forms of different jointed rock slopes under seismic load and can quantitatively evaluate the dynamic stability of jointed rock slopes (dynamic safety factor and critical failure surface). In addition, the results also emphasize that the slope failure degree significantly affects the dynamic response of the slope and suggest that the combination of the cut-through of the failure surface and the increase of kinetic energy should be used as the criteria for slope dynamic instability simulation by FDEM. This study provides scientific and technological support for revealing the failure mechanism of jointed rock slopes and disaster reduction and prevention. [ABSTRACT FROM AUTHOR]

The combined finite-discrete element method (FDEM) is an advanced finite and discrete element coupling method. This study further applies it to the problem of stress wave propagation in the jointed rock mass. Firstly, the fundamental theory of FDEM is briefly described and the methods commonly used in FDEM to characterize structural planes are comprehensively analyzed. Then, based on the viscous boundary condition (VBC), the viscous-spring boundary condition (VSBC) is added to absorb the reflected wave at the artificial boundary and restore the residual displacement to meet the actual engineering. In addition, the application range of the VBC and VSBC is verified, which indicates that the VBC and VSBC in the improved FDEM can be well applied not only to the continuum range but also to the case of the new fracture formed. Finally, several classic models are solved to verify the stress wave propagation across different types of joints by FDEM. The results reveal that results from FDEM agree well with analytical solutions based on the displacement discontinuity model and numerical solutions from universal distinct element code (UDEC), which means that the improved FDEM can capably and accurately simulate wave propagation in the jointed rock mass. [ABSTRACT FROM AUTHOR]